Insight Editions · No. 268
The Laws of Thought
The Quest for a Mathematical Theory of the Mind
Tom Griffiths
Tom Griffiths traces three centuries of attempts to write the mind in mathematics: rules and symbols from Boole to Chomsky, networks from perceptrons to deep learning, and probability from Bayes to modern cognitive science.
A study guide to The Laws of Thought by Tom Griffiths. Shared free for personal reading; not for sale and not endorsed by the authors or publishers. Please read and buy the original.
The Book on One Page
Griffiths tells three centuries of attempts to do for the world inside us what physics did for the world outside: find mathematical Laws of Thought. Three traditions each captured part of the mind: rules and symbols (logic, grammars, search), spaces and networks (similarity, perceptrons, backpropagation, LLMs) and probability (Bayes' rule over structured hypotheses). The traditions are not rivals but levels: logic and probability define the ideal at Marr's computational level, minds approximate it with fallible algorithms, and neural networks implement it.
The argument in eight steps
- Thought as a formal system. From Leibniz (1679) to Boole (1854), logic turned truth into rule-following, so reasoning could run on paper or in machines; Turing, Shannon and von Neumann built the machines, and on 11 September 1956 Newell and Simon, Chomsky and Miller launched cognitive science under logic's banner.
- Rules and symbols triumph and stall. Heuristic search, the Logic Theorist and production systems modeled problem solving, and Simon's ant on the beach put much complexity in the environment; Chomsky's generative grammars and hierarchy captured productivity, hierarchy and compositionality, but rich rules seemed unlearnable, forcing innate knowledge (the poverty of the stimulus).
- Logic meets induction. Vision, categories, speech and language acquisition are inductive problems: data underdetermine the answer, so success depends on inductive biases. Hume and Goodman show logic cannot justify them; Mill asks why one instance sometimes suffices.
- Spaces and features. Rosch's fuzzy categories, Shepard's psychological spaces and Tversky's feature sets replaced definitions with similarity to prototypes.
- Networks learn. Hebb, Rosenblatt's perceptron, Minsky and Papert's critique (1969), Quillian's semantic networks and spreading activation, then backpropagation (Rumelhart, Hinton and Williams, 1986) let networks learn their own representations; ImageNet and GPUs made deep learning win with AlexNet in 2012.
- Language as prediction. From the past-tense war (Rumelhart and McClelland against Pinker and Prince) through Elman's next-word networks, LSA and word2vec to transformers and ChatGPT: prediction alone learns much of language, but needs about 1,000 times a child's data and still fails novel wug tests.
- The missing question. Marr's levels (his cash register: addition is forced by shoppers' expectations) show what both traditions lack: a computational-level theory of why. Probability supplies it: over possible worlds, Bayes' rule extends logic to uncertainty, answers Mill (high priors), and explains network data hunger (conservation of learning), despite Kahneman and Tversky's evidence that people are not Bayesian.
- Synthesis. Shepard's universal law (1987), Anderson's rational analysis, Bayesian readings of the Wason task and other "biases", Pearl's Bayesian networks and probabilistic grammars combine structure with statistics; Horning's proof and the size principle dissolve the logical problem of acquisition, and LLMs are probability distributions whose missing priors, perhaps instillable by meta-learning, are the frontier.
Chapter map
| Ch | Title | One-line payload |
|---|---|---|
| Intro | Introduction | Three frameworks each capture part of the mind and are converging |
| 1 | Turning Aristotle into Arithmetic | Leibniz fails, Boole succeeds: logic reduces truth to syntax |
| 2 | Computing a Cognitive Revolution | Turing, Shannon, von Neumann and 1956 make logic the first theory of mind |
| 3 | Solving Problems | Thinking as heuristic search and production rules; the ant on the beach |
| 4 | Language as a Formal System | Chomsky's grammars and hierarchy; rich rules imply innate knowledge |
| 5 | The Limits of Logic | Minds face inductive problems logic cannot solve; Mill's question |
| 6 | Categories, Spaces, and Features | Fuzzy categories, Shepard's spaces, Tversky's features, prototypes |
| 7 | Computing with Spaces | Hebb, the perceptron, and Minsky and Papert's 1969 limits |
| 8 | Brain Models and Mind Models | Semantic networks and spreading activation: thought as a trajectory |
| 9 | The Plot Deepens | Backprop (1986), PDP connectionism, ImageNet and GPUs: AlexNet wins in 2012 |
| 10 | Language as a Prediction Problem | Past-tense war, Elman, LSA, word2vec, GPT: prediction learns language, at huge data cost |
| 11 | The Missing Question | Marr's three levels; networks and logic lack a theory of why |
| 12 | Probability Is the New Logic | Possible worlds and Bayes' rule extend logic; priors answer Mill; Kahneman-Tversky doubt it |
| 13 | Universal Laws of Cognition | Shepard's law, rational analysis, Bayesian networks, structure plus statistics |
| 14 | Language as a Probability Distribution | Dice on grammars, Horning's proof, meta-learned priors and LLM "embers" |
| 15 | Putting It All Together | Logic and probability compute, algorithms approximate, neurons implement |
Introduction
Griffiths sets out to tell the missing half of the scientific story: how mathematics has been used to explain the world inside us, the Laws of Thought, as physics explains the world outside. Three mathematical frameworks (rules and symbols, neural networks, Bayesian models) each capture part of the mind, and after three centuries their insights are starting to converge.
The world inside us
School gives everyone a working grasp of the Laws of Nature (force, acceleration, gravity), but almost nobody leaves with a comparable vocabulary for the mind. Most people know Newton's universal law of gravitation; few have heard of Shepard's universal law of generalization, which Griffiths presents as a simple principle that should describe any intelligent organism anywhere in the universe. The history of mathematics applied to the external world is familiar; the history of mathematics applied to thought is not, and that second history is the book's subject.
Three hundred years of trying
The founders of modern science, a little over three hundred years ago, combined careful observation, experiment and new mathematical notation, and that mix has since mapped the physical universe from the smallest scales to the largest. The same people wanted to turn those tools on the mind:
- Thomas Hobbes. He proposed that reasoning ("ratiocination") is a kind of calculation, in which thoughts could be added and subtracted.
- René Descartes. He imagined assigning numbers to thoughts, the way numbers are assigned to collections of physical objects.
- Gottfried Wilhelm Leibniz. He spent his life trying to describe human reason with arithmetic and, Griffiths stresses, ultimately failed.
The first real success came in the mid-nineteenth century, when George Boole invented a new kind of algebra that solved the problem Leibniz could not. It led to formal logic and to computers. Testing such theories against human behavior began only in the mid-twentieth century, with the Cognitive Revolution that founded cognitive science, the interdisciplinary science of mind. Cognitive scientists then found the limits of logic as a model of thought and built two new mathematical approaches: artificial neural networks (continuous representations, statistical learning) and Bayesian models (prior knowledge, uncertainty). The term "Laws of Thought" itself is nineteenth-century: it is what Boole and his contemporaries called the mathematical principles of mind.
Griffiths argues that in the twenty-first century knowing these laws is as much a part of scientific literacy as knowing the laws of physics. AI systems keep showing that one more slice of thought and language can be emulated by machines, which forces us to rethink ourselves, our careers and our children's world. He promises that by the end the reader will understand the basic principles of modern AI and know where it is likely to keep falling short of humans.
1600s Hobbes, Descartes and Leibniz imagine thought as calculation.
1850s Boole's algebra of logic: the first working mathematics of thought.
1900-1950 Behaviorism bans the mind from psychology.
1950s Cognitive Revolution: testable mathematical models of mind.
1980s-2020s Neural networks and Bayesian models challenge pure logic.
Why a field without consensus
Unlike physics, cognitive science has no single accepted account of its subject, and Griffiths says this openness is what drew him in. At high school in Australia in the 1990s, physics and chemistry felt like settled formulas, so at university he chose subjects that still seemed mysterious: philosophy, psychology, anthropology and ancient mythology. A chapter at the back of a philosophy book introduced him to artificial neural networks (mathematical systems that emulate the brain); he spent a summer sketching huge networks, then talked his way into a professor's lab on the first day of semester. Graduate work at Stanford and MIT followed, then faculty posts at Brown, Berkeley and now Princeton (he tags each stage with a soundtrack, from the Smashing Pumpkins to Taylor Swift). The mysteries remain: researchers still disagree on how the mind works and instead hold several frameworks that each capture part of it, though in recent years their insights have begun to converge.
From behaviorism to the Cognitive Revolution
Thought cannot be seen or touched, so how can science study it? Psychologists of the early twentieth century answered by not trying: behaviorism applied scientific method only to observable behavior and the stimuli that produce it, holding that a rigorous science of mind should never mention the mind. In the 1950s researchers recovered the Hobbes-Descartes-Leibniz insight and borrowed the physicist's move for unobservables: mathematical theories that make precise predictions about observable phenomena. Formal logic showed how a small system of rules and symbols could produce complex intelligent behavior, and the new digital computer made that idea concrete. Mathematics became the way to state hypotheses about the mind rigorously enough to test against behavior.
The three frameworks and their questions
- Rules and symbols. Logic applied to reasoning, categorization, problem-solving and language. Noam Chomsky showed simple formal grammars generate complex sentences and used ordinary speakers' judgments to choose between grammars. Appeal: infinite behavior from finite rules, plus the hierarchies and combinations central to intelligence. Question: What are the rules behind human cognition?
- Spaces and networks. Thought is fuzzier than logic (is a lamp furniture? is an olive a fruit?) and rule systems are so hard to learn that some, Chomsky included, called it nearly impossible. So concepts become points in a psychological space with continuous dimensions; semantic networks in psychology merged with artificial neural networks in computer science. Networks capture rule-like regularities of language without explicit rules and, given enough data, can in principle learn anything (AlphaGo, ChatGPT). Question: How much of cognition is explained by the input data?
- Bayesian models. Networks need huge amounts of data, but a child can learn a new word from a single use. Bayes' rule (eighteenth century) says what a rational agent should conclude from data given its biases, and lets logic's structure combine with statistical learning. Question: How can we characterize the biases of human learners?
| Framework | Key mathematics | Strength | Weakness | Driving question |
|---|---|---|---|---|
| Rules and symbols | Logic, formal grammars | Infinite behavior from finite rules; hierarchy and combination (Chomsky) | Thought is fuzzier than logic; rules hard to learn | What are the rules behind cognition? |
| Spaces and networks | Continuous spaces, artificial neural networks | Handles fuzziness; general-purpose learning (AlphaGo, ChatGPT) | Needs huge amounts of data | How much is explained by the input data? |
| Probability | Bayes' rule (18th century) | Learning from little data; combines logic with statistics | Must specify the learner's biases | How can we characterize human learners' biases? |
Each framework's failure motivates the next: logic's rigidity and learning problem led to spaces and networks; networks' data hunger led to Bayesian models, which explain how a child can learn a word from one hearing.
What the book promises
Griffiths will take the frameworks in turn: the key ideas, the insights each offers, the challenges that lead to the next, the experiments comparing models with people, and finally how the three must combine for a fuller account. A running thread is the gap between humans and AI systems that need massive data: studying how people learn from little data shows what is still missing from machines and what remains distinctively human.
For anyone building with AI, the book's frame is practical: today's systems sit almost wholly in the neural network tradition, so their weak spots (data efficiency, explicit structure, calibrated uncertainty) map onto what the symbolic and Bayesian traditions do well.
Turning Aristotle into Arithmetic
The first mathematical framework for the mind treats thought as a formal system of rules and symbols. Griffiths traces it from Leibniz's failed attempt (1679) to turn Aristotle's syllogisms into arithmetic through Boole's algebra of logic (1847, 1854) to modern propositional logic. Logic's great achievement is reducing a semantic question (what is true) to a syntactic procedure (what follows by rule), so that thought can happen on paper or in a machine.
Leibniz struggles
Gottfried Wilhelm Leibniz was born in Leipzig in 1646. His father, a professor of moral philosophy, died when he was six, and his mother let the boy loose in the father's library. He entered university at fourteen, had a bachelor's degree in philosophy a little over a year later, then advanced degrees in philosophy and law, and went on to contribute to almost every field of his day: mathematics, astronomy, physics, chemistry, geology, botany, psychology, medicine, history, linguistics, ethics, law and theology. In mathematics alone he developed the calculus in parallel with Isaac Newton, formulated binary numbers and invented a new mechanical calculator. Yet, Griffiths stresses, he never found a way to describe human thought mathematically.
Logic was an early love. Leibniz recalled that before reaching the logic class he lived among historians and poets, but once he met logic at about thirteen he was thrilled by the ordering and division of thoughts it revealed. As his mathematics matured he suspected that the structure of logic might be mirrored by the structure of mathematics; his university dissertation at twenty studied the mathematics of combinations, building concepts out of simpler parts.
A decade later the problem returned. A run of notes from 1679, when Leibniz was thirty-two and newly in Hanover as counselor to Duke Johann Friedrich, lets us watch a great mind fail. In the same burst he planned a new encyclopedia and a scientific society to support it, worked on philosophy, and devised a new mathematics of physical forces; mental forces resisted him. Each note follows one pattern: a new scheme, a promising test, a snag, and the text stops abruptly, only for the next note to begin again and end the same way. Griffiths uses this struggle to introduce the framework that explains the mind as rules and symbols, and the man who eventually solved Leibniz's problem, George Boole.
Aristotle and the syllogism
Leibniz started small: not all of thought, only Aristotle's syllogisms. Logic was developed independently in India and China, but for Leibniz and the West the canon was Aristotle (student of Plato, tutor of Alexander the Great), whose Prior Analytics gave the first systematic account of the structure of arguments, two thousand years before Leibniz.
A syllogism is an argument with two premises (statements assumed true) and one conclusion that may follow from them. Leibniz's example: every wise person is pious; some wise person is wealthy; therefore some wealthy person is pious. It is valid: accepting the premises forces you to accept the conclusion. Aristotle wanted to know which syllogisms are valid and whether they could be characterized systematically.
- Four statement types. Replace the terms with letters, and every premise and conclusion must be one of Every A is B, No A is B, Some A is B, Some A is not B.
- Recurring terms. The terms reappear: Leibniz's argument becomes Every A is B; Some A is C; therefore some C is B.
- Three Figures. The order in which A, B and C appear defines a Figure. Swap "Every A is B" for "Every B is A" (every pious person is wise) and the argument becomes invalid, since the wealthy wise people might not be the pious ones.
- Combinatorics. One of 3 Figures times one of 4 types for each of three statements gives 3 × 4 × 4 × 4 = 192 possible syllogisms.
- Result. Aristotle found 14 valid ones, by intuition for obvious cases (Every A is B, Every B is C, so every A is C) and proofs reducing doubtful cases to obvious ones, plus general rules such as no valid syllogism has two negative premises (No A is B, Some A is not B).
The reason for letters instead of real things is Aristotle's key insight: validity depends on form, not content. Whether a conclusion follows is a matter of which of the 192 patterns is used, not of what the terms mean or whether the premises are true. Lewis Carroll, in his introduction to logic, gives the same form with absurd premises: every cat understands French; some chickens are cats; therefore some chickens understand French. The premises are false, the form is still valid.
Leibniz's arithmetic of terms
Leibniz wanted a mathematical theory that automatically picked out all the valid syllogisms, and in 1679 he thought arithmetic would do it. Each term received two numbers, one positive and one negative, and a statement was checked by divisibility. If "wise person" is (+70, −33) and "pious" is (+10, −3), then "Every wise person is pious" holds because 70 is divisible by 10 and −33 by −3. He hoped rules on these numbers would identify valid conclusions from premises.
Using properties of prime numbers, he found schemes that sometimes worked: he could prove "some wealthy person is pious" from the first example. But they also certified invalid arguments, for instance: every pious person is happy; some pious person is not wealthy; therefore some wealthy person is not happy. Aristotle would reject it, and so would any reader. Arithmetic could not replace Aristotle.
The universal characteristic
Leibniz's logic was really an attempt to give mathematical form to another seventeenth-century obsession: a universal language. Language fascinated the age. The first alphabetical English dictionary appeared in 1606, encyclopedias were popular, scholars compared languages (Leibniz studied Chinese), and thinkers who were remaking the world began to imagine remaking language.
Descartes's letter to Mersenne
René Descartes (of Cartesian coordinates and Cartesian dualism) took up the idea in a 1629 letter to the mathematician Marin Mersenne. Mersenne had forwarded a scheme to rationalize language, which Descartes doubted, but the general idea excited him. His argument ran in three steps:
- The number analogy. Number words follow a few simple rules, so in a single day one can learn to name infinitely many numbers; you instantly know what "two thousand seven hundred and nineteen" means without having heard it. If all thoughts were ordered like the natural order of the numbers, a whole language could be learned as quickly.
- The precondition. Such a language needs the "true philosophy": someone must identify the simple ideas out of which all thoughts are compounded, and have that analysis widely accepted.
- The payoff. Judgment would become so clear that error would be almost impossible. Just as you can see at once that 2,719 is less than 3,000,012, two thoughts in the ideal language could be compared directly, exposing false claims; ordinary words, by contrast, have confused meanings.
Wilkins's Real Character
The project's most serious attempt came from the Reverend John Wilkins in An Essay Towards a Real Character, and a Philosophical Language (1668). Wilkins headed colleges at both Oxford and Cambridge, was a founding member of the Royal Society, wrote the first comprehensive English book on cryptography and is credited with planning a scientific mission to the moon. The Essay ran past 600 pages of text and tables, surveyed the written and spoken properties of natural languages, and proposed a new language to "remedy the Curse of Confusion", easing learning, trade and religious disputes. Most of it is lists of concepts organized by his scheme.
- Genus. 40 broad classes (Element, Bird, Beast, Measure), each written as a horizontal line with one of 40 distinct marks.
- Difference. 9 subclasses per Genus (Element: Fire, Water...), marked on the line's left end.
- Species. 9 finer kinds per Difference (Fire's first Species: Flame), marked on the right end; extra marks show connecting words.
- Capacity. 40 × 9 × 9 = 3,240 concepts.
- Speech. Genus = consonant + vowel, Difference = consonant, Species = vowel: Heaven is "Dad" ("Da" for World, "d" for the 2nd Difference), Earth "Dady" ("y" for the 7th Species), and the prefix "o-" makes an opposite, so Hell is "Odad".
In the Lord's Prayer the relationships jump out: "Heaven" and "Earth" both carry the horizontal-plus-vertical line of the Genus World; Heaven has the left mark for the second Difference, and Earth adds a right mark for the seventh Species. Written or spoken, the scheme has a unique property: to know the word is to know the thing. A newcomer can place an unfamiliar word in the system at once, related things are visibly related, and some false statements become impossible to make innocently: "a cat is a beast" is fine, but "a cat is a fish" is visibly false to someone who knows nothing about cats or fish beyond their names.
Leibniz's version: reason as calculation
Descartes said thoughts must be numbered; Wilkins in effect numbered them by Genus, Difference and Species. But Wilkins's numbers were arbitrary apart from encoding hierarchy. Leibniz's universal characteristic sought numbers that captured something intrinsic about each concept, so that a claim like "every wise person is pious" could be checked by a quick calculation and valid conclusions computed from true premises. The decisive corollary: if reason were arithmetic, it could be automated.
That is why Leibniz's calculator belongs in the same story. Blaise Pascal had built a gear-driven machine that added and subtracted; Leibniz extended the idea to multiplication and division, showed a prototype to the Royal Society in 1673, and eventually, despite the limits of contemporary engineering, built a device that could multiply two eight-digit numbers. In the draft introduction to a planned book, he promised that characteristic numbers would give humanity a tool boosting the mind far more than lenses boost the eye, letting people judge arguments and draw conclusions without effort or risk of error.
The book was never completed. In notes of 1686 and 1690 he moved past arithmetic, using abstract symbols for classes and new operations for their relations, anticipating later mathematical logic, but no system satisfied him. Near the end of his life he told a friend he might still have finished it had he been less distracted, younger, or had talented young men to help. What Descartes, Wilkins and Leibniz did leave was a new way of seeing the mind: as a system of rules and symbols.
Formal systems
These three thinkers introduced ideas that recur throughout the book (numbers for concepts, concepts arranged in hierarchies), but their deeper move was to reduce a mysterious process, thought, to a familiar one, arithmetic. In modern terms they claimed the mind is a formal system, a precise name for a system of rules and symbols. On a popular definition it has three properties: it is a token manipulation system, it is digital, and it is medium independent. Anyone who has played a board game has used one.
- Token manipulation. The system is fully specified by a set of tokens, their initial positions and rules for manipulating them. In chess the tokens are the pieces, the initial position is the familiar setup, and the rules say how a pawn or knight may move; following them generates every possible game.
- Digital. There are no in-between states: a piece is on a square or it is not, and a move is legal or not, with no slow-motion replay needed. That is why a game played a century ago can be re-created exactly.
- Medium independent. Nothing in the rules fixes the pieces' form: sets with South Park characters (the pawn is, fittingly, Kenny), exotic birds or baseball players work equally well, and so do silicon chips, or the neurons of experts playing blindfold.
Fencing is the counterexample. It has rules and deep strategy ("physical chess"), but outcomes hinge on continuous positions, ambiguity needs a human referee, replays help, and no bout can be reproduced perfectly; fencers argue over whether an action was a feint or an attack. Historical fencers reconstructing old styles from archaic manuals face the same gap that early-music performers face between notation and sound.
| Property | Meaning | Chess example | Counterexample |
|---|---|---|---|
| Token manipulation | Tokens + initial positions + rules for moving them | Pieces, starting setup, legal moves generate all games | |
| Digital | No in-between states; legality is yes or no | A century-old game can be replayed exactly | Fencing: positions continuous, needs referees, never perfectly reproducible |
| Medium independent | Physical form of tokens is irrelevant | South Park pieces, computers, blindfold play in neurons |
Arithmetic is a formal system too. Its initial positions are well-formed problems built from numbers, operators (+, −, ×, ÷) and parentheses, such as 4 × (3 + 2); its rules fix the order of operations and the results (do the parentheses, 3 + 2 = 5), giving a new problem 4 × 5, and so on until any problem is solved.
Tokens become symbols when they are taken to stand for something else: the pawn for a foot soldier, the king for the leader, the character 3 for three peaches or the three dollars paid for them. The system then gains new uses: chess teaches battlefield strategy, arithmetic predicts the checkout total. Much of science, Griffiths argues, is establishing correspondences between formal systems and the world. Newton's physics succeeded by linking Euclidean geometry to the motion of planets and comets: in calculus class x is a token; down the hall in physics it is a planet's location. In 1679 Leibniz tried to map an existing formal system, arithmetic, onto thought. His failure showed only that arithmetic was the wrong formal system, not that no formal system fits. The next breakthrough took more than a century.
A formal system is defined by tokens, starting positions and rules, with no in-between states and no dependence on physical medium. If thought is such a system, then it can in principle run on paper, a board, a brain or a chip; the open question after Leibniz was only which system.
An algebraic insight
Walking through a field at seventeen, a young schoolteacher named George Boole, his head full of algebraic methods for hard calculus problems, had a sudden vision: thought itself might be expressed in algebra.
Boole was born at four in the afternoon on 2 November 1815 in Lincoln, England, first child of John Boole, a shoemaker more likely to be grinding telescope lenses than making shoes, and Mary Ann. He studied classics and mathematics, learned flute and piano, became fluent in Latin and Greek, and at fourteen published translations of classical poems in the local paper (one reader refused to believe a schoolboy wrote them). When his father's business faltered a couple of years later he became a teacher in Doncaster and read mathematics more deeply, partly because slow-going mathematics books stretched his book budget.
His vision owed something to an unusual education. He learned calculus from the French (Lacroix, Lagrange and Laplace), who followed Leibniz's abstract, algebraic approach rather than the geometric approach of Newton favored in England, which inclined him toward systems richer than arithmetic. He could not pursue it at once: his job led to the headship of a school nearer his family, then to a school of his own in Lincoln at nineteen. With the family finances steadier he published a string of papers in the Cambridge Mathematical Journal applying algebra to calculus, on topics such as analytical transformations and linear differential equations with constant coefficients. The schoolteacher who had never attended university was soon corresponding with leading mathematicians, and "On a General Method in Analysis", a technique for tricky differential equations published in the Proceedings of the Royal Society in 1844, won him the Society's first medal for mathematics. These papers made his name and gave him the tools for his life's work, a mathematical theory of logic.
1815 Born 2 November in Lincoln.
c.1832 Vision in a field: thought as algebra (age 17).
1834 Opens his own school in Lincoln (age 19).
1844 "On a General Method in Analysis"; Royal Society medal.
1847 The Mathematical Analysis of Logic.
1849 First professor of mathematics at Cork.
1854 An Investigation of the Laws of Thought.
1864 Dies 8 December, aged 49.
The Laws of Thought
In hindsight Boole was perfectly equipped to arithmetize Aristotle: steeped in algebra and able to read Aristotle in Greek. Yet he came to the problem independently of Leibniz, learning of their shared passion only later. The spur was a feud between his friend Augustus De Morgan, a mathematician, and the philosopher William Hamilton. Each had extended logic and scorned the other's methods: De Morgan wanted mathematics, Hamilton verbal argument, and Hamilton even accused De Morgan of plagiarism. Boole saw a shortcut: a mathematical approach in De Morgan's spirit that formalized Hamilton's intuitions.
His sister MaryAnn described an obsession: he moved through the day like a sleepwalker, lay awake for hours, and on winter nights got up wrapped in a cloak to write down ideas in the family sitting room. The Mathematical Analysis of Logic appeared as a short book only weeks after he began. Aristotle's syllogisms take pride of place: it gives an algebra of relations between classes, derives conclusions by checking whether equations follow from others, and shows that this recovers the valid syllogisms. Boole felt it rushed and spent seven more years on the much longer An Investigation of the Laws of Thought. Its opening paragraph states the program: find the laws of the mind's reasoning operations, express them in the symbolic language of a calculus, build logic on that foundation, use it as the basis for probability, and draw from the inquiry hints about the nature of the human mind.
- Symbols as classes. Let x = white things and y = sheep; then xy = white sheep, so multiplication is the overlap (intersection) of classes.
- Commutation. Sheep that are white are white things that are sheep: xy = yx.
- Idempotence. White things that are white are white things: xx = x, i.e. x² = x. In ordinary algebra this holds only for x = 0 or x = 1, and Boole noted his logic is just familiar arithmetic restricted to 0 and 1.
- Universe and nothing. 1 = the class of all things (1x = x); 0 = the empty class (0x = 0).
- Addition and subtraction. If x and y share no members (xy = 0), x + y is the class of things in either; x − y is things in x but not y; they combine as usual: x(y + z) = xy + xz.
- Statements as equations. Each Aristotelian statement type gets an equation; "Every A is B" becomes a = ab (the things that are A and B are just the things that are A).
- Proof as algebra. For Every A is B, Every B is C: write a = ab and b = bc; substitute bc for b to get a = abc, then replace ab by a to get a = ac, "Every A is C". Boole gave similar proofs for the other syllogisms.
The Laws of Thought went further, building a parallel algebra for statements not about classes. The system had flaws. The logician and economist William Stanley Jevons objected that x + y applied only to disjoint classes and devised a revised system; some of Boole's proofs contained errors found only years later. Even so, Griffiths judges it the first meaningful realization of what Descartes and Leibniz had dreamed of: a formal system describing aspects of human thought.
Why logic matters
To show why logic deserves excitement, Griffiths skips a century of work by Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein and Emil Post and lands on the simpler modern form, propositional logic: the logic of statements (propositions, written P, Q) that are either true or false. Let P be "it rained in the night" and Q "the grass is wet". Instead of Boole's arithmetic symbols, it uses connectives whose meanings are fixed by truth tables, which list the truth value of the compound for every combination of its parts.
| P | Q | P ∧ Q (and) | P ∨ Q (inclusive or) | P → Q (if) | P ↔ Q (iff) |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | T | F | T | T | F |
| F | F | F | F | T | T |
- And (∧). P ∧ Q is true only when both parts are true, matching "it rained and the grass is wet".
- Or (∨). P ∨ Q is true when either or both are true: an inclusive or. English often means an exclusive or: offered cake or ice cream, you should not take both.
- Not (¬). ¬P is true when P is false and vice versa, the relation between "it rained" and "it didn't rain".
- If (→). P → Q is false only when P is true and Q false, so it is true whenever P is false. Griffiths's gloss: "if it rained the grass is wet" rules out rain with dry grass and says nothing about the grass on dry nights.
- If and only if (↔). P ↔ Q is true when both have the same value. "I'll go to the party if you go" usually means this: both go or neither does.
Compound statements build bigger tables. (P ∨ Q) ∧ R is true when P ∨ Q is true and R is true; with three propositions the table has eight rows, true only in the three rows where R is true and at least one of P, Q is true. Each new proposition doubles the table (2 propositions, 4 rows; 3, 8 rows; n, 2ⁿ), and since a person knows countless facts that could be true or false, enumerating rows quickly becomes unmanageable.
Checking arguments with possible worlds
Each row of a truth table is a possible world. Number the rows of P → Q as 1 (T,T), 2 (T,F), 3 (F,T), 4 (F,F). If P → Q is true we are in world 1, 3 or 4; if P is also true only world 1 remains, where Q is true. Since no world makes P → Q and P true with Q false, the inference is valid for any propositions related this way (if R → S and R, then S). Written with a line between premises and conclusion, like the "therefore" of a syllogism, this is modus ponens: from P → Q and P, infer Q.
Not every pattern survives. From P → Q and Q, worlds 1 and 3 remain, and P is true in one and false in the other, so P does not follow: the famous fallacy of affirming the consequent. Truth-table checks validate other short arguments, called inference rules because they license valid inferences:
- And-elimination. From P ∧ Q, infer P.
- Or-introduction. From P, infer P ∨ Q.
- Modus tollens. From P → Q and ¬Q, infer ¬P.
Semantics reduced to syntax
Any intelligent system must decide what to believe: which facts hold, and which further facts follow from them. Logic answers this without surveying the vast space of possible worlds, by chaining simple pieces already known to be valid. Suppose we know (P ∨ Q) → R and P ∧ S. Is R true? The brute-force route writes the table and checks all sixteen worlds. The logical route takes three steps:
This is what Griffiths finds thrilling. Logic reduces a semantic question (what is true in the world) to a syntactic procedure defined purely by the formulas, just as Descartes and Leibniz imagined. Deciding truth becomes following the rules of a game whose pieces are P, Q, ∧, ∨ and whose moves are inference rules like modus ponens.
Reducing semantics to syntax, discovering truth by following rules, was the first big step in demystifying thought. With the right rules written down, thinking stops being something only heads do and becomes something that can happen on a page, a game board or inside a machine.
The remarkable Booles
His publications took Boole out of his Lincoln school to become the first professor of mathematics at Cork (1849). Shortly after The Laws of Thought he married Mary Everest, niece of the surveyor after whom the highest mountain is named; he was thirty-nine, she twenty-three, bound by shared interests in mathematics and spirituality. Mary helped prepare his Treatise on Differential Equations and attended his Cork lectures on the subject, possibly the first woman to attend a university course in Ireland. They had five daughters between 1856 and 1864.
Boole died on 8 December 1864, aged forty-nine. Lifelong ill health, worsened by juggling teaching and research, ended with pneumonia after a long walk home in the rain from a lecture (one of Griffiths's students later worried when he arrived at class in the rain without a coat). Griffiths notes that men take most leading roles in his story because of social, economic and institutional barriers that kept women and other groups out of science; he promises to highlight less-known figures, starting with the Booles, and to introduce modern cognitive scientists at the end.
- Mary Everest Boole. Librarian, then tutor in arithmetic and geometry at Queen's College London, where she taught George's logic; later secretary to the surgeon James Hinton. Wrote on spirituality and mathematics education, corresponded with Darwin, and used George's logic to explain psychological phenomena, making her one of the first mathematical psychologists; her collected works fill four volumes. She raised the five girls alone.
- Ethel (youngest). Novelist of The Gadfly, a success in England and a phenomenon in Russia; married Wilfrid Voynich, discoverer of the Voynich manuscript, a fifteenth-century illustrated book in a still-undeciphered code.
- Lucy. Research chemist; first woman fellow of the Royal Institute of Chemistry and first woman professor at the London School of Medicine for Women.
- Alicia (middle). An intuitive mathematician who learned to visualize four-dimensional space, catalogued and built models of the regular 4-D polytopes (analogues of regular polygons and polyhedra) and characterized them before conventional mathematicians did.
- Margaret. Married the artist Edward Ingram Taylor; their son Sir Geoffrey Ingram Taylor, mathematical physicist on the Manhattan Project, received the Royal Society's Copley Medal exactly a century after his grandfather's medal.
- Mary Ellen (eldest). Married Charles Howard Hinton, son of James Hinton, a fourth-dimension enthusiast who had taught Alicia to visualize it, later a Princeton professor (and designer of a gunpowder-powered pitching machine for the baseball team). Descendants include nuclear physicist Joan Hinton, one of few women on the Manhattan Project; more Hintons appear later.
Boole showed that reason could be captured by a formal system. Testing logic as a theory of human thought would wait another hundred years, until psychologists and linguists began using formal systems to explain behavior.
Some details need care. The Royal Society's 1844 award to Boole was the Royal Medal, the first given for mathematics; "Gold Medal" is the book's loose label. Griffiths gives no dates for Boole's books: The Mathematical Analysis of Logic is 1847 and The Laws of Thought 1854. The count of 14 valid syllogisms is Aristotle's for his three figures; later traditions with a fourth figure list 19, or 24 if conclusions with existential import are added, so the number depends on the framework. Mary Boole's "first woman" status is, as Griffiths says, only possible.
Computing a Cognitive Revolution
Griffiths tells how psychology escaped behaviorism, which banned the mind from science, by borrowing the physicist's strategy: state hypotheses about unobservable mental processes mathematically and test them against behavior. Two strands met at MIT on 11 September 1956: the machinery of computation (Babbage, Turing, Shannon, von Neumann) and experiments on thought (Bruner, Goodnow, Austin, Miller). Logic supplied a single answer at three levels: what is true (possible worlds), how to derive it (inference rules), and how brains might do it (logical neurons).
An invisible subject
The people filing into an MIT conference room on the morning of 11 September 1956 for the Symposium on Information Theory did not know they were about to witness one of the defining days of cognitive science. The chapter explains why the day mattered by first showing how hard it had been to study the mind for the previous half-century.
Psychology was launched at the end of the nineteenth century by Wilhelm Wundt in Germany and William James (brother of the novelist Henry James) in the United States, and its practitioners soon grew anxious about their subject. Physics studies things very far away or very small, yet telescopes, microscopes, space probes and particle accelerators let physicists see or touch them; no instrument will ever let us see a thought or touch a feeling. The first psychologists therefore measured behavior, the downstream effect of thoughts and feelings, and inferred backward to the mind.
- Wundt's reaction times. A participant sat in a quiet room with a button; an electrical signal triggered a sound and started a timer, which the button press stopped. The measurement of reaction time was precise enough to catch a strange effect: participants sometimes responded at the exact instant of the sound, anticipating it. Wundt described the accompanying experience as hearing, reacting and feeling the reaction all at one moment.
- James's attention. James built on such results a theory of attention, the mind taking vivid possession of one of several possible objects or trains of thought, grounded in the finding that attending to a signal speeds the response to it.
For rigor-minded psychologists, inferring consciousness and attention from button presses was a stretch too far. This was the seed of behaviorism: psychology should study only what can be seen and touched, environments and the behaviors they produce. Some behaviorists went further and treated consciousness and attention as illusions a fuller science would dissolve. On the morning of the symposium behaviorism dominated psychology; by evening, Griffiths says, a new way of theorizing rigorously about the mind had appeared.
The rise of behaviorism
On a winter day in 1913, forty years before the symposium, John B. Watson addressed the New York branch of the American Psychological Association at Columbia University. He was a thirty-five-year-old Johns Hopkins professor who had risen from a poor, strictly religious upbringing in South Carolina. His audience, heirs of Wundt and James, relied on participants' reports of their thoughts and feelings (introspection). Watson told them, in the version printed that year in Psychological Review as "Psychology as the Behaviorist Views It", that psychology is a purely objective experimental branch of natural science whose goal is the prediction and control of behavior; introspection has no essential place in it, and the behaviorist recognizes no dividing line between humans and animals.
The argument was partly about rigor and partly temperament. Watson's work with animals made behavior seem sufficient, and his autobiography admits he never wanted to use human subjects, hated being one, and felt at home only with animals, close to biology.
- Little Albert. With Rosalie Rayner he taught a nine-month-old boy to cry and show distress at a white rat (and, it turned out, other white furry objects) by making a loud noise whenever the rat appeared. Even fear, a basic emotion, seemed reducible to the learning mechanisms seen in animals.
- Scandal and second career. In 1920 Watson's affair with Rayner and his wife's divorce suit became so scandalous that Johns Hopkins asked him to resign. He married Rayner and built a successful career in advertising, where he found a new product's sales curve as thrilling as an animal's learning curve.
Skinner and radical behaviorism
The torch passed above all to Burrhus Frederic ("Fred") Skinner, who first wanted to be a writer. In college he published poems in the campus magazine and at the Bread Loaf workshop in Vermont was encouraged by Robert Frost. After graduating he spent a year failing to write, then "six months of bohemian living" in Greenwich Village. There he read Bertrand Russell describing Watson's view that everything knowable about humans comes from external observation; he bought Watson's book on behaviorism and enrolled in Harvard's psychology department.
Skinner, like Watson, studied animal learning, and transformed its methods. He standardized experiments with the operant conditioning chamber, later called the "Skinner box", a cage where animals perform behaviors and receive rewards or punishments, and used it to analyze operant conditioning (learning to produce a behavior to gain a reward or avoid a punishment); his training techniques are still used on animals. He believed the principles of animal learning could explain all of human psychology.
Radical behaviorism went beyond Watson. Watson said thoughts and feelings need no place in psychology; Skinner granted that they exist but held they must never be used to explain behavior. They are themselves behaviors, to be explained by the same learning principles as everything else. In "Why I Am Not a Cognitive Psychologist" (1977), written long after many had left behaviorism, he redefined the mental vocabulary:
- Knowledge. Watching a football game or reading a book does not deposit facts inside us; it changes our behavior (we can now describe the game or summarize the book). Being "in possession of the facts" means having been affected by them, not containing them.
- Concepts. Changes in the contingencies between environment and behavior.
- Memory. A repertoire of established behavior patterns.
- Language. The product of associations between the environment and verbal behavior.
In 1956 Skinner was a Harvard professor, a mile from the MIT symposium. But behaviorism's grip was loosening, and its sharpest critics were closer still.
Glimpses of mind
Jerome Bruner, who would expose behaviorism's limits through studies of perception, was born blind; surgery removed congenital cataracts at two. He later said that not seeing made him construct a model of the visual world in his mind, and Griffiths sees a link to his lifelong argument that the internal worlds we build matter as much as the environment in explaining behavior. Griffiths met him in 2013, a spry ninety-seven-year-old in a book-filled New York apartment an hour from his childhood home in Far Rockaway, Queens, still in the thick spectacles he had worn since the surgery.
Bruner studied psychology at Duke and began a Harvard doctorate, but the war intervened. His family were Polish Jews, and his first trip to Europe in summer 1939 was cut short by the declaration of war; he sailed home from Bordeaux on the SS Shawnee feeling he was deserting those left behind. He made his research useful, writing his thesis on propaganda broadcasting by belligerent nations (partly at Princeton, where a group monitored foreign broadcasts), then worked for the FCC's Foreign Broadcast Monitoring Service in Washington and the Office of Public Opinion Research in Princeton.
Wanting a more direct role, he tried to enlist and was rejected for "insufficient vision". Through the Office of War Information he became a major in the Anglo-American Psychological Warfare Division of SHAEF, Eisenhower's Supreme Headquarters, arriving in London just before D-Day. In liberated France he was responsible for "cultural relations with the French university world", which meant dining and drinking with intellectuals, including Simone de Beauvoir and Jean-Paul Sartre (he and Sartre bonded over thick spectacles).
He came home with bigger ambitions and frustration that they did not fit behaviorism. In 1946 a group split from Harvard's psychology department to form the Department of Social Relations; Bruner joined it and found room for his work. With colleagues he began making cracks in behaviorism's walls, starting with perception.
With Cecile Goodman, a Radcliffe student writing her senior thesis, Bruner had children turn a knob that changed a lens aperture so a circle of light matched the size of a coin. All children overestimated the coins, more for valuable ones: a quarter by over 35%, a penny closer to 15%. Children came from two groups, "rich" ones from schools serving Boston's professional class and "poor" ones from a nearby slum, and the poor children overestimated by more than twice as much. The study made national news and has over 2,000 citations, probably making Goodman's one of the most influential senior theses in psychology, though later studies have struggled to reproduce it.
The point of this and other late-1940s studies was radical: people do not all experience the same environment the same way, so perception can differ in ways that cannot be measured from outside, a problem for behaviorism. Bruner's inspiration was literary. In a Faulkner novel a boy paid a quarter for chores feels it grow in his hand. The deeper idea, that experienced and actual worlds differ, he traced to a student performance of Hamlet, where Hamlet gets Polonius to agree a cloud looks like a camel, then a weasel. Bruner's date called it Shakespeare's theory of drama; Bruner said it was a theory of perception; she reconciled them: if life imitates art, perception imitates drama. Literature showed the richness of inner experience but did not answer the behaviorists' demand for a scientifically sound method. For that Bruner needed another muse, John von Neumann, which requires a detour into where thinking machines came from.
A mechanical interlude
Leibniz's calculators had taught him that if thought reduces to something like arithmetic, a machine can do it. Boole supplied the first half (a logic, later refined into modern mathematical logic); the second half, a machine that automates reasoning, remained open.
It nearly closed in Boole's lifetime. Charles Babbage, after years on his Difference Engine for approximating mathematical functions, turned in 1833 to the far more ambitious Analytical Engine. Inspired by the Jacquard loom, which wove patterns specified by punched cards, he planned to feed it punched-card instructions for any arithmetic calculation. Babbage saw calculating machines as aids for executing arithmetic. His collaborator Ada King, Countess of Lovelace, the estranged daughter of Lord Byron, saw further: the engine might act on things besides number whenever their relations could be expressed in the abstract science of operations, an idea that anticipates Boole's treatment of logic. She also wrote instructions for computing the Bernoulli numbers, the first published algorithm designed for a machine. She died in 1852, aged thirty-six.
Ten years later, at the 1862 International Exhibition in London, Boole met Babbage, who demonstrated his machine to him; a witness felt the two men together had taken steps toward a true calculating machine, Boole having shown reasoning could be mathematical and Babbage having mechanized mathematics. Nothing came of it, and Babbage died with the engine unfinished.
Today's digital computers needed more innovations. Just as ordinary arithmetic was the wrong formal system for thought, an ordinary calculator was the wrong machine for automating it. What was required was a machine that could implement any formal system, which in turn needed a link between logic and electricity.
The Turing machine
In summer 1935, after a long run along the River Cam, Alan Turing lay in a meadow and saw how to build a machine that could think. He was a fellow of King's College, Cambridge, a year past his mathematics degree. A question from Max Newman's course on the foundations of mathematics (the effort to express mathematics in logic) had stuck with him: can a mechanical process decide whether a mathematical statement is provable? Newman meant an abstract procedure a mathematician could follow; Turing wanted a concrete machine that does mathematics.
He began with a mathematician at work: paper in front of her, pencil in hand, reading a line, changing her mental state, writing the next line, sometimes going back to replace a line or forward to add a note. Then he replaced each part with a simpler abstraction.
- Paper → tape. Stretch the page left and right into an infinitely long tape.
- Writing → few symbols. Allow only a small alphabet, such as 0 and 1.
- Hand → head. A read/write head moves along the tape like a typewriter or printer head.
- One symbol at a time. Read a single symbol, write a single replacement.
- One step at a time. Move at most one cell left or right per write.
- Mind → finite states. Replace mental states with a finite set of numbered states.
- Knowledge → rules. If-then rules of the form (symbol read, state) → (symbol written, new state, move).
The mathematician's knowledge becomes rules too, but rules about the machine's states and actions. Inference rules from Chapter 1 can be written as if-then rules ("if you see x × (y + z), write x × y + x × z", i.e. x × (y + z) → x × y + x × z). For a Turing machine, the "if" names the symbol read and the current state; the "then" names the symbol to write, the next state and the direction to move. (1, 2) → (0, 3, R) means: reading 1 in state 2, write 0, go to state 3, move right.
- A zeroing machine. Reading left to right, it replaces every 1 with 0: (0,1) → (0,1,R); (1,1) → (0,1,R); ([],1) → ([],H,R), where [] is a blank and H is the halt state.
- A parity machine. Starting in state 1, it writes 0 at the end if the number of 1s is even, 1 if odd: (0,1) → (0,1,R); (0,2) → (0,2,R); (1,1) → (1,2,R); (1,2) → (1,1,R); ([],1) → (0,H,R); ([],2) → (1,H,R). Here the state records whether the count so far is even (1) or odd (2).
- Richer machines. Both examples make one pass; sophisticated machines move back and forth, using the tape to store intermediate results.
Turing argued that such a machine could produce any proof a human mathematician could: translate a branch of mathematics into machine rules and it gets an answer whenever a human could. More surprising, a single universal Turing machine can do this for every system: given a tape describing any other Turing machine's rules, it emulates that machine and gives the same answers.
The universal machine, changing its behavior when handed new instructions, founds computer science: it is the distinction between hardware (the physical substrate) and software (the program), which is why the field's Nobel equivalent is the Turing Award. Griffiths adds that such a machine should not surprise us, since the human brain is one: Boole's same brain followed the rules of differential calculus and of his new logic.
In the 1936 paper Turing also answered Newman's question, showing that no machine can determine whether a statement is provable. The paper that defined what a computer is also proved there are questions no computer can answer. Still, the universal machine meant one device could answer every question any machine can. Building one with an infinite paper tape was impractical; a physical computer needed a different route.
Claude Shannon and Boolean circuits
As Turing wrote, Claude Shannon, twenty, was finishing dual degrees in mathematics and engineering at Michigan. A postcard on a bulletin board advertised a job running a differential analyzer, a kind of analog computer, while doing a master's at MIT; he got it. Shannon had loved gadgets since childhood, building with erector sets and rigging a telegraph with a friend. Before digital computers, hard calculations ran on analog machines that physically simulated the mathematics (for example, tracing out a curve to find the area under it). Shannon's machine was a room-sized system of gears with an electrical controller, and his main job was connecting the controller's one hundred switches, which, unlike light switches, were themselves controlled by electricity.
He had taken a mathematical logic class at Michigan, and during a summer at Bell Telephone Laboratories in New Jersey after his first MIT year he saw the connection: a switch is on or off; a proposition is true or false. Could circuits correspond to logical formulas?
- The base circuit. A lamp, a battery and two wires; cut either wire and the lamp goes dark. A switch on one wire cuts and reconnects it.
- AND = series. Put two switches on the same wire and the lamp lights only if both are on; the on/off table is exactly the truth table of P ∧ Q.
- OR = parallel. Add a third wire duplicating one connection and put a switch on each duplicate; current flows if either switch is on, the table of P ∨ Q.
- Relays. An electrically controlled switch: current through a coil creates magnetic force that pulls one metal strip against another (often with a click), closing the switch, so one circuit can "pass the baton" to the next.
- Composition. Replace the lamp with a relay and it represents P ∧ Q or P ∨ Q; let it control another circuit, and circuits nest as formulas do.
- NOT. Wire a relay to break a connection when current flows and complete it otherwise.
- Result. With analogues of ∧, ∨ and ¬, any logical formula can be built as a circuit.
These Boolean circuits gave engineers a way to design circuits for complex functions and forged the link between logic and electricity. Shannon later said he never had more fun in his life; his first wife Norma (later a Hollywood screenwriter) recalled him repeating "Boolean" for the sound. He was modest: no one else happened to know both fields at once. Back at MIT he wrote it up in what has been called one of the greatest master's theses ever. When he signed up for a flying course, the instructor asked MIT's president whether it was wise to risk "the genius"; the president approved the class, judging that grounding him would do more harm.
From abstract machines to digital computers
Both men came to Princeton: Turing finished a Princeton PhD in 1938, and Shannon was a National Research Fellow at the Institute for Advanced Study (IAS) in 1940, among Alonzo Church, Kurt Gödel and Hermann Weyl. Brightest of all was John von Neumann, whose former student Israel Halperin remembered him radiating excitement in lectures on Hilbert space and operator rings, holding court at afternoon tea, and sometimes standing apart, lips moving silently, while nobody dared disturb him. Griffiths's summary: a brilliant mathematician, charming talker, sharp dresser, terrible driver, and a designer of the modern computer.
- Early life. Born in Hungary in 1903, gifted in languages and mathematics, son of a banker. His father insisted on a practical profession, so at seventeen he studied chemical engineering in Berlin while secretly enrolled in a mathematics PhD at Budapest; he had both by 1926.
- Output. Between 1927 and 1929 he published thirty-two major papers, from mathematical logic to quantum mechanics.
- Emigration. He began visiting Princeton in 1930 and moved for good by 1933, as anti-Semitism made Germany untenable for a Jewish academic.
- Turing. He probably met Turing in Cambridge in 1935, backed his Princeton fellowship (calling him "quite promising"), knew his computing ideas by summer 1938, and offered him a job; Turing went home to England.
Having seen totalitarianism firsthand, von Neumann threw himself into military consulting on explosive weapons, which led him into the Manhattan Project in 1943. Its calculations strained the limits of the possible. On a 1943 visit to England he had seen calculating machines used on similar problems; on the project he used IBM accounting machines, fed by punch cards like Jacquard's and programmed crudely by wiring a plugboard. They helped, but he knew something more powerful was needed.
So he was primed when, in late summer 1944, Army captain Herman Goldstine (a mathematics PhD) approached him on a train platform. Goldstine was liaison to the Moore School of Electrical Engineering at the University of Pennsylvania, which was building the ENIAC (Electronic Numerical Integrator and Computer). Once von Neumann learned the machine would do 333 multiplications per second, Goldstine recalled, the chat turned into something like a doctoral oral exam.
The room-filling ENIAC grew directly out of the analog analyzer's shortcomings: the Moore School had one, engineers kept speeding it up with electronic parts, and they realized the answer was a fully electronic computer. Its weakness was that each new computation meant physically rewiring cables and plugboards, done by the highly skilled "ENIAC Six": Kathleen McNulty, Betty Jean Jennings, Elizabeth Snyder, Marlyn Wescoff, Frances Bilas and Ruth Lichterman. Set-up could still take weeks.
The cure, worked out by von Neumann with the Moore School team, was the stored program: as a universal Turing machine emulates any machine by reading its tape, a stored-program computer runs any computation by changing the instructions in its memory, with no rewiring. The new machine was the EDVAC (Electronic Discrete Variable Automatic Computer), described in von Neumann's "First Draft of a Report on the EDVAC", dated 30 June 1945, months before the war ended.
- Arithmetic organ. Performs the arithmetic operations.
- Control organ. Decides which operation comes next and follows the program; with the arithmetic organ it forms the central processing unit.
- Memory. Stores intermediate results and the program itself, playing the role of Turing's infinite tape.
- Input organ. Reads program and data into memory (from punch cards), as the cortex has areas for the senses.
- Output organ. Returns results to the user, as the brain has areas for motor action.
Von Neumann was deeply inspired by the brain and modeled the five "organs" on it; he had in effect designed a practical universal Turing machine. After the war he returned to the IAS and began building one.
Mathematics and the mind
While building his computer, von Neumann unwittingly lit the fuse of the Cognitive Revolution. Bruner spent 1951 to 1952 at the IAS at the invitation of Robert Oppenheimer, whom he had known during the war (they stayed with the same friends in Washington) without realizing Oppenheimer ran Los Alamos. As IAS director after the war, Oppenheimer tried to spark ideas by mixing disciplines, and one success was putting Bruner, a psychologist keen to widen his field, next to von Neumann. Von Neumann was then designing a system to store and retrieve information from memory, which required specifying how facts are represented and how to search for them. Inspired, Bruner went back to Harvard, asked the dean for space for a new Cognition Project, and set out to show psychologists could study thought, not just perception. At its center was logic.
Boole, for all his interest in the Laws of Thought, was no psychologist: he wanted logic's abstract principles, not how people actually think, and wrote that knowing the laws of mind needs no extensive collection of observations. Bruner saw that logic could characterize thought precisely and decided to test it experimentally.
The project team:
- Jerome Bruner. Founder; brought logic into the lab after his IAS year with von Neumann.
- Jacqueline Goodnow. Born in Toowoomba, Australia; entered the University of Sydney at sixteen and won a University Medal. Sydney had no psychology doctorate, so she went to Harvard's Department of Social Relations, enrolled through Radcliffe as a woman, and finished her PhD in just over two years. Her thesis with Bruner showed people choose differently when they think they are solving a tricky problem rather than gambling, even with identical reward odds: same environment, different behavior. She returned from work in England and Germany to join the project.
- George Austin. Fresh Michigan PhD who had been at the IAS with Bruner; he died of cancer before the project was finished.
Concepts as logical rules
How the mind divides the world into categories is a question as old as Aristotle, and logic gives a neat answer: each category is a logical rule. For example Cat ↔ (Small ∧ Furry ∧ Domestic ∧ Carnivore): something is a cat if and only if it is a small, furry, domestic carnivore. The rule does two jobs. Knowing something is a cat, you can infer its properties; wondering whether something is a cat, you check the four properties, and if any is missing it is not a cat.
This contrasts sharply with behaviorism, which sees a label like "cat" as associated with features like small or furry, the association strengthening a little each time they co-occur, as a rat or pigeon slowly links stimulus and outcome. Logic is all or none: propositions are true or false, rules satisfied or not. Bruner's team asked whether human categories share that property.
- Materials. Cards varying on four dimensions with three values each: shape (cross, circle, square), color (white, black, gray), number of figures (one to three) and number of border lines (one to three). Many rules can be defined, such as Black ∧ Circle or White ∨ Square.
- Procedure. Harvard undergraduates were told the experimenter had a concept in mind, picked cards, heard whether each matched, and could guess the concept and learn whether they were right.
- Finding 1. People did not slowly accumulate associations; they explicitly formed and tested hypotheses, with deliberate strategies such as exploring one dimension or varying a card already known to belong.
- Finding 2. Performance depended on the rule's logical structure: disjunctions (White ∨ Square) were far harder than conjunctions (Black ∧ Circle). Bruner observed that members of a disjunctive class share something with each other but not everything, making such categories "intellectual nightmares".
They published the work as A Study of Thinking (1956). Bruner called it a "protest" book, an escape from behaviorism's anti-intellectual corset through rigorous logic, more a logician's than a psychologist's approach. Oppenheimer's review said it had the flavor of the opening of a new science: one using mathematics, here logic, to build precise theories of mind.
George Miller and information theory
The work left its mark on Bruner's Harvard colleague George Miller. Raised in West Virginia, Miller found psychology by accident at the University of Alabama: he was interested in a young woman who was interested in psychology, met a professor who took him under his wing, and became a psychologist while she became his wife. He took his Harvard PhD just after Bruner left for the war, joined the psychology faculty, and was there when Bruner returned to Social Relations.
Miller cared about language, which led him to the mathematics of communication. Wartime radio had raised a basic question: what is the most efficient way to use a channel? Every message must be turned into a code the channel can carry; in Morse code each letter is a sequence of long and short sounds, with E = · and Z = −−··. Efficiency means finding the best code. The answer came from Shannon, who after the IAS moved to AT&T's Bell Laboratories, worked on wartime code-breaking, and wrote "A Mathematical Theory of Communication", explaining not only why Morse gives E a short signal and Z a long one but giving a general theory of efficient communication: information theory. Miller saw it could describe how the mind processes information and said his life was never the same.
He spent a few years at MIT applying the theory to speech perception, then returned to Harvard. Hearing Bruner's category results (information processing par excellence), he urged Bruner to read Shannon. Bruner declined: the logical hypotheses of A Study of Thinking were as far as he would take mathematics, and he moved on to language, cognitive development and education. Miller in turn grew disenchanted with information theory as an account of language and began seeking an alternative. The stage was set for the day Miller later called the conception of cognitive science.
11 September 1956
The symposium convinced Miller that mathematics could study the mind rigorously, and he wrote a first-person account of the day. It takes a village to start a revolution, Griffiths remarks, and three more villagers appear here, sketched briefly before later chapters: Allen Newell, Herbert Simon and Noam Chomsky. Day one did what the name promised: Shannon headlined, followed by talks such as "A Linear Circuit Viewpoint on Error-Correcting Codes" and "Theory of Information Feedback Systems". Day two turned cognitive.
| Talk | Speakers | What it did |
|---|---|---|
| The Logic Theory Machine: A Complex Information Processing System | Allen Newell, Herbert Simon (fresh from the first academic AI workshop) | A program that discovered proofs in mathematical logic by emulating human problem solvers, relying on heuristics rather than systematic algorithms; opened the way to computer models of thought |
| Three Models for the Description of Language | Noam Chomsky | Demolished the information theorists' (and implicitly behaviorists') model by exhibiting acceptable English sentences it cannot generate; proposed two new formalisms; showed grammaticality judgments can test sophisticated formal hypotheses, setting linguistics' agenda for the rest of the century |
| Human Memory and the Storage of Information | George Miller | Used information theory to study memory capacity; an ingredient of "The Magical Number Seven, Plus or Minus Two" |
Miller's famous paper observed that the number of stimuli on one scale people can tell apart (such as tones differing in pitch) roughly equals the number of items they can hold in mind at once (such as the digits of a phone number): about seven. Bruner thought it deserved a retrospective Nobel for the mid-1950s on that one article. Miller was humbler: asked for an hour-long talk, he had two unrelated thirty-minute talks and tied them together with the only thing they shared, the number seven.
Another day of talks followed, but Miller had seen enough. He left with a conviction, "more intuitive than rational", that experimental psychology, theoretical linguistics and computer simulation of cognition were pieces of one whole that would be progressively coordinated.
The common strategy answered the behaviorists' worry about rigor: express theories of mind mathematically, then test them against behavior. That is the founding method of cognitive science.
1913 Watson's behaviorist manifesto at Columbia.
1920 Watson leaves Johns Hopkins for advertising.
1936 Turing's paper on computable numbers.
1937-38 Shannon's master's thesis on Boolean circuits.
1943 McCulloch and Pitts's logical neurons; von Neumann joins the Manhattan Project.
1945 First Draft of a Report on the EDVAC (30 June).
1946 Harvard's Department of Social Relations founded.
1948 Shannon's "A Mathematical Theory of Communication".
1951-52 Bruner at the IAS with von Neumann.
1956 A Study of Thinking; the MIT symposium (11 September).
1977 Skinner's "Why I Am Not a Cognitive Psychologist".
A compelling answer
Many kinds of mathematics were on show at the symposium, neural networks and probability theory among them. But the two papers with the greatest later impact both used logic or its relatives: Newell and Simon took logic as their subject, and Chomsky's language models were close cousins of the systems used to define logical formulas. Add Bruner's logical concepts and 1956 was a banner year for the idea that thought is a formal system based on logic. Part of logic's appeal, Griffiths argues, is that it answers questions about the mind at three levels at once.
- Abstract level. Logic's language (propositions, conjunctions, disjunctions) states facts and concepts precisely, and its theory says when a statement is justified by others we accept. By relating possible worlds to propositions, it turns working out what we know into a solvable math problem, a good basis for rigorous theories of knowledge, concepts and language.
- Procedural level. Inference rules such as modus ponens say which statements follow from which, and chains of them make longer proofs with unexpected consequences. Because the procedure uses syntax only, it could be automated, giving digital computers, and it suggests that thinking is applying inference rules to statements in the head. Bruner began here (how people revise logical hypotheses given new information); Newell and Simon would build a whole theory of problem-solving on it.
- Neural level. In 1940 Warren McCulloch, a distinguished psychiatry professor at the University of Illinois, met Walter Pitts, a homeless teenage polymath who had read mathematical logic since twelve. Their 1943 paper "A Logical Calculus of the Ideas Immanent in Nervous Activity" showed how neurons could build logical circuits.
In the McCulloch-Pitts model each connection is excitatory (a positive signal) or inhibitory (a negative one). A neuron sums its inputs and fires when the sum reaches a threshold, sending a signal to every neuron it connects to. In their figures each excitatory dot adds 1, an inhibitory circle subtracts 1, and the threshold is 2: one neuron can simply relay another; a neuron can fire if either of two inputs fires (∨) or only if both do (∧); and inhibition implements ¬, so neuron 3 fires when neuron 1 fires and neuron 2 does not. The paper was the neural counterpart of Shannon's circuits and influenced von Neumann's EDVAC design.
Rules and symbols, formal systems built on logic, thus became the first candidate for the Laws of Thought. The next chapters follow how the approach was used to explain reasoning and language, and then the problems it ran into.
Griffiths's heroic compression omits some credit disputes. Alonzo Church proved the undecidability of provability with the lambda calculus months before Turing's 1936 paper (Turing then did his PhD with Church). The "von Neumann architecture" drew heavily on ENIAC designers J. Presper Eckert and John Mauchly, who resented the single-author First Draft. Watson's Little Albert study is now a textbook case of unethical research, and the child's identity remains disputed. Griffiths does flag the shaky replication of the coin study, which matters because it was a founding crack in behaviorism. He also gives no dates for Shannon's thesis (1937, published 1938) or his communication paper (1948); the timeline above supplies them.
Solving Problems
Newell, Simon and Shaw turned the rules-and-symbols view into working programs, showing that chess, geometry and logic are all search through a tree of choices and that humans tame that tree with heuristics. Their path ran from the Logic Theorist (1956) through the General Problem Solver and production systems to the physical symbol system hypothesis (Turing Award lecture). Simon's ant on the beach adds the twist: much of the complexity of behavior comes from the environment, not from the rules.
A thinking machine
The war had concentrated enormous intellectual effort on military problems, and the RAND Corporation was created afterward to keep it going, with staff scientists running ambitious projects and university professors joining as consultants. One project, the Systems Research Laboratory, studied how people solve problems together by building a model of military decision-making "made of metal, flesh, and blood": college students played the crew of an air defense radar station, facing simulated scenarios while scientists recorded their actions and interactions.
A consultant was Herbert Simon of the Carnegie Institute of Technology in Pittsburgh (later merged with the Mellon Institute into Carnegie Mellon), a political scientist and economist known for studying organizations, an ideal fit for a lab that manufactured organizations. Visiting in February 1952, four years before the MIT symposium, he watched RAND scientists Allen Newell and Clifford Shaw generate imitation radar maps on a computer. The machine was producing map locations, output that could be read as abstract symbols, and Simon had Ada Lovelace's thought of a century earlier: computers could be general symbol systems, able to process symbols of any kind. The IBM punch-card calculators he had used only crunched numbers; machines that crunched symbols changed his thinking.
Feb 1952 Simon sees Newell and Shaw's computer-generated radar maps at RAND.
Summer 1952 Von Neumann lectures at RAND on computer chess (already explored by Turing and Shannon).
1954 On a long drive to an air force base, Newell becomes convinced he can program good chess, as a means to studying "ultracomplicated" problems.
1955 Work on RAND's JOHNNIAC moves from chess to geometry to logic; by December a logic prover is in sight.
Jan 1956 Simon tells a class he and Newell invented a thinking machine over Christmas.
Jun 1956 Program described at the first AI workshop, where "artificial intelligence" (AI) became the field's name.
9 Aug 1956 The automated Logic Theorist completes its first proof, a month before the MIT symposium.
Chess was never the real goal; Newell's chess paper called it a way to learn what computers, mechanisms and programs are needed for "ultracomplicated problems". He recruited Shaw, the radar-map maker, and they began on JOHNNIAC, the computer von Neumann had designed for RAND. Newell moved to Pittsburgh to do a PhD under Simon, though Simon always insisted they were equal partners. Meeting on Saturdays to maximize time, they shifted from chess to geometry and then to logic.
The machine lagged the announcement by months. The first run of the Logic Theorist was a model of "card stock, flesh, and blood": Simon gave his wife, their children aged nine, eleven and thirteen, and some graduate students each a card holding part of the program, and by following its rules together they proved theorems in logic. Meanwhile Newell phoned program descriptions to Shaw, running up hundreds of dollars a week in bills, and Shaw implemented them on JOHNNIAC.
Making choices
Pawns, triangles and conjunctions look unrelated, so how did a chess project become a logic prover? All three are formal systems. Chess was Chapter 1's first example; logic exists to express thought as rules about which statements follow from which; and Euclid showed geometry's properties can be derived from a few axioms about points and lines. Every formal system poses the same problem the moment you use it: which rule do you apply next?
In chess, White opens with 20 legal moves (each of 8 pawns forward one or two squares, each knight to two squares), and books have been written on which to choose. The difficulty is that you know where you want to end up (a win) but not the sequence of moves that gets there.
- 20 opening options for each side, so 20 × 20 = 400 positions after one move each.
- 8,902 positions after White's second move; 197,281 after Black's reply.
- 2,015,099,950,053,364,471,960 positions further on (the book says after eight moves each).
- Over 60 trillion years to try them all at one move per second.
Griffiths pictures this as a tree: the trunk is the starting position, the first branches the 20 moves, each splitting into 20 replies, and so on until a leaf, a finished game, sprouts from a tiny twig. Playing chess means choosing branches to reach a winning leaf. Proofs are the same: each step applies an inference rule such as modus ponens, and the prover must choose branches leading to the leaf that is the desired result. Because the possibilities cannot be enumerated one by one, this is what Newell called an ultracomplicated problem, and it needs a smarter way to consider paths.
- Allen Newell. Grew up in San Francisco, summers in the Sierra Nevada, football at Lowell High School. In the Navy he served on a ship carrying scientific observers to the Bikini nuclear tests, which drew him to science. Studied physics at Stanford, where he took George Pólya's class (Pólya wrote How to Solve It on discovering proofs). Began a Princeton mathematics PhD but wanted problems he could go out and solve, not pure mathematics; RAND suited him.
- Herbert Simon. Grew up in Milwaukee; father an engineer turned patent attorney, mother a music teacher. At the University of Chicago he meant to major in economics until he found it required accounting, so he chose political science. Took classes from Rudolf Carnap and dived into mathematical logic; his dissertation, on the logical foundations of administrative science, became Administrative Behavior. Directed a municipal administration study at Berkeley, taught at the Illinois Institute of Technology, joined weekly meetings of the Cowles Foundation, where Merrill Flood brought him into RAND, then moved to Carnegie Tech.
- Clifford Shaw. Grew up in Fullerton, California, around his family's paint and decorating store. Studied mathematics, became an actuary, served as a wartime navigator (chart plotting, he said, is modeling), worked at an insurance company in Los Angeles, then joined RAND the same year as Newell and wrote the code that ran JOHNNIAC.
The Logic Theorist
The program combined three inheritances: mathematical logic from Simon's time with Carnap, proof strategies from Newell's class with Pólya, and programming from Shaw's work on JOHNNIAC. Its theorems came from Principia Mathematica, by Bertrand Russell (the philosopher who had sent Skinner into psychology) and Alfred North Whitehead, an attempt to derive all mathematics from a few axioms. Simon met it in college, and its simplicity made it the natural place to start automating proof.
- Three connectives. Or (∨), not (¬) and if (→).
- Five axioms. Basic logical truths such as (P ∨ P) → P and P → (Q ∨ P).
- Two inference rules. One of them modus ponens.
- Task. Chain the rules to prove Principia's theorems one after another.
The catch is that each rule can apply to any statement derived so far, axioms or earlier results, so the tree is extremely bushy. The aim was to find proofs without walking every branch, and here Pólya's study of how human mathematicians actually work helped. He called their rules of thumb heuristics; by using some of them, the Logic Theorist could claim to be a model of human cognition.
- Goal-directed heuristics. Both main heuristics exploit knowing the statement you want to prove.
- "Do you know a related problem?" Pólya's question: match the target against already-proved statements and use their proofs as starting points.
- Working backward. Pólya's advice to solve a related problem first: to prove A, it suffices to prove B and B → A, then apply modus ponens; if either is already proved (or something similar is), you are well on the way.
- List processing. Even with heuristics the program searched a lot. Shaw found JOHNNIAC's language "far too low-level", so he and Newell built an information-processing language for searching trees, whose basic unit is the list (like a shopping list), easy to add to and delete from. A proof is a list of rules and intermediate results; explored paths are stored as lists too.
- Formula for success. A simple mathematical domain, a smart set of heuristics and the right programming language.
The Logic Theorist proved some of Principia's theorems. In October 1956 Simon wrote to Russell, enclosing the symposium paper. Russell replied with good humor that he wished he and Whitehead had known of this before they wasted ten years doing it by hand, and later declared himself delighted that the machine had found a more elegant proof* of one theorem than theirs. The editor of The Journal of Symbolic Logic was less charmed and *rejected a paper listing the Logic Theorist as co-author because the result was not new.
From problems to productions
The Logic Theorist was inspired by human cognition but missed something. When Newell and Simon had people think aloud while solving logic problems, a different heuristic dominated: means-ends analysis. You usually know where you are and where you want to be, but only the moves available from here; so make the move that brings you closest to the goal.
Griffiths's example is a rail trip from Princeton, New Jersey, to Berkeley, California. From Princeton Junction you can go northeast to New York or southwest to Philadelphia; means-ends analysis says Philadelphia, which is closer to the West Coast, and it works, continuing through Pittsburgh (home of Newell and Simon) and Chicago to California. In general form you have a goal (Berkeley), find differences between the current state and the goal (distance), and take actions that reduce them (the Philadelphia train). Define goals, states, differences and actions for any domain and the method applies. Newell, Shaw and Simon built a program on it, the General Problem Solver, and showed it solving problems in several domains, including logic and algebra.
Cryptarithmetic and the limits of GPS
They kept attacking fiendish puzzles, favorite among them cryptarithmetic: sums with digits replaced by letters, a different digit per letter, to be decoded. The classic SEND + MORE = MONEY was first published in 1924. Brute force means trying assignments of digits 0 to 9 to the eight letters D, E, M, N, O, R, S, Y: 1,814,400 possibilities. People instead use arithmetic constraints: in the leftmost column, M must be 1, because two digits sum to at most 8 + 9 = 17.
Working through a solver's verbal report, Newell saw that the General Problem Solver did not capture it. People were not searching the tree; they were setting goals and manipulating symbols, like running a simple computer program in their head. A 1967 memo of his gave the needed formal system: the production system.
Production systems
A production system is a set of if-then rules: the "if" describes a situation and the "then" says what to do; the rule is a production because the condition produces the action. A fire-alarm sign saying "In case of fire, break glass" is a production. Newell and Simon's thermostat:
- (temperature > 70°) ∧ (temperature < 72°) → stop.
- temperature < 32° → call for repairs; turn on electric heater.
- (temperature < 70°) ∧ furnace off → turn on furnace.
- (temperature > 72°) ∧ furnace on → turn off furnace.
Following these four rules the thermostat holds the temperature and even detects emergencies.
The idea is simple but powerful. An early use was computing on strings of letters and numbers. The rule a → b means: scan the string left to right, replace the first a with b, return to the start, and repeat until a full pass uses no rule, then stop; so aaa → baa → bba → bbb. With several rules you try them in order: check rule one against the string from the left, then rule two, then three; after any match, make the change and start again at rule one; halt when nothing matches any rule.
Griffiths's puzzle uses three rules: X0 → 0XX, 1 → 0X, and 0 → (delete the 0). On input 1: rule one fails, rule two gives 0X, rules one and two then fail, rule three deletes the 0, leaving X. On input 10: rule two gives 0X0, rule one now matches to give 00XX, and rule three deletes both 0s, leaving XX.
| Rules (in order) | Input | Output | What it computes |
|---|---|---|---|
| X0 → 0XX; 1 → 0X; 0 → (delete) | 1 / 10 / 11 / 100 / 101 / 110 / 111 | X / XX / XXX / XXXX / XXXXX / XXXXXX / XXXXXXX | Binary number → that many Xs, for infinitely many inputs |
Three rules capture a computation that gives infinitely many correct outputs for infinitely many inputs.
- Universality. A production system can do anything a Turing machine can, so production systems are an alternative definition of computation; each is a little program, exactly what Newell wanted.
- Means-ends as a special case. Put the differences on the "if" side and the reducing actions on the "then" side.
- More general. The "if" side can also list goals and the state of the analysis so far, and the "then" side can hold complex operations. One of Newell's cryptarithmetic rules: if you know one letter's value, find a column containing that letter and do the calculation there. A few such rules captured people's complex solution processes.
- Landmark. Productions were central to Human Problem Solving (1972, 920 pages), whose conclusion confessed a "strong premonition" that human programs are organized like production systems.
Twenty years after Simon's insight at RAND the work had come full circle: from using human minds to design better computer programs to using computer programs to describe how human minds work.
Physical symbol systems
Turing Award winners give a lecture to a broad computing audience, and Newell and Simon (the book dates the award to 1976) used theirs for the clearest, boldest statement that rules and symbols are all you need for intelligence. They defined a physical symbol system: a physical machine producing symbol structures that describe objects and their behavior, structures that are themselves physically instantiated and acted on by processes that create, destroy and modify them. This is a formal system with one more requirement, physical realization. Propositional logic and Turing machines are not physical symbol systems, being abstract; von Neumann's computer is, since it turns rules and symbols into atoms and electrons.
Physical symbol system hypothesis: such a system has the necessary and sufficient means for general intelligent action. Necessity (every intelligent system is one) is tested by cognitive science; sufficiency (rules and symbols are enough to build intelligence) by AI.
Each test runs as research proceeds: cognitive scientists modeling human thought with rules and symbols on computers find out how much of our intelligence fits, and AI researchers building intelligent systems probe the limits of physical symbol systems. Newell and Simon saw no limit. Simon told his daughter in a letter that research would eventually settle how many human thinking processes can be simulated, that he believed all of them can, and that he felt no urge to convince doubters; problem after problem had yielded so far.
Writing down common sense
The idea that intelligence is a matter of writing down the right rules shaped AI. Doug Lenat left a Stanford faculty post in 1984 to devote industrial-scale resources to it, and it became his life's work. After decades of effort by computer scientists and philosophers, the system (Cyc) held more than twenty-five million rules aimed at common sense, the background knowledge that guides action without explicit reasoning: for instance, that most people sleep at night, for hours, lying down with eyes closed, and can be woken by a loud noise but dislike it.
Chatbots and the Turing test
Large rule sets proved surprisingly effective for chatbots, systems that converse in typed text. In 1950 Turing proposed testing machine intelligence by whether a machine can act indistinguishably from a human: a judge questions a human and a machine and must say which is which, a perfect task for a chatbot. Turing predicted that in about fifty years machines would play the imitation game well enough that an average interrogator would have no more than a 70% chance of a correct identification after five minutes.
At a Turing test held for the sixtieth anniversary of Turing's death, the chatbot "Eugene Goostman", built by Russian and Ukrainian programmers to simulate a thirteen-year-old Ukrainian boy, convinced a third of the judges it was human, in line with Turing's prediction. Its core was a massive set of if-then rules anticipating likely questions (the if) and supplying responses (the then); some gave good answers to common questions, others sidestepped harder ones. When computer scientist Scott Aaronson asked whether fooling unsophisticated judges exposed a flaw in the test, Eugene replied with a fake server-maintenance error; asked whether Turing failed to foresee judges accepting evasion, it produced garbled banter ending in "Anything else?". Griffiths's point: a remarkably simple system of rules can go a long way in emulating intelligent behavior.
Ants on the beach
The hypothesis looks ambitious because human behavior looks so complex. Part of Newell and Simon's optimism was a belief that the underlying rules might be simple. In The Sciences of the Artificial Simon asks us to picture an ant on a beach: it moves ahead, angles right to climb a small dune, detours round a pebble, pauses to exchange information with another ant, and so weaves its halting way home. The path is hard to describe mathematically, yet the complexity belongs to the beach, not the ant: its behavior largely reflects the complexity of its environment.
The parable offers another route to rules and symbols: humans behave complexly but also live in complex environments, so how much of our behavior is simple rules applied to a complicated world? It also explains Eugene Goostman's success: in a conversation, as long as one party supplies complexity, the other can get by on simple rules.
Griffiths sums up their legacy in three parts:
- Computers as theory-testers. They showed how computers could formulate and test hypotheses about human thought.
- Heuristic search. They explained how people handle ultracomplicated problems in logic, geometry and chess by using heuristics to search the tree of possibilities efficiently.
- Goals and subgoals. They saw that people break hard tasks into goals and subgoals that can be expressed and pursued with simple rule sets such as production systems.
Their partnership adds a human lesson: for hard problems, the best heuristic can be finding another mind that complements yours. With different experiences and rules, Newell and Simon injected complexity into each other's behavior. Simon, expert on organizations and the organisms in them, wrote in a book on automation and jobs that people are problem-solving, skill-using, social animals, with two deep needs once fed: to apply their skills to challenging tasks and to have warm relations with a few others. Griffiths ends on that note: humans build environments for one another; sometimes we are the ants, but just as often we are the bumps and furrows in the sand that guide other beachcombers.
Several details are loose. The 2,015,099,950,053,364,471,960 figure is the count after 15 half-moves (7.5 moves each), not "eight moves each". Newell and Simon received the 1975 Turing Award (the lecture appeared in print in 1976). "Artificial intelligence" was coined in McCarthy's 1955 Dartmouth proposal, then adopted at the 1956 workshop. Eugene Goostman's 2014 "pass" (at the Royal Society, London) was widely dismissed as exploiting a child persona and broken English, as Aaronson's transcript shows; it says little about intelligence.
Search plus learned heuristics is now the frontier. Reasoning models (OpenAI o1, Sept 2024; DeepSeek-R1, Jan 2025) are trained by reinforcement learning to search through long chains of thought, and in July 2025 Google DeepMind and OpenAI models reached gold-medal level at the International Mathematical Olympiad in natural language. DeepMind's AlphaProof (Nature, Nov 2025) proves theorems in the Lean formal system with RL-guided search, a direct descendant of the Logic Theorist. In August 2026 OpenAI reported that an internal model, Astra, produced Lean 4-checked proofs on ten long-open problems. On imitation, Jones and Bergen (UC San Diego, March 2025) found GPT-4.5, with a persona prompt, was judged human 73% of the time in three-party Turing tests. Doug Lenat died in August 2023; his hand-coded Cyc is now cited as the road not taken.
Language as a Formal System
Noam Chomsky recast linguistics as the study of formal systems that generate languages, and he tested candidate systems against the one instrument every speaker carries: judgments of what is and is not a sentence. He showed that word-to-word statistics (finite-state grammars) cannot capture English, proposed phrase structure grammars plus transformations, and arranged all languages into the Chomsky hierarchy. The richer the rules needed to describe language, the harder they are to learn: the poverty of the stimulus led Chomsky to innate, genetically endowed knowledge of language, a conclusion that modern AI now puts under pressure.
A twenty-seven-year-old at the podium
When Chomsky rose to speak at the 1956 Symposium on Information Theory at MIT he was 27 and had been on the MIT faculty for just one year. By his own wry account he was teaching language "cram courses" that helped graduate students scrape through PhD reading exams in French and German. His magnum opus, the nearly 1,000-page The Logical Structure of Linguistic Theory, had just been turned down by MIT Press; one reviewer quipped that it was the kind of book that could only appear once its author was famous. The talk he was about to give became one of the first steps toward exactly that fame.
The room was full of elite mathematicians who thought about communication every day, the most eminent being Claude Shannon. Chomsky's message to them was blunt: their way of thinking about language was wrong from the ground up.
Life, language and logic
Language had surrounded Chomsky from childhood. His parents, Russian émigrés, taught in several Hebrew schools before settling at the school of the Mikveh Congregation in Philadelphia. His father was a Hebrew scholar who wrote Hebrew: The Eternal Language and a grammar of medieval Hebrew, and Noam read his father's historical work at ten. That same year he published his first article, an editorial on the Spanish Civil War in his elementary school's newspaper. Politics and language would stay braided together for the rest of his life.
He entered the University of Pennsylvania at sixteen and soon tired of formal education. He took Arabic, his anarchist and left-wing politics deepened, and after two years he was ready to leave, with vague plans of going to Palestine, perhaps to a kibbutz, to work for Arab-Jewish cooperation in a socialist setting. What changed his course was meeting the linguist Zellig Harris, who came from the same Russian émigré community as his parents and shared his political causes. Their relationship began over politics and quickly turned to linguistics.
Structuralism: linguistics under behaviorism
Linguistics had absorbed the same behaviorist impulse as psychology. The dominant paradigm studied only the observable side of language (sounds, words, sentences), characterizing them within each language and comparing them across the world's languages. Harris led the effort to find rigorous methods for justifying conclusions about a language's structure from the utterances its speakers actually produce, summarized in his Methods in Structural Linguistics. Before Chomsky had taken a single linguistics class, Harris asked him to proofread it; the preface, dated January 1947, thanks "N. Chomsky" for much-needed help with the manuscript.
From describing a language to generating it
Chomsky's undergraduate thesis, under Harris, was a grammar of Hebrew, but Harris's methods gave results he found dull and unsatisfying. So he flipped the question. Instead of rules that relate a language's structure to observed data, could one write down rules that describe the language itself? He had learned logic from reading Bertrand Russell and Rudolf Carnap and from classes with the philosopher Nelson Goodman (who returns in Chapter 5). Logic uses a system of rules to generate proofs; perhaps a similar system could generate a language.
- First attempt. The thesis grew into his master's thesis, Morphophonemics of Modern Hebrew, the first instance of treating a language as a formal system in which a whole language is produced from a simple rule set, with many tools borrowed from logic.
- Harvard, 1951. He left Penn for Harvard's Society of Fellows, a post giving him complete research freedom, but he kept struggling to square his approach with Harris's methods, which he characterized as discovery procedures for extracting a language's structure.
- The shipboard insight, 1953. Seasick on a listing, salvaged ship that the Germans had once sunk, making its first crossing after repair, he suddenly saw the obvious reason why years of work on discovery procedures had gone nowhere while his nearly solitary work on generative grammar and explanatory theory kept producing results.
- The dissertation, 1955. He spent two years working out the consequences in The Logical Structure of Linguistic Theory and submitted one chapter of it as his Penn dissertation. A committee member reportedly said he did not know what field it belonged to, but it certainly was not linguistics. Chomsky got the PhD in 1955 and moved to MIT.
Through the Society of Fellows he had met researchers across many disciplines, including Peter Elias, who organized the 1956 symposium. That gave Chomsky an audience with far more mathematicians than linguists, one likelier to welcome a heavily mathematical account of language.
Colorless green ideas
Information theorists cared about language because they wanted better codes. Suppose each English letter must be sent as a string of 0s and 1s. There are 2 one-symbol strings, 4 of length two, 8 of length three, 16 of length four, so with 26 letters some must get longer codes. An efficient code keeps messages short by giving the shortest strings to the most frequent letters (E, T) and the longest to the rarest (Q, Z). Real codes do exactly this: in Morse, E is · and T is −, while Q is − − · − and Z is − − · ·. Part of Shannon's achievement was to turn this link between code length and probability into a general theory of efficient coding.
To build such codes you need the probabilities of letters and words, and the information theorists estimated them by counting occurrences in text. Counting pairs of letters or words did even better, which suggests a simple generative model: start with a word that has begun a sentence before, then choose each next word from the words that have followed the previous one. The output looks surprisingly like English.
The model also fit behaviorism neatly. Words are observable, and links between them can be learned as associations: each time one word follows another, their association strengthens. Animals learn this way in many settings, so behaviorists could explain language without any mechanism beyond those already seen in other species.
The famous nonsense sentence
Chomsky's first move was to say this would not work, not even for English, and not with sequences of any length. He offered what became the world's best-known nonsense sentence, "Colorless green ideas sleep furiously." Its meaning is obscure, yet any speaker recognizes it as grammatical, unlike "Furiously sleep ideas green colorless." A theory of language must explain the difference, and counting cannot. Chomsky had built the sentence deliberately from improbable word pairs: in 1956 his listeners had almost certainly never heard "green" after "colorless", "ideas" after "green", "sleep" after "ideas" or "furiously" after "sleep". Both strings are equally unattested, so no count of word sequences can tell them apart.
The method mattered as much as the result. Chomsky joined a mathematician's models to a linguist's data: a single sentence that violates a model refutes it, so rival formal systems could be tested quickly and anyone who speaks the language can check the verdict.
Three models
His paper, "Three Models for the Description of Language", played out in three acts, each introducing a formal system: a villain, a hero and the hero's helper. First he set the stage by recasting the linguist's problem as a mathematician's one, the construction and analysis of a system of rules and symbols.
- Language, defined. A language is a set (finite or infinite) of sentences, each of finite length, built from a finite alphabet of symbols. Nothing in the definition mentions communication or meaning.
- Grammar, defined. A grammar of L is a device that produces all the strings that are sentences of L and only these. This is the core of generative grammar: the linguist should build a device, usually a rule system, that generates the language, just as logic's inference rules derive all and only the consequences of a set of propositions.
The task then became clean: evaluate formal systems for their adequacy in capturing human languages.
- Finite-state grammar (the villain). A board game from START to FINISH that writes the word on each path taken; it formalizes the information theorists' and behaviorists' idea that each word depends only on the one before, and it fails for English.
- Phrase structure grammar (the hero). Rewrite rules such as Sentence → NounPhrase VerbPhrase build hierarchical trees, explaining why phrases can be swapped and how questions and answers relate to statements.
- Transformations (the helper). Rules that turn one sentence into a related one (active into passive, statement into question), borrowed from Harris but used generatively, so phrase structure need only produce a small kernel.
Act one: the finite-state villain
Griffiths returns to his board-game picture of formal systems. Put a piece on START, move along the arrows, write the word on each path, and stop at FINISH: you have a sentence. The first board yields only "The dog runs" and "The dogs run". Add one looping move labelled "hairy" and the same board yields infinitely many sentences: "The hairy dogs run", "The hairy hairy dogs run" and so on. A few rules, unlimited output: what Chomsky called making "infinite use of finite means". Because the board has a finite set of positions (states), he called this a finite-state grammar and its output a finite-state language.
His argument that English is not finite-state starts from a toy language. The apple-banana language has two words; a sentence is any number of apples followed by the same number of bananas ("apple banana", "apple apple banana banana", but not "apple apple banana"). No finite board can generate it. To guarantee the right count of bananas you would need a separate path for every number of apples, and since any number is allowed the board would need infinitely many states, contradicting the definition.
He then found the same structure in English through center-embedding:
- "The dog runs." A grammatical base sentence.
- "The dog the cat chases runs." One embedded clause.
- "The dog the cat the mouse fears chases runs." Two: the cat frightens the mouse and makes the dog run.
Such sentences get hard to parse, but Chomsky cared only that a speaker with time and paper would accept them. Allow them to grow without limit and each noun (dog, cat, mouse) must pair with its own verb (runs, chases, fears) just as each apple needs a banana: to choose "runs" over "run" at the end, you must remember that the first noun was "dog" and not "dogs". His conclusion: any finite-state grammar that produces only English sentences will fail to produce infinitely many of them, including infinitely many true sentences, false ones and sensible questions. A model that fails for English cannot describe all human languages.
Act two: the phrase structure hero
Griffiths admits the phrase "phrase structure grammar" sends some readers to sleep and gives others flashbacks to English class, but insists it is one of the best demonstrations of what rules and symbols can do. The idea is to capture sentence structure directly. "Colorless green ideas sleep furiously" splits into a noun phrase naming an agent ("colorless green ideas") and a verb phrase naming an action ("sleep furiously").
- Substitution. Either part can be swapped for another phrase of the same type and the result stays grammatical: "Involuntary beige liberties sleep furiously", "Colorless green ideas run motionlessly".
- Questions and answers. The parts are the units we question and answer with: "What slept furiously?" "Colorless green ideas."
The machinery is the production rule from Chapter 3, an if-then rule, here used as a rewrite rule saying which symbols may replace which. Sentence → NounPhrase VerbPhrase lets the symbol Sentence be replaced by a noun phrase and a verb phrase, and each application is drawn as branches down from the symbol it came from. Add more rules:
| Phrase rules | Word rules |
|---|---|
| NounPhrase → Adjective NounPhrase | Adjective → colorless, green (later hairy, brown) |
| NounPhrase → Noun | Noun → ideas (later dogs) |
| VerbPhrase → Verb Adverb | Verb → sleep (later run); Adverb → furiously (later happily) |
Keep rewriting until only words remain and you derive "colorless green ideas sleep furiously". The grammar is already infinite, because the recursive NounPhrase → Adjective NounPhrase rule can stack any number of "colorless" or "green". Add the bracketed word rules and the same derivation yields "hairy brown dogs run happily", with an identical tree in which only the words differ.
That shared structure is the explanation. It does not matter that the words have never been neighbors before, because the underlying structure is one English allows; "Furiously sleep ideas green colorless" has no such derivation. Grammaticality comes from structure, not from word co-occurrence. A phrase structure grammar is a formal system like any other: the start state is the symbol Sentence, the pieces are symbols (NounPhrase, VerbPhrase) and words ("green", "dogs"), and the rules are the legal moves.
It can also do what the villain could not. Two rules generate the apple-banana language exactly:
- Sentence → apple Sentence banana
- Sentence → apple banana
From Sentence you either stop with "apple banana" or recurse, getting "apple Sentence banana", and so on as long as you like. Each rewrite of Sentence adds one apple and one banana together, so the counts always match.
Act three: transformations
Chomsky wrote out simple phrase structure grammars for parts of English but saw that they could not do the whole job alone. Many sentences are systematically related: questions and answers derive from statements, and "The man ate the food" and "The food was eaten by the man" are different word strings with an obvious link. His third model made such links explicit with transformations, rules that take one sentence and turn it into another.
Harris had used transformations to analyze complex sentences into simpler parts; Chomsky used them to generate. Phrase structure produces a basic sentence, and a transformation reshapes it. This captures higher-level regularities and also lightens the phrase structure grammar, which now only has to cover a core set of sentences. His closing summary pictured a language as a small, possibly finite kernel of basic sentences with phrase structure, plus transformations applied to kernels or to earlier transforms to build ever more complex sentences, an approach he hoped would cut the immense complexity of real language down to size and illuminate how language is used and understood.
The Chomsky hierarchy
George Miller was hooked. He spent months in intense conversation with Chomsky about formal models of language and finally borrowed the heavily annotated manuscript of The Logical Structure of Linguistic Theory, which nearly ended in disaster.
Miller was reading the manuscript one afternoon in his basement office in Harvard's Memorial Hall when colleague Eddie Newman shouted that the building was on fire. Flames were rising from the clock tower; the Cambridge fire station sat across the street, but the water pressure could not reach the blaze. Each man grabbed his most valued possession: Newman took the laboratory cat and Miller took Chomsky's manuscript, and they watched until the fire burned low enough for water to reach it.
The two became friends, and their families spent the summer of 1957 together in California, connecting Chomsky's theory with Miller's psychology of language. The same year others started to notice. Alongside his French and German courses, Chomsky taught an undergraduate course on language at MIT, and its lecture notes were published by a Dutch press as Syntactic Structures. MIT graduate student Robert Lees sent the leading journal Language a 34-page review, about a third the length of the book, hailing it as one of the first serious attempts by a linguist to build a comprehensive theory of language in the manner of chemistry or biology. Linguists began to pay attention, and Chomsky made a string of contentious, influential conference appearances over the following years.
Languages nest
The idea that most gripped linguists was that kinds of languages form a hierarchy, and that human languages might be located within it. Chomsky had shown more than that phrase structure grammars can produce languages finite-state grammars cannot: every finite-state language is also a phrase structure language. The dog board, for instance, rewrites as:
- Sentence → the theLast
- theLast → dog dogLast, or theLast → dogs dogsLast
- dogLast → runs; dogsLast → run
The helper symbols theLast, dogLast and dogsLast simply remember the last word. Any finite-state grammar converts this way, so phrase structure grammars cover all finite-state languages plus others such as apple-banana.
He also split phrase structure grammars in two:
- Context-free grammars. A rule X → Y lets X be replaced by Y wherever it appears; context is irrelevant. All the grammars above are of this kind.
- Context-sensitive grammars. A rule such as Z X W → Z Y W lets X become Y only between Z and W. Any context-free grammar is a context-sensitive one with the context left out, and some languages need context: apple^n banana^n cherry^n (equal numbers of all three) is context-sensitive but not context-free.
The result is the Chomsky hierarchy, an ordering of language classes in which each is a subset of the next.
| Class | Rule form | What first needs this class | Human languages? |
|---|---|---|---|
| Finite | A list of sentences | Too weak: language is infinite | |
| Finite-state | Next word depends on current state | Loops ("hairy hairy dogs") | Too weak: center-embedding |
| Context-free | X → Y anywhere | apple^n banana^n; center-embedding | Good first pass for English |
| Context-sensitive | Z X W → Z Y W (only in context) | apple^n banana^n cherry^n; copies; crossing links | Needed, but only "mildly" |
| Turing-generable | Any computable rule system | Far too strong |
Where human languages sit
Chomsky had shown human languages are at least context-free. In the 1970s and 1980s linguists hunted for structures beyond that.
- Reduplication. A language whose sentences are a word sequence followed by an exact copy of it is not context-free. One linguist argued that Yiddish-inspired contemptuous reduplication makes English non-context-free, as in "context-free, shmontext-free", where "shm" precedes a copy of the phrase.
- Cross-serial dependencies, the clincher. English says "Al saw Bo make Cy let Di help Ed shave", but Dutch and Swiss German allow the order "Al Bo Cy Di Ed saw make let help shave": the first noun goes with the first verb, the second with the second, and lines joining each pair cross. No context-free grammar produces this (again assuming unboundedly long sentences).
So human languages are context-sensitive, and later work refined this to mildly context-sensitive: finer divisions within the hierarchy's classes showed that many context-sensitive patterns never appear in human languages. Context-free grammars remain a reasonable first approximation for English, but the world's languages need a step beyond.
The hierarchy became the backbone of computer science: regular expressions are finite-state machines, and nearly every programming language is specified with a context-free grammar that a compiler's parser follows. Engineers building validators or data-format schemas still choose a grammar class first, because it fixes what can be checked cheaply.
The allure of formal systems
Chomsky's turn has been compared to the move from alchemy to chemistry or from Aristotelian to Newtonian physics. Whether or not the analogies hold, it redirected linguistics. Treating a grammar as a device for generating a language, not merely describing it, also opened doors across cognitive science: a generative grammar sounds like exactly what you would need inside your head to produce sentences.
The revolution was not bloodless. One prominent linguist called generative grammar a theory "spawned by a generation of vipers"; others openly called Chomsky a fraud and a liar; and bitter fights over the approach's scope drove some former disciples to found rival frameworks. Its influence spread well beyond linguistics:
- Music. Leonard Bernstein, composer of West Side Story, rethought musical analysis in grammatical terms.
- Morality. Researchers proposed a moral grammar that generates morally acceptable actions as a syntax generates sentences.
- Computing. Generative grammar shaped the design of programming languages.
Part of the appeal is how cleanly phrase structure grammars show the power of formal systems, explaining three traits of language and of human behavior generally.
- Productivity. Language is endlessly creative: we easily produce sentences never said or thought before, and generative grammar explains how finite rules give infinite variety. The same holds for action, since almost nothing we do exactly repeats an earlier act.
- Hierarchy. Sentences break into phrases that break into smaller parts, and so do music and breakfast: making breakfast means tea and toast, which means putting on the kettle, which means filling it with water. Even this book nests chapters, sections, paragraphs and sentences.
- Compositionality. Parts are reusable: the steps for making tea recur in making coffee or soup. A grammar says how a small set of parts combines in new ways, yielding a rich vocabulary of actions for reaching goals.
Newell and Simon had relied on the same three properties. Their problem solver broke new problems into solvable subgoals, giving them hierarchy, and used productions to express flexible strategies, landing on essentially Chomsky's answer to generating compositional behavior. The two lines of work, which met on 11 September 1956, together make the strongest case for rules and symbols as a theory of mind.
The same success created a new problem. By showing that language needs something like phrase structure grammar, Chomsky proved it far more complex than behaviorists or information theorists had assumed, and so he now had to explain how such rules could ever be learned.
The poverty of the stimulus
While Miller, Simon, Newell and Chomsky were launching the cognitive revolution at MIT, B. F. Skinner was a few miles away at Harvard finishing Verbal Behavior. He had started it more than twenty years earlier, in 1934, after a dinner at which Alfred North Whitehead challenged him to give a behaviorist account of Whitehead himself announcing that no black scorpion was falling on the table. Skinner went home and began planning a book. Published in 1957, it offered an ambitious account of word meaning and syntax in terms of environmental contingencies.
Two years later Chomsky's review destroyed it. The review attacked every part of the theory, from Skinner's definition of a stimulus to his idea that sentences reuse simple grammatical frames. For those already drifting toward cognitive science it was a fatal wound to behaviorism; Jerome Bruner praised it as Chomsky "mercilessly out for the kill" and said he wanted to cheer when he finished.
The child as theorist
The review's most revealing passage is about learning. Given how complex grammar turned out to be, children's acquisition of it is all the more astonishing. Chomsky's argument, paraphrased:
- The child builds a grammar. She constructs it herself from sentences and nonsentences, the latter meaning corrections from those around her.
- That grammar is abstract. Speakers' ability to tell sentences from nonsentences and to spot ambiguities implies a grammar that is extremely complex and abstract.
- So the child is a theorist. Formally, she has carried out a remarkable act of theory construction.
- And does it under odd conditions. She does so very fast, largely independent of intelligence, and in much the same way as every other child; any theory of learning must account for this.
In effect the child faces the linguist's problem of documenting a new language, a problem Chomsky already doubted any simple rules could solve.
Plato's problem
Chomsky later named this Plato's problem. Russell framed it as the puzzle of how humans, whose contact with the world is brief, personal and limited, nonetheless know as much as they do. For language: how do children with limited exposure come to know so much about it? Philosophers had offered versions of one answer.
| Thinker | Answer to Plato's problem | Image |
|---|---|---|
| Plato | Knowledge is inborn; experience prompts us to recollect it | Learning as remembering |
| Descartes | Some ideas arise without outside input and are innate as a tendency, not as finished facts at birth | Generosity runs in some families, gout or stones in others |
| Leibniz | Minds favor some ideas; they must still be uncovered by work | A block whose veins outline Hercules, needing a sculptor's labor |
| Chomsky (1965) | A child cannot help building a particular kind of transformational grammar, as she cannot help seeing solid objects | Language grows like an organ, such as the liver, from the genes |
In Aspects of the Theory of Syntax (1965) Chomsky argued that the general features of language structure may reflect less the course of experience than the character of our capacity to acquire knowledge: innate ideas and principles in the traditional sense. His twentieth-century twist was to locate that knowledge in our genes, with language developing in the mind as an organ develops in the body, tied to the genetic endowment all humans share.
The argument began with syntax but extends further. Children must learn meanings too, and meanings carry subtle constraints: you cannot have six ounces of an idea or be chased by a building (except metaphorically), because only physical objects have mass and only intentional agents chase. Perhaps children also need innate ideas about objects, agents, intentions and goals.
From intuition to proof: the subset problem
This is the poverty of the stimulus argument: the language a child observes (the stimulus) is too thin to explain what she comes to know. Chomsky's version rested on the intuitive gap between a grammar's complexity and the child's limited input, and it would be far stronger if one could prove that certain languages are unlearnable from such input. His 1959 review had already floated the idea, saying that in principle one might work out what built-in structure a hypothesis-forming system needs to reach a grammar from the available data in the available time.
That question spawned a large body of learnability mathematics, but the core intuition is simple. Picture the true language and the learner's current hypothesis as two regions, with observed sentences as dots:
- (a) Overlap. Neither contains the other; some observed sentence will fall outside the hypothesis and reveal the error.
- (b) Hypothesis too narrow. The hypothesis sits inside the true language; again, sentences outside it expose the mistake.
- (c) Hypothesis too broad. The true language sits inside the hypothesis; every observed sentence fits, so a learner who sees only grammatical sentences can never deduce she is wrong.
English supplies an instance. You can say "I gave the book to her" and "I gave her the book", and also "I said no to her". Yet "I said her no" is out, although it relates to "I said no to her" exactly as "I gave her the book" relates to "I gave the book to her". Nobody ever told you it was ungrammatical, so how do you know your language is the smaller one that excludes it? A learner reasoning logically from positive examples is stuck whenever the truth is a subset of her guess.
This is the logical problem of language acquisition, and it drove a great deal of twentieth-century theory and experiment on how children learn language. Griffiths's closing twist: seen from a world where AI systems learn language remarkably well with weak constraints on what they can learn, it no longer looks like much of a practical problem. Explaining why occupies much of the rest of the book, starting in Chapter 5 with the possibility that language learning should not be framed as logic at all.
Griffiths states the classic arguments without their main rebuttals. Fernando Pereira (2000) showed that a smoothed statistical model rates "colorless green ideas sleep furiously" roughly 200,000 times more probable than its reversal, so the sentence defeats raw counts, not probabilistic models as such. Center-embedding beyond two levels is rarely produced or understood, so the finite-state refutation leans on idealized infinite competence. The subset problem is formalized in E. Mark Gold's 1967 learnability theorem, which the book does not name, and "infinite use of finite means" is Wilhelm von Humboldt's phrase, quoted by Chomsky.
LLMs reopened the debate. Steven Piantadosi (2023) argued that large language models refute Chomsky's approach; Chomsky, Ian Roberts and Jeffrey Watumull replied in the New York Times (March 2023) that such systems cannot tell possible from impossible languages. Kallini and colleagues (ACL 2024 best paper) found GPT-2 learns synthetic "impossible languages" (reversed or count-based rules) less readily than natural ones, partial support for both sides. The data gap remains: frontier models train on trillions of tokens (Meta's Llama 3: 15 trillion, 2024) against roughly 100 million words heard by a child, the budget of the BabyLM Challenge (2023 onward), whose best small models still trail large ones. That gap is the hook for Griffiths's later Bayesian chapters.
The Limits of Logic
By the 1960s rules and symbols handled theorems, games, puzzles and sentence generation well but stalled on learning language, forming categories, interpreting images and recognizing speech. Griffiths's diagnosis is a mismatch of tool and task: logic solves deductive problems, where the data fix the answer, while these are inductive problems that go beyond the information given. A theory of mind needs a second kind of mathematics, one that explains which uncertain guesses people favor and why.
A roadblock, not a delay
In the 1950s systems of rules and symbols looked like the best route to describing and rebuilding intelligence. The new field of artificial intelligence built expressive programming languages and aimed them at harder and harder problems. As the decade turned, trouble appeared. Some problems (proving theorems, playing games, solving puzzles, generating sentences) yielded readily. Others (learning language, forming categories, interpreting images, recognizing speech) resisted. The canonical misjudgment came in 1966, when a group of MIT students was given the summer to build "a significant part of a visual system". They did not.
A researcher who hits such a wall faces two possibilities:
- Not trying hard enough. The approach is right but needs more effort: more than a few students for one summer to solve vision, more than a few production rules to capture commonsense reasoning.
- The wrong approach. The more chilling option: a fundamental mismatch between the problem and the tools. Griffiths argues this was the position of the rules and symbols camp.
Logic is powerful and beautiful, but it fits problems of one shape: you know some things are true and derive what else must be true. Many problems an intelligent system meets lack that shape; we know some facts but not enough to be certain of anything further.
Vision is the example. A famous drawing (Figure 5.1) shows a young woman with her head turned away, or, on a second look, an old woman with a scarf over her hair. One image, one set of facts, two interpretations. Logic cannot pick between them because the image does not contain enough information, yet the visual system settles happily on a single reading from the many that the light on our retinas allows. Problems of this kind are solved by human minds constantly and without effort, and how we do it has puzzled philosophers for as long as there has been logic.
Deduction, induction, abduction
Aristotle's other argument
The book's account of logic began with the deductive syllogisms of Aristotle's Prior Analytics, where premises fix the conclusion with certainty. In the same work Aristotle described a second kind of reasoning, an argument from the particular to the universal, from instances to a general rule. Besides rhetorical argument (the origin of the syllogisms) he cared about what we would call scientific argument, and his attempts to understand nature laid foundations for biology and physics.
His example: he divided animals into those with "clean" blood and those without (resting on a complicated theory about bile in the blood), observed that clean-blooded humans, horses and mules seemed to live longer, and concluded that clean-blooded animals are longer-lived. This is induction: we accept a conclusion the facts support, but which remains far from certain. Deduction starred in the Prior Analytics; induction had to wait two thousand years for its moment.
Bacon, Darwin and Boole
That moment was the Scientific Revolution (sixteenth to eighteenth centuries, the era of Descartes and Leibniz, who cared as much about the physical world as about the mind). The period changed both what was known and how knowledge was acquired: detailed catalogs of natural phenomena and experiments designed to test hypotheses. Early scientists collected many particulars and wanted to turn them into universals.
- Francis Bacon. Politician, lawyer and philosopher, Lord Bacon argued that the new science needed a new kind of reasoning. Aristotle's logical works were known as the Organon (Greek for "tool"); Bacon's Novum Organum (1620), "The New Organon", set out to replace that tool. He charged syllogistic logic with entrenching errors built on popular notions instead of seeking truth, and so doing more harm than good. His alternative put induction at the center of science, strengthened by collecting large, systematic data across the widest possible range of cases.
- Charles Darwin. The meticulous cataloguer of barnacles later said he had worked on the true principles of Baconian induction.
- George Boole. At nineteen, in an address on the "Genius and Discoveries of Sir Isaac Newton", Boole described the summit of a science as reducing its secondary laws to a higher, more general one, and called the mental process that discovers general truths induction, the highest exercise of our intellectual powers and the trait that most separates humans from other animals.
Induction, then, deserves explaining as much as deduction does.
The detective's inference
The nineteenth century also gave taxonomists of thought a new puzzle. A few years after Boole's address, Edgar Allan Poe published what is usually called the first detective story, "The Murders in the Rue Morgue" (1841).
Monsieur Auguste Dupin solves the brutal murders of two women from four clues: a voice no one could identify by language, entry by climbing a lightning rod and through a window, bodies placed in a way requiring great strength, and some unusual hair. He concludes the killer was not human but an orangutan. Dupin talks repeatedly of "deductions", but written as logic one step reads: if the culprit were an orangutan it could climb to the window; the culprit climbed to the window; so the culprit was an orangutan. That is P → Q; Q; therefore P, the fallacy of affirming the consequent from Chapter 1 (the arrow fits because Dupin allows that a human could make the climb, though only with unusual agility). Calling it a fallacy misses the point: taken together the inferences work well enough to solve the case, and they resemble neither deduction nor Bacon's induction.
Charles Sanders Peirce agreed that this was a third kind of reasoning. Born a couple of years before Poe's story and later an admirer of Poe, he came from a distinguished family: his father was Perkins Professor of Astronomy and Mathematics at Harvard, and his mother was a U.S. senator's daughter. Drawn to logic early, he split his career between writing on the mathematics of reasoning and working for the U.S. Coast and Geodetic Survey, which his father had helped found. He taught logic at Johns Hopkins for five years but was fired after starting a relationship with a much younger woman before his divorce was final (Griffiths notes the parallel with John Watson's fate some forty years later). He spent his last years in poverty, helped by his close friend William James.
On a Survey trip Peirce took a boat from Boston to New York, left in a rush and forgot an expensive watch in his room. Back on board, he found it gone and had the crew line up for questioning. After walking the row he told himself he had not the slightest clue to go on; then an inner voice insisted he must name someone anyway, reason or no reason. He took a short loop of a walk, and when he turned back all doubt had gone. He took the man he had fixed on into the stateroom, his hunch proved correct, and he got the watch back, by a process nothing like deduction.
Peirce called the detective's inference abduction, to set it apart from deduction and induction, and worked for much of his life to define it. In Harvard lectures he described it as studying facts and devising a theory to explain them, and claimed it is the only logical operation that introduces a new idea: induction merely fixes a value, and deduction merely unfolds what a pure hypothesis already implies. On that description abduction is central to science too, since it is how a scientist gets from curious observations to a theory that explains them.
Beyond the information given
Griffiths groups induction (inferring a rule from examples) and abduction (forming a theory from observations) under one label, inductive problems, because both go beyond the information given: they make an uncertain inference from the available data to a plausible hypothesis. In a deductive problem, all the information needed is present and we just work out its consequences.
- Deductive. Solving x + 2 = 5: follow the rules of algebra to a unique answer.
- Inductive. Solving x + y = 5: the sum carries some information but not enough to fix x and y.
Even so we are willing to guess. Given x + y = 5, x = 3, y = 2 seems more plausible than x = −1 and y = 6, x = 3.1 and y = 1.9, or x = π and y = 5 − π. The factors that make us favor one solution over another are inductive biases. Bias usually sounds bad, but where information is incomplete, biases that line up with the world are the only way to make progress; good detectives and scientists are people whose biases produce the right hypotheses from limited evidence.
There are two kinds of problems in the world, and the cognitive revolution was using tools for only one. Theorems, games, puzzles and sentence generation can all be cast as deductive problems; language learning, categorization, vision and speech are inductive problems, where logic is not enough.
Griffiths recasts each hard problem as inferring hypotheses from data:
| Problem | Data | Hypotheses | Why logic falls short |
|---|---|---|---|
| Acquiring syntax | Utterances the child hears | Possible grammars | Data rarely fix the grammar; the "logical problem" of acquisition came from forcing an inductive problem into deductive form |
| Forming categories | Objects labelled in or out of the category | Candidate rules (Bruner, Goodnow, Austin) | The rule is undetermined until every possible object has been seen |
| Vision | Light on the retina | 3D structure of the scene | One observation, several unknowns |
| Speech | Noisy sound waves | Intended words | Noise on the way from mouth to ear creates ambiguity |
- Language. Focusing on Chomsky's problem, acquiring syntax, the utterances a child hears rarely settle which grammar is correct (Chapter 4). The logical problem of language acquisition arose from demanding that the data identify a single hypothesis; treat acquisition as inductive and the conclusions about learnability may change.
- Categories. Bruner, Goodnow and Austin explicitly framed concept formation as induction: people get data on which objects do and do not belong, and evaluate hypotheses about the category. In A Study of Thinking they assumed the hypotheses were logical rules, a classic move from particulars to universals. The section title comes from Bruner's collection of papers on learning, Beyond the Information Given.
- Vision. The light on the retina cannot distinguish among the possible three-dimensional worlds that produced it, and illusions exploit that gap.
- Speech. Words in a speaker's head become pressure waves that reach our eardrums, with plenty of room for noise along the way. Misheard lyrics prove the point: Jimi Hendrix pausing to "kiss this guy" (for "kiss the sky"), Kurt Cobain's teen spirit "in containers", Taylor Swift's "all the lonely Starbucks lovers" (for "got a long list of ex-lovers").
Vision scientists offer even sharper cases than the old drawing. In Edward H. Adelson's checkerboard, squares A and B are exactly the same gray (cover everything but a thin vertical strip through both letters to check), yet B looks much lighter. The brain is solving an inductive problem: the light from both squares is identical, but it notices that B lies in the cylinder's shadow and compensates by inferring that B is a lighter surface. Like x + y = 5, there is one observation (light at the retina) and two unknowns (surface color and amount of shadow), and a visual system with good inductive biases solves it instantly.
The rules and symbols approach had a tool for deduction but not for induction, and induction is much of what brains do. The natural hope was to understand induction with the logic built for deduction. Two philosophers, David Hume in the eighteenth century and Nelson Goodman in the twentieth, gave reasons for doubt.
Hume's riddle
David Hume was born in Scotland in 1711 and entered the University of Edinburgh at ten. He meant to study law like his father, but found he could stand nothing except philosophy and general learning, and set out to live as an intellectual.
- Collapse at eighteen. A few months in, his enthusiasm vanished and he could no longer lift his mind to its former pitch. His physician diagnosed "the Disease of the Learned" and prescribed, among other things, an English pint of claret a day and horse rides of eight to ten Scotch miles. Griffiths reads it as what we would call depression; Hume recovered from the first episode but struggled for years.
- Bristol. Unable to follow his planned scholarly path, he tried life as a merchant in Bristol, hoping to toss about the world until he left the illness behind. It suited him badly and lasted a few months.
- France and the Treatise. In France he recovered, thought and wrote, returning to London three years later* with the manuscript of A Treatise of Human Nature, published in *1739.
The Treatise asks when we are entitled to believe that cause and effect are linked. Hume's answer: only through experience. We remember many instances of one kind of object regularly followed by another, contiguous and in sequence. Every time we have seen a flame we have felt heat, so we expect heat from the next flame.
That inference needs a further assumption: that cases we have not experienced resemble those we have, and that nature stays uniform. Why believe the future will be like the past? No deductive argument establishes it. The only defence Hume could find was inductive: the future has resembled the past in the past. But that answer raises the same question again, and again, without end, so the reasoning has no just foundation.
The Treatise disappointed him; his autobiography says it "fell dead-born from the press", without even stirring a murmur from zealots. He spent eight years reworking it into An Enquiry Concerning Human Understanding (1748), which made the skeptical argument plainer and added an explanation of why induction nonetheless feels unobjectionable: we infer by habit, an irresistible natural instinct, as unavoidable as feeling love when we receive benefits or hatred when injured, something no reasoning can produce or prevent.
Habits, though, can still be analyzed. When we infer causes or generalize, there are patterns in which answers we find acceptable. What makes some inductions better than others was the question Goodman took up two centuries later.
Goodman's new riddle
Nelson Goodman was an unusual Harvard philosophy graduate student in the 1930s: not for his birthplace (nearby Somerville) or his undergraduate degree (also Harvard), but because he supported himself by running an art gallery in Boston, since his Jewish heritage made him ineligible for graduate fellowships. His PhD took twelve years, finished just in time for him to join the army in World War II as a psychological tester, the start of a lifelong interest in how people see and think. After a short spell at Tufts, he became a professor at Penn, where he taught and befriended a seventeen-year-old Noam Chomsky. While Chomsky built generative grammar, Goodman wrestled with induction.
Regularity is not enough
Hume's habit explanation left Goodman unsatisfied. A habit needs an underlying regularity, but regularity alone cannot license an induction:
- Copper. Finding that one piece of copper conducts electricity makes it more credible that other pieces do, so it confirms "all copper conducts electricity".
- Third sons. Finding that one man in this room is a third son does nothing to make it likelier that the other men here are third sons.
In deduction, an argument's quality depends on its form, not its content, which is why arbitrary propositions can be written P and Q. Inductive arguments are sensitive to what the hypothesis actually says.
Mill and the black swans
The point was not wholly new. While Boole was formalizing deduction, his contemporary John Stuart Mill tried to reduce induction to strict rules and a scientific test, as the syllogism is for deduction, in his bestselling A System of Logic. He found it hard. An Englishman's plausible inference that all swans are white had recently been overturned by black swans in Australia (Griffiths, who grew up there, notes that black swan events were routine). Yet Mill judged some inductions sturdier: we should still doubt reports, going back to the naturalist Pliny, of men whose heads grow beneath their shoulders, because a bird's color is more credibly variable than the arrangement of a human's principal organs. The question is what makes one hypothesis more credible than another.
Grue
- The setup. Every emerald you have seen is green, which seems to support "all emeralds are green".
- The rival. An object is grue if it is observed to be green before some date (say 1 January 2050) or observed to be blue after it. Every green emerald so far fits "all emeralds are grue" equally well, since both hypotheses predict exactly what you have seen.
- Not complexity. Define bleen as blue until 2050 and green thereafter. To someone raised speaking of grue and bleen, green is the awkward word: grue until 2050, then bleen. So simplicity, an arbitrary date or a change of observed color cannot separate the two, because from the other side green has those features too.
- Entrenchment. What makes a hypothesis a good target for induction is that it is entrenched: established by earlier successful inductions with similar predicates. We trust "green" because such inductions have worked before; Mill's color-versus-organs intuition is justified because past inductions about organ arrangement have done better than those about color.
The lesson is that observing a regularity is only half of induction; deciding which hypotheses are the kind that regularities can support matters at least as much, and is hard to specify. Goodman's answer still justifies induction by induction (past inductions teach which future ones will succeed), but he accepted that this is simply how our inductive habit works.
Hume (1739, 1748). The problem is justification: no non-circular argument shows the future will resemble the past. Answer: we induce by habit, a natural instinct beyond reason's reach.
Goodman (1950s). The problem is selection: the same evidence confirms "green" and "grue" alike, so habit cannot say which regularity to project. Answer: we project entrenched predicates, those with a track record of successful inductions.
The falling-out with Chomsky
Goodman's view of learning was far more optimistic than Chomsky's account of acquisition, and the tension ended their friendship. Chomsky recalled that they stayed close until Goodman realized he was serious about innate structures, which for Goodman was almost a religious impossibility, and they had to break relations. Goodman then published a scathing critique in dialogue form, explaining that the format suited a theory that only his respect for its advocates let him take seriously at all. Jason (a thin disguise for Chomsky) returns from the nomads of "Outer Cantabridgia" with something Anticus (Goodman) suspects is "more fleece than golden". Jason argues for innate ideas, Anticus scoffs, and the dialogue closes with Anticus dismissing innate ideas as unsubstantiated conjectures propped on implausible and untestable hypotheses.
Understanding induction
The story so far: rules and symbols struggled with inductive problems because their methods were built for deduction, and the two kinds of problem are deeply different. Since inductive problems such as learning language and perceiving the world are central to human minds, describing thought mathematically requires a new kind of mathematics, a new component of the Laws of Thought, for induction.
Goodman also suggested how to find it. Logic is now accepted as describing valid deduction, but it earned that status: deductive principles are justified by conformity with accepted deductive practice, and a rule that yields unacceptable inferences is dropped as invalid. Do the same for induction: study the inferences people make, propose mathematical frameworks, and revise them when they produce conclusions people reject. Philosophers call the target state reflective equilibrium, where intuitions and mathematics agree.
Griffiths takes the cognitive scientist's step further: compare mathematical theories with people's behavior, not just with intuitions. Joining psychology's experimental methods to the rigor of mathematical theory offers a way to extend the Laws of Thought beyond deductive logic. That was the path cognitive science took in the 1960s and 1970s, exploring new representations and processes for what had defeated rules and symbols:
- Spaces. Treating objects and mental states as points in a space, the subject of the next two chapters.
- Learning. Deriving new models of learning, notably in neural networks.
- Uncertainty. Accepting uncertainty as part of inductive inference, the road to probability.
The path runs through the rest of the twentieth century and into the twenty-first, toward an account of how we might learn complex languages and make sense of the world.
Mill reduced the whole challenge to one question, which frames the rest of the book: why does one instance sometimes justify a complete induction while myriads of agreeing instances, with no known exception, barely support another? Whoever answers it, Mill said, has solved the problem of induction.
Poe's 1841 story is usually, not universally, called the first detective story (E. T. A. Hoffmann's 1819 Das Fräulein von Scuderi is a rival claimant). Folding abduction into "inductive problems" is Griffiths's choice; philosophers often keep it apart as inference to the best explanation, and Peirce's own definitions shifted over his career. The "grue" definition here (observed green before 2050) is the popular simplification of Goodman's Fact, Fiction, and Forecast (1955). The 1966 Summer Vision Project is often retold as hubris, but its memo set a limited goal (figure/ground segmentation), so the moral is sharper in retelling than in the document.
Few-shot induction remains the hardest test for AI. On François Chollet's ARC-AGI puzzles (infer a grid transformation from two or three examples), OpenAI's o3 scored 75.7% at standard compute and 87.5% at very high compute in December 2024, after years in which systems scored far below humans. The ARC Prize Foundation responded in March 2025 with ARC-AGI-2, on which leading models initially scored in the single digits while every task had been solved by human testers. Mill's question, why one example sometimes suffices, is still an open engineering target.
Categories, Spaces, and Features
Logic presumes that categories have crisp edges defined by necessary and sufficient conditions. In the 1970s Eleanor Rosch showed that everyday categories have good central examples and fuzzy boundaries, and two Stanford colleagues supplied rival mathematics of resemblance: Roger Shepard's psychological spaces and Amos Tversky's feature lists. Categories became regions around a prototype, judged by similarity, and that opened a "West Coast" set of Laws of Thought built on spaces, features and networks.
East Pole and West
By the 1960s rules and symbols offered a clear, coherent account of the Laws of Thought, spanning logic, problem-solving and language, and implying that our nature sharply constrains what we can learn. Because so many of its champions sat at Harvard and MIT, it became known as "East Coast" cognitive science. Griffiths credits Jerry Fodor with calling MIT the "East Pole": as every direction from the North Pole is south, every direction from Cambridge, Massachusetts, is ideologically west. And in the West a different view of the mind was forming.
One of the hardest blows against rules and symbols began about as far from Cambridge as one can get, in the highlands of Western New Guinea, where a young psychologist met a people who carved up the world differently. Her findings undercut assumptions about categories that were built into the idea of understanding the mind through formal systems like logic, and making sense of them led to new mathematics of spaces, features and networks.
Things get fuzzy
A childhood lesson in boundaries
Eleanor Rosch grew up in the 1940s in San Fernando, California, just north of Los Angeles, in a family recently arrived from New York and puzzling over a culture of owned houses and driving everywhere. Houses, unlike apartments, have unclear boundaries. When a new neighbor proposed stringing shared clotheslines between the two lots, Rosch's mother was furious; nobody knew where the property line ran, a surveyor was hired, and the street discovered that families had been tending one another's gardens. The episode stuck: some things were definite (your house was the center), while boundaries were indefinite, and she began noticing indefinite boundaries everywhere.
A bookish child, she went to Reed College in Portland, Oregon, to study literature, argued her way into an advanced poetry class and found it baffling, full of poets she had never heard of and an idiosyncratic teacher. She switched to philosophy and psychology and wrote her thesis on Ludwig Wittgenstein.
Wittgenstein, the cure for philosophy
- Early career. Born into a wealthy Jewish family in Austria, Wittgenstein first trained as an engineer and designed a new kind of airplane propeller. The mathematics of flight led him to the foundations of logic and to Cambridge to study with Bertrand Russell.
- The Tractatus. Delayed by World War I and a spell as an Italian prisoner of war*, his work appeared in English as Tractatus Logico-Philosophicus, which used logic to characterize *thought and language rigorously.
- Grief and exile. As the book's fame grew he was mourning a brother's suicide and his lover, by coincidence a relative of David Hume, killed in a plane crash. He worked as a schoolteacher and a gardener.
- Return, 1929. Back at Cambridge as lecturer and later professor, he taught Alan Turing, who called him a "very peculiar man". His thinking had flipped: he now stressed the limits of logic for describing thought and language.
- The cure. He actively discouraged students from becoming professional philosophers. Rosch told Griffiths she was the only person she had ever met whom he cured.
Harvard, color words and memory
With philosophy abandoned, Rosch worked as a social worker, then began a clinical psychology PhD in Harvard's interdisciplinary Department of Social Relations, alongside Jerome Bruner and other figures of the cognitive revolution. Her dissertation on language and color perception was with the psycholinguist Roger Brown. Brown had shown that Harvard undergraduates recognized colors judged good matches for the eight basic English color words (red, orange, yellow, green, blue, purple, pink, brown) more easily than colors fitting no word: a perfect blue beats a muddy yellow. The implication was that learning color words makes those colors easier to store in memory. Rosch would invert it.
In Social Relations Rosch met and married the anthropologist Karl Heider, who worked with the Dani of the Western New Guinea highlands, and she joined his 1968 field expedition (so uncomfortable, she joked, that her work there had to be important). The Dani farmed sweet potatoes and raised pigs with mostly wood and stone tools, wore little clothing and lived in grass-roofed huts. Most relevant for Rosch, they used essentially two color words: mola, roughly light plus warm shades such as red and yellow, and mili, roughly dark plus cool shades such as green and blue.
- Memory test. Running Brown's experiment, Rosch found the Dani, like Harvard students, remembered the best examples of English color words more easily, though they had no names for them. Memorability looked like a fact of human perception, not of language or culture.
- Learning test. She taught Dani groups new color words. Categories centered on the most recognizable (focal) example of each English color were learned more easily than categories shifted so their examples sat at the English boundaries.
Conclusion: Brown had the arrow backwards. Perception shapes categories, not the reverse: we may have the word "blue" because a perfect blue is easy to recognize, and categories may be built around particular examples.
Rosch had rediscovered her childhood suburb in New Guinea: color categories had a definite center (the blue everyone agrees on) and an indefinite boundary, just as neighbors knew whose house was whose but argued over the edges. Back in the US, divorced from Heider and a professor at UC Berkeley, she asked whether all categories work this way.
Why categories matter to logic
The question went to the root of logic-based accounts of mind. Categories, and the assumption that membership can be clearly decided, are the foundation of rules and symbols:
- Aristotle labored to classify the world (animals with and without bile in their blood), and his syllogisms are about sets, presupposing that each object's set can be determined.
- John Wilkins's universal language required knowing whether a thing was an "Element", "Bird" or "Beast" before you could name it.
- George Boole's logic assumes clean, meaningful sets such as sheep or white things, and knowledge of when a proposition applies.
Typicality effects
In a series of 1970s papers Rosch demolished that foundation. Ask people to rate how good an example something is of an everyday category and you find stars, misfits and a large disputed middle.
| Category | Good examples | Bad examples | Disputed cases |
|---|---|---|---|
| Furniture | Chair | Telephone | Rug, clock |
| Fruit | Orange | Squash | Coconut, olive |
| Sport | Football | Sunbathing | Hunting |
| Toy | Doll | Books | A deck of cards |
Typicality is not just an opinion; it shows up throughout behavior:
- Speed. People confirm that a robin is a bird faster than a penguin.
- Instability. Asked twice whether an olive is a fruit, people are more likely to change their answer for less typical items.
- Production. Asked to name an example of a category, people tend to give typical members.
So categories are not simply sets of objects. What, then, are they?
Definitions fail; family resemblance
Rosch found the answer in her old thesis subject. One virtue of logic was explicit definition, such as Cat ↔ (Small ∧ Furry ∧ Domestic ∧ Carnivore): something is a cat if and only if it is a small, furry, domestic carnivore. Bruner, Goodnow and Austin relied on this in their concept-learning studies, where subjects inferred a category's defining rule. Such a formula states necessary and sufficient conditions. Wittgenstein argued that natural categories, unlike the artificial ones of logic, have none.
- Not small. Himmy, a grossly overweight Australian tabby, weighed 46.8 pounds and was still a cat.
- Not furry. The sphynx breed is hairless.
- Not domestic. Feral cats remain cats.
- Not carnivore. An elderly cat fed intravenously eats no meat and remains a cat.
- A false positive. Ferrets are small, furry, domestic carnivores and are not cats.
Patch the definition and Wittgenstein showed you can always find new counterexamples for ever finer definitions. Instead categories have a family resemblance structure: relatives share overlapping features but no defining set (one has a long nose and small ears, another small ears and brown hair, a third brown hair and a long nose; nobody has all three, yet all resemble one another). Cats tend to be small, furry, domestic and carnivorous, but no feature is necessary.
This matched Rosch's data: categories have core cases everyone accepts, and other items are judged by similarity to that core. Chairs, tables and chests of drawers resemble one another and differ from ashtrays, fans and telephones, which would stand out at any furniture family reunion. But family resemblance only moves the problem: how do we judge that two things resemble each other?
Similarity and psychological spaces
Similarity has troubled psychology from the start. In The Principles of Psychology (1890), the first American psychology textbook, William James made it as basic as Mill's logical methods: even to list instances by Mill's methods, we depend on the brain's spontaneous association by similarity to bring resembling facts to mind.
Shepard finds his problem
Roger Shepard grew up in Palo Alto, next to Stanford, son of a materials science professor specializing in metallurgy. A mixed student, he skipped social studies homework to build remote-control robots and 3D models of four-dimensional objects (the kind Alicia Boole studied). Admitted to Stanford, he was suspended after two quarters for poor grades, and spent the enforced leave as a research assistant at the Stanford Research Institute and in the naval reserve, which convinced him to return and finish.
He loved geometry, especially the non-Euclidean geometry Einstein had built into physics, but doubted he had the mathematics for physics; he was equally drawn to perception, mental imagery and dreams. A philosophy professor discouraged him (echoing Rosch), and his father suggested psychology. Psychology 1 did not connect math and mind; his second class, a graduate course on perception, did, when the instructor mentioned the dissertation of Stanford student Fred Attneave.
Attneave's city blocks
Attneave studied how people judge the similarity of figures varying in size, shape and color (the kind of stimuli Bruner, Goodnow and Austin later used) by testing spatial representations. Describe a shape's size and redness as coordinates (x, y), so two stimuli sit at (x₁, y₁) and (x₂, y₂).
- Euclidean. The school-geometry distance, the square root of the summed squared differences: d = √((x₁ − x₂)² + (y₁ − y₂)²). It is how far a bird flies over the buildings.
- Manhattan (city-block). The sum of absolute differences on each dimension: d = |x₁ − x₂| + |y₁ − y₂|. It is how far you walk in a gridded city, counting east-west blocks plus north-south blocks because you cannot cut through buildings.
- Attneave's result. People's similarity judgments for stimuli with identifiable, separable dimensions (size, redness) fit Manhattan distance, not Euclidean; being on the same street or avenue as your destination makes the walk shorter even when the map distance is equal.
Yale, Hovland and the law of generalization
Hooked on using non-Euclidean geometry to model the mind, Shepard went to graduate school at Yale, found it gripped by behaviorism, was assigned to shock rats, and promptly switched to Carl Hovland, already thinking about information theory and concept learning. Among scientists building mathematical models of behavior, mostly of learning, Shepard judged similarity more fundamental: since we never meet exactly the same situation twice, no theory of learning is complete without a law of how learning generalizes from one situation to another. That problem occupied the rest of his graduate work and much of his career.
Behaviorism said to study measurable stimulus properties, but physical measurement and experience come apart. Color depends on wavelength: violet shortest, red longest, blue, green, yellow and orange between. Yet people rate red and violet as more similar to each other than either is to green: maximally far apart physically, close perceptually. The resolution is that internal representations can have a different geometry from the external world. Physically colors lie on a line; perceptually they form a circle, as Isaac Newton, one of Shepard's heroes, sketched in Opticks.
Multidimensional scaling
To recover such psychological spaces from judgments, Shepard used multidimensional scaling (MDS): given the distances between points, reconstruct their positions, as the distances between major US cities would let you redraw a map. In his dissertation he mapped colors, using how often people confused two colors as a proxy for closeness. With enough colors and judgments, the color wheel emerges.
- Miller's spell. At Yale Shepard heard George Miller give his magical number seven talk and was captivated: Miller, he told Griffiths, was a "natural born actor" (he had acted in Greek plays) who delivered it as if memorized. Shepard spent two years at Harvard with Miller analyzing similarity judgments of speech sounds.
- Bell Labs and non-metric MDS. At Bell Labs, home of Shannon's information theory, he built the next generation of MDS. The old method needed exact distances, daunting when distances come from confusion rates. Shepard saw that the rank order suffices: you need not know the mileage from Boston to San Francisco, only that it exceeds Boston to Murray Hill, New Jersey. Because the number of distances grows with the square of the number of points, with enough points the orderings contain enough information to fix the locations.
The method took psychology by storm. Anyone could now visualize mental representations just by collecting similarity judgments, which in many ways fulfilled the cognitive revolution's promise of turning hypotheses into rigorous mathematical theory: MDS turned fuzzy judgments into objective coordinates. Rosch used it with the Dani, comparing their color space with Harvard undergraduates'; her earlier experiments showed how language affected representations, but MDS let her see the difference.
By revealing such structures Shepard made spatial representations a compelling alternative to rules and symbols. After eight years at Bell Labs he had a brief professorship at Harvard, then returned to Stanford, where he would share a department with the man who rejected his central idea.
Features of similarity
The paratrooper philosopher
While Shepard was starting with Miller, Amos Tversky was jumping out of planes over Israel. A native Israeli, son of a politician and a veterinarian, he volunteered as a paratrooper after high school, so slight that he drank large amounts of water to make the weight minimum. His bravery verged on the pathological: when one of his men fainted beside an armed explosive with a twenty-second fuse, Tversky ran in and dragged him clear, taking shrapnel. The army gave him a medal and told him not to do it again.
- Hebrew University, 1957. After service he studied philosophy and psychology. He enjoyed Hume but despaired of philosophy, telling a classmate that Plato had solved too many problems, leaving too many clever people and too few soluble problems. Psychology offered open problems and data to solve them.
- Ward Edwards. In his second year he read "The Theory of Decision Making" by Ward Edwards of Michigan, which surveyed economists' theory of rational choice and psychologists' tests of it (preferences, uncertainty, strategic interaction) and called it a rich field where the theory was built and the experiments remained to be done. Tversky was sold.
- Michigan, 1961. He flew to Michigan for a doctorate, the first time he had been on a plane he did not intend to jump from. Michigan was then a center of mathematical psychology; Tversky, from the humanities track and a writer of poetry and plays, taught himself mathematics and became one of the leading practitioners of the axiomatic approach.
The axiomatic approach
An axiom is a basic assumption from which a mathematical theory is derived; Euclid's include "a straight line can be drawn between any two points" and "all right angles are equal". Mathematicians explore what axioms imply; scientists ask whether the resulting system describes the world. Newton held that Euclidean geometry describes physical space; Einstein argued it needs non-Euclidean geometry (the challenge that inspired Shepard). Tversky applied the method to psychology, testing which sets of axioms fit behavior.
- Dissertation. He switched from Edwards to Clyde Coombs, another mathematical psychologist of decision-making, and asked when behavior is an additive function of environmental variables. For decisions: is a slice of bread plus a piece of cheese worth the value of the bread plus the value of the cheese? For similarity: do we just add up differences along dimensions, as Attneave's Manhattan result suggests?
- Barbara Tversky. He met his wife, a fellow psychology graduate student, in Coombs's class; she became a distinguished researcher on how people represent and reason about space. After a year at Harvard's Center for Cognitive Studies, the couple went to teach at Hebrew University.
- Krantz. Coombs had worked on the foundations of MDS, and with Michigan professor David Krantz Tversky published papers on the conditions objects must meet to be represented in a psychological space, testing them on perceptions of faces and rectangles.
Best known for decision-making (Chapter 12), Tversky ran this parallel line on similarity, culminating in "Features of Similarity" (1977). His strategy: distances obey the metric axioms, so if similarity judgments violate them, similarity cannot be distance. Some axioms are obvious (a thing is at zero distance from itself; distances are positive). Others are subtler.
- Strike 1, symmetry. Distance must be symmetric: Israel to Michigan is as far as Michigan to Israel. But people prefer "an ellipse is like a circle" to "a circle is like an ellipse", and "a camel is like a horse" to the reverse.
- Strike 2, triangle inequality. For any A, B, C, d(A,C) ≤ d(A,B) + d(B,C): the sides of a triangle, with equality only when B lies on the line between A and C, otherwise a shortcut exists. Yet, as William James noted in 1890, the moon is like a gas-jet and like a football, while gas-jet and football are not alike at all.
- Strike 3, nearest neighbors. "Fruit" should be highly similar to apple, banana, cherry and so on, perhaps the most similar item to each; but in a low-dimensional space no single point can be nearest to many others, since they crowd each other out. Distance, Griffiths says, has to start the long walk back to the dugout.
- The alternative. Tversky wrote his own axioms for similarity and proved a simple model satisfies them: treat objects as lists of features (a horse has four legs, short hair, runs fast) and compute similarity by adding shared features and subtracting distinctive ones, with weights that can depend on the direction of comparison (the camel's hump counts more one way). In the 1977 paper this is the contrast model, S(a,b) = θ f(A∩B) − α f(A−B) − β f(B−A).
James had resolved his own example the same way: moon and gas-jet share luminosity and nothing else, moon and football share roundness and nothing else, and football and gas-jet share no attribute at all. Features also connected to Rosch. With Carolyn Mervis she had people list the features of objects and counted how many each shared with other category members: more typical members share more features (a chair has much in common with other furniture). That fit Tversky's emerging theory exactly.
After pages of proofs and experiments, the paper closes on similes, with a joke reflecting Tversky's literary side: an essay seems nothing like a fish until you recall that both have a head and a body, and an essay sometimes ends with a flip of the tail.
Shepard. Similarity is inverse distance in a psychological space recovered by MDS. Fits continuous, hard-to-name dimensions such as color and pitch.
Tversky. Similarity is a weighted count of shared and distinctive features. Fits discrete, conceptual items such as fruits, faces and countries, and explains asymmetry.
Redefining categories
Amos and Barbara Tversky moved to Stanford's psychology department in 1977, meaning to stay a sabbatical year, and stayed for good after a bidding war among Stanford, Harvard and Michigan. The strongest advocate and the strongest critic of similarity-as-distance ended up at opposite ends of a short third-floor corridor. They got on fine. Shepard said people imagined them at odds, but both recognized that different sets of things need different representations: colors, continuous and with hard-to-describe dimensions, suit a Euclidean space; kinds of fruit suit features.
When Griffiths was a Stanford graduate student, social psychologist Lee Ross (instrumental in hiring Tversky) told him of a Shepard talk with Tversky in the audience. Tversky asked whether the result, clear in two or three dimensions, held in eleven. Shepard closed his eyes for a moment, opened them and said "Yes." Ross recalled two thoughts: what had gone on in Shepard's head with his eyes closed, and how many anxious graduate students he would be reassuring that week that they were smart enough for psychology.
Prototypes
At Berkeley and Stanford, Rosch, Shepard and Tversky had laid foundations for a new theory of categories, since spaces and features can represent categories as well as objects.
Prototype theory unites the strands: a category is represented by its most typical member (the perfect blue; for furniture, a chair), which is a point near the members in a psychological space or a list of features they share. A new object joins the category with the most similar prototype, so to decide cat or dog you pick the closer prototype.
This explains what defeated logical definitions:
- Fuzzy boundaries. Some objects lie far from every relevant prototype.
- Graded typicality. Better examples are closer to the prototype.
- Spillover. Computing such distances is so natural that typicality colors other reasoning, as in verification speed and naming.
The formal models also yielded a surprise. If objects are points and classification goes to the nearest prototype, the boundary between two categories is always a straight line: everything on one side is closer to the cat prototype, everything on the other to the dog prototype, and closeness to the line sets how hard the decision is.
Thinking in spaces and features was a major advance. Rosch made clear that logical rules could not describe how people organize the world; Shepard and Tversky supplied rigorous alternatives. Spaces and features became the foundation of a new set of Laws of Thought for turning a fuzzy world into precise categories, and the straight-line limitation is what the next chapter's learning models confront.
Modern AI search and recommendation systems represent words, documents and products as points in a learned vector space and score relevance by distance or cosine similarity, a direct descendant of MDS. Tversky's strikes still bite: a single symmetric metric scores "query like document" and "document like query" identically, which is one reason many retrieval systems use separate query and document encoders (dual encoders such as DPR) instead of one shared space.
The "East Pole" label is usually traced to Daniel Dennett's 1986 essay "A View from the East Pole", which named Jerry Fodor among its residents, rather than to Fodor himself as Griffiths states. Rosch published the Dani work as Eleanor Rosch Heider (1972), building on Berlin and Kay (1969) on basic color terms; later studies of the Berinmo of Papua New Guinea (Roberson, Davidoff and colleagues, 2000) found language does shape color memory, so the universalist reading Griffiths gives is now contested. Non-metric MDS is usually co-credited to Shepard (1962) and his Bell Labs colleague Joseph Kruskal (1964).
Computing with Spaces
Spaces and features described mental states but said nothing about how thought moves between them. The alternative to copying a mathematician's steps (the Turing machine) was to copy her brain: networks of simple units that transform one space into another and learn their connection weights from experience. Griffiths follows the arc from Hebb's correlation rule to Rosenblatt's error-driven perceptron and its takedown by Minsky and Papert in 1969. A single-layer perceptron could learn whatever it could represent but could represent too little, while multilayer networks could represent more but had no reliable way to learn.
Electronic "brain" teaches itself
Part of logic's power was that it created ideas as well as describing them: start with facts, follow inference rules, derive new facts. Spaces and features gave psychologists new descriptions of the inner world, but no account of how mental states change over time, of thought itself. Logic had grounded computation in the Turing machine; using spaces and features as representations required a new model of computation.
The Turing machine idealized the behavior of a mathematician: reading a problem, calculating, writing intermediate results. Look past skin and bone and all of that is neurons firing, connected into a neural network. Griffiths's question: instead of copying the mathematician's thoughts and actions, why not copy her brain?
Americans met artificial neural networks through The New York Times of 13 July 1958, under the headline "Electronic 'Brain' Teaches Itself." The Navy had demonstrated the "embryo" of a computer called the Perceptron, expected within about a year to become the first non-living mechanism able to perceive, recognize and identify its surroundings without human training or control; the officers said they hesitated to call it a machine because it was so like a "human being without life."
Brain-like circuits were not new. In the 1940s Warren McCulloch and Walter Pitts had shown how neurons could be wired to compute Boolean logic, which led John von Neumann to think of computers organized like brains (Chapter 2). Networks were present at cognitive science's birth: at the 1956 MIT Symposium on Information Theory, Newell and Simon's talk was followed by a paper on simulating model neurons on the IBM Type 704 Electronic Calculator. One coauthor, Lois Haibt, was the only woman whose work was presented at the symposium, and later the only woman on the team that built FORTRAN.
- What was new. The perceptron was not just built of neuron-like circuits; it used experience to learn how its neurons should be connected.
- The salesman. Its inventor, Frank Rosenblatt, told the Times it would be the first electronic device to think as the human brain. The reaction to his work would expose a deep obstacle to building any such device.
Meet the perceptron
Rosenblatt was a psychologist, with bachelor's and doctoral degrees from Cornell, then a research psychologist at the Cornell Aeronautical Laboratory in Buffalo. His 1956 dissertation on the statistical analysis of behavioral tests had two unusual parts for psychology at the time: computer code for the statistics and the design of an electrical circuit he built so colleagues could analyze behavioral data automatically. Whatever his starting interest in minds, he had become devoted to building better computing machines.
The laboratory let him pursue both through a Navy-funded effort, Project PARA (Perceiving and Recognizing Automaton). His first report traced its ideas to his own research in physiological psychology, aimed at a brain analogue useful for analysis. In short, Rosenblatt wanted to build an artificial brain, specifically an artificial visual system.
- Sensory units. An artificial retina; each unit switches on when light falls in its region of the image.
- Association units. Each is wired to the sensory units in its own way, sums its incoming signals, and outputs 1 if the sum exceeds a threshold, otherwise 0.
- Weights. Each association-to-response link has its own strength (weight), so detecting a triangle can use different weights from judging left versus right; these weights are what the machine learns.
- Response unit. Sums the association activations times their weights: input_r = activation₁ × weight₁r + activation₂ × weight₂r (+ …), and outputs 1 if input_r > 0, else −1 (say, square versus triangle).
- Worked case. Activations (1, 0) with weights (0.2, −0.3) give 1 × 0.2 + 0 × (−0.3) = 0.2, so 1: "square". With weights (−0.1, 0.4) the same input gives −0.1, so −1: "triangle".
Griffiths says this bit of math is worth learning because it recurs in the next chapters. It also shows what computing with spaces means: a network transforms one space into another.
- Input space. The association units define axes (activation₁, activation₂); every retinal pattern becomes a point, our example being (1, 0).
- Output space. The response defines a one-dimensional space with positions 1 or −1.
- The mapping. The perceptron as a whole is the map between the spaces, and the weights determine it: change the weights from (0.2, −0.3) to (−0.1, 0.4) and (1, 0) moves from 1 to −1.
Rosenblatt's first description supplied hand-set weights that solved a simple classification. His real goal was for the perceptron to learn them, and for that he turned to the newest neuroscience.
Firing and wiring
In the 1950s neuroscientists had hypotheses, not answers, about how brains learn. One of the most provocative came from Donald Hebb, a psychologist at McGill.
Hebb's crooked path
- Childhood. Born in 1904 in Saint John, New Brunswick, to two doctors (his mother was the third woman to earn a medical degree in Nova Scotia), he was homeschooled until eight, then jumped from second to fourth grade in his first school year and to sixth in his second. Entering ninth grade at twelve, the youngest by far, he cared about peers' respect more than about algebra, wide reading or outspelling the teachers. Bored, he failed eleventh grade and left with a low opinion of school.
- The principal. After a degree from Dalhousie University, he became principal of his home village's school. It went badly; the provincial inspector consoled him that he had not been worth anything his first year either.
- Drift to psychology. He worked as a teamster in the wheatfields and a laborer in Quebec, read Freud, and started a graduate degree at McGill while teaching elementary school in Montreal.
- Applied learning theory. In class he made schoolwork a reward for good behavior and punished unruly pupils by sending them out to play. At McGill he ran Pavlovian conditioning experiments training dogs to salivate to cues.
His master's thesis proposed how such cue-outcome links might live in the brain: an excited neuron reduces its discharge to inactive neurons and increases it to active ones, forming a route to them, with repetition making this tendency dominant. Neurons strengthen their links to others active at the same time: the idea that made him famous seventeen years later.
In between, Hebb did his doctorate in Chicago, followed his adviser Karl Lashley to Harvard, studied how human brain damage changes behavior at the Montreal Neurological Institute, studied chimpanzee brains in Florida, and returned to McGill as a professor. He came to see a concept as a group of cortical neurons exciting and re-exciting each other. How would such a group get wired up?
Hebb's postulate (The Organization of Behavior, 1949): when an axon of cell A is close enough to excite cell B and repeatedly or persistently helps fire it, some growth or metabolic change raises A's efficiency at firing B. The axon is the fiber leading from a cell body to its connections, so the rule says that nearby neurons active together get more strongly linked: in Carla Shatz's motto, "neurons that fire together, wire together."
Stripped of neurons, it is plain association: we link sunlight with warmth because they repeatedly co-occur, and a favorite food's smell makes us salivate because it usually comes with eating, like Pavlov's dogs. Hebb pushed the principle down to single cells. It had two virtues: it proved basically correct (neuroscientists found a closely similar process in the brain), and it translates cleanly into mathematics.
The Hebbian rule in symbols
For the toy perceptron, Hebb suggests: increase each weight by the product of the activations it connects.
- Both firing (1 and 1): the weight rises.
- Association unit silent (0): no change.
- Association unit firing, response at −1: the weight falls.
So units that tend to be active with positive responses gain weight, and those that coincide with negative responses lose it. Written out, change in weight_ir = activation_r × activation_i, where i indexes association units; adding a learning rate to control step size gives the Hebbian learning rule. It captures correlations in data: weights track how often the things two units stand for co-occur. When we know the target response, a modified version, change in weight_ir = learning rate × target_r × activation_i, makes weights track the correlation between inputs and the correct answer.
Learning from mistakes
Hebbian learning has powered many network models, especially of how neuron populations store and retrieve memories through associations. It did not satisfy Rosenblatt, who had visited Hebb's lab as a graduate student. Hebb captured one kind of learning, Pavlovian conditioning by co-occurrence; Rosenblatt wanted learning from mistakes. A trainer can tell when the perceptron errs, and errors seemed the key to finding the right weights.
His idea: update weights only when the perceptron makes a mistake, and then nudge the weights of active association units toward the correct answer.
- Should be 1, said −1 (a square called a triangle): raise weights from active units, pushing the response up.
- Should be −1, said 1 (a triangle called a square): lower them.
- Correct: leave everything alone.
The intuition is everyday: finding out that the "cat" you saw was actually a dog makes you reweight the features that separate cats and dogs, while confirming it was a cat barely changes your idea of cats. In symbols, the perceptron learning rule is change in weight_ir = learning rate × (target_r − activation_r) × activation_i. If response equals target, the bracket is zero and nothing changes; if target is 1 and response −1, weights rise; if the reverse, they fall.
| Rule | Weight change Δw_ir | What it learns | Character |
|---|---|---|---|
| Hebbian | learning rate × activation_r × activation_i | Correlation between unit activities | Pure association, like Pavlovian conditioning |
| Supervised Hebbian | learning rate × target_r × activation_i | Correlation between inputs and the correct answer | A teacher supplies the target |
| Perceptron rule | learning rate × (target_r − activation_r) × activation_i | Correlation between error and input | Changes weights only on mistakes, in the right direction |
Set side by side, the perceptron rule is also a correlation, but between the error (target minus activation) and each input's activation. That was Rosenblatt's key innovation: learning driven not by co-occurrence but by how responsible a unit was for the mistake, which concentrates change where it most reduces error.
The convergence theorem and the Mark I
With this rule Rosenblatt proved a striking result: a perceptron can learn any input-output relationship it can represent. If some setting of the weights gives correct answers on a set of examples, applying the rule to those examples often enough will find weights that give those answers.
That theorem fed his confidence. He believed the sensor-to-association wiring could be arranged so the rule would learn any meaningful task. The 1958 Times demonstration was a computer simulation; afterwards his team built hardware, the Mark I Perceptron:
- Retina. 20 × 20 photocells, so images of up to 400 pixels.
- Wiring. A plugboard to configure connections from sensors to association units.
- Weights. 512 stepping motors turning potentiometers that set the resistance on links from association units to up to eight response units.
New Yorker readers who missed the Times got an interview in Talk of the Town later in 1958, titled "Rival" because the machine struck the magazine as the first serious rival to the human brain. Asked what a perceptron could not do, Rosenblatt threw up his hands: love, hope, despair, human nature in short; if we do not understand the human sex drive, why expect a machine to? His own rivals had a more fundamental limitation in mind.
What perceptrons can't do
An outsider in AI
The AI community received Rosenblatt coolly. In Principles of Neurodynamics he offered three reasons:
- Rigor. The early reports admittedly lacked mathematical rigor.
- The press. Coverage of the 1958 announcement showed the exuberance and discretion of a pack of happy bloodhounds, with headlines like "Frankenstein Monster Designed by Navy Robot That Thinks" (Tulsa, Oklahoma Times).
- Motive, the most important. Critics missed that the program aimed to study the physical structures and neurodynamic principles of natural intelligence, not to engineer pattern recognizers or "artificial intelligence": a perceptron is first a brain model.
That last point marked him as an outsider with a different perspective. He set about winning over AI researchers one by one. Charles Rosen of the Stanford Research Institute, an AI and robotics pioneer, recalled an unimpressive first sight (a short man in very heavy glasses, more psychologist than engineer) transformed by a deep voice and absorbing ideas, ending in deep respect. Woody Bledsoe of UT Austin said Rosenblatt irritated many but charmed as many, and just when you thought he was out of tricks he would produce another, so convincing that he just had to be right.
Minsky
One researcher he never convinced was Marvin Minsky, a fellow New Yorker and his classmate at the Bronx High School of Science.
- Marvin Minsky. Served in the Navy electronics program, 1945 to 1946; as a Harvard undergraduate (per physicist Jeremy Bernstein) studied composition with Irving Fine, ran his own labs in psychology and biology, and wrote a brilliant topology thesis. Encouraged by George Miller, he did a Princeton math PhD, "Theory of Neural-Analog Reinforcement Systems and Its Application to the Brain-Model Problem", thanking John Nash, John von Neumann and Claude Shannon for arguments. Then Harvard's Society of Fellows, Tufts, co-organizer of the 1956 meeting that named AI, and MIT in 1957, where he cofounded the AI Group and saw symbolic methods as central.
- Seymour Papert. South African finishing a second mathematics PhD at Cambridge; at a 1960 London information-theory conference he presented limits on learning in perceptron-like systems. He came to MIT at Warren McCulloch's invitation in 1963; had written on visual perception, worked with Jean Piaget in Geneva, shared Minsky's interest in logic and topology, and later created the LOGO language for teaching geometry.
Minsky's dissertation analyzed McCulloch-Pitts networks more deeply and asked how networks could learn. Inspired by Skinner, and consulting him, Minsky designed a network whose connections strengthened with experience. Miller said to just try it and found a couple of thousand dollars from the Office of Naval Research. In summer 1951 Minsky and Dean Edmonds built it at Harvard: 300 vacuum tubes, many motors, hand-machined electric clutches, a memory stored in the positions of 40 control knobs that the machine turned itself while learning, all driven by a surplus B-24 bomber gyropilot.
He tried adding memory loops so the network could predict, half hoping a big enough net might "get lucky" and learn to envision things. Despite early success he quit, concluding the idea was either bad or would need thousands or millions of neurons, which he could not afford. His dissertation had also posed a question at the heart of both AI and cognitive science: whether a system of humanlike behavioral complexity could be simple enough in structure to understand.
Told of Rosenblatt's convergence theorem, Minsky at first did not believe it. Once convinced, he asked about limits. The perceptron could tell E from F and 5 from 6, but uncorrelated clutter near the figures destroyed recognition; he thought its proponents had been misled by very clean examples. He wanted a rigorous treatment and found his partner in Papert, met in London in 1960; they began collaborating after 1963 and became close friends, their shared mathematics giving them unusual tools for the job.
Linear boundaries
The analysis starts from the response unit's input, a linear function of the association activations (activations times weights, summed). Plotted over the activation space it is a flat surface tilted by the weights, and where it crosses zero is a straight line. Since the unit outputs 1 above zero and −1 below, it answers 1 on one side of that line and −1 on the other. Like the prototype model of Chapter 6, the perceptron classifies with a linear boundary; unlike it, the perceptron learns where to put the line.
More association units add dimensions, turning the line into a plane or hyperplane, but the fact stands: any perceptron with one layer of learnable weights can represent, and so learn, only distinctions that a linear boundary can draw in its association-unit space.
XOR
Return to Chapter 1's truth tables, with one association unit coding P (1 if true, 0 if false) and one coding Q.
| P | Q | P ∧ Q (and) | P ∨ Q (inclusive or) | P ⊕ Q (exclusive or) |
|---|---|---|---|---|
| T | T | T | T | F |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | F | F |
- And and or. Plot the four combinations of P and Q; many lines put the true points on one side and the false on the other. McCulloch and Pitts had in effect shown this with their and/or circuits.
- Exclusive or. XOR is true when exactly one input is true (the English "or" in "you can have cake or ice cream"). Its true points sit on one diagonal and its false points on the other, and no straight line separates them.
Perceptron builders knew this and had a fix: wire the sensory-to-association connections so the target becomes linearly separable. Add a third association unit coding P ∧ Q. Since XOR is true when P ∨ Q is true unless P ∧ Q is, take OR's weights for P and Q and give the P ∧ Q unit a negative weight. With complete freedom over what sensory units detect and how they connect, perceptrons could in principle solve any classification. So Minsky and Papert needed problems demanding surprisingly complex sensory wiring, and their topology training supplied one.
Connectedness
A figure is connected if it cannot be split into non-touching parts, an intuitive idea with deep roots in topology. Minsky and Papert proved that no perceptron whose sensory units cover local regions of an image can decide connectedness: the area those units must cover grows with the image. The intuition: take a figure whose connectedness depends on two short side bars; it is connected only if one bar is at the top and the other at the bottom, and disconnected if both are top or both bottom. That is another XOR, so some association unit must see both sides (the analogue of P ∧ Q). Because the figure can be stretched as wide as you like without changing the problem, that unit's input area must be arbitrarily large. Minsky's summary: a perceptron cannot put together things that are visually nonlocal.
Connectedness itself is not an everyday need. The point was that designing the right sensory units can be hard and may require those units to do heavy processing themselves, echoing Minsky's undergraduate finding that a useful network would be far larger and more complex than expected. Papert's later verdict: networks, like brute-force search programs, scale very badly.
There was a pointed irony. Geometric problems are easy for rules and symbols: Newell and Simon considered geometry proofs before turning to logic and the Logic Theorist, and Papert would build LOGO to teach geometry in schools. Minsky and Papert had shown that a class of problems natural for symbols could be hard for perceptrons. Their book, Perceptrons (1969), hit AI immediately, both for the limits it proved and for showing that network capacity could be analyzed mathematically. Allen Newell's review in Science opened by simply calling it a great book.
Block's rebuttal
Not everyone agreed. Rosenblatt had largely moved on: he spent years testing whether transplanting brain tissue between animals transferred memories, and took up astronomy and the search for extraterrestrial intelligence, buying a house near Cornell and building his own observatory. He died in a boating accident in 1971, on his forty-third birthday, without answering. His collaborator, the mathematician David Block, did.
Minsky and Papert. The simplest perceptron provably fails tasks that rules handle easily; perceptrons can learn what they represent but represent too little, and multilayer networks have no reliable learning procedure.
David Block (review, 1970). They analyzed a perceptron no enthusiast would call typical, ignoring the two-layer, feedback and within-layer (lateral) versions Rosenblatt built. Humans find connectedness hard too (Figure 7.5's spirals), so a brain model should; Rosenblatt would even distrust a machine that beat human perception. His image: a mohel who burns the baby and hands over the foreskin, saying it will never amount to much.
Block was right in the long run: later chapters show multilayer perceptrons solving an impressive range of problems. But in 1970 nobody could learn the weights of a multilayer perceptron, or even show it would work; Rosenblatt's theorem covered only a single layer of weights. That gap deterred many researchers, and for the next decade attention turned to a different problem: representing the complex structure of the world.
The convergence theorem in its rigorous form is usually credited to Block (1962) and Albert Novikoff (1962), building on Rosenblatt. Minsky's 1951 machine was the SNARC (Stochastic Neural Analog Reinforcement Calculator). The popular story that Perceptrons "killed" neural networks is overstated: the authors acknowledged multilayer nets could represent more, funding shifts and the hardware limits of the era mattered as much, and Minsky and Papert's 1988 expanded edition defended the book against the revival. Griffiths presents Block's case fairly, but his own account still leans on the dramatic version.
Hebb's rule had a second life. John Hopfield's 1982 network stored memories as attractors using Hebbian weights, and on 8 October 2024 Hopfield shared the Nobel Prize in Physics with Geoffrey Hinton "for foundational discoveries and inventions that enable machine learning with artificial neural networks", the first Nobel to honor work in Rosenblatt's tradition.
Brain Models and Mind Models
Told by Minsky and Papert to go away and work out knowledge representation, researchers built networks of words and concepts to capture meaning, and in modeling how such networks are searched they rediscovered the perceptron's mechanics under a new name: spreading activation. Griffiths traces the line from Quillian's semantic network through Collins and Loftus to McClelland and Rumelhart's interactive activation model. A mental state became a point in activation space, and thought a trajectory through it shaped by weighted connections.
Why meaning needs networks
Perceptrons could learn hard tasks only if handed good representations of them. As Minsky and Papert later put it, network researchers would have to pause and develop adequate ideas about the representation of knowledge, so computer scientists began exploring ways to capture the rich knowledge people bring to the world.
- Implicit knowledge. People know many things never explicit in what they see or hear: that canaries have skin, that fish eat. Representing and using such knowledge became a central AI problem.
- Context. The same word means different things in "The bank has a high interest rate" and "Despite the fast-flowing water he made it to the bank." Chomsky had focused on syntax, how words assemble into sentences; world knowledge belongs to semantics, what words and sentences mean.
To understand a word we must weigh the words around it, so word meaning reflects the connections between words. Treating words as nodes in a network became the route by which neural-network-like structures returned: studying how information spreads through word networks led cognitive scientists to rediscover ideas from perceptrons. What began as models of the brain became models of the mind.
Semantic networks
Quillian's dictionary
Ross Quillian had an odd pedigree. He never earned a bachelor's degree, studying English at UCLA for two and a half years, yet the University of Chicago, which occasionally admitted students without one, took him for a master's. In 1961 he joined MIT's Research Laboratory of Electronics, which had grown from wartime radar work into a powerhouse of communication and computation research; Claude Shannon joined it on moving to MIT in 1956, and it was an intellectual home for Chomsky and his colleagues.
There Quillian worked on representing word meanings, especially the relations between words. His 1962 paper gave the example that "death" leaves no trace in the symbol "murder", yet every English speaker knows the first concept is part, but not all, of the second. Relations also solved ambiguity. In the bank sentence, "interest" and "rate" signal a financial institution, but so could "account", "savings" or "mortgage". Nothing in their spelling or syntax marks them as clues; what they share, Quillian wrote, is a reference to money, which lives solely in their meaning.
His plan was an automated dictionary in which each word's meaning is its relationships to other words, a twentieth-century update of John Wilkins's scheme from Chapter 1. Merriam-Webster defines "wheel" as a circular frame of hard material, solid, partly solid or spoked, able to turn on an axle; but "circular" and "axle" have entries too. Draw lines from "wheel" to them and you get a network of words, each linked to its semantic parts.
- Semantic network. A network of words or concepts joined by links that state their relationships.
- Familiar today. Browsing the web is exploring such a network; Wikipedia's "wheel" page links to "axle" and "rolling". In 1962, with no World Wide Web, it was a radical idea.
- The cost. Quillian warned that building such a memory meant a gigantic amount of "very tedious and dirty work", but nothing beyond a concerted effort.
He made that effort over the next four years in a PhD at Carnegie Mellon with Allen Newell and Herbert Simon, producing a working system that simulated the inference and disambiguation he saw as central to understanding language.
- Nodes and IS-A links. Canary → Bird → Animal; Shark, Salmon → Fish → Animal; each node also links to its features ("can sing", "is yellow" at Canary).
- Cognitive economy. A property shared by a whole category ("has skin") is stored once, at the most general node it fits, saving the space redundant copies would take.
- Search by activation. To check "a canary can fly", activate canary and can fly and let activation spread along links: from canary to can sing, is yellow and bird; from can fly to bird. When bird is reached from both sides, the statement is confirmed.
- Prediction. Spreading takes time, like waiting for a clicked web page to load, so more links means slower answers: "a canary has skin" (three links) should take longer than "a canary can sing" (one).
From storage trick to psychology
Quillian's 1967 report closed by noting that anyone inclined to treat the model as a psychological theory faced many new problems: whether it suffices for other behavior, how variations in its structure affect performance, and how its features correlate with individual differences. That was his next step. While finishing the PhD he had joined Bolt, Beranek and Newman (BBN), a Cambridge research company spun out of MIT faculty work, where he met Allan Collins, a psychologist finishing a Michigan PhD and an expert on human memory and on reaction time as a measure of how accessible information is.
Collins and Quillian tested the link-counting prediction on BBN colleagues. People took longer to confirm "a canary has skin" than "a canary can sing", and longer for "a canary is an animal" than "a canary is a canary" (less surprising). Each time, response time tracked the number of links to traverse. Later work found that answering "can a canary sing?" right after "can a canary fly?" was faster, as if the words linked to "canary" were still active. The network was turning from an efficient fact store into a theory of how people evaluate statements, and a new way to think about thought.
Spreading activation spreads
Loftus and the yellow fruit
Beth Loftus stood out in Stanford's mathematical psychology program. Gordon Bower, one of its founders, praised her mathematical training and her motivation, enthusiasm and efficiency; she was also often the only woman in a male field. And she was bored.
Not by the math. After her mother died suddenly when Loftus was a teenager, math was one of the few ways she could connect with her father; she excelled, majored in mathematics at UCLA, and took psychology only as electives. What bored her was the math's shortfall: she disliked constantly simplifying assumptions about behavior to keep equations tractable. Her dissertation, on computers in teaching, was a holdover from plans to teach math. The spark came in a social psychology class taught by Jonathan Freedman, who introduced her to semantic memory. Together they timed answers to questions like "What is the name of a fruit that is yellow?"
- Order effect. People answered faster when the category (fruit) came before the property (yellow) than the other way round.
- Priming. Responses sped up when the same category had appeared in a recent question.
- Decay. The speedup depended on the gap: fastest for two fruit questions in a row, slower with one intervening question, slower still with two.
Association strength
Griffiths runs a mini-experiment on the reader: write the first word that comes to mind for hive, meow, omelet, oak, quench. His guesses: bee, cat, eggs, tree, thirst. He bets on a few hits not by mind reading but because research on semantic association shows some words automatically summon others.
At the New School for Social Research in New York, Loftus used those associations. Students saw a category and an object on an index card and confirmed membership, while she varied the strength and direction of association:
| Card | Category → object | Object → category | Result |
|---|---|---|---|
| Tree-oak | Strong | Strong | Fast |
| Seafood-shrimp | Strong | Weak | Fast |
| Insect-butterfly | Weak | Strong | Slow |
| Cloth-orlon | Weak | Weak | Slow |
(Orlon is a synthetic fiber popular in the 1970s, like nylon; Griffiths admits he had to look it up.) What mattered was the link from category to object: strong category-to-object associations produced faster confirmations.
Collins and Loftus
When Allan Collins saw these results he knew at once they fit spreading activation; more than that, they explained how it worked. Quillian had invented spreading activation as a bookkeeping trick, marking each word visited until paths met. Collins and Loftus made it a complete psychological theory.
Collins and Loftus (1975) made activation graded: each word carries a real-valued activation that decays without input (the repeated-category results), receives activation along links of varying strength (more flows down stronger links, so seafood-shrimp beats insect-butterfly), sums what arrives, and becomes active when the total passes a threshold. That is the perceptron's arithmetic, rediscovered from the study of semantic memory and freed to flow in any direction.
They also dropped the strict hierarchy, using a plain network of words joined by associative links. Take one corner: "flowers" has links from "violets" and "roses", with weights weight_violets,flowers and weight_roses,flowers, and each word has an activation. See the word "roses" and its activation spreads, so at the next moment:
- input_flowers = activation_roses × weight_roses,flowers
- input_violets = activation_roses × weight_roses,violets
If those inputs trigger "flowers" and "violets", they feed activation back and onward, so activation flows back and forth among words.
The theory covered Loftus's findings and much else in semantic memory. It explains the yellow fruit result: "fruit" has fewer associates than "yellow", so activating it first lights up fewer words and makes the answer easier to pick out. Their paper became one of the most cited in psychology, and any textbook account of what happens in people's heads now talks about one thing "activating" another. Spreading activation went from a computer scientist's linking trick to our best guess at how what we see and hear changes our mental state.
Interactive activation
Two kinds of lecture
In 1974 Jay McClelland, fresh from a Penn PhD and new at UC San Diego, was terrified to co-teach with David Rumelhart, who had been a professor for seven years, had graduated from Stanford's math-psych program shortly after Loftus arrived, had just won tenure, was reputed extremely smart, and had written the course textbook. McClelland survived, but never forgot Rumelhart's two lecture styles: barely prepared, slow, mumbled three-hour monologues, or animated, confident and mind-expanding ones.
Cascades and the editor's demand
McClelland's research was on how people perceive words. Reading fascinates cognitive psychologists because words are built from letters as objects from features, so it exposes how minds split the world into parts and reassemble them. Theories then held that reading, like everything else, runs in discrete stages: letters first, then words. McClelland proposed instead that processes cascade, letter recognition gradually shading into word recognition. He sent the idea to Psychological Review; the editor replied, in effect, nice idea, now do the math.
Math was not in his toolkit, but the person building the tools was down the hall, and taught a class on it.
Rumelhart's mathematical psychology class had no textbook and no reading of existing models. Each week a student brought data and a half-formed interpretation; Rumelhart used the Socratic method to steer it toward ideas expressible mathematically, then wrote equations on the board, simplifying and teaching the tricks by example. By the end of a session the class often had a complete mathematical account that fit the data well. McClelland's lesson: there was nothing mysterious about building theory; lay out the ideas, put them in equations, and work out the implications.
He also drew on a talk by Jim Anderson, showing how simple Hebbian networks capture aspects of human memory. Anderson, trained in physics and physiology at MIT, had settled at Brown, moving among Applied Math, the Center for Neural Science and Cognitive and Linguistic Sciences. (He was still there thirty years later when Griffiths took his first faculty job at Brown; his neural networks class was notorious for ambitious projects that seniors had to finish over sleepless nights to graduate.)
McClelland's cascade model treated reading as the gradual accumulation of activation: glimpsed letter parts begin to activate letters, which gradually activate words, through a network linking letter parts to letters and letters to words. It reproduced phenomena used to argue for discrete stages and made new predictions that matched data. The paper, now full of math, was accepted, and it sparked a collaboration, since Rumelhart had been thinking along very similar lines.
Words talk back to letters
Rumelhart's frustration with stage theories was different. The problem was not discreteness but no interaction: letter recognition finished, then informed word recognition, with no way for knowledge of words to influence which letters you see. That seemed wrong. In a four-letter word starting W, O, R, very few fourth letters are possible, and an intelligent system should use that constraint.
Steeped in semantic networks, Rumelhart saw that McClelland's model could capture this if links ran in both directions: letters activate words, and activation flows back to candidate letters. The price was the math. Unlike the cascade model, no simple equations described activation sloshing back and forth, so they ran endless computer simulations. Rumelhart recalled building models they did not understand, not knowing why they worked or failed or what mattered, struggling until they gained insight and became obsessed with making it work. About two years later they published a series of papers on the interactive activation model (McClelland and Rumelhart 1981; Rumelhart and McClelland 1982).
A general model of mental process
The model turned spreading activation from an account of how ideas influence one another into a general description of mental processes.
- Units can be anything. A network need not represent words; its units can stand for whatever the mind must represent, and its weighted connections for association strengths between them.
- Reading phenomena. Spreading activation in such a network captured many reading effects, such as context shaping the reading of ambiguous letters.
- Induction solved by constraints. It gave an elegant answer to reading's inductive problem: with incomplete information, spreading activation combines constraints from many sources into a solution.
| Logic (rules and symbols) | Network (spreading activation) | |
|---|---|---|
| Mental state | A set of facts (P, Q, P → Q) | A point in activation space, e.g. (0.4, 0.1, 0.9) |
| Thinking | Applying inference rules to derive new facts | Spreading activation moves the point along a trajectory |
| What directs thought | The rules | Connections and their weights |
| Handles | Deduction | Combining partial constraints (induction) |
More fundamentally, it offered a different picture of thought. In logic, your mental state is the set of facts you know, and thinking applies inference rules to derive more. In a network, your mental state is the pattern of activation across units, and thinking is spreading activation. With three units, give the activations as coordinates on three axes: a state of 0.4, 0.1 and 0.9 is the point (0.4, 0.1, 0.9). Spreading activation changes the values and moves the point; repeat and join the points, and thought traces a curve through activation space whose shape is set by the network's links and weights.
That is the new model of thought: a mental state is a point summarizing the activation of every concept in your head, thought is a curve through activation space, and what steers it is the set of connections you have formed. But in rediscovering brain-model ideas, psychologists also rediscovered that approach's biggest obstacle. The interactive activation model was effectively a multilayer neural network, exactly the kind with no effective algorithm for finding good weights; its weights were set by hand. To go further, mind models had to learn.
Quillian's design survives in knowledge graphs (entities linked by typed relations) used in search and enterprise data catalogs, and cognitive economy is the same move as inheritance in object-oriented software: define a property once on the parent class and let exceptions override it locally, as "can't fly" does at Ostrich.
The hierarchy's link-counting predictions did not hold up cleanly: Carol Conrad (1972) showed property frequency explained much of the effect, and Rips, Shoben and Smith (1973) found typicality violations (people verify "a chicken is an animal" faster than "a chicken is a bird"), which is why Collins and Loftus dropped strict hierarchy. Griffiths omits that Loftus became famous mainly for her later work on eyewitness testimony and false memories. The interactive activation model was built to explain the word superiority effect (letters are identified better inside words; Reicher, 1969), which Griffiths alludes to without naming.
The Plot Deepens
Neural networks survived their "dark ages" in psychology, away from the newspapers, until David Rumelhart and Geoff Hinton treated learning as optimization: follow the gradient of the error and pass error signals backward through the same weights that carry predictions forward. Backpropagation (published in Nature in 1986) launched connectionism and the PDP volumes, a full West Coast challenge to rules and symbols. Data (ImageNet) and graphics chips (GPUs) finally let deep networks win in 2012, and AI has been a contest of data and compute ever since.
Ideas that move between fields
Enthusiasm for artificial neural networks has risen and fallen several times since the 1950s, and each swing passed the baton between disciplines. A psychologist, Frank Rosenblatt, showed what perceptrons could do; two computer scientists, Marvin Minsky and Seymour Papert, showed what they could not; now another psychologist, David Rumelhart, began working out how to get past those limits.
Griffiths reads this traffic as the payoff of the Cognitive Revolution's model of research. Psychologists borrowed formal tools from mathematics and computing, then returned new approaches that mathematicians and computer scientists could study in turn. Because the fields judged ideas by different standards, psychologists, whose goal was to understand human minds, could keep working on ideas that were unfashionable in computer science. That is what happened to networks.
- Jim Anderson's summary. During the dark ages, he said, the ideas had moved out of highly visible big science into fields that never made the newspapers.
- The turn. Those quiet efforts were about to yield visible results that would pull computer scientists back to networks and lay the foundation of today's AI breakthroughs.
Layers upon layers
Rumelhart the persistent
David Rumelhart grew up in South Dakota and finished high school in 1960 in Wessington Springs, population 1,488. His family prized hard thinking: Jay McClelland told Griffiths they would bend their attention to working out things like the best possible hand of gin rummy. At the University of South Dakota he worked as a janitor in a church-run student center in exchange for free rent and took an extra class each semester so he could double major in mathematics and psychology. When he won several prestigious fellowships for a Stanford PhD, his senator, George McGovern, announced it proudly to the US Senate.
The persistence carried into science. Robert Bjork, the learning researcher known for "desirable difficulties" and a close friend from graduate school, remembered Rumelhart getting keyed up before a hard problem like an actor before a show, and growing more animated the more puzzling it became; once puzzled, Bjork said, he was "like a dog with a bone." Making learning work in networks with several layers of weights was one such bone, and he chewed on it for years.
Why hidden layers help
Adding layers mattered for the psychology Rumelhart cared about, but it was also the obvious answer to Minsky and Papert. A network is a system that transforms one space into another. In modern terms (replacing Rosenblatt's association and response units), the simple perceptron maps input units to output units; insert hidden units between them and the mapping gains a step. The input-to-hidden weights carry the input space into a new intermediate space, and the hidden-to-output weights carry that into the output space. A problem a one-layer perceptron cannot solve can be broken into steps that it can.
Exclusive-or in two layers
- The obstacle. "And" (∧) and "or" (∨) give linearly separable sets of true and false points, so one perceptron computes them; exclusive or (⊕), true when exactly one input is true, does not.
- Rewrite in logic. From the truth tables, P ⊕ Q is true whenever P ∨ Q is true except when P ∧ Q is true, so P ⊕ Q = (P ∨ Q) ∧ ¬(P ∧ Q), with ¬ meaning "not".
- Hidden layer. Choose weights so one hidden unit computes P ∨ Q and another computes P ∧ Q, each outputting 1 when true and 0 when false.
- Output layer. Feed both into a unit with a positive weight on P ∨ Q and a larger negative weight on P ∧ Q. Its input exceeds 0 only when the "or" is true and the "and" false; thresholded at 0, like Rosenblatt's units, it outputs 1 for true and −1 for false.
| P | Q | P ∧ Q | P ∨ Q | P ⊕ Q |
|---|---|---|---|---|
| T | T | T | T | F |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | F | F |
More layers in general make more complex functions easy to compute. A deep unit takes input from every unit beneath it, which take input from the units beneath them, so higher units can draw on a wider range of the input and on the transformations computed below. The condition is a nonlinearity in the intermediate layers: each unit must alter its weighted sum somehow rather than pass it on, or the stacked maps collapse into one. Rosenblatt's version was a sharp threshold on each unit's input.
Rumelhart knew all this. He had read Perceptrons when it appeared and taught a class on it; he later said Minsky, more than anyone, was the reason he read about the perceptron at all. As his work with McClelland began using multilayer networks, he turned to learning, and he would not solve it alone. With a Sloan Foundation grant, he and colleagues hired a team of young researchers in cognitive science, among them a British polymath.
Following the gradient
Hinton's long road
Geoffrey Everest Hinton belongs to the family met in Chapter 1: he is the great-great-grandson of George Boole and Mary Everest Boole, through their eldest daughter Mary Ellen Boole and her husband Charles Howard Hinton. His father, Howard Everest Hinton, was a University of Bristol entomologist and a staunch Stalinist; Geoffrey went to a British public school with a strong Christian ethos, the opposite of home. He later counted that good training for science, since it accustomed him to the idea that at least half the people are completely wrong.
His path to the field ran through a string of false starts:
- Cambridge physics and physiology. He aimed to be a biophysicist but found the equations too hard; several integral signs side by side, he joked, seemed to be ganging up on him.
- Philosophy. Still hoping to find the meaning of life, he tried it for a year, but disliked that you could never tell whether you were right.
- Psychology. In his third year he expected models of how the mind worked and found people studying rats in mazes.
- Carpentry. He then spent a year as a carpenter, until he met a real carpenter so much more skilled that academia looked easier.
- Back to Bristol. As a research assistant on a study of children's language learning, he regained interest in science; a mathematician friend introduced him to neural networks, a field mixing mathematics, physiology, psychology and even philosophy (only carpentry was missing). He asked around and went to Edinburgh for a PhD on them.
At Edinburgh Hinton was gripped by the contrast between two kinds of learning. Supervised learning is what a perceptron does: it uses a signal about right and wrong answers. Unsupervised learning is what Hebb's rule does: it captures correlations between events without any training signal. Hinton was convinced unsupervised learning loomed large in human brains.
San Diego
He applied for the posts Rumelhart and colleagues had advertised at UC San Diego, was at first passed over, and accepted a job at the University of Maryland. Hours after calling to accept it, he learned San Diego wanted him; he said yes and phoned Maryland to unaccept. The Maryland professor who had hired him never spoke to him again. California felt liberating: far less critical than England and pluralist, accepting that MIT and Berkeley could hold rival views at once, where England's small, closed community recognized a substantive view and nonsense.
Hinton's arrival fed Rumelhart's fixation. McClelland and Hinton organized a San Diego conference of like-minded people. Rumelhart took a sabbatical at Stanford to write a book on symbolic representations of semantic information and could not do it: his mind kept drifting back to networks. He worked in committed blocks, so he made a five-year plan for neural networks and assembled a group: Hinton, McClelland, Francis Crick (Nobel laureate for the structure of DNA, deeply interested in neuroscience) and graduate students who became leaders themselves. For six months they met for hours every day.
Take the gradient
The decisive idea was Hinton's: step back from neurons and treat networks as mathematical systems. McClelland recalled him telling the group that people were inventing learning rules by imagining what a pre- and post-synaptic neuron might do together, when the right move was to take the gradient of whatever you want and let that dictate the weight changes. That is the classic optimization method of gradient descent. (Griffiths keeps the math light in the text and puts the full calculus in the endnotes.)
Gradient descent. Express the network's total error as a function of its weights. The gradient is the slope of that function at the current weights: negative means error falls as the weight rises, positive means it climbs. So move each weight against the slope: if raising it increases error, lower it; if lowering it increases error, raise it. Stop where the slope is zero, a minimum. Computing slopes is elementary calculus (Newton and Leibniz), and it can be done one data point at a time, like the perceptron rule.
Two obstacles and a trick
A standard error measure is the squared difference between what the network should output and what it does. With a single output unit r:
- error = (target_r − activation_r)²
The weights feed into activation_r, but here the first snag appears: a perceptron's threshold is not differentiable. Because the output jumps when input crosses the threshold, a tiny weight change can cause a huge change in error, and the gradient can be infinite.
After their reading models, McClelland and Rumelhart had been trying other activation functions, especially the sigmoid, which keeps the idea of thresholding but smooths it so the output does not leap between values. They still could not make the networks learn until they found a shortcut. In Rumelhart's words, they would pretend the units were linear, compute derivatives as if they were, and train on that basis.
Applied to one output-layer weight, weight_3r, gradient descent gives:
- Δweight_3r = learning rate × (target_r − activation_r) × activation_3
That is exactly Rosenblatt's perceptron learning rule from Chapter 7. The only difference is that Rosenblatt's association units took discrete values while Rumelhart's group worked with real-valued activations. Linearity made the gradient easy; adding back a sigmoid needed just one more step in the derivation. By re-deriving a classic rule as gradient descent, Rumelhart had a new tool for the multilayer problem.
(Re)discovering backpropagation
The plane ride
Rumelhart had other duties, among them a meeting of the Office of Naval Research to report on the grants funding his work, and the flight gave him hours to do the math. He had been rereading Rosenblatt, who stated the problem clearly. In a multilayer perceptron we already know how to learn the last layer (hidden to output): use the output errors. The hard part is the layer below, where nothing says what the error is, because there is no target value for a unit inside the network.
Rosenblatt had proposed "back-propagating" the error from output units to hidden units, building an error signal for each hidden unit from the errors of the outputs it fed. His heuristic worked on simple problems but was finicky, and Rumelhart judged that he had the idea of sending errors back across layers without a principled way to do it. On the plane Rumelhart reused his earlier move: pretend the network is linear, work out how to train it, then put the sigmoids back.
The rule
Linearity made the derivation easy. For weight_13, from input unit 1 to hidden unit 3, differentiating gave:
- Δweight_13 = learning rate × (target_r − activation_r) × weight_3r × activation_1
This is almost the output-layer rule. The only change is that the output error is multiplied by weight_3r, the strength of hidden unit 3's connection to the response, which reads naturally as unit 3's contribution to the error.
Each unit's responsibility for the error equals its influence on the response. Rosenblatt's notion of back-propagating errors was right; he had missed only that the share of error sent back to a unit is set by the weight linking it to the output. Predictions flow forward through the network's weights, and error signals flow backward through exactly the same weights.
Underwhelmed, then shelved
Having the insight from the linear case, Rumelhart restored the nonlinearities, wrote code, and tried exclusive or first. It worked, but it needed a thousand iterations, and his private verdict was that this was pretty bad. When he landed and told Hinton, Hinton was unimpressed too: nothing guaranteed a global optimum, the method was bound to stick in local minima, so it hardly seemed worth pursuing.
That worry is inherent in gradient descent. Following the slope gets you to a minimum, usually one near your starting point rather than a particularly good one (the smaller dip in Figure 9.2). Hinton wanted an algorithm with a guarantee of finding a good solution. So Rumelhart did the last thing anyone would expect of the man who had just found the algorithm that would revive networks and eventually power AI's big breakthroughs: he sat on it, for years.
Early 1980s On a flight to an Office of Naval Research meeting, Rumelhart derives backpropagation and tests it on XOR (a thousand iterations).
1982 He works instead on language learning and competition between neurons.
1983 He nearly gives a talk on backpropagation, then changes his mind.
1984 He rewrites the code and shares it with a few researchers.
1985 Hinton, now a professor at Carnegie Mellon, had coded it as a baseline for his own training method, was shocked at how well it did, and messages that the algorithm seems to actually work.
1985-86 Rumelhart and Hinton train networks on problem after problem, add postdoc Ron Williams, and publish in Nature (1986).
Who got there first
Griffiths is careful on priority. Rumelhart was not the first to propagate errors backward through a system of equations to optimize it by gradient descent: Seppo Linnainmaa did that in a 1970 Finnish master's thesis. Nor was he first to propose it for multilayer perceptrons: Paul Werbos did so in 1981. What Rumelhart, Hinton and Williams did first was demonstrate what the algorithm could achieve, and that excited a new generation about networks both as theories of the mind and as tools for building AI.
Making connections
Parallel, distributed
Rumelhart's five-year plan became a fifteen-year plan. With McClelland he edited two volumes of collaborators' network research titled Parallel Distributed Processing (PDP). The title named the difference: computation runs in parallel as information passes through the many units of each layer, and representations are distributed across units rather than each unit standing for one concept, as in a semantic network.
For perhaps the first time, networks underpinned a complete theoretical framework able to challenge rules and symbols as an account of mind. West Coast cognitive science, radiating from San Diego, now faced East Coast orthodoxy.
Rules and symbols (East Coast). Discrete representations that are either true or false; inference rules update a set of facts; learning is shaped by strong domain-specific innate constraints.
Connectionism (West Coast, San Diego). Continuous representations with graded values; functions implemented by weights transform activations; powerful domain-general learning rules update the weights from experience.
The framework was called connectionism, a word Rosenblatt had used when aligning himself with "the empiricist, or 'connectionist' position." He took it from Edward Thorndike's behaviorist idea that learning forms connections between stimuli and responses. The new connectionism was richer: with multilayer perceptrons, stimulus and response are linked through representations the system discovers for itself by learning the statistical structure of its environment.
Ten years of network models of mind
Inspired by the PDP books, researchers over the next decade applied networks to many aspects of cognition.
- What categories are. Present objects as features or as points in psychological space, and a network learns a new representation of them and finds category boundaries automatically.
- How learning changes representation. The models' internal representations suggested how learning a category might shift which features people attend to, and the progress of gradient descent predicted the timecourse of human learning.
- Rosch's family resemblance. Networks capture it easily, since features combine continuously to give graded category membership, a problem case for rules.
- Variation and interpolation. Mappings between spaces tolerate noisy variants of an input and generalize to novel inputs by interpolating between known ones (Figure 9.4).
The generality was the point. One model type, the multilayer perceptron, explained learning across many settings. A big enough one can approximate any continuous function from one space to another, and starting from small random weights imposes only a weak inductive bias on which functions are easy to learn. So the explanations were driven mostly by the data available to learners: behavior explained by the statistics of the learning environment plus a general-purpose learner.
Real brains, and Rumelhart's end
Artificial networks also gave a new angle on real brains. Computational neuroscientists asked whether brains might perform something like backpropagation, or built alternative learning algorithms from the latest biology. Influence ran both ways: thinking about the hippocampus led McClelland and colleagues to a fix for catastrophic forgetting, where a network trained on a new task quickly loses an old one.
Rumelhart joined all of these directions. He moved to Stanford in 1987 and built network models of everything from understanding stories to sound localization in owls. The end of the fifteen-year plan was also the end of his career: he developed Pick's disease, a rare neurodegenerative disorder causing dementia, retired in 1998 and died in 2011. Former student Bob Glushko endowed the David E. Rumelhart Prize for work on the theoretical foundations of cognition; awarded annually, it is among the field's biggest honors and has gone to many people in this book.
Another path to AI
Turing's blank notebook
Just as networks offered an alternative account of the mind, they offered another route to building AI. Meeting Minsky and Papert's challenge, by showing multilayer perceptrons could learn hard problems, drew people to using them for intelligent machines. Machine learning as a field arose at about the same time, and the idea of learning as the road to AI goes back to Alan Turing in 1950: rather than program an adult mind, simulate a child's and educate it, since a child's brain is presumably like a fresh stationer's notebook, with little mechanism and many blank sheets. Networks were the ideal mechanism, their weights the blank sheets where knowledge would be written.
As networks rose, machine learning became entwined with cognitive science and neuroscience. Two late-1980s creations show it: the Conference on Neural Information Processing Systems (NeurIPS), which became the premier machine-learning conference, and the Connectionists email list, still a lively venue for debate and self-promotion in the era of the platform formerly called Twitter.
Griffiths's favorite Connectionists post came from Hinton on 27 January 2014. Someone had argued brains are a poor guide to intelligent machines because evolution never found good engineering solutions such as the wheel. Hinton, contrarian as ever, replied that evolution did invent a time-shared wheel: rough ground needs a wheel about 6 feet across with very soft suspension, and the light, compact way to get one is to time-share two small sections of rim, each joined to the axle by spokes in compression that change length easily. Swapping spokes in and out costs energy, he conceded, but it solves the problem of feeding the rim nutrients. He was describing legs.
Convolutional networks
Collaboration between machine learning and neuroscience also produced new architectures. The most influential is the convolutional neural network, built to learn efficiently from images.
- The naive design. One input unit per pixel (its brightness, for a black-and-white image), one output unit per label, and learned weights in between.
- What it misses. Shift a dog one pixel left and it is still a dog, but every input number changes, so the network sees a new picture. Labels should be invariant to translation.
- Pattern detectors. The first hidden layer holds simple patterns (an eye, a nose), each a set of weights over a square patch of pixels, with one hidden unit per pattern per patch, so activations show which patches contain which patterns.
- Pooling. The next layer pools across locations, signaling how much a pattern appears in a larger region.
- Stacking. Further layers look for patterns among those activations, building up to bigger configurations (eyes and noses together) recognized wherever they appear.
- Weight tying. Each pattern uses the same weights at every position, so one number sets many weights: fewer numbers to learn and less data needed.
The design drew on neuroscience. In the 1950s, David Hubel and Torsten Wiesel found that neurons in cat visual cortex respond to simple patterns in specific parts of the visual field, with more complex patterns over larger areas detected deeper in the system. That work had also inspired Rosenblatt, who proposed perceptrons joined by "templates" replicated across an image, much as in convolutional networks. The efficiency of weight sharing made real problems tractable: in the late 1980s and early 1990s, Yann LeCun and colleagues at Bell Labs built convolutional networks for handwritten digits, eventually used to read zip codes on letters and numbers on bank checks.
The second winter
Interest in networks within machine learning nonetheless waxed and waned. Researchers realized networks worked partly because they were very flexible statistical models, and turned to other ideas from statistics. Others wanted methods with stronger theoretical foundations, returning to single-layer perceptrons and making them more expressive by sophisticated feature expansion of their inputs.
- The rejection signal. NeurIPS organizers computed which words best predicted a paper's acceptance or rejection. In the early 2000s, when Griffiths began attending, "neural" and "network" were often among the top 10 predictors of rejection.
- The mailing list. The same turn shows in traffic on the Connectionists list (Figure 9.5).
A dedicated core kept going, Hinton among them, now focused on deeper networks with more hidden layers. Backpropagation works in principle at any depth, but in practice the error signal was diluted as it passed back layer by layer. Among Hinton's tricks was a return to unsupervised learning: build the network one layer at a time, each discovering patterns in the activations below, then fine-tune the output with supervised learning. That showed deep networks' potential, but cracking the problem took two more breakthroughs.
Breakthrough one: data
Surprisingly, George Miller laid the groundwork, earning credit in a second revolution. In 1979 he moved to Princeton and founded a cognitive science lab whose main project was WordNet, a large-scale semantic network reconstructing part of English's semantic structure: over 100,000 words linked by relations of same meaning ("plank" and "board"), specific instance of a general class ("dog" and "canine") and part of a whole ("foot" and "leg").
In 2006 Fei-Fei Li was thinking about a new computer vision dataset; while others chased algorithms, she saw data as what would move the whole field. Li had come to the US from China at fifteen, studied at Princeton while working weekends in her parents' dry cleaning shop, took a computer-vision PhD at Caltech, and became a professor at the University of Illinois Urbana-Champaign. On a visit to Princeton she met linguist Christiane Fellbaum, one of Miller's WordNet partners, and saw she could use WordNet's structure to organize an image dataset. She moved to Princeton and, with graduate students including Jia Deng and Olga Russakovsky (both later Princeton professors), built ImageNet: more than 14 million images in 21,000 categories, far more data than any vision system had used and far more categories to learn.
Breakthrough two: graphics chips
Rosenblatt had built a physical perceptron, but as general-purpose computers got faster and easier, network research moved onto ordinary machines. In 2006 Rick Szeliski, a former Hinton student, told Hinton networks should run much faster on graphics hardware.
- CPU and GPU. Von Neumann's architecture produced the CPU, which crunches numbers and interprets symbols. As graphics grew, drawing the screen moved to a graphics processing unit (GPU), long the less famous cousin that merely displayed what the CPU did.
- Why it fits. To work out what goes where on screen, GPUs specialize in multiplying and adding matrices, and computing hidden-unit activations or back-propagating their errors is exactly that computation.
- 2009. Vlad Mnih, a Hinton graduate student, wrote software that made it easy to run networks on GPUs; two other students used it for speech recognition, and with heavy engineering by Google it sharply improved recognition on Android phones in 2012.
- The hard part. Getting several GPUs to cooperate, and running convolutional networks efficiently on them, were very tricky programming problems.
Alex Krizhevsky, another Hinton graduate student, solved both problems. On a purpose-built machine with two high-end gaming GPUs in his bedroom, he trained a convolutional network with seven hidden layers on an ImageNet subset of 1.2 million images and 1,000 categories; it took six days. The previous best top-5 error (five guesses, correct if any is right) was just over 25%; AlexNet scored 17%, and further tweaks reached 15.3%. Vision researchers Griffiths knew had to replicate it to believe it, and the NeurIPS presentation drew a crowd split evenly between enthusiasts and skeptics. Part of the trick was the ReLU activation, which stopped error signals shrinking as they passed back through many layers.
The paradigm shift
AlexNet's influence went beyond vision, which spent the next ten years exploring convolutional variants: it proved deep networks could be trained by backpropagation. Connectionists traffic surged again from 2012, and researchers built tools such as software that automatically calculates the derivatives backprop needs, so trying a new architecture became a matter of specifying hidden units, weights and activation functions and letting the software learn the weights.
It also shifted who did AI and where:
- Google hired Hinton, Krizhevsky and AlexNet's third author, Ilya Sutskever.
- A small company, DeepMind, used convolutional networks as a key part of an AI playing Atari games at superhuman level.
- Mark Zuckerberg and Elon Musk turned up at NeurIPS to recruit network researchers.
- Demand for data and compute only rose. Networks remain a technology for learning mappings from data, and more complex mappings need more data; companies saw that investing in data and compute directly solved hard AI problems.
For cognitive science, the new models were tools too. Two members of Griffiths's lab, Josh Peterson and Josh Abbott, showed that representations in convolutional networks' hidden layers predict human similarity judgments for images. Those representations can then feed richer cognitive models that work directly on real images, not the simple stimuli of Bruner, Goodnow and Austin in the 1950s and many later studies. Yet as with rules and symbols, networks' most striking results would come in language, the subject of the next chapter.
The scale curve AlexNet started has steepened. Two gaming GPUs for six days in 2012 compare with Grok 4, which Epoch AI (September 2025) estimated used about 246 million H100-hours at roughly $490 million, the largest known training run at the time. Nvidia, maker of those GPUs, became the first company worth $5 trillion on 29 October 2025. Hinton, who left Google in 2023 to warn about AI risk, shared the 2024 Nobel Prize in Physics; Sutskever founded Safe Superintelligence in 2024; Fei-Fei Li's World Labs released its first commercial world model, Marble, in November 2025.
Priority is messier than even Griffiths's corrected account. Werbos described backpropagation for networks in his 1974 Harvard PhD thesis, before his 1981 paper; gradient-trained multilayer networks also trace to Shun-ichi Amari (1967), and Jürgen Schmidhuber has long argued deep-learning history undercredits such work. Hinton's layer-wise pretraining (deep belief nets, 2006) was soon made unnecessary by ReLU, better initialization and GPUs. The source's reference to XOR "in Chapter 8" should read Chapter 7. Hubel and Wiesel's classic cat recordings date from 1959 to 1962, so "the 1950s" stretches it. AlexNet won ILSVRC 2012 against a runner-up at about 26% top-5 error, which matches Griffiths's "just over 25 percent".
Language as a Prediction Problem
Neural networks offered an alternative to rules and symbols and, in their boldest form, a replacement: Rumelhart and McClelland proposed that language behavior could be describable by rules without any rule being written down in the mechanism. Griffiths traces that bet from the past-tense war of the 1980s through Elman's next-word prediction, Latent Semantic Analysis and word2vec, to transformers and ChatGPT. Predicting the next word turned out to be enough to learn much of the structure of language, but large language models still need superhuman data and still generalize poorly beyond it.
The challenge to East Coast cognitive science
In a PDP chapter that set off a firestorm in the late 1980s, Rumelhart and McClelland suggested that lawful behavior and grammaticality judgments could come from a mechanism containing no explicit representation of any rule: its performance would be characterizable by rules, but the rules themselves would be written nowhere inside it. The test case was how people form the past tense of verbs. The topic sounds harmless, but it hit the core of East Coast cognitive science. Since Chomsky's 1950s work, language had been the best demonstration of rules and symbols; showing it could be represented, and even learned, without explicit rules undercut one of the strongest arguments for treating the mind as a formal system. The reply was quick and fierce.
The past-tense fight was only the opening. Over the following decades networks captured more of syntax and semantics:
- Predicting the next word let simple models tell nouns from verbs.
- Tracking how word frequencies vary across documents gave a handle on word meaning.
- More training data made the networks steadily more powerful.
Today's large language models descend from these early network models and inherit insights from generations of cognitive scientists. They show how much networks can learn from data, and they also expose how modern AI differs from human minds.
The past gets tense
Jean Berko and the wug
When Jean Berko bought five-by-seven-inch index cards and colored pencils at the Harvard bookstore, she had no idea she was laying the ground for one of cognitive science's bitterest debates. The child of Transylvanian immigrants, she grew up in Cleveland with a brother who had cerebral palsy; others said they could not understand his speech, but she always could, and acting as his translator made her unusually alert to language. She entered Radcliffe in 1949, studied every language she could, and found what was missing in "The Psychology of Language," taught by new professor Roger Brown (Eleanor Rosch's adviser, met in Chapter 7). It covered how language is acquired, processed, produced and understood, and she decided on graduate school in linguistics.
Working with Brown, she asked whether children are simply told "one dog, two dogs; one book, two books," or at some point internalize the rules. Her tool was a picture book drawn on those index cards.
Card one showed a cute birdlike creature: "This is a wug." Below were two of them and a sentence to complete: there are two ___. Children had never heard "wug", so saying "wugs" meant applying the regular plural rule (add "s"; "regular" means rule-following, and nouns like mice and geese that break it are irregular). Other cards probed other relations, including the past tense: a man who knows how to spow did it yesterday, so yesterday he ___ (presumably "spowed"). Berko tested children aged four to seven at a preschool and an elementary school. 76% of preschoolers said "wugs", rising to 97% by first grade; only 36% of preschoolers said "spowed" (59% in first grade), but most preschoolers happily said someone glinged or ricked yesterday. The work became her dissertation; at a conference the editor of the prestigious journal Word grabbed the paper and published it almost exactly as written, minus the first chapter.
Over-regularization in three stages
Berko showed children regularize novel words, apparently applying a rule. The next finding was that they over-regularize, using the rule where it does not belong; her famous example is a child reporting that the teacher "holded" the baby rabbits. Such errors are a window into how language is learned.
Explaining them took new technology. Susan Ervin, a Michigan PhD who was an instructor at the Harvard School of Education during Berko's graduate years and part of the Brown group, took a faculty post at UC Berkeley in 1958 to study how children acquire Chomsky's rules. With linguist Wick Miller she tape-recorded preschoolers at home, following the same children for months. Plurals and past tenses passed through three stages:
- Correct irregulars. The child says "held".
- Over-regularization. The child switches to "holded".
- Recovery. The child returns to "held".
The rules interpretation: children first imitate what they hear, then grasp an abstract rule and apply it everywhere, then learn the exceptions. For Chomskyans the irregulars are not even true exceptions but other rules: they fall into classes that behave alike, as drink/drank and sing/sang do.
The Rumelhart-McClelland model
Rumelhart and McClelland were discovering that networks could explain things previously credited to rules. Their interactive activation model of reading behaved differently depending on whether a letter string obeyed English spelling rules, which led them to note that the model was lawful though it contained no explicit rules. The past tense looked like an ideal test, and their model was deliberately simple.
- Coding. Each sound in a word was represented by linguistic features (roughly, how air moves through the mouth to make it) plus the features of the sounds before and after it.
- Mapping. Input was the present tense ("hold"); output the past tense ("held").
- Learner. A one-layer perceptron trained with the perceptron learning rule; it did not use backpropagation.
- Training schedule. Verbs split into high-, medium- and low-frequency groups, on the assumption that children first learn the most frequent verbs and later a much larger set with a higher share of regulars: train on high, then add medium, then test on low.
- Results. Correct irregulars rose, fell, then rose again (the U-shaped curve), and the network seemed to learn the similarities within irregular classes.
Their conclusion was strong. They claimed a real alternative to the view that children learn past-tense rules in any explicit sense: a "rule" is no more than a description of the language, and in this case, they wrote, "there is no induction problem." The child need not work out what the rules are, nor even that rules exist. Shots had been fired at rules and symbols as the basis of language.
Ambush at MIT
The shots were returned. In 1985 Steve Pinker heard McClelland talk and invited him to the seminar series of MIT's Center for Cognitive Science, where a speaker presents and commentators respond. Pinker would be one commentator and invited Alan Prince to be the other; both were committed to rules and symbols.
- Steve Pinker. Grew up in Montreal; at McGill studied psychology shaped by Donald Hebb and philosophy taught by a friend of Chomsky; Harvard PhD under Roger Brown on vision and language, cross-registering in Chomsky's MIT course and Joan Bresnan's on linguistics and computation; an MIT postdoc on language acquisition; by then an MIT professor.
- Alan Prince. Also from McGill, a few years before Pinker, but in linguistics; MIT PhD; specialist in phonology (sound structure) and morphology (how sounds form words); had just moved to Brandeis.
Even Pinker later admitted the event was not quite fair: McClelland was double-teamed on the other side's home field. McClelland spoke for 20 minutes; the two commentators then spoke for about an hour each. Pinker began warmly, praising the effort to engage with data, then laid out basic problems the model could never address. Prince attacked from phonology, arguing the chosen sound representations could not capture English words properly, and pointed out that on new, untested items the model scored only about 80%. He called that unacceptable, said models must capture competence before learning and processing, hammered his fist on the table, raised it skyward, and demanded the model inflect the past tense perfectly before anything else was discussed. For McClelland, 80% from experience alone was exciting. Grueling as it was, the onslaught convinced him: if they cared enough to hit this hard, he might be onto something.
- Stages. The model's account of children's stages of regularization is demonstrably wrong.
- A key speech error. Its account of one striking type of childhood error is wrong too.
- Hollow successes. Its other developmental successes either owe nothing to PDP architecture (symbolic models match them) or rest on confounds.
- Unrepresentable words. Some kinds of words cannot be represented at all.
- Similarity. It cannot explain psychological similarity among words.
- Unnatural rules. It readily learns rules found in no human language.
- English sound patterns. It misses central phonological generalizations.
- Morphology. It mispredicts derivation, compounding and novel words.
- Homophony. It fails the basic problem of words that sound the same but inflect differently.
- Error rate. It gets a large percentage of test words wrong.
- Silence. For some words it produces no past tense at all.
- Rules vs exceptions. It mispredicts the reality of the regular/irregular distinction in children and languages.
Griffiths picks out two as decisive.
- The U-curve was an artifact of training. Many high-frequency verbs are irregular, so the first phase let the network concentrate on irregulars. Adding the medium-frequency batch sharply raised the proportion of regulars, causing the dip that looked like over-regularization; more training then let irregulars recover.
- The model never really got the past tense. It handled most verbs but made bizarre errors: past of "mail" → "membled", of "tour" → "toureder". Such mistakes undercut the claim that the past tense needs no explicit rules.
The past tense stayed a battlefield for twenty years, producing more than a hundred papers and dozens of models, now using more sophisticated networks (the original had not even used backprop) and answering specific objections. Pinker and Prince stayed unconvinced but each absorbed connectionist ideas. Pinker settled on a learned rule for regular verbs, with irregular generalizations based on features like Rumelhart and McClelland's. Prince worked with Paul Smolensky, from the UCSD connectionist group, on a new phonology combining symbols with graded computation. Yet the largest gains for networks came not from the past but from predicting the future.
What comes next
Elman the phone hacker
The past-tense model framed learning as mapping one word form to another, sensible for tenses but no help with how children arrange words into sentences or interpret them. The next step came from Jeff Elman.
While Ervin was taping children, Elman was a disaffected Harvard sophomore, a transplant from Southern California who had dropped history (too many dates) for the Department of Social Relations. Reading the campus paper instead of textbooks, he spotted a cryptic classified ad seeking a 121-page notebook from Harvard-MIT "Fine Arts 13" and a 40-page reply, with stray references to 2,800, a battery, "m.f." and a line of Spanish. Elman had taken Fine Arts 13, so on a whim he wrote that he had the notebook.
A postcard came back: anyone with the information should be able to complete the sequence 6 0 4 2 3 1 1 2 1, say what M.F. means, and name the telephone equipment MIT students had been playing with. Guessing the sequence ended in 2 and looked like a phone number, Elman dialed it; a woman answered with "Cleaner clean inward." Area code 604 was western Canada, including Kleena Kleene. A roommate's girlfriend, a telephone operator, explained that an inward is an operator who completes calls and should not be dialable; the glitch let Elman make free calls by claiming to be the test board. She also decoded M.F.: multifrequency, the operators' tone system. An old news story revealed Harvard and MIT students caught making free calls, whose notebook and 40-page report the phone company had impounded. Elman got a warning letter written backward in Cyrillic, replied with his number, and finally heard from B. David, who knew he had no notebook but admired the bluff and belonged to an underground phone-hacking community. Elman spent three months building a multifrequency tone generator, reading the Bell System Technical Journal and mapping the network for glitches, until AT&T security and the FBI arrived at his dorm.
His energy then went into protests against the Vietnam War, including building occupations and arrests. As a senior he hacked Harvard's computer system, was caught, and was asked what he planned after graduating; when he said he didn't know, they offered him a job, which he took for a while.
He then taught history in Spanish to recent immigrants, and bilingual children sparked his interest in language. He bought Roger Brown's Words and Things, which he found lucid, and Chomsky's Syntactic Structures, which he could not understand at all; he reasoned that a field whose books he couldn't follow was one worth a graduate degree. With no linguistics courses, he entered the University of Texas at Austin, found research was like solving a mystery, as phone hacking had been, and joined UC San Diego in 1977.
The problem of time
Elman worked on speech perception, a corner of linguistics driven more by data than by Chomsky's latest pronouncements. He fell in with McClelland and Rumelhart and, with McClelland, built a speech version of interactive activation: features activate phonemes, which activate words. Speech exposed a difference from print: time. A written word arrives all at once; a spoken one unfolds sound by sound. Their model let unit activations persist, leaving a trace of each sound, but Elman could not see how to do the same for sentences, and spent a couple of years banging his head against a wall.
Rumelhart's advice had two parts: if you can't do what you want, do the things you can, stay occupied and out of trouble; and also keep banging your head against the wall. Elman did both, and a new way of applying networks to language emerged.
Predict the next word
Training by an error-driven method like backprop needs a target output. The past-tense model's target was the correct past form, but for language in general there seemed to be none.
Elman's answer was to make the network predict the next word from all the words so far. He saw it as an acid test: if you have really figured out the hidden generator of a time series, you can keep generating forward, and when your predictions fail, that failure is the error signal. Language supplies its own targets.
Prediction alone was not enough. A network whose input is the current word and output the next merely rebuilds the associative chain model Chomsky had demolished (Chapter 4), and conditioning on the last two or three words falls to the same critique. It had to use information from all previous words.
His fix was the simple recurrent network (SRN). Input is the current word and output the next word, as before, but the hidden-unit activations are copied and fed back as extra input to the hidden units when the following word arrives; then copied again, and so on. The hidden units thus carry information forward from every earlier word.
Elman built a miniature English with noun and verb classes and simple combination rules. One template, NOUN-HUM VERB-EAT NOUN-FOOD, yields "woman eat cookie"; there were 15 templates over six kinds of noun and six kinds of verb. He generated 10,000 sentences and ran them together with no breaks ("woman smash plate cat move man break car..."), each word activating one input node, and asked the SRN to activate the output node of the next word. It learned to predict, and even detected where sentences ended. Inspecting the hidden activations, Elman found that a network never told about nouns, verbs, humans or animals had placed nouns and verbs in separate regions of its hidden space, and within the noun region put humans in one place and animals in another. Prediction alone produced the linguistic categories.
Elman saw deep parallels between the phone system and the brain: both just grew, and the brain did not evolve to solve partial differential equations and is not optimally designed. Representations, too, could grow without being designed to internalize rules; learning the statistical patterns that rules produce was enough to find the key distinctions. Hacking phones had got him into trouble, but hacking brains gave two ideas that grew more powerful as data grew: train networks by predicting the next word, and treat word meanings as points in space.
Words in space
Dumais and vocabulary mismatch
Elman's networks learned from tens of thousands of sentences in a toy language; scaling to English was beyond late-1980s computers. A few years later others learned word meanings from tens of thousands of real English documents.
Susan Dumais grew up in Lewiston, Maine, the first in her family to attend college, which meant walking down the street to Bates College. Bates had no computer science department, but as a statistics teaching assistant she learned to solve problems by visualizing distributions and data, and saw endless possibilities for simulation alongside analysis. A mathematical psychology class showed her the same tools could explain minds; she realized she cared less about computing itself than about what it enabled, understanding how humans learn and decide. She did a math-psych PhD at Indiana University on how practice makes effortful tasks automatic, enjoying above all the mathematical models of learning, then joined Bell Labs in one of the first human-computer interaction groups.
The group was run by Tom Landauer (Harvard Social Relations PhD 1960; taught at Dartmouth, Stanford and Princeton; Bell Labs from 1969). There the researchers moved from how people retrieve information from their own memory to how they retrieve it from other sources, and hit the vocabulary mismatch problem: the people who create information and those who seek it describe things differently. One document says "doctor", another "physician", and systems that matched exact words were brittle and literal.
Latent Semantic Analysis
The breakthrough came from math, not psychology. Two words for the same thing rarely appear together ("doctor" is informal, "physician" formal), but both turn up alongside "nurse", "patient" and "hospital". Represent words by the company they keep and doctor and physician become very similar.
- Matrix. A giant table: rows are words, columns documents, each entry how often that word appears in that document.
- Prediction problem. Given a word, predict how often it occurs in each document.
- Network form. Input units for words map to a hidden layer with few dimensions, which maps to output units giving frequencies per document.
- Shortcut. Assuming linear activation functions lets a linear-algebra method find the weights quickly, avoiding slow training on large text.
- Result. Each word becomes a point in the hidden space, a vector of values (echoing Leibniz's numbers for words, Chapter 1); words used similarly across documents land close together.
- Search. A query becomes a vector and is compared with document vectors, so matching no longer depends on exact words.
LSA did more than improve retrieval; it suggested how human minds retrieve information. Chomsky had answered Plato's problem (knowing so much from so little) with innate knowledge. In 1997 Landauer and Dumais offered another answer: we fill the gaps in experience by building efficient, compressed representations of our environment. Doctor and physician may never share a document, but reduced to points in a low-dimensional space, their shared correlations with other words reveal their similarity.
- The test. The synonym section of the Test of English as a Foreign Language: pick which of four words (trivial, minor, substantial, irrelevant) is closest to "significant"; LSA picks the nearest point.
- 30,473 encyclopedia articles trained the model.
- 300 dimensions scored 64.4% on retired TOEFL items, almost exactly the average of a large group of international applicants.
- All 30,473 dimensions (raw frequencies) did much worse: the documents alone hide that doctor and physician are related.
- Fewer than 300 dimensions made performance fall again, as compression began discarding real differences between words: a sweet spot.
Word embeddings and analogies
Words-as-points have only grown in impact. Machine learning now calls them word embeddings, and faster machines and smarter algorithms learn them from ever larger datasets. Google's word2vec took an embedding from the hidden layer of a network that predicts a word from its neighbors, simple and fast enough to train on more than a billion words.
Its embeddings had a surprising property: they solve analogies. For man:king::woman:?, take the difference between "man" and "king" and add it to "woman", and you land near "queen" (king − man + woman ≈ queen). David Rumelhart and Adele Abrahamson had proposed exactly this spatial model of analogy in 1973, using spaces from multidimensional scaling; word2vec recovered such spaces just by predicting word co-occurrence in a large text. Dumais herself finds the moment thrilling for its data, representational richness, speed, scale and societal impact.
Tversky's objections return
Spatial representations still carry the constraints Amos Tversky exposed for similarity (Chapter 6), and they show up in language:
- Asymmetry. In free association, "keg" evokes "beer" far more than "beer" evokes "keg"; distances in space are symmetric.
- Triangle inequality. "Asteroid" is close to "belt", and "belt" to "buckle", yet asteroid and buckle are unrelated; in space both must sit near belt while far from each other.
- Analogies too. In Griffiths's lab Dawn Chen and Josh Peterson found asymmetric analogies (hairdresser:comb::pitcher:baseball is decent; reversed it is not) and triangle violations (nurse:patient::mother:baby and mother:baby::frog:tadpole are good; nurse:patient::frog:tadpole is not).
As in Tversky's cases, these exploit different senses of words or different kinds of relation between pairs, subtleties that points and vector differences cannot capture.
Language models get large
From rules to networks in translation
The step from LSA (millions of words) to word2vec (billions) began a rapid growth in both training data and model size, and with it ever more impressive feats. Machine translation, a dream for centuries, came first, its methods evolving with ideas about language. In 1954 IBM demonstrated a system that took a Russian message punched into a card and rendered it in English, using a vocabulary of about 250 words and hand-built rules. As connectionist models grew stronger, networks replaced the rules. On Berkeley dissertation committees Griffiths watched computational linguistics students build elaborate systems from formal grammars, then join Google and switch to neural networks, made feasible by huge data and compute.
Large networks let translation become just another prediction problem. Several groups converged on the idea, among them a Google team led by Ilya Sutskever (part of the AlexNet team Google acquired). A recurrent network read an English sentence and was trained to predict the words of its French translation; trained on 12 million sentence pairs, it beat traditional translation systems, and networks now underpin Google's translation. McClelland sees vindication of his MIT talk: there is no explicit rule system anywhere in Google Translate, and that is exactly what he and Rumelhart had been aiming for.
Better recurrent networks
- The SRN's weakness. Its whole hidden layer is rewritten with each input, so holding information for long is hard.
- LSTM (1995). Sepp Hochreiter and Jürgen Schmidhuber's long short-term memory networks add subnetworks that decide whether information should enter the hidden layer, whether stored information should be kept, and whether it should be used in processing the input; they learn patterns spanning longer sequences.
- Climbing the Chomsky hierarchy. With Paul Rodriguez and Janet Wiles, Elman showed SRNs could learn simple context-free languages such as the apple-banana language (Chapter 4): each "apple" pushed hidden activations one way, each "banana" pulled them back toward the start, so a context-free language is learnable when it can be represented spatially. In general, though, SRNs find only regular languages easy.
- Controllable memory. Giving a recurrent network a memory it can read and write, like a Turing machine's tape, makes context-free and even some context-sensitive languages easy to learn.
Transformers and GPT
The newest architecture is the transformer, developed at Google in 2017. Instead of updating a representation word by word, it considers all the words at once and, using their embeddings, learns which words to attend to to solve the prediction problem.
2017 Google researchers introduce the transformer.
2018 OpenAI's Alec Radford, Karthik Narasimhan (now Griffiths's Princeton colleague), Tim Salimans and Ilya Sutskever (a founding member of OpenAI) show that training a transformer to predict the next word gives hidden representations good for other language tasks: the first generative pretrained transformer (GPT).
2019 GPT-2, trained on about 6 billion words of internet text, writes internally coherent passages. Studying it, OpenAI finds a reliable relation between training data and next-word accuracy: scaling laws predicting gains from bigger models and more data.
2020 GPT-3, trained on about 400 billion words.
Nov 2022 Tuning GPT-3 to respond to human prompts yields ChatGPT, an interactive language agent.
Mar 2023 GPT-4 scores at the 90th percentile on the Uniform Bar Exam and the 99th on the graduate-school verbal test.
ChatGPT was a striking breakthrough. The production-system chatbots of Chapter 3 depended on humans writing response rules; ChatGPT learned its responses from experience. GPT-4 can also combine obscure knowledge into novel output. At the suggestion of Jean Berko Gleason, now in her nineties, Griffiths asked it for a haiku about kishkas, stuffed intestines. The poem described them as savory delight that "kishkas gently hold" in a warm embrace. One can quibble with its choices, but it understood the assignment and produced a phrase with no Google hits, a modern "colorless green ideas".
How fast the data grew is clearest set side by side:
| Model | Year | Training data | What it learned from |
|---|---|---|---|
| Past-tense net | 1986 | Verbs in three frequency bands | Present form → past form |
| Elman SRN | c. 1990 | 10,000 toy sentences | Next word |
| LSA | 1997 | 30,473 articles | Word counts per document |
| word2vec | 2013 | 1 billion+ words | Word from its neighbors |
| Seq2seq | 2014 | 12 million sentence pairs | French words from English |
| GPT-2 | 2019 | ~6 billion words | Next word |
| GPT-3 | 2020 | ~400 billion words | Next word |
What LLMs tell us about minds
Griffiths draws four lessons.
- Elman vindicated. Predicting the next word suffices to learn much of the structure of language, and even of the wider world: predicting what Elizabeth will say to Fitzwilliam requires knowing about human nature, character and social mores.
- Language carries much of intelligence. Learning language alone yields intelligent behavior in a surprisingly wide range of settings, but not all: next-word predictors do relatively poorly at planning and search, the tasks Newell and Simon saw as central. One remedy is agents that pair Newell-Simon search strategies with LLM representations.
- Minsky's scale objection, paid for. Minsky shunned networks because making them work might take thousands or millions of neurons he could not afford. AI companies decided they could afford such a machine: GPT-4 is estimated at more than a trillion weights and tens of millions of dollars to train.
- Pinker and Prince still bite. Humanlike performance needs superhuman data: GPT-3's training text equals 5,000 years of someone talking nonstop, about 1,000 times what a child needs. And the models generalize poorly beyond their data: on a recent wug test, ChatGPT applied the regular past tense correctly only 60% of the time.
Pinker. Rule-governed processing handles any number of edge cases outside the training set, items people manage easily and networks stumble on. Berko Gleason's verdict: fine for kishka haiku, but for facts, don't count on it.
McClelland. Google Translate contains no explicit rule system anywhere, and it works; that was the whole aim of the 1986 model.
Network models of language have widened what we think can be learned from data and how language contributes to intelligence. They leave two questions: why can networks overcome the logical problem of language acquisition, and why do they need so much more data than humans? Answering them, Griffiths says, requires another approach to the mind.
Griffiths's GPT-4 is now three generations back. GPT-5 (August 2025), Gemini 3 (November 2025) and Gemini 3.1 Pro (February 2026) ship with built-in reasoning modes, the RL-trained chain-of-thought search described in Chapter 3's update, which narrows the planning gap Griffiths identifies. The transformer remains the core architecture, but next-word pretraining is now only the first stage, followed by instruction tuning and large-scale reinforcement learning, so "trained to predict the next word" is less complete a description than in 2023.
GPT-4's "over a trillion weights" is a press estimate; OpenAI never disclosed its size. The Rumelhart-McClelland claim was always about explicit rules: Pinker never denied networks approximate rule behavior, only that humans work that way, and his dual-route compromise (Words and Rules, 1999) is closer to today's consensus than either 1986 position. Griffiths does not name it, but Prince and Smolensky's hybrid is Optimality Theory (1993). The LSTM's "forget" gate was actually added in 2000 (Gers, Schmidhuber and Cummins); the 1995-97 original had input and output gates only. The sequence-to-sequence translation paper is Sutskever, Vinyals and Le (2014). The data gap Griffiths cites has widened since (see Chapter 4's update).
The Missing Question
Griffiths uses David Marr's three levels of analysis to name what both rule-free networks and neuron-by-neuron neuroscience leave out: the question of function, of why the mind works as it does. Algorithmic and implementation theories say how; a computational-level theory says why. Networks learn from data but cannot explain why one example sometimes suffices for induction; that needs a computational theory of inductive inference, and the candidate is probability theory.
The mysterious object
Griffiths opens with a thought experiment he runs in every cognitive science course. Two agents in suits hand you a phone; the US president asks for help on a secret project; you are escorted through checkpoints to a guarded room at a local university. An archaeological dig has turned up an object roughly spherical, about six inches across, apparently some 100,000 years old. It seems to process information: electrical stimulation at some places ("inputs") produces patterns of activity at others ("outputs"). You may use every resource the university has, but you may not damage it. How would you work out what it does? (He asks readers to write down steps before reading on.)
His students' typical answers, and the three categories they always fall into:
| Category | Typical plans | Field that asks it |
|---|---|---|
| Physical structure | Put it in an MRI or X-ray it; find what material it is made of | Neuroscience |
| Internal process | Time how long outputs take for different inputs; check whether outputs are always the same | Psychology |
| Function | Record input-output pairs and look for patterns; see how it interacts with other objects from the dig | Neither, traditionally |
The object matters only as a stand-in for another roughly spherical, six-inch, 100,000-year-old device: the human brain. When we study brains and minds, our questions fall into the same three bins. Structure belongs to neuroscience and process to psychology; questions about function have played little part in either field, yet may be the key to both.
David Marr
Both observations, that information processors can be analyzed at several levels and that function is neglected, come from the computational neuroscientist David Marr. Born in 1945 and raised in Essex, he went to Rugby School on a scholarship and told a friend as a schoolboy that he wanted "a mathematical theory of the brain." At Cambridge he studied mathematics, then physiology, and published three influential papers giving mathematical theories of the cerebellum, the neocortex and the archicortex. An invitation to talk at MIT became a longer stay at the AI Lab and then a faculty post.
There he built a group studying vision, how the brain makes sense of light on the retina. Seeing feels effortless but is a very hard inductive problem (Chapter 5): the three-dimensional world is projected onto a two-dimensional image of light intensities at different wavelengths, and the brain must clean that signal up and extract what action requires. Marr and his collaborators developed a new approach to vision, and ultimately to the rest of the brain.
Marr's levels
With Tomaso Poggio, then at the Max Planck Institute in Tübingen and later an MIT professor, Marr stated the core idea in 1976: complex systems such as a nervous system or a developing embryo must be analyzed and understood at several different levels. The canonical version is in Marr's 1982 book, aptly titled Vision. Griffiths shortens the three labels to computational, algorithmic and implementation, and they map onto his students' three categories.
- Computational theory (function). What is the goal of the computation, why is it appropriate, and what is the logic of the strategy that achieves it?
- Representation and algorithm (process). How is the computational theory carried out: what represents the input and output, and what algorithm transforms one into the other?
- Hardware implementation (structure). How are the representation and algorithm realized physically?
Marr's cash register
Marr's own example reverses the archaeology story: thousands of years from now, diggers unearth a well-preserved mechanical cash register. That is less far-fetched than it sounds: the Antikythera object, recovered from the sea off a Greek island and about 2,000 years old, is thought to be an astronomical calculator with more than fifty gears. Understanding the register needs answers at all three levels.
Computational, part one: what. The archaeologists tabulate which sequences of button presses produce which symbols in the window, realize the symbols are base-10 digits, and discover the machine performs addition. They have described it mathematically.
Computational, part two: why. Studying the era, they discover commerce. Shoppers expect that an item you don't bring to the checkout costs nothing; that the order of items doesn't change the price; that two separate checkouts total the same as one combined; and that a refund deducts an item's price. Translated into math: adding zero changes nothing; order of addition doesn't matter; adding two subtotals equals adding everything; adding then subtracting a number equals adding zero. These are the axioms that characterize addition: any operation obeying them is equivalent to ordinary adding. So the goal forces the solution.
Algorithmic. Numbers can be written in base 10 ($12 is a 1 then a 2, meaning 10 + 2), binary (1100, meaning 8 + 4 + 0 + 0) or Roman numerals (XII). Each makes some operations easy and others hard (testing divisibility by 10 is trivial in decimal, harder in binary), and the addition algorithm differs with the representation.
Implementation. One algorithm can be realized many ways: this register is mechanical, but the same operations run in silicon or neurons. The theory here explains how the procedure becomes gears, springs and other parts.
The cash register is only the warm-up; the payoff is using these distinctions on a far more complicated information processor, the human mind.
Having our goals in mind
Mathematics can express theories at any of the three levels, so whenever we model the mind it helps to ask which level we are working at. Different levels have different goals, which tells us which data to attend to and what insights to expect. Griffiths's example is memory:
- Implementation. Model how a system of neurons stores, maintains and retrieves information, which is what Marr did in his theory of the archicortex.
- Algorithmic. Model how people represent information and the strategies they use to retrieve it. Lynne Reder at Carnegie Mellon argues people decide whether to try retrieving a fact based on how accessible it is.
- Computational. Explain why memory is as it is. Reder's colleague and husband John Anderson treats memory as an optimal solution to predicting which information will be needed in the future (see Chapter 4 of Griffiths and Christian's Algorithms to Live By).
Theories at different levels need not compete. Reder's mechanisms could produce behavior that satisfies Anderson's theory (which, Griffiths notes, keeps domestic harmony) and be implemented as Marr described the archicortex. We usually assume only one rival explanation can be right; when they sit at different levels, they may all be compatible.
The same holds for building intelligent machines inspired by minds, which can borrow from every level:
- Neuron models show how intelligent behavior can emerge from highly parallel computation.
- Human algorithms for solving problems suggest efficient strategies machines can copy.
- Characterizing the abstract problems humans face, and their solutions, guides machines meant to solve the same problems.
Which level is most important?
Scientists in practice stake their careers on all three levels, choosing the methods they think most likely to pay off. Marr, though, favored one. He began with neurons, but thinking about vision convinced him a different approach was needed. Studying perception only through neurons, he wrote, is like studying bird flight only through feathers: it cannot be done. You must first understand aerodynamics; only then do the structure of feathers and the shapes of wings make sense.
Marr knew aerodynamics firsthand. He earned a pilot's license to overcome a fear of flying and came to love flying small aircraft; he and Poggio flew together while discussing research (Griffiths notes that Claude Shannon had already set the precedent of MIT geniuses risking their lives in flight). The aerodynamics of the mind, for Marr, would be a mathematical theory at the computational level: principles explaining the fundamental forces at work, derived from the abstract problems minds must solve. In Griffiths's terms, the Laws of Thought.
Marr I. The Cambridge theorist of neural circuitry and what it might compute: cerebellum, neocortex, archicortex.
Marr II. The MIT "AI phase": functional, focused on the theory of the process and possible algorithms, with much less attention to realistic implementation.
Griffiths qualifies Marr. His stress on function suited his moment, which had plenty of neurons and little research driven by function; fully understanding minds and brains will require theories at all three levels. Marr's friend Francis Crick split his work into the two phases above and believed that, had Marr lived, he would have moved to a synthesis of them.
He did not get the chance. Marr died of leukemia in 1980, aged thirty-five. He spent his last years writing Vision through chemotherapy and other treatment; in a 1979 letter to his wife, Lucia Vaina, he wrote that he felt the book strong within him, was sure its message mattered, and that his task was to fulfill its potential. It was published posthumously, his last gift to cognitive science.
Asking why
Marr's legacy is the recognition that models of the algorithms minds follow and of their neural implementation may not add up to a complete explanation. Algorithmic and implementation theories tell us how minds and brains work; computational theories tell us why.
The previous chapters showed networks reproducing key parts of human intelligence and learning complex representations from data. But even a network that perfectly mimicked human cognition would leave questions open. As networks grow, it gets harder to see why they do what they do from their weights and representations. Paraphrasing Marr, studying cognition only through artificial neurons is studying flight through feathers. We need aerodynamics.
The candidates so far:
- Logic. A genuine computational-level theory: a mathematical system solving a real human problem, inferring new truths from established ones. But it fell short on induction.
- Neural networks. They show how to learn from data and handle fuzzy representations, but they do not answer Mill's question: why is a single example sometimes enough for a complete induction? Indeed they need large amounts of data to learn anything.
To explain why networks can learn language and why it takes them so much data, we need a computational-level theory of inductive inference. There is one, Griffiths announces: probability theory.
Marr's levels are widely taught but contested. Critics note real systems blur them (Churchland and Sejnowski argued for many levels of organization in the brain, not three), and Marr's own priority on the computational level has been challenged by neuroscientists who find that implementation constraints shape what is computed. Griffiths acknowledges the need for all three levels but, as a computational-level theorist, he is an interested party. The "100,000 years" figure for the brain refers roughly to anatomically modern humans; current estimates put Homo sapiens at about 300,000 years (Jebel Irhoud fossils).
Probability Is the New Logic
Probability is the third thread, wound through both logic and networks: Leibniz, Boole, Turing, Minsky, Tversky, Rumelhart, Hinton and McClelland all used it, and the second half of Boole's Laws of Thought is about it. Read as degrees of belief spread over possible worlds, probability extends logic: it agrees on every certain case and says what to believe in the uncertain ones. Bayes' rule answers Mill's question (a single instance suffices when the prior is high) and explains why flexible networks need so much data. Being able to learn many things comes at the cost of needing more data to learn each one: a conservation law of learning.
What probability is for
We talk loosely about the probability of events as their propensity to occur: coin flips and dice, but also a rain forecast, a job application, anything uncertain. And we update those numbers as information arrives, after a glance out the window or an encouraging email. Working out the rules of uncertain reasoning proved as hard as working out logic, and turning probability into a practical tool for studying the mind was harder still, not achieved until the late twentieth century.
Griffiths promises a lot from it. Probabilistic reasoning extends logic, explains why networks behave as they do, offers a way to link the strengths of both and strike a compromise between them, and leads to new accounts of similarity, categories and language, with clues for analysis at all three of Marr's levels. First, though, the meaning of probability needs pinning down.
Two kinds of probability
Chances at the gaming table
Probability theory began with a roll of the dice. Girolamo Cardano, born in Pavia in 1501, was a mathematician, physician, astrologer, influential author and inveterate gambler; his autobiography confesses he gambled for decades, every day, he admitted with shame, at the cost of thought, money and time. Away from the table the dice fascinated him, and he calculated the right bets for events involving several dice.
- Cardano's rule. Count the ways a favorable outcome can occur and divide by the total number of possible outcomes: the method still taught in school.
- Consequences. Every probability lies between 0 and 1, and the probabilities of all outcomes sum to 1.
- Delay. His work appeared only in 1663, almost a century after his death.
Blaise Pascal was a prodigy despite his father, who, his sister Gilberte wrote, kept mathematics from him for fear he would neglect other subjects. Pascal worked out the basics of geometry on his own at twelve, and his father relented. By sixteen he had proved a new theorem on conic sections; by eighteen he had invented a mechanical calculator (forerunner of Leibniz's).
Pascal travelled with Antoine Gombaud, Chevalier de Méré, a lover of mathematics and gambling, who in 1654 asked how to split the stakes when a game is interrupted. If players flip coins until someone has six heads, and one has five while the other has one, the leader clearly deserves more, but how much? Mathematicians, Cardano among them, had worried at this for over a century. Pascal's correspondence with Pierre de Fermat (of the Last Theorem) solved it and in doing so laid the foundations of modern probability theory.
Degrees of belief
Later in 1654 Pascal had an intense religious experience, felt himself in God's presence, and renounced mathematics. Religion led him back to probability anyway. His notorious wager, published in the Pensées (1670), runs: if God exists, belief may win infinite gains in an infinite afterlife; if not, there is nothing to gain; belief costs only something finite, one observant life. So belief is rational for any probability of God's existence above zero.
That argument quietly assumes one can assign a probability to God's existence. Others did similar things:
- John Wilkins. The creator of the Real Character (Chapter 1) argued that an orderly universe needs an all-powerful designer, so God is the most probable explanation; in ordinary life, he said, people act on what is most probable when certainty is out of reach.
- David Hume. Used probability skeptically against miracle reports: told that someone saw a dead man revived, he weighs whether the witness's deception or error is more probable than the event. Only if the testimony's falsehood would be more miraculous than the miracle should it win belief.
Chances (Cardano, Pascal and Fermat). The propensities of outcomes in games of chance: dice, coins, divided stakes.
Degrees of belief (Pascal's wager, Wilkins, Hume). The certainty attached to a conclusion, applied to God, design and miracles.
The shift looks subtle but is radical. It suggests the mathematics built for dice might capture an aspect of human thought: how people reason when uncertain. That makes probability a natural complement to logic, which governs reasoning with certainty.
Leibniz, Bayes and Laplace
Leibniz saw the connection. His work on combinations made him at home counting ways outcomes arise, and at nineteen he wrote about assigning fractional degrees of belief in legal cases. Scholars argue he regarded probability as a "new kind of logic," part of the inference system he sought for his universal characteristic. But, as with logic, he never built the mathematics.
That fell to the eighteenth century, to a British minister and a French mathematician. Rev. Thomas Bayes asked how someone watching a lottery should update the probability of winning from the wins and losses observed so far. His friend Richard Price polished the notes after his death and presented them to the Royal Society in 1763. The Marquis Pierre-Simon Laplace independently posed the same problem about ten years later, solved it, and contributed much more. Such reasoning is now called Bayesian inference. Laplace hardly lacks credit: his solution is Laplace's Law, and he also has a transform, the Laplacian, an equation, a series and a demon.
Probability over possible worlds
The machinery
As a degree of belief, a probability is the same mathematical object as in dice games, a number from 0 to 1 with all outcomes summing to 1, but it now measures the chance a person gives a possible world being the true one. The total is 1 because we have a single unit of belief to distribute; a world with probability 1 is certain. Reasoning means recomputing the chances of each world as information arrives, and that, decisively, is a way to solve inductive problems.
Griffiths returns to Dupin's inference from Chapter 5: if the culprit were an orangutan, it could climb to the window; the culprit climbed to the window; so the culprit was an orangutan. Its form, P → Q, Q, therefore P, is the fallacy of affirming the consequent: with P → Q and Q both true, we could be in world 1 (P true) or world 3 (P false). Yet sometimes, perhaps in Dupin's case, it seems reasonable. Probability explains why.
- Assign. Give each row of the truth table (each possible world) a probability, 0 to 1, summing to 1. Knowing P → Q is true means world 2 (P true, Q false) gets 0.
- Query. The probability of any proposition is the sum over worlds where it holds: P(P) = world 1 + world 2.
- Observe. Data rule out worlds; set their probability to 0.
- Renormalize. The survivors no longer sum to 1, so divide each by their total.
- Re-query. Sum again to get the updated probability.
| World | P (orangutan) | Q (climbed) | P → Q | Prior A | After Q (A) | Prior B | After Q (B) |
|---|---|---|---|---|---|---|---|
| 1 | T | T | T | 0.15 | 0.75 | 0.15 | 0.158 |
| 2 | T | F | F | 0 | 0 | 0 | 0 |
| 3 | F | T | T | 0.05 | 0.25 | 0.80 | 0.842 |
| 4 | F | F | T | 0.80 | 0 | 0.05 | 0 |
| P(P) | 0.15 | 0.75 | 0.15 | 0.158 |
Assignment A: Dupin's assumptions
In the first assignment the most likely world is world 4 (neither orangutan nor climbing, 0.8), with world 1 (orangutan that climbed) at 0.15 and world 3 (non-orangutan that climbed) at 0.05. Then P(P) = 0.15 + 0 = 0.15 and P(Q) = 0.15 + 0.05 = 0.2. Discovering that Q is true rules out world 4; dividing the survivors by 0.2 gives world 1 0.15 / 0.2 = 0.75 and world 3 0.05 / 0.2 = 0.25. So P(P) rises from 0.15 to 0.75. Climbing does not make the orangutan logically certain, but it makes it far more probable: the inference makes sense.
These numbers fit Dupin's assumptions: an orangutan culprit is fairly unlikely, but would be guaranteed to reach the window; a human culprit is far more probable but unlikely to manage the climb.
Assignment B: a different world
Now keep P(P) at 0.15 but suppose a human culprit could almost always reach the window (world 3 high, world 4 low). Run the same steps and, after observing Q, P(P) is about 0.158, exactly 0.15 / 0.95 = 3/19. Seeing a good climber now tells us very little.
The strength of an induction depends on exactly how probabilities are assigned, and those depend on the semantics of the case: what P and Q mean, what orangutans can do, how climbable lightning rods are. That is why induction is harder to formalize than deduction. Certain inference needs only which worlds are possible; uncertain inference needs their probabilities, so no argument form can justify every induction by syntax alone.
Running a valid argument
Apply the same method to modus ponens (P → Q, P, therefore Q), with assignment A. Observing P leaves only worlds where P is true with nonzero probability; world 2 is already 0, so world 1 alone survives and must take probability 1. Q is true there, so P(Q) = 1: Q follows with certainty.
This is no coincidence. Whatever the assignment, as long as world 1 is nonzero (we don't already know ¬P) and world 2 is zero (we know P → Q), observing P gives Q probability 1. The same holds for every valid logical argument: whatever satisfies logic also satisfies probability theory. In that sense probability really is the new logic, extending it by agreeing on all certain cases and telling us what to think in uncertain ones. Griffiths, warning that probability is unintuitive, urges readers to work the examples with pencil and paper before going further.
Understanding induction
The murky Reverend Bayes
Bayes did more than suggest updating beliefs with probabilities: he proved a theorem giving a simple rule for it, which illuminates induction and why some inferences (and learning algorithms) need more data than others. He kept a low profile, so his life story carries fitting uncertainty: historian Stephen Stigler gives a probability of 0.8 that he was born in 1701.
- Background. Son of a Nonconformist minister (outside the Church of England), so barred from Oxford and Cambridge; studied mathematics and theology at Edinburgh; assisted in his father's London chapel, then took his own congregation at Tunbridge Wells for the rest of his life.
- Mathematics on the side. In 1736 he defended Newton's calculus against the Irish philosopher George Berkeley, which seems to have won his election to the Royal Society in 1742.
- Posthumous paper. He published nothing on probability in his lifetime; his will named Richard Price literary executor, and after Bayes died in 1761 Price recovered, edited and introduced the manuscript for the Royal Society.
His motive is unknown, but there are clues to why Price valued the work. The paper was published as "An Essay Towards Solving a Problem in the Doctrine of Chances," but its original title promised a method for calculating the exact probability of all conclusions founded on induction. And Price later used Bayes's arguments in his own theology against Hume's case against miracles, noting that several independent witnesses can be strong evidence. These clues suggest, inductively, that Bayes was answering Hume with a mathematics of induction. (Bayes's specific lottery solution is covered in Chapter 6 of Algorithms to Live By.)
Bayes' rule, step by step
Reframe Dupin's inference as hypotheses (culprit is orangutan or human) and data (climbed the lightning rod or not). The four worlds become: orangutan and climbed 0.15; orangutan and didn't climb 0; human and climbed 0.05; human and didn't climb 0.8. The goal is now explicit: update belief in the hypotheses from the data.
- Prior probability, P(h). Belief before data, summed over matching worlds: P(orangutan) = 0.15 + 0 = 0.15.
- Posterior probability, P(h | d). Belief after data, read "h given d": P(orangutan | climbed).
- Conditional probability. The world where both hold divided by the total for worlds where the data hold: P(h | d) = P(h, d) / P(d), here 0.15 / (0.15 + 0.05) = 0.75.
- Likelihood, P(d | h). The same formula reversed, P(d | h) = P(h, d) / P(h): P(climbed | orangutan) = 1, P(climbed | human) = 0.05 / 0.85 = 1/17. This gap is what swings belief toward the orangutan.
- Bayes' rule. Since P(h, d) = P(d | h) × P(h): P(h | d) = P(d | h) × P(h) / P(d). Posterior = prior × likelihood ÷ probability of the data.
Bayes' rule gives exactly the same answers as the possible-worlds procedure, but phrasing it as hypotheses, data and a formula linking prior to posterior makes it a sharper tool. Griffiths recasts the four inductive problems of Chapter 5:
- Learning language (syntax). Hypotheses are grammars, data are utterances. The prior encodes the learner's inductive biases: higher-prior grammars are easier to learn, needing less evidence. Chomsky's strong innate constraints become priors that put high probability on a few grammars, but tolerating uncertainty escapes the logical problem of language acquisition and opens other possibilities (Chapter 14).
- Learning categories. Hypotheses are possible categories, data the examples, updated one by one; the prior expresses expectations about category form, so low-prior hypotheses like White ∨ Square need more compelling data than Black ∧ Circle.
- Vision. Priors over 3-D configurations of the world, data the light on the retina, likelihood the chance of that light given a configuration. Ambiguous images (Figure 5.1) balance prior and likelihood so two hypotheses get similar posteriors.
- Speech. Hypotheses are words, data the sounds at the eardrum, and the prior should track word frequency, predicting errors toward commoner words. It fits: "kiss this guy" beats "kiss the sky", "in containers" beats "entertainers", "Starbucks" beats "star-crossed".
Why a single instance
Prior times evidence
Bayes' rule says just two things should set belief in a hypothesis: the prior and the evidence from the data. Moving the prior outside the fraction makes this plain:
- P(h | d) = [P(d | h) / P(d)] × P(h)
The posterior is the prior multiplied by how probable the data are under the hypothesis relative to how probable they are in general. That ratio is the evidence.
Copper and third sons
Take Nelson Goodman's pair from Chapter 5. One piece of copper conducting electricity makes us believe all copper conducts; one man in a room being a third son does not make us believe everyone there is a third son. Both observations are certain under their hypotheses (likelihood 1 in each case). Conductors may be rarer than third sons, so P(data) may be a bit lower for copper, but the evidence ratio is high for both. The difference lies in the priors: a kind of metal conducting is far more plausible than a room full of third sons. With a high enough prior, one observation lifts the posterior to acceptance.
More evidence shows the difference is the prior, not the evidence. We could be convinced about the third sons: the first is unremarkable, the second curious, the third suspicious, and by the fourth or fifth we suspect we have stumbled into the Annual Meeting of the Society for Thirdborn Sons. Each new third son fits the hypothesis but is increasingly improbable under the alternatives, so the evidence ratio grows until it overwhelms the low prior.
This matches Goodman's own answer. He said induction is supported by overhypotheses, knowledge of which hypotheses are plausible targets for induction. He never formalized them; in Bayesian terms, overhypotheses are knowledge that raises the prior of some hypotheses over others.
Mill answered. A single instance suffices for a complete induction when it gives strong evidence for a hypothesis whose prior is already high, so the posterior ends up high. That defines a research program: map the priors that encode the inductive biases guiding human inference.
Why networks need so much data
Bayes also answers the network question. Treat each configuration of weights as a distinct hypothesis about the input-output mapping. Networks are powerful because they can entertain so many hypotheses: one architecture can learn mappings for language, vision and much else. The cost is that with so many hypotheses sharing probabilities that must sum to 1, each gets a low prior, and being convinced of any one requires a lot of data.
That is Griffiths's conservation of learning, modeled on physics' conservation of momentum: with one unit of prior probability spread over everything a learner can consider, learners who consider fewer hypotheses give each, on average, a higher prior, and so learn each from less data.
All of this sits at Marr's computational level. Griffiths does not claim brains or networks calculate Bayes' rule whenever they learn. Because it describes the ideal solution to inductive problems, it lets us analyze any system facing them, just as birds need not solve the Navier-Stokes equations on each wingbeat for aerodynamics to explain flight.
Pessimism about probability
Conservative Bayesians
The Cognitive Revolution displayed the power of rules and symbols but also encouraged psychologists to test probability as a theory of thought. Ward Edwards, whose decision-making papers had drawn Amos Tversky into graduate school, ran experiments suggesting people update beliefs broadly as Bayes' rule prescribes. Follow-ups agreed, and an influential 1960s paper called people good intuitive statisticians, if slower to update than Bayes would like: "conservative Bayesians."
When Tversky presented this work in Daniel Kahneman's seminar at the Hebrew University in 1969, he began a collaboration that would badly damage the idea.
Kahneman was born in Tel Aviv and raised in Paris; the family fled after the German occupation and his father died during the escape. After the war he and his mother moved to Jerusalem. He studied psychology and mathematics at the Hebrew University, briefly led an army platoon, then joined the army's psychology unit, where he built a test of recruits' aptitude. A Hebrew University fellowship took him to Berkeley for a PhD finished in 1961 under Susan Ervin (Chapter 10); back on the Hebrew University faculty, he spent 1965 to 1967 at Michigan and Harvard, mixing with Bruner and Miller at the Center for Cognitive Studies. He did not expect Tversky's seminar visit to launch a research program for both of them.
Kahneman doubted people did anything like Bayesian inference. As with Tversky's critique of spaces, the method was to identify a principle of probability theory and build clever examples that people violate. The program ran two decades, produced dozens of papers, a Nobel Prize and the bestseller Thinking, Fast and Slow; Griffiths picks three famous cases.
- Randomness. Told that in a city survey 72 families of six children had the birth order G B G B B G, students estimated how many had B G B B B B. 75 of 92 undergraduates judged the second less likely, yet if boys and girls are equally likely the two sequences are exactly equally probable.
- Base-rate neglect. Subjects read sketches such as Jack: forty-five, married with four children, conservative, careful, ambitious, uninterested in politics, fond of home carpentry, sailing and mathematical puzzles, and judged lawyer or engineer. Half were told the sketches came from 30 engineers and 70 lawyers, half from 70 engineers and 30 lawyers. In Bayesian terms the sketch is the data and the pool sets the prior, which should shift judgments even when the description is strong. People were far less sensitive to the priors than they should have been, driven mainly by the description.
- Conjunction fallacy. Linda is thirty-one, single, outspoken, very bright, a philosophy major who cared about discrimination and social justice and joined antinuclear demonstrations. Asked which is more likely, 85% of undergraduates chose "bank teller and active in the feminist movement" over "bank teller".
The Linda result breaks the possible-worlds picture itself. With "bank teller" and "feminist" as propositions there are four worlds. "Teller and feminist" is world 1; "teller" is worlds 1 + 2, so it can only be greater than or equal to world 1. No assignment of probabilities to worlds can satisfy people's intuition.
The results looked damning. Kahneman and Tversky concluded that people were apparently not conservative Bayesians but not Bayesian at all. Just as Minsky and Papert had cooled enthusiasm for perceptrons, Kahneman and Tversky cooled interest in probability as an account of cognition. It would be more than a decade before psychologists made a strong case for Bayesian probability as part of the Laws of Thought.
The source has slips worth knowing. It calls Bayes's editor "Sir Robert Price" once; he was the Rev. Richard Price, as Griffiths says elsewhere. In the renormalization example the leftover probability goes to world 3, not "world 2". The printed tables also appear shuffled: the first table lists world 3 at 0.8 and world 4 at 0.05 while the text says the reverse, the "updated" table repeats the prior (0.15, 0, 0.05, 0.8), and the "second assignment" table shows the updated values (0.75, 0, 0.25, 0), which would give P(P) = 0.75, not the stated 0.15. The table above uses the values the text implies (assignment B: 0.15, 0, 0.80, 0.05, giving 3/19). The modus ponens display reprints the affirming-the-consequent form. The heuristics-and-biases verdict has itself been contested: Gigerenzer showed base-rate neglect and the conjunction fallacy shrink sharply when problems use natural frequencies, and Chapter 13 recasts several of these "biases" as good answers to a different question.
Universal Laws of Cognition
Probability could unite structured representations with statistical learning, because Bayes' rule works on any kind of hypothesis, from logical formulas to connection weights. First it had to survive Kahneman and Tversky. The way through was to stop studying human quirks and ask what any mind, anywhere in the universe, must solve: Shepard's universal law of generalization (1987), Anderson's rational analysis, and the reinterpretation of "biases" as good answers to a different question. Pearl's Bayesian networks then made rich probabilistic models practical. Rules and symbols can describe what minds learn while probability explains how they learn it.
What probability offers, and what stood in the way
Probability is the best available theory of induction, able to explain both why people learn from little data and why machines currently need so much. It can also join two big ideas: the structured representations of rules and symbols and the statistical learning of networks. Because Bayes' rule applies to any kind of hypothesis, from logical formulas to connection weights, it lets researchers explore the consequences of different assumptions about how the world is represented and which inductive biases guide learning.
First, though, cognitive scientists had to overcome the belief that probability and human minds were incompatible. Kahneman and Tversky had shown compelling departures from probability theory and from Bayes' rule in particular; probabilistic accounts would be taken seriously only if they could explain those departures. The key, Griffiths says, was to stop focusing on human minds and ask about principles of intelligence that would hold for any mind, anywhere in the universe.
Psychology's first universal law
Newton's template
In 1687 Isaac Newton published the universal law of gravitation in the Principia Mathematica: the gravity we feel on Earth acts on every object in the universe, with any two objects attracting by a force proportional to their masses and inversely proportional to the square of the distance between them. From it he derived the motions of the visible celestial bodies, one principle unifying astronomy.
Exactly 300 years later, in 1987, Roger Shepard proposed psychology's first universal law, the universal law of generalization. Like Newton's, it was meant to hold anywhere: by starting from an abstract problem any intelligent organism must face and deriving its solution, Shepard aimed at a principle for minds of all kinds, not just humans or animals.
Shepard's law also concerns pairs of objects, but its subject is generalization: given that one object has a property, say being good to eat, what is the probability that another object has it too? Every intelligent organism faces this, whether weighing whether a mushroom is worth the risk or whether a planet might be habitable. And like Newton, Shepard expressed his law as a function of distance, but in psychological space (Chapter 6).
Shepard's universal law of generalization: the probability of generalizing a property from one object to another falls off as an exponential function of their distance in psychological space, roughly g(d) = e^(−k·d).
The derivation
Shepard derived the law with Bayesian inference:
- Objects are points in psychological space.
- Properties are regions. The set of objects with a property (the edible mushrooms) is some unknown region of the space; each possible region is a hypothesis.
- Data. Learning that one object has the property (this mushroom is edible) shows it lies inside the true region, which updates beliefs about the hypotheses.
- Generalization. The probability that a second object has the property is the sum of the posterior probabilities of all regions containing it, just like summing over possible worlds in Chapter 12.
- Robustness. Shepard's observation was that the prior over regions hardly matters: across reasonable assumptions the result falls off roughly exponentially with distance.
The law held for terrestrial creatures at least: generalization in humans, rats and pigeons fell off exponentially. Shepard's Science paper ended bullishly: psychology had lagged physics by at least 300 years, and behavior would never be predicted with celestial precision, yet psychology need not be confined to describing particular species; behind the varied behavior of humans and animals, as behind the motions of planets, we might discern universal laws.
Asimov reads Shepard
A Science paper reaches a wide audience, including Isaac Asimov. He made a valiant attempt but found the mathematics too tough, and even the prose left him with only a hazy grasp. The idea still struck him. His novel Foundation imagines Hari Seldon inventing psychohistory, a mathematics for predicting human behavior. Asimov felt Seldon would understand Shepard easily, and worried that psychohistory might arrive within the next century, when he had set it 20,000 years in the future: another case, he wondered, of his imagination falling ludicrously short.
Generalizing generalization
A boy from Perth
Shepard's paper inspired a generation of cognitive scientists, Griffiths among them, and here he steps into his own story.
- Childhood. Raised in Perth, Western Australia, a beautiful city far from anywhere. His father, an electrical engineer turned systems analyst and management consultant, kept computers at home; Griffiths taught himself to program on a machine whose screen showed four lines of text and saved to cassette tape. His mother was a psychotherapeutic social worker, so minds were constant talk.
- Probability through games. Like Cardano, he learned probability calculating odds for cards and dice, but for collectible card games and tabletop role-playing games, not gambling. He also spent much time at the beach and took up fencing, partly because he was bad at traditional Australian sports.
- Illness. In high school he caught a cold, recovered, caught another and did not return for two years: a post-viral fatigue syndrome of the kind now familiar from long COVID. On good days he played text-based online role-playing games, where advanced players became "wizards" who could modify the game by writing code; that is how he learned object-oriented C and UNIX at fourteen. Years with an undiagnosed illness drew him to fields full of mysteries, and so to psychology.
- University of Western Australia. He learned statistics, read the PDP books on his own over a summer, and found the campus network researcher, Mike Kalish, trained at UC San Diego in connectionism's heyday; he cornered him at 9 a.m. on the first day of class and became his research assistant. In Kalish's class on similarity and categorization he met Shepard's law, and from then on built models combining networks with Shepard's generalization.
- Thesis. He showed that errors in mental multiplication follow exponential generalization in a psychological space of multiplication problems: a rules-and-symbols task described by a spaces-and-features model. He applied to Stanford to work with Shepard and Rumelhart.
Tenenbaum
On the other side of the world, Josh Tenenbaum grew up near Stanford, where his father did a computer science PhD and ran AI research labs and tech companies; his mother was an educational psychologist evaluating curricula. Though he majored in physics at Yale, he spent the summer after sophomore year in Shepard's lab, trying to reconcile the universal law with neural networks. That set his course: a PhD in MIT's Brain and Cognitive Sciences department, with a dissertation on how generalizations should change as you learn that more and more objects share a property. In Bayesian terms, more examples consistent with a hypothesis raise its posterior, which explains how people learn new words and concepts quickly.
By the time Griffiths applied, Shepard and Rumelhart had both retired, but Stanford psychology had just hired Tenenbaum (Shepard, who usually avoided departmental politics, had left a letter strongly endorsing him in every colleague's mailbox). Tenenbaum picked the application from Western Australia, even though much of its research statement enthused about Stanford's fencing facilities. Barbara Tversky phoned early one morning with the admission.
Spaces and features reconciled
One of their first projects extended Shepard's analysis. Shepard had treated generalization in psychological spaces, in line with his view of similarity; Griffiths and Tenenbaum showed the theory bridges Shepard's spatial account and Tversky's feature-based theory.
- Shepard's version. Hypotheses about which objects share a property are regions in space; Bayes yields the exponential law.
- Tversky's version. Make the hypotheses sets defined by objects' features, and Tversky's feature-based similarity falls out of the same calculation.
- The setting. Griffiths worked on it in a tiny room in the short corridor between the old offices of Shepard and Tversky.
- The general lesson. Bayes' rule says how to update beliefs, not what they are about: hypotheses can be regions, network weights, features or logical rules, so Bayes lets us ask what can be inferred from data whatever the learner's representations.
Making probability respectable again
Anderson's beach reading
On a South Pacific beach in 1983, John Anderson opened his vacation book, Marr's Vision. A Carnegie Mellon professor of psychology and computer science, he was famous for turning Newell and Simon's symbolic approach into a complete model of human cognition. The book turned him toward the abstract computational problems behind the memory, category learning, causal learning and problem solving his symbolic models had captured.
Over the next few years he found each problem involved inductive inference with an ideal solution from probability. Like Shepard, he used Bayes' rule to explain how the brain might judge which information will be useful later, whether something belongs to a category, what causal relations data support and how to approach new problems. That went far toward Marr's demand to explain why minds work as they do. He named the method rational analysis: the rational solutions to abstract computational problems can illuminate behavior.
Optimal agents versus bad statisticians
With Shepard's law, Anderson's results implied that much of cognition fits Bayesian models, seemingly at odds with Kahneman and Tversky. Both went further, expecting people and other organisms to act like optimal Bayesian agents elsewhere too. Shepard later wrote that as organisms evolve more powerful ways of representing the world, their cognition and behavior will increasingly approximate the optimum for that world.
Kahneman and Tversky. Studied people's explicit probabilistic reasoning on paper-and-pencil puzzles, and found people fail.
Shepard and Anderson. Used Bayesian inference implicitly, to characterize the form of the optimal solution to problems organisms face in their environment. Shepard noted that abstract paper tasks bear little resemblance to the concrete challenges of our Pleistocene ancestors, so there was little selective pressure for solving them.
The Wason selection task
Support came from reanalyzing another kind of irrationality. Psychologists had cooled on logic partly because people seemed bad at it. Peter Wason of University College London found a compelling example in 1960 and developed it during a 1963 visit to Harvard's Center for Cognitive Studies at George Miller's invitation.
Each card has a letter on one side and a number on the other; the table shows D, 3, B, 7. Which must you turn to test the rule "if a card has a D on one side, it has a 3 on the other"? In logic the rule is P → Q, false only when P is true and Q false: a D without a 3. So you should turn the D (to check for a 3) and the 7 (to make sure there is no D behind it). People instead turn the D and the 3. Wason and many successors took this as a failure of logic, held up as proof of human irrationality.
In 1994 Mike Oaksford and Nick Chater, friends since graduate school at Edinburgh and inspired by rational analysis, argued people do a good job on a different problem. Thinking in probabilities, they proposed people choose the cards that would most change the probability that the rule is true, under one assumption: the properties rules mention are rare. If few cards have a D, or a 3, then turning the 3 can be more informative than turning the 7, which almost surely has no D behind it. Griffiths's everyday version: to test "if a saucepan falls, it clangs," you look around whenever you hear a clang, but checking the kitchen floor every time you don't would cost a lot for little payoff. Logic is a poor fit for the world we live in. Oaksford and Chater built this into a full Bayesian theory of human deductive reasoning.
Biases as answers to a different question
Their stance, that cognition is shaped by the problems people face in their environment rather than being a general tool for arbitrary tasks, suggested people do badly on probability puzzles because they are not solving the experimenters' problem but one closer to what minds evolved for. Griffiths and Tenenbaum set out to re-evaluate probability as a theory of mind.
- Randomness. Equal-chance births make P(data | random) identical for G B G B B G and B G B B B B. But people may be judging P(random | data): did a random process produce this? Bayes says that depends on the evidence ratio P(data | random) / P(data); if some other process could easily produce the sequence, P(data) rises and the evidence for randomness falls. If doctors suspected a chemical in a town's water affecting births, B G B B B B would worry them more, because it is more probable under a nonrandom process. So it is weaker evidence of randomness, and P(random | data) is higher for G B G B B G, which is what people track. A simple model of evidence for a random generating process captured people's randomness judgments of binary sequences very well.
- Diagnosing causes. People implicitly infer the underlying causal process instead of explicitly computing an outcome's probability, which is useful for understanding the environment and finding exploitable patterns.
- Jack and Linda. People answer with the evidence the description gives for a hypothesis, not its probability: Jack's sketch is strong evidence for engineer, Linda's even stronger evidence for feminist bank teller than for bank teller. Griffiths and Tenenbaum showed this measure of evidence can also be written in Bayesian terms.
The takeaway is not that people use probability theory correctly: they plainly give wrong answers to Kahneman and Tversky's questions. It is that probability theory explains their behavior. People are good at picking up evidence about the causes of events; that instinct leads them to answer a different question from the one asked. For inductive problems Bayes gives the ideal solution; the only challenge is building models rich enough to capture people's beliefs about the world.
Graphs and generative processes
Pearl reads Rumelhart
Judea Pearl had an idea while reading a paper by David Rumelhart. He had already switched fields once, from the electrical components of computers to algorithms that let computers solve harder problems, and leaping again came naturally. He grew up in newly founded Israel, where people constantly reinvented themselves: his grandfather went from merchant to dairyman, his high school had not existed a year before he enrolled, and his teachers were college professors who had escaped Europe. After the Technion he went to the US, earning at once a physics master's at Rutgers and a PhD from the Polytechnic Institute of Brooklyn (1965). He worked at a firm building computer memories, then joined UCLA, exploring heuristic strategies for AI in the spirit of Newell and Simon.
The paper came from Rumelhart's work on word perception before his network collaboration with McClelland. It shared the later idea of interaction among strokes, letters and words, but described that interaction differently: Rumelhart proposed as his measure the "Baysian" [sic] probability that a hypothesis is true given the evidence. Like Leibniz with logic, a brilliant mind grasped an idea it could not quite make work. Rumelhart saw reading as Bayesian inference, strokes as data and words as hypotheses, but could not work out how to compute the probabilities as evidence flowed up from strokes to letters to words and beyond. He drew a figure of the connections and sketched a theory that never came together.
The figure electrified Pearl: uncertainty resolved by messages passed between nodes of a graph. Reading it, he became convinced that AI would have to model itself on human intelligence, and that machine reasoning under uncertainty would need a similar message-passing architecture. But what are the messages? Pearl, Griffiths writes, was about to become the Boole to Rumelhart's Leibniz.
The answer: conditional probabilities
After a few months he had it: the messages should be conditional probabilities. His 1982 paper, "Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach," introduced Bayesian networks.
- Write the generative process. To produce a word: choose a word, then write each of its letters, then write each stroke of those letters.
- One distribution per step. P(word), P(letter | word), P(stroke | letter). The full distribution over words, letters and strokes is defined just by each step's conditional probability given the step before.
- Draw it as a graph. Arrows trace the generative process: choose the word, follow an arrow, write a letter with the matching conditional probability.
- Inference runs backward. The conditional probabilities work in both directions: observed strokes send evidence up the arrows to letters; letters collect it and send it up to words. Generation flows down the graph; inference flows up.
The discovery made probability far easier to work with:
- Before Pearl. Defining a distribution over variables meant listing every combination of values and giving each a probability, as with Chapter 12's possible worlds. Like logical truth tables, the count grows exponentially with the number of variables: unsustainable.
- After Pearl. Complex distributions are broken into simple steps, each needing only a conditional probability, and inference lets probabilities flow through the graph instead of painfully updating a distribution over all outcomes after each observation.
Impact on AI and cognitive science
The first impact was in AI. In the 1980s AI used production systems, which could not separate generation from inference.
"If the sprinkler was on last night, the grass is wet" sounds reasonable; so does "if the grass is wet, the sprinkler was on." The first describes the generative process (sprinkler causes wetness), the second an inference (wet grass, so infer sprinkler). Mixing them causes trouble: the grass could be wet because it rained, so the inference rule must become "if the grass is wet and it didn't rain, the sprinkler was on," and listing every exception gets unwieldy. Probability was the right fix but impractical until Bayesian networks.
The insight spread to machine learning: Griffiths attributes the dip in neural-network interest in the late 1990s and early 2000s to Bayesian methods taking over the field. In cognitive science, Bayesian networks were a decisive tool for probability's revival: they made expressive models easy to define and probability more intuitive, so researchers could model the complexity of the problems minds face and test whether people's inferences match. In one notable case Jay McClelland answered Rumelhart's old question by showing the interactive activation model has a Bayesian interpretation.
Most significantly, Pearl's approach put generative processes at the center. Observers try to reconstruct the process that produced their data; moving through the world, we draw on a rich internal model specifying the generative processes behind events, which lets us interpret them. Thinking in generative processes also gives a new view of how rules and symbols relate to neural networks.
Structure and statistics
The combination
The two traditions differ on two dimensions:
| Rules and symbols | Neural networks | Bayesian synthesis | |
|---|---|---|---|
| Representation | Discrete, composable into new ideas, plans, sentences | Continuous, fuzzy and graded | Structured hypotheses (formulas, programs) with probabilities |
| Learning | Innate constraints | Learned from data | Statistical inference over structured hypotheses |
| Strength | Productivity, hierarchy, compositionality | Graded concepts, learning from experience | Rules describe what is learned; probability explains how |
A third possibility uses rules and symbols to characterize what minds learn while explaining how learning happens. Bayesian inference supplies it: define generative processes from rules and symbols, put probabilities on hypotheses such as logical formulas or even computer programs, and you combine structured representations with statistical learning. What people learn can then be described in a rich language with productivity, hierarchy and compositionality.
Categories as distributions
Categories, a thread through the whole book, show the power. Logic described categories by membership rules; networks allowed fuzzier boundaries learned from experience. Generative processes connect Bayes to both and allow richer category forms. Meeting a hairy creature, we ask whether it is a cat or a dog; the features are the data, the categories the hypotheses:
- P(cat | features) = P(features | cat) × P(cat) / P(features)
The critical term is P(features | cat), how likely those features are if it is a cat. From the Bayesian view a category is a probability distribution over features, and different generative processes give different theories of categories.
- Independent features → perceptron. Assume each feature is generated independently given the category (a cat's hair length says nothing about its nose length). Minsky, exploring perceptrons, saw that the Bayesian two-category decision then has a linear boundary, so a one-layer network implements Bayes' rule: its weights pass up how much evidence each feature gives. Prototype models also give linear boundaries, so they too have a Bayesian reading. Such equivalences are expected: good solutions to inductive problems often equal Bayes under some assumptions.
- Probabilistic rules. Cats are described by a logical formula, small furry domestic carnivores, applied with variability: each feature may be missing with small probability. This revives Bruner, Goodnow and Austin's hypothesis testing with fuzzy edges, explaining how people learn rules while allowing fuzziness in both inference and rule.
- Causal structure. Birds fly because they have wings. Seeing an airborne animal in the distance, flight suggests bird; but for an animal on the ground already seen to have wings, learning whether it flies adds little. Bayesian networks capture such dependencies between features (Figure 13.5: Bird → has wings → can fly; Bird → has beak).
- Generative growth. Trees range from saplings to sequoias. Rather than modeling height and width variation, recognize that trees grow predictably, taller and broader, adding branches and leaves: take a random number of steps, adding branches and leaves each time, and you get a distribution expressing rich, abstract knowledge of what trees are.
- Learning the form itself. Which generative process fits a category? Also Bayes. Charles Kemp, a fellow Australian now at the University of Melbourne, showed with Tenenbaum as a graduate student that hypotheses about structural form can be written as Pearl-style generative models and chosen by Bayes' rule: US Supreme Court justices fit points on a line (political leaning), colors varying in hue a two-dimensional space, and animal properties a taxonomic tree.
The fuller picture
Categories are not just prototypes or linear rules; those are effective approximations to Bayesian inference for the simplest categories, those defined by independent features. Interpreting something new, we use hypotheses that include logical rules, causal relations and knowledge of how things change over time. The Bayesian view captures that knowledge as rich generative processes and, more generally, describes the models minds build of the world: human minds, and those of intelligent organisms anywhere in the universe.
Griffiths is a principal of this school, and the chapter is partly memoir. Critics (Bowers and Davis, 2012; Jones and Love, 2011) argue Bayesian models are too flexible: with free choice of priors and likelihoods, almost any behavior can be fitted after the fact, and "people answer a different question" can rescue any bias. Shepard's exponential also has exceptions: he himself noted more Gaussian-shaped curves under some conditions, such as highly confusable stimuli. The claim that Bayesian methods caused the neural-net dip understates the role of kernel methods such as support vector machines (Chapter 9's "sophisticated feature expansion"). The source says production systems were "introduced in Chapter 4"; they appear in Chapter 3.
Language as a Probability Distribution
Griffiths returns to language with probability in hand and offers a third view: language is neither just a formal system nor just a prediction problem but a probability distribution over sentences, which can be built by adding dice to grammars. That view defeats Chomsky's "colorless green ideas" argument, dissolves the logical problem of language acquisition, and explains why LLMs need so much data: their training stands in for evolution as well as learning. Training a network to predict the next word makes it treat language as a probability distribution, and probabilistic thinking explains both its powers and its "embers".
Four open questions
Language was the best demonstration of rules and symbols (Chomsky's grammars) and the key testing ground for neural networks (learning without explicit grammars). Network success left four questions:
- Statistics versus Chomsky. How do networks capture so much structure when Chomsky argued statistical approaches could not?
- The logical problem. How do they decide that unseen sentences do not belong to the language?
- The data gap. How do children learn from so much less data?
- Improvement. How can networks be made better learners?
The answers matter beyond language: they show how rules and symbols, neural networks and probability fit together, and so complete the picture of the Laws of Thought.
P(colorless green ideas)
Markov's model
In his talk of 11 September 1956, Chomsky argued that a sentence's probability cannot measure its grammaticality: "Colorless green ideas sleep furiously" is recognizably grammatical and "Furiously sleep ideas green colorless" is not, though no two adjacent words in either had likely ever appeared together. That hit the information theorists in the room, who computed sentence probabilities exactly that way: P(colorless), then P(green | colorless), then P(ideas | green), and so on.
This is a Markov model, named for the Russian mathematician Andrey Markov, who first analyzed its properties as pure mathematics and later applied it to language, painstakingly tabulating which letters follow which across 20,000 letters of Pushkin's Eugene Onegin. Claude Shannon did the same for English letters and words (Chapter 5). It is arguably the simplest probabilistic model of language, and once models get even slightly richer, Chomsky's argument collapses.
Hidden Markov models
Chomsky's grammars tie each word to one or more syntactic classes: "colorless" and "green" are adjectives, "ideas" a noun, "sleep" a verb, "furiously" an adverb. Compute conditional probabilities over classes instead of words: P(Adjective), P(Adjective | Adjective), P(Noun | Adjective) and so on.
- Sample a class for the first word.
- Sample the word from that class's distribution over words.
- Sample the next class conditioned on the previous class.
- Sample the next word from the new class, and repeat until the sentence ends.
- Why "hidden". The classes are never observed, only the words they emit.
Moving the chain from words to classes is enough to defeat Chomsky's argument. Unseen word pairs no longer matter so long as their classes have appeared side by side. "Colorless green ideas sleep furiously" follows a common English class sequence, the same as "Hairy brown dogs run happily" (Chapter 4); the reversal follows a rare one. So an HMM trained on newspaper text gives the first sentence a probability more than 200,000 times higher than the second: a probabilistic model can detect grammaticality.
Hidden information is also why large language models succeed. Geoff Hinton named networks' intermediate layers "hidden units" after the hidden Markov model. Instead of a single discrete class per word, an LLM represents everything conveyed by all previous words as a point in space, an embedding, letting predictions transcend relations between individual words and meet Chomsky's challenge.
Dice on grammars
HMMs are also a probabilistic finite-state grammar: the board game of Chapter 4 with dice added. Each board position is a hidden class; the choice of next position follows the conditional probabilities.
- Coin flip. At the "dog" or "dogs" choice of Figure 4.2, flip a coin: "the dog runs" and "the dogs run" each come out with probability 1/2.
- Die roll. Add hairy dogs (Figure 4.3) and roll a die among "hairy", "dog" and "dogs". The result is a probability distribution over infinitely many sentences.
- Context-free version. The same trick works for context-free grammars: choose the rewrite rule at random. For the apple-banana grammar (Sentence → apple Sentence banana; Sentence → apple banana), a coin flip per rewrite gives "apple banana" 1/2, two apples and two bananas 1/4, three and three 1/8, and so on.
This is Griffiths's third view of language: not a formal system, not a prediction problem, but something with elements of both, a probability distribution. Every sentence gets a probability that can be used to judge grammaticality, and probability can be layered onto any representation, formal grammars included. That link between structure and statistics matters most for learning.
The probabilistic solution to the logical problem
The argument and the reply
A central case for strong innate constraints was the logical problem of language acquisition: a learner who sees only sentences from a language can never logically rule out a larger language of which the true one is a subset, and so risks accepting ungrammatical sentences. Probability answers it. If a language is a distribution, a learner grows ever more confident in the right hypothesis as samples accumulate. Certainty never comes (the logical problem still holds), but the chance of being wrong becomes vanishingly small.
Feldman and Horning
That view gave the first positive learnability results.
- Jerome Feldman. A Carnegie Mellon PhD who worked partly with Newell and Simon, he had just joined Stanford in 1967, after work on formal languages for computers and associative memories. His memo "First Thoughts on Grammatical Inference" carried a front-page note that it was a draft not to be reproduced, copied or believed without the author's permission.
- Jim Horning. A graduate student in programming languages, he read it and told Feldman he wanted permission to believe.
- The result. Feldman supervised Horning's dissertation (and, a few years later, that of Marty Tenenbaum, Josh's father). Horning proved that languages as complex as those of a probabilistic context-free grammar can be learned purely from randomly generated example sentences by Bayesian inference.
Griffiths has two dice: a normal six-sided one and a twelve-sided one borrowed from his daughter's Dungeons and Dragons kit (tabletop games run in families). He picks one at random and rolls: 5. Which die? The priors are equal, so the likelihoods decide: 1/6 for the six-sided die, 1/12 for the twelve-sided. The 5 makes the six-sided die (1/6) / (1/12) = 2 times more likely. Each further roll (1, then 3) doubles the odds again. After ten rolls, 5, 1, 3, 6, 4, 6, 5, 2, 3, 6, all consistent with six sides, you have doubled ten times: 2¹⁰ = 1,024, so be about a thousand times more confident in the six-sided die. Not certainty, but compelling.
The dice are the language learner's predicament. 1 to 6 is a subset of 1 to 12: the twelve-sided die is never ruled out, only made vanishingly improbable. A learner likewise can never be sure that constructions never heard are not in the language, but the more of it they hear, the bigger the coincidence of never having met them.
Later learnability results
Horning's was the first mathematically sophisticated blow to Chomsky's case for strong innate knowledge: a plausible setting in which language is learnable. Later work strengthened it:
- Tenenbaum. His dissertation showed that random sampling favors smaller hypotheses, including subsets, and used the Figure 14.2 mechanism to explain children's word learning, later tested with developmental psychologist Fei Xu.
- Chater and Vitányi. Nick Chater, with computer scientist Paul Vitányi, proved that a preference for simplicity suffices to make certain kinds of languages learnable given enough data.
- Perfors, Tenenbaum and Regier. As a graduate student, Perfors (now at Melbourne), with Tenenbaum and Berkeley linguist and computer scientist Terry Regier, showed simplicity-biased Bayesian models learn specific structures that Chomsky and others had singled out as especially hard to learn.
The same results explain GPT-4's success: by predicting the next word, LLMs implicitly treat language as a distribution and, given enough samples, converge on it. But they need far more data than any child could ever experience, which Bayes also explains.
Learning language from less data
Two levers
Write Bayes' rule with candidate languages as hypotheses:
- P(language | data) = [P(data | language) / P(data)] × P(language)
There are two ways to learn from less data: data that are more diagnostic (a higher evidence ratio), or a learner biased toward languages of that form (a higher prior). Griffiths thinks both operate in children.
- Richer data. LLMs train on massive text. Children learn from less language but interact with people amid the things language describes; caregivers tailor their speech to scaffold learning, other children offer simplified language, and learners choose their questions and how to engage with the world.
- A better starting point. That cannot be the whole story, since LLMs see thousands of times more data. Children are also more disposed to learn human-like languages than current networks: not Chomskyan strong innate knowledge, but a meaningful difference from today's networks.
Human child. Starts from a brain shaped by evolution, so ordinary experience is enough for learning to reach full knowledge of a language.
Large language model. Starts from a much more distant state; most of its extra data just brings it near the human starting point. Training its weights stands in for both learning and the evolution that made brains receptive to language.
What is in the human starting state
One way to describe it is a prior distribution over languages.
- Yang and Piantadosi. Yuan Yang and Steven Piantadosi built, for the first time, a system that learns formal languages (like apple-banana) from very few examples without a prior tailored to each language. It does Bayesian inference with a probabilistic context-free grammar defining the prior over hypotheses (formal languages), and learned new languages fairly well from as few as ten grammatical examples; off-the-shelf networks needed hundreds.
- McCoy and Griffiths. That does not mean networks cannot have such biases, or that explicit Bayesian inference over symbols is required. With Tom McCoy, a computational linguist and crossword constructor now a professor at Yale, Griffiths simulated the split between learning and evolution. Using meta-learning, they optimized a recurrent network's initial weights so it would quickly learn formal languages drawn from a distribution like Yang and Piantadosi's prior. The simulated evolution left a network whose starting weights encoded the prior. It learned from limited data as well as the Bayesian model, using only weights and backpropagation, no symbols or explicit Bayes, and it also learned better from small amounts of English, having apparently absorbed that languages use recursion.
Such demonstrations suggest networks can be given humanlike inductive biases; even complex, structured biases such as grammars over programs can be reproduced by choosing the right weights. Studying those networks hints at what to build into models to make them learn more like people.
Neural networks as probabilistic models
The lesson is that LLMs make sense in probabilistic terms: built from networks, but trained to predict the next word, they treat language as a probability distribution.
- Weights as hypotheses. A configuration of weights is a hypothesis, and a network has an implicit prior favoring weights near its initialization: each backprop step moves it away, but with limited data or training it stays biased toward its starting point.
- Architectures as biases. LSTMs, transformers and others differ in expressive capacity and inductive bias. Humanlike learning will require explicitly tailoring those biases to people's, recapitulating the evolution that produced human brains.
Sparks and embers
The probabilistic view also exposes limits, which matters as LLMs spread through everyday AI. GPT-4's release drew great enthusiasm; a Microsoft group declared it showed "sparks" of artificial general intelligence. But these systems are, fundamentally, trained to predict language, and predicting language well is not the same as being a general intelligence.
With Tom McCoy and a team of Princeton graduate students, Griffiths showed that GPT-4 also shows "embers" of its training problem:
- Counting. Asked to count the letters in a sequence, GPT-4 is more often right with 30 letters than 29, because "30" is commoner in internet text. Its prior over numbers, learned from next-word prediction, intrudes where it is irrelevant.
- Breadth. The same pattern appeared across tasks given to GPT-4 and other LLMs, from simple ciphers to forming acronyms.
Centuries after the death of the Rev. Thomas Bayes, thinking in probabilities still illuminates intelligent systems, even when they are machines built from massive networks trained on vast data: just what one would expect of a component of the Laws of Thought.
Xie and colleagues (ICLR 2022) cast LLM in-context learning as implicit Bayesian inference over latent concepts, with prompts as data updating a pretraining prior. Embers persist in reasoning models: McCoy, Griffiths and coauthors (2024) found OpenAI's o1 much improved but still better on high-probability outputs. The meta-learning work appeared in Nature Communications (2025).
The source names "Andrew Perfors"; the Melbourne cognitive scientist is Amy Perfors. The 200,000× figure is Fernando Pereira's (2000; see Chapter 4). Griffiths's "evolution" framing is a hypothesis: BabyLM-style results suggest part of the data gap is training efficiency, not missing priors. Horning's learnability proof (1969) assumes the learner can enumerate grammars and has unlimited computation, so it shows learnability in principle, not a practical algorithm.
Putting It All Together
A little over three hundred years after Leibniz, Griffiths offers his finished picture: rules and symbols, neural networks and probability are three threads woven in layers, one per Marr level, so they stop competing. Logic and probability are the Laws of Thought at the computational level; minds approximate them with fallible algorithms; and at the implementation level it is neural networks all the way down. The open frontier is inductive bias: what lets humans learn so much from so little.
Three threads, three levels
Marr's insight that information processors can be analyzed at different levels is what dissolves the old wars. Instead of arguing for rules or networks or probability, all three get roles to play in explaining minds.
| Marr level | What lives there | Human mind | AI systems |
|---|---|---|---|
| Computational | Logic (deduction: truth to certain truth) and probability (induction: data to uncertain hypotheses) | Same laws, judged leniently | Same laws, held to a higher standard |
| Algorithmic | Approximations: fallible inference rules, tree search, heuristics, sampling; implicit Bayesian intuitions of similarity, categories, words | Fixed by evolution | Far more options, e.g. explicit symbolic reasoning and search added to networks |
| Implementation | Neural networks, natural or artificial | Neurons; how they encode symbols and probabilities is unknown | Weights; limits of their approximations still open |
The computational level
Logic says how to solve deductive problems, from truth to certain truth; probability says how to solve inductive ones, from data to uncertain hypotheses. These principles really are the Laws of Thought: they say how any intelligent agent should behave, biological or machine, on Earth or anywhere in the universe.
- Logic's second job. It also expresses the productivity, hierarchy and compositionality of problem solving and language. That language has these properties, even approximately, makes it a strong vehicle for expressing and generating ideas, and LLMs show how powerful a substrate for intelligent behavior language is.
- The data clue. That people learn language from far less data than networks implies a predisposition toward human-language structure that current models lack.
- Probability as the probe. Bayesian inference lets us explore those predispositions by trying different priors. Grammars and logical formulas help define priors matching human inductive biases, but that does not mean people literally perform Bayesian inference or store symbolic hypotheses.
The algorithmic level
Here sits everything brains do to approximate the optimal solutions of logic and probability:
- Explicit, conscious strategies. Applying logical inference rules or Bayes' rule fallibly; searching trees of possibilities as well as we can; falling back on heuristics or sampling from probability distributions.
- Implicit approximations. Brains also approximate Bayesian solutions to environmental problems without conscious thought, giving an intuitive sense of similarity, of category structure and of the right word.
The implementation level
Here, Griffiths says, it is neural networks all the way down. Artificial networks show that systems of simple neurons can approximate logical and statistical inference well enough to capture much of what minds do. Open questions remain: where those approximations fail and whether brains use the same ones; how to build networks that form abstractions and encode rich world models like people's; how such abstractions are realized in neurons, a topic the book has largely left aside; and how to decode the representations of natural and artificial networks and relate them to behavior. As Marr found, systems of neurons are hard to understand.
Birds and jets
The Laws of Thought should also guide building intelligent systems: aerodynamics explains birds and also builds jets, which fly faster and higher. Our ambitions for AI likewise exceed our own abilities. The computational level should be the same for humans and machines, since logic and probability define intelligence; the differences come below it.
- A higher bar. We forgive human reasoning's foibles but want machines to do better. Networks that only learn to approximate logic and Bayes may need explicit symbolic processes, such as algorithms for symbolic reasoning or search. Machines have far more algorithmic and implementational options than humans, a vast space to explore.
- Data efficiency. Humans have inductive biases tuned to our world, from physics to the structure of language; networks have none of that head start. Efficient machine learners will require understanding and reproducing those biases, a major project for cognitive science needing the same back-and-forth between studying human inference and formalizing it that drove the Cognitive Revolution.
- Breadth. One human brain diapers babies, plays chess, proves theorems, cooks dinner, writes novels and composes symphonies. AI systems are usually trained for one thing: ChatGPT can give diaper tips but cannot hold a squirming infant. Our breadth comes from biases formed by evolving in a world posing all these challenges, leaving us just equipped enough to learn what one lifetime demands. Engineers may see little reason to recreate that evolution; specialized systems are easier. Jets don't catch worms, build nests, find mates or lay eggs, but they fly well.
Human intelligence was shaped by finite lives, finite brains and a limited capacity to communicate, constraints machines do not share. So even as machines get smarter, Griffiths expects human minds to stay "a little bit special".
Open mysteries and the next generation
Much of cognitive science's progress came in the three decades since Griffiths entered the field, but mysteries remain: what exactly are the inductive biases that let humans learn so much from so little? How do brains implement intelligence? How do neurons represent symbols and probabilities? What role does consciousness play? To show what drives the next generation, he asked early-career researchers for the field's biggest questions for the next thirty years.
- Anna Schapiro (assistant professor, Penn). Grew up near Boston; hooked by a high school psychology class and Hofstadter's Gödel, Escher, Bach. Asks what architecture, or constellation of architectures, will let networks simulate the human mind.
- Ishita Dasgupta (research scientist, Google DeepMind). Steered toward engineering and physics at IIT Bombay, found cognitive science during a Harvard physics PhD. Sees AI as a proof-of-concept system to inspire new theories of emerging intelligence, and wants to learn how to augment human cognition while defending people against manipulation.
- Kelsey Allen (assistant professor, University of British Columbia, in her home city). Trained as a physicist, turned to how humans and animals achieve complex goals. Thinks the field studies phenomena too simple; asks what mix of experience, innate representations and algorithms supports playing Minecraft or inventing the first microscope.
- Julian Jara-Ettinger (associate professor, Yale). Studied math at Mexico's Universidad Michoacana de San Nicolás de Hidalgo, disillusioned by AI's narrow range, found cognitive science in a textbook chapter. Notes theories linking computation to consciousness all assume computation produces experience; asks whether a complete theory of mind can rest on computation alone.
- Jessica Hamrick (senior staff research scientist, Google DeepMind, UK; raised in Charlottesville, Virginia). Came from robotics and NLP. Asks of Bayesian models where the priors come from: LLMs are undifferentiated architectures trained on the whole internet with detailed knowledge of much; brains are highly structured by evolution, with strong general priors but little specific knowledge at birth. Reconciling the two is a major challenge.
- Judy Fan (assistant professor, Stanford). Daughter of parents from Taiwan, raised near Phoenix; came via cosmology and evolutionary biology. Asks how humans alone evolved symbolic systems, the basis of writing, mathematics, art and cumulative culture: why us, why this way, and why not other species.
The answers show there are mysteries enough to keep the field busy, and that the next generation brings a wide range of perspectives. Understanding cognition takes many kinds of thinkers, alert to how minds are alike and how they differ. As human intelligence is joined by new kinds of artificial intelligence, Griffiths closes, we have a unique chance to truly understand the Laws of Thought.
Griffiths's call for models spanning human cognition got a test in July 2025: Nature published Centaur, a Llama-based model fine-tuned on Psych-101 (about 10 million choices from 160 experiments), which predicted held-out human behavior better than classic cognitive models. Critics answered that it predicts without explaining, which is Marr's feathers problem again.
The synthesis is elegant but tidier than the field. Many connectionists (McClelland among them) reject the idea that networks merely implement an independently specified computational theory, arguing that emergent network behavior can itself be the explanation. Griffiths's own related work on resource rationality (with Falk Lieder, 2020), which treats algorithms as optimal under limited time and computation, would sharpen the algorithmic level but goes unmentioned here.
Master Takeaways
Griffiths's book reduces to a division of labor: logic for certain inference, probability for uncertain inference, neural networks as the machinery that approximates both. What humans have and machines lack is not more computation but better priors: inductive biases tuned by evolution that let a child learn from one example what a network needs millions to learn.
- Three traditions, three levels. Logic and probability say what intelligence should compute; algorithms approximate it; neurons, natural or artificial, implement it.
- Probability extends logic. Every valid argument also holds in probability theory, which additionally says how strongly to believe uncertain conclusions.
- Priors answer Mill. One example suffices when it is strong evidence for a hypothesis that already has a high prior.
- Flexibility costs data. One unit of prior spread over vast hypothesis spaces leaves each hypothesis little, so flexible networks need enormous data: a conservation law of learning.
- Prediction learns language. Next-word prediction, from Elman to GPT, recovers syntax and semantics without explicit rules, but needs about 1,000 times a child's input.
- Generalization is universal. Shepard (1987): generalization decays exponentially with psychological distance, in humans, rats and pigeons alike, and it follows from Bayes.
- Logic reduces truth to syntax. Boole's algebra (1847, 1854) let reasoning run by rule, which made computers and the first theory of mind possible.
- Thinking as search. Newell and Simon cast problem solving as search through a tree of choices tamed by heuristics such as means-ends analysis; much behavioral complexity comes from the environment, as with Simon's ant.
- Formal systems have three signatures. Productivity, hierarchy and compositionality: what rules and symbols capture, what make language a strong vehicle for ideas, and what pure networks struggle to guarantee.
- Structure beats word statistics, but not probability. Chomsky showed word-to-word chains cannot judge grammaticality; a hidden Markov model over syntactic classes rates "colorless green ideas" 200,000 times above its reversal.
- Most hard problems are inductive. Vision, categories, speech and language acquisition go beyond the data; success depends on inductive biases matched to the world, which logic cannot justify (Hume, Goodman).
- Categories are fuzzy. Rosch's typicality effects defeat definitions; similarity to a prototype, measured in spaces or features, does better.
- Similarity resists one geometry. Shepard's spaces suit continuous dimensions; Tversky's asymmetric feature sets suit concepts; one Bayesian calculation yields both.
- Error-driven learning is the core trick. From the perceptron rule to backpropagation, each weight changes in proportion to its share of the error.
- Representation limits learning. Single-layer perceptrons learn only linearly separable functions (no XOR); hidden layers with nonlinearity fix that, at the price of local minima.
- Meaning as a network. Semantic networks and spreading activation made a mental state a point in activation space and thought a trajectory through it.
- Backprop sends blame back through the forward weights. A hidden unit's error is the output error scaled by its weight to the output; Rumelhart found it, then sat on it for years.
- Data and compute bought capability. ImageNet's 14 million images and gaming GPUs let AlexNet cut top-5 error from about 25% to 15.3% in 2012; scaling laws then drove GPT-3 and ChatGPT.
- Words are points in space. LSA and word2vec embed words by the company they keep, with a dimensionality sweet spot, but inherit Tversky's asymmetry and triangle-inequality problems.
- Ask why, not only how. Marr's computational level explains behavior by the problem solved; weights and neurons alone are feathers without aerodynamics.
- The goal forces the math. Marr's cash register adds because shoppers' expectations are exactly the axioms of addition; computational theories derive structure from function.
- Biases may be right answers to other questions. Wason's card choices fit rare-property information gain; Kahneman and Tversky's errors fit judging evidence for causal processes.
- Generative models invert into inference. Pearl's Bayesian networks factor distributions into conditionals; generation runs down the arrows and inference back up.
- Structure plus statistics. Probabilistic grammars and Bayesian inference over rules, causes or programs unite symbolic representations with learning from data.
- Categories are distributions. Independent features make Bayes a perceptron; probabilistic rules, causal links and growth processes give richer categories, and Bayes can pick the structural form itself.
- The logical problem is not practical. If language is a distribution, never seeing a construction becomes mounting evidence against it: Horning's proof and the size principle (2¹⁰ for ten rolls).
- Priors can live in weights. Meta-learning distilled a symbolic Bayesian prior into a network's initial weights, simulating evolution and learning from ten examples.
- LLM training stands in for evolution. Much of a model's extra data only brings it to the starting point a human brain gets from evolution.
- LLMs show embers of their training. Prediction-shaped priors leak into unrelated tasks (counting 30 letters beats 29), so predicting text is not general intelligence.
- Machines face a higher bar. We forgive human foibles; AI may need explicit search and symbolic reasoning on top of networks, and humans will likely stay a little special.
- An insider's map. Griffiths is a leading Bayesian cognitive scientist, and the narrative climaxes in his own school (Chapters 13 and 14 are partly memoir); critics who find Bayesian models too flexible to falsify (Bowers and Davis; Jones and Love) get little space.
- Tidy convergence. Assigning each tradition a Marr level is elegant, but many connectionists and symbolists dispute their assigned roles, and Marr's levels themselves are contested.
- History by heroes. Priority is sometimes simplified (Church before Turing, Werbos and Amari before 1986, Kruskal with Shepard, Block and Novikoff on perceptron convergence).
- Source slips. "Sir Robert Price", "Andrew Perfors", a world 2/world 3 mix-up and shuffled probability tables in Chapter 12, and chapter cross-reference errors in Chapters 9 and 13.
- Already dated on AI. Reasoning models (2024 to 2026) narrowed the planning gap Griffiths ascribes to LLMs, though their data hunger and embers remain.
- Thin on brains. The implementation level ("neural networks all the way down") is asserted more than explored, as Griffiths admits.
Connect it
- AI engineering. Frontier models sit almost wholly in the network tradition; their known weak spots (data efficiency, systematic generalization, calibrated uncertainty) are exactly what symbolic and Bayesian methods supply, which is why tool use, search and verifiers keep getting bolted on.
- Evaluating models. Marr's levels give a checklist for any claim about an LLM: is it about the objective (computational), the mechanism (algorithmic) or the hardware and weights (implementation)? The "embers" result is a template for probing where training priors leak into answers.
- Enterprise search and data. The vocabulary mismatch problem Dumais solved with LSA is the same one embedding-based retrieval solves today, and semantic networks survive as knowledge graphs; both inherit the asymmetry limits Tversky identified.
- Risk and banking. Base-rate neglect and the conjunction fallacy are standing hazards in credit, fraud and audit judgments; Bayes' rule with explicit priors is the corrective, and Bayesian networks remain a practical tool for causal risk models.
Frameworks & mental models
| Name | What it says | Use it when |
|---|---|---|
| Formal system | Tokens, positions and rules; digital and medium independent | Deciding whether a task can be mechanized by rules |
| Possible worlds / truth tables | A statement's truth is the set of worlds where it holds | Checking validity; grounding probability as belief over worlds |
| Heuristic search / means-ends analysis | Pick the action that most reduces the gap to the goal | Planning, problem solving, designing search-based AI |
| Physical symbol system hypothesis | Symbol manipulation is necessary and sufficient for intelligence | Weighing claims that intelligence needs explicit symbols |
| Chomsky hierarchy | Finite-state ⊂ context-free ⊂ context-sensitive ⊂ Turing | Judging what structure a language or model can express |
| Poverty of the stimulus | Rich rules plus sparse data imply innate knowledge | Assessing nativist arguments; spotting where priors are needed |
| Deduction vs induction | Data fix the answer vs data underdetermine it | Diagnosing whether rules suffice or priors are needed |
| Prototype theory | Category = most typical member; join the most similar prototype | Explaining typicality and fuzzy category edges |
| Tversky's contrast model | Similarity = common minus distinctive features, weighted | Asymmetric or non-metric similarity judgments |
| Perceptron / backpropagation | Change weights by error times input; propagate blame backward | Understanding how networks learn and where they fail |
| Spreading activation | Activation flows along weighted links, sums, decays, thresholds | Modeling priming, association and constraint satisfaction |
| Next-word prediction (SRN to GPT) | Language supplies its own targets; hidden state carries context | Training language models; explaining emergent categories |
| Latent Semantic Analysis / embeddings | Compress word-document counts; words become vectors | Search with vocabulary mismatch; semantic similarity |
| Marr's three levels | Computational (why), algorithmic (how), implementation (in what) | Placing any theory or AI explanation; avoiding false rivalry |
| Bayes' rule | P(h given d) = P(d given h) × P(h) / P(d) | Any uncertain inference; quantifying evidence vs prior |
| Conservation of learning | More hypotheses means lower priors and more data needed | Trading model flexibility against data efficiency |
| Shepard's universal law | Generalization decays exponentially with psychological distance | Predicting how far a lesson transfers to new cases |
| Rational analysis | Derive behavior from the optimal solution to the environment's problem | Explaining "why" a cognitive process has its form |
| Bayesian networks | Factor a distribution into conditionals; generate down, infer up | Modeling causal structure and reasoning under uncertainty |
| Probabilistic grammar | Add dice to grammar rules to get a distribution over sentences | Combining structured language models with learning |
| Size principle | Smaller consistent hypotheses gain likelihood with each example | Learning from positive examples only |
| Meta-learning priors | Optimize initial weights over many tasks to instill a prior | Making networks learn from few examples |
Who's who
| Name | Role | Why they matter |
|---|---|---|
| Gottfried Leibniz | Philosopher, mathematician | Dreamed of reducing reasoning to arithmetic; saw probability as a new logic |
| George Boole | Mathematician, Cork | Algebra of logic; The Laws of Thought (1854) |
| Alan Turing | Logician | Universal machine; definition of computation; the child-machine idea |
| Claude Shannon | Engineer, Bell Labs | Logic as circuits; information theory |
| Allen Newell and Herbert Simon | AI and cognitive science pioneers | Logic Theorist, heuristic search, production systems |
| Noam Chomsky | Linguist, MIT | Generative grammar, hierarchy, poverty of the stimulus |
| Eleanor Rosch | Psychologist, Berkeley | Fuzzy categories and typicality |
| Roger Shepard | Psychologist, Stanford | Psychological spaces, MDS, universal law of generalization |
| Amos Tversky and Daniel Kahneman | Psychologists | Feature similarity; heuristics and biases against Bayesian minds |
| Frank Rosenblatt | Psychologist, Cornell | The perceptron and its convergence theorem |
| Marvin Minsky and Seymour Papert | MIT AI | Perceptrons (1969): limits of single-layer networks |
| Ross Quillian, Allan Collins, Elizabeth Loftus | Semantic memory researchers | Semantic networks and spreading activation |
| David Rumelhart, Jay McClelland, Geoffrey Hinton | Connectionists | Interactive activation, backpropagation, PDP |
| Fei-Fei Li, Alex Krizhevsky | Computer vision | ImageNet and AlexNet, the 2012 turn to deep learning |
| Steven Pinker and Alan Prince | Rules-and-symbols linguists | The past-tense critique of connectionism |
| Jeff Elman | Linguist, UCSD | Simple recurrent network and next-word prediction |
| Susan Dumais and Tom Landauer | Bell Labs | Latent Semantic Analysis |
| David Marr | Neuroscientist, MIT | Three levels of analysis |
| Thomas Bayes and Pierre-Simon Laplace | Probabilists | Bayesian inference |
| John Anderson | Psychologist, Carnegie Mellon | Rational analysis |
| Mike Oaksford and Nick Chater | Cognitive scientists | Bayesian account of the Wason selection task |
| Judea Pearl | Computer scientist, UCLA | Bayesian networks |
| Jim Horning | Computer scientist | First proof that probabilistic grammars are learnable |
| Josh Tenenbaum | Cognitive scientist, MIT | Bayesian generalization and structure learning; Griffiths's adviser |
| Tom McCoy | Computational linguist, Yale | Meta-learned priors; "embers of autoregression" |
About this book
The Laws of Thought: The Quest for a Mathematical Theory of the Mind by Tom Griffiths, 2026. This is a chapter-by-chapter study of the book, with summaries, maps, tables and diagrams; short quotations keep their page references. It is shared free for personal study, is not for sale, and is not endorsed by the author or publisher. Please read and buy the original.
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