Insight Editions · No. 46
The Cambridge Companion to Bertrand Russell
Nicholas Griffin (ed.)
A collection of essays examines Bertrand Russell’s life and work in logic, mathematics, analytic philosophy, philosophy of science, metaphysics, and ethics.
A study guide to The Cambridge Companion to Bertrand Russell by Nicholas Griffin (ed.). Shared free for personal reading; not for sale and not endorsed by the authors or publishers. Please read and buy the original.
The Book in Brief
The argument in brief
The argument map: from Hegelian beginnings to the ethics of an anti-realist moralist.
How the book is built
The volume runs roughly in chronological order. Griffin's Introduction sets out the life and the developmental thesis. Chapters 1 to 5 cover origins: Grattan-Guinness on the mathematics behind logicism, Griffin on the neo-Hegelian apprenticeship of 1894-98, Cartwright on the revolt with Moore, Beaney on Frege, and Godwyn and Irvine on what logicism claimed. Chapters 6 to 8 explain the logical machinery (descriptions, Landini's archive-based substitutional theory, and the theory of types). Chapter 9 identifies the method behind it all. Chapters 10 to 14 follow the metaphysics and epistemology after 1910: logical atomism, neutral monism, structuralism, and two accounts of Russell's theory of knowledge. Chapter 15 rehabilitates his ethics.
Three recurring tensions the chapters keep returning to.
| No. | Title | In one line |
|---|---|---|
| Intro | Russell's Life and the Shape of His Philosophy (Griffin) | Many phases, one project; the narrow canon hides most of it. |
| 1 | Mathematics in and behind Russell's Logicism, and Its Reception (Grattan-Guinness) | Logicism was mathematics first, owed much to Cantor and Peano, and captured less than claimed. |
| 2 | Russell's Philosophical Background (Griffin) | The contradiction of relativity sank the Hegelian plan; 1898 turned it into a reductio. |
| 3 | Russell and Moore, 1898-1905 (Cartwright) | The new philosophy of propositions and terms was less settled than Russell later claimed. |
| 4 | Russell and Frege (Beaney) | Paraphrastic analysis, not decomposition, marks the turn. |
| 5 | Bertrand Russell's Logicism (Godwyn and Irvine) | Logical axioms are supported inductively by their consequences. |
| 6 | The Theory of Descriptions (Hylton) | Its real gain was eliminating denoting concepts, not escaping Meinong. |
| 7 | Russell's Substitutional Theory (Landini) | A type-free logic of propositions is the missing link to Principia. |
| 8 | The Theory of Types (Urquhart) | A failed foundation whose hierarchy shaped computing. |
| 9 | Russell's Method of Analysis (Hager) | One two-way method throughout: analysis back, synthesis forward. |
| 10 | Russell's Neutral Monism (Tully) | A monism of events that still talks like a dualist. |
| 11 | The Metaphysics of Logical Atomism (Linsky) | Metaphysics guided by logic; construction for theorems. |
| 12 | Russell's Structuralism and the Absolute Description of the World (Demopoulos) | Newman shows pure structural knowledge is trivial. |
| 13 | From Knowledge by Acquaintance to Knowledge by Causation (Baldwin) | A causal, reliabilist epistemology from 1921. |
| 14 | Russell, Experience, and the Roots of Science (Grayling) | One aim, five strategies, ending in animal postulates. |
| 15 | Bertrand Russell: Moral Philosopher or Unphilosophical Moralist? (Pigden) | Six phases; pioneer of emotivism and the error theory. |
Reading tips
- Read chapters 6 to 8 together. Descriptions, substitution and types are three answers to one question: how to keep logic universal without paradox.
- Pair 12, 13 and 14. Demopoulos, Baldwin and Grayling read the same late books (Analysis of Matter, Human Knowledge) and reach different verdicts.
- Watch the dates. Several authors place the same event in different years; the critical lenses flag each conflict rather than silently choosing one.
What "deepen" added
Earlier Insight passes aimed at a ~10% band. Priority went to Beaney, Hylton, Landini and Tully (thin relative to source), then Griffin's Introduction and the science/ethics close. Web checks for Oct 2026 update facts (SEP revisions, Rocq rename, AlphaProof, deaths already noted in progress) stay in critical-lens or people boxes.
Russell's Life and the Shape of His Philosophy
Influence and life to 1914
Griffin claims that virtually every strand of twentieth-century analytic philosophy either began with Russell or was transformed by passing through him. With Frege and Peano he created modern formal logic, and more than they he showed philosophers its use. Carnap, the positivists, Quine, Ayer and Wittgenstein all built on him or against him; even the ordinary language school "would have been inconceivable without him".
Born in 1872 into the radical Whig aristocracy, Russell was the grandson of Lord John Russell, architect of the 1832 Reform Bill. Both free-thinking parents died before he was four; his grandparents overturned their will, and his grandmother raised him at home with tutors, apparently grooming him for prime minister. His brother called him "an unendurable little prig". John Stuart Mill, a secular godfather, rescued the adolescent: Russell accepted almost all of Mill except his empiricist philosophy of mathematics, and hid his doubts in a notebook written in Greek letters.
- Cambridge. Trinity from 1890; a first in the Mathematical Tripos of 1893, one year of philosophy, and the Apostles, then dominated by McTaggart's idealism.
- First books. A fellowship thesis became An Essay on the Foundations of Geometry (1897); German Social Democracy (1896) was his first, and lasting, critique of Marxism.
- Marriage. He married Alys Pearsall Smith, an American Quaker five years older, in December 1894; his grandmother tried to stop it by raising fears of hereditary insanity.
- The 1901 crisis. Seeing Evelyn Whitehead in apparent heart failure, he felt "the loneliness of the human soul is unendurable" and, by his account, became a pacifist; the marriage collapsed within a year, though no divorce came until 1921.
War, Russia, schools and money
From 1910 Russell held a five-year Trinity lectureship, his capital having run down while he finished Principia. World War I turned him into a full-time campaigner for conscientious objectors in the No-Conscription Fellowship. He was fined, lost his passport, was dismissed by Trinity in 1916 and later jailed. His affair with Lady Ottoline Morrell (from 1911) was the most passionate of his life; from 1916 he was with Constance Malleson ("Colette O'Niel").
- Bolshevism. He visited Russia in 1920 and wrote The Practice and Theory of Bolshevism, the first book from the left to warn of communist dictatorship.
- China and Dora. A year at the University of Peking with Dora Black, whom he married on return; a son and, in 1923, a daughter.
- The radical's paradox. Only people as ruthless as the Bolsheviks could force change, and their ruthlessness would spoil the result (letter of 24 July 1920). Hope moved to psychology and education.
- Beacon Hill. The school he founded with Dora in 1927 drew many "problem children"; he concluded "only government can prevent brutal violence", and his son John was badly bullied. He left it, and the marriage, in 1932.
- Lean 1930s. Freedom and Organization (1934), Power (1938), The Amberley Papers (1937); marriage to Patricia Spence in 1936, plus alimony to Dora and £400 a year to his late brother's second wife.
Respectability and the nuclear campaigns
Back at Trinity in 1944, Russell finished Human Knowledge (1948). Postwar Britain caught up with him, and his hostility to Stalin made him a useful cold warrior. Honours followed: the Order of Merit (1949), the Nobel Prize for Literature (1950), the first BBC Reith Lectures (1949).
- Baruch and coercion. He first argued that Russia might be coerced, by threat of war, into international control of the bomb; the H-bomb (USA 1952, USSR 1955) ended that view.
- Back channel. An open letter to Eisenhower and Khrushchev (1957) began a correspondence that ran through the Cuban Missile Crisis.
- Campaigns. "Man's Peril" (BBC, Christmas 1954); the Russell-Einstein Manifesto (1955) and Pugwash; CND (1958); the Committee of 100, which got him jailed again in 1961.
- Last years. The International War Crimes Tribunal (1967) on Vietnam; he condemned the 1968 invasion of Czechoslovakia on the same grounds. His last statement, on the Middle East, is dated 31 January 1970, two days before his death.
Russell's public life in six turning points, as Griffin tells it.
One project, many phases
Broad joked that Russell produced a new system every few years. Griffin answers that the phases grow from one aim: unlike rationalists who start from principles and empiricists who start from sensation, Russell began in the middle, with the sciences, "at no moment quite right" but "seldom quite wrong". His final position was a "synthesis of four sciences": physics, physiology, psychology and mathematical logic. A discipline becomes science when its hypotheses can be confirmed or refuted; "science is what we know, and philosophy is what we don't know", which Griffin notes is said tongue in cheek. Russell was never a positivist: he kept metaphysics and rejected verificationism.
- Neo-Hegelian start. Separate the a priori from the a posteriori in each science by Kantian transcendental argument.
- 1899 break. That method could not escape psychologism; with Moore, Russell adopted "absolute realism" (Principles of Mathematics, Principia Ethica, both 1903), his "one genuine revolution".
- Terms. Almost every word means a term: objective, neither linguistic nor mental; propositions are complexes of terms; every term has being, if not existence.
- Analysis and logicism. After meeting Peano in Paris (1900), Russell wrote "The Logic of Relations" (1901), defined the cardinal of a class u as the class of all classes similar to u, and by early 1902 had twenty-two logical axioms.
The phases Griffin distinguishes, each carrying the same science-first project forward.
Paradox, denoting and the new analysis
Russell found his paradox in May or June 1901; Principles states it as unsolved and offers an early theory of types in an appendix. Before adopting the ramified theory of Principia, he spent years trying to avoid types, notably with the substitutional theory, because types restrict variables and so destroy his account of what is special about logic: its truths stay true under any substitution of terms.
General propositions posed a parallel problem. Under the principle of acquaintance one must be acquainted with every constituent of a proposition one understands, so "every number is odd or even" cannot contain all the numbers. Principles used denoting concepts ("all", "every", "any", "some", "a", "the") that shift aboutness to their denotation. Fregean quantification replaced all but "the"; definite descriptions survived because "Russell met Quine" and "Russell met the author of Quiddities" differ, and because some descriptions denote nothing. Griffin, against most commentators, holds that Russell always allowed non-denoting concepts.
The result was "a positive mania for analyzing away": numbers (as classes), classes (by propositional functions in Principia, or by substitution), and propositions themselves. Under the multiple relation theory of judgment (1910-13), "Othello believes that Desdemona loves Cassio" becomes a four-place relation, and truth becomes correspondence to a fact, replacing the earlier identity theory. Wittgenstein's 1913 objection, perhaps that judgment would need prior type judgments, made Russell abandon it and the Theory of Knowledge manuscript.
Science in the middle, and the demarcation criterion
Griffin's developmental thesis rests on a constant aim. Unlike rationalists who start from principles and empiricists who start from sensation, Russell began in the middle, with the sciences: "at no moment quite right" but "seldom quite wrong" (MPD 17). As an undergraduate he already wrote that epistemology should "make a self-consistent whole of the various Sciences" (Papers 1, 121); at the end he described his final position as a "synthesis of four sciences": physics, physiology, psychology and mathematical logic (MPD 16). The line between science and philosophy was never sharp for him. Psychology was emerging as an independent science at the start of his career; he regarded his own work and Frege's as having achieved the same independence for mathematical logic.
Griffin's demarcation criterion, drawn from Russell: a discipline becomes science when its hypotheses can be confirmed or refuted. "Science is what we know, and philosophy is what we don't know" is said tongue in cheek. Russell was never a positivist: he kept metaphysics and rejected verificationism. Broad's joke that Russell produced a new system every few years (Contemporary British Philosophy, 1924) ignores how each phase carries the same science-first project forward.
Griffin: one project through six phases, science in the middle.
Absolute realism, acquaintance, and the 1901-1905 crises
The 1899 break with idealism, made with Moore, is Griffin's "one genuine revolution". Almost every word means a term: objective, neither linguistic nor mental; propositions are complexes of terms; every term has being, if not existence. After meeting Peano in Paris (1900), Russell wrote "The Logic of Relations" (1901), defined the cardinal of a class u as the class of all classes similar to u, and by early 1902 had twenty-two logical axioms.
The paradox of May or June 1901 forced years of work before the ramified theory of Principia. Substitution was tried because types restrict variables and so destroy what is special about logic: its truths stay true under any substitution of terms. Parallel problem: under the principle of acquaintance one must be acquainted with every constituent of a proposition one understands, so "every number is odd or even" cannot contain all the numbers. Principles used denoting concepts; Fregean quantification replaced all but "the". Griffin, against most commentators, holds that Russell always allowed non-denoting concepts.
After "On Denoting" (1905), analysis becomes paraphrase: Fs are analysed into Gs when every sentence about Fs can be reparsed as one about Gs only. That is a linguistic criterion for ontological reduction, not a proof that Fs do not exist. Expressions so eliminated are "incomplete symbols". The result was "a positive mania for analyzing away": numbers, classes, and propositions themselves. Under the multiple-relation theory of judgment (1910-13), "Othello believes that Desdemona loves Cassio" becomes a four-place relation. Wittgenstein's 1913 objection made Russell abandon it and the Theory of Knowledge manuscript.
Prison, naturalism, and the five postulates
In prison in 1918 Russell rewrote belief: "On Propositions" (1919) has word and image propositions sharing structure with facts, plus belief-feelings. It marks the turn to naturalism and neutral monism. The five-minute hypothesis of 1921 Griffin finds "rather banal". The Analysis of Matter (1927) makes physics reveal only structure; Newman (1928) showed structure alone is nearly empty. Griffin links this to Putnam's later model-theoretic argument.
Late Russell abandons strict empiricism ("The Limits of Empiricism", 1936). Human Knowledge (1948) offers five postulates for non-demonstrative inference: quasi-permanence, separable causal lines, spatio-temporal continuity, the structural postulate, and analogy. They are truth-conducive, not truth-preserving. Enumerative induction Russell judged bad non-demonstrative reasoning seven years before Goodman's "new riddle"; Wesley Salmon (1974) noticed the anticipation. Vagueness (1923) treats meaning as one-many; the "problem of the many" was reinvented by Unger (1980). The Inquiry (1940) anticipates Dummett's realism test by a quarter-century.
Contributors' map and the narrow canon problem
Griffin's Introduction also serves as a prospectus for the fifteen chapters. Grattan-Guinness handles mathematics behind logicism and reception; Griffin himself the idealist background; Cartwright the Moore years; Beaney Frege; Godwyn and Irvine logicism's two faces; Hylton descriptions; Landini substitution; Urquhart types; Hager analysis as method; Tully neutral monism; Linsky atomism's metaphysics; Demopoulos structuralism; Baldwin acquaintance to causation; Grayling experience and science; Pigden moral philosophy. The selective bibliography and index close the book; politics, education and religion get Introduction coverage but no dedicated chapter, a gap the Companion's own A closer look (synthesis) flags.
The narrow-canon problem is quantitative in Griffin's hands. From 1945 to about 1970, he estimates at least 70% of citations to Russell in the analytic literature drew on the thirteen years 1905-1918. "On Denoting", Principia, the 1914 Lowell lectures, the 1918 atomism lectures and Mysticism and Logic essays dominate. Human Knowledge (1948), Analysis of Mind (1921) and Analysis of Matter (1927) were under-read. The Companion is an explicit corrective: Landini, Tully, Demopoulos, Baldwin and Pigden pull weight outside the narrow band.
Griffin's prospectus: who covers which band.
Politics, pacifism, and why philosophy chapters omit them
Griffin's biographical half records the public Russell the philosophy chapters mostly skip. Liberal candidate at Wimbledon (1907) and for the Women's Suffrage cause; principled opposition to World War I costing Trinity lectureship; Brixton 1918; Beacon Hill school with Dora; run for Parliament again; Barnes Foundation appointment and dismissal (lectures became A History of Western Philosophy); Einstein-Russell Manifesto 1955; Committee of 100 and second imprisonment 1961. Griffin's point for philosophers: the same mind that wrote Principia wrote Which Way to Peace? and Has Man a Future?. The Companion's thematic limit is not Russell's.
Inheritance, frugality, and the 1910 lectureship
Griffin's life narrative includes economic detail philosophers skip. Russell supported himself from inheritance after giving away the small stipend of his 1895 Trinity Fellowship; he opposed inherited wealth in general but allowed it for art and learning. He and Alys lived frugally and gave much away. By 1910, with Principia complete, capital was depleted; he took a five-year Trinity lectureship in logic and the principles of mathematics, no longer seeing moral justification for unearned income alone.
War as vocation
August 1914: "I never had a moment's doubt as to what I must do... when the War came I felt as if I had heard the voice of God" (Auto. 2, 17-18). Too old for conscription, he joined young radical conscientious objectors, lost his lectureship, was fined, and in 1918 imprisoned in Brixton. Griffin presents this not as a hobby beside philosophy but as the same absolutist energy that drove the logicist programme, redirected. The emotional prison of the failed marriage to Alys (she remained devoted; he escaped) is narrated as the private counterpart to public rupture.
Alys, Ottoline, Colette, Dora
Griffin's Introduction names the companions who structure the emotional biography: Alys Pearsall Smith (marriage 1894; loveless after 1901); Lady Ottoline Morrell; Constance Malleson ("Colette"); Dora Black (marriage 1921; Beacon Hill; divorce). Later Patricia (Peter) Spence and Edith Finch complete the four marriages. The philosophical chapters rarely need these names; the Introduction insists the life is one piece.
Leibniz lectures and the 1901-2 logic course
Griffin notes occasional Cambridge lecturing before the 1910 post: 1899 on Leibniz (the book A Critical Exposition of the Philosophy of Leibniz followed), and 1901-2 on mathematical logic, the course that seeded Principia's workshop culture (Hawtrey among the hearers). The Leibniz work matters philosophically: internal relations, sufficient reason, and the continuum are foils for Russell's external-relations realism. Heroes Leibniz and Spinoza supply the "system of the world" ambition; science-in-the-middle supplies the method that separates Russell from them.
Griffin: teaching before the lectureship.
Auto-citations Griffin leans on
MPD, the Autobiography, Papers 1, and Broad 1924 are the Introduction's narrative spine for development claims. Readers should treat autobiographical reconstructions as evidence, not gospel; Griffin usually does. Where Newman is mistimed (1927 vs 1928) or names slip (Sydney/Sidney Webb), A closer look already flags the print.
Narrow canon quantitatively
"On Denoting", PM, Lowell lectures, 1918 atomism lectures, and Mysticism and Logic essays crowd out AMi, AOM, and HK. The Companion's chapter mix is an editorial reply: Landini, Tully, Demopoulos, Baldwin, Pigden pull weight outside the band. Readers trained only on the narrow canon will misread late Russell as decline rather than continuation of the science-first project.
Griffin: citation funnel 1945-1970.
Science demarcation and anti-positivism
A discipline becomes science when hypotheses can be confirmed or refuted. Philosophy keeps what is not yet so settled. Russell's quip that science is what we know and philosophy what we do not is tongue in cheek. He never accepted verificationism; metaphysics of facts, events and postulates remains. Broad's "new system every few years" joke misses the constant aim: a system of the world consistent with the best science, begun in the middle rather than from pure reason or pure sense.
Five postulates listed with Griffin's gloss
Griffin summarises HK's five postulates as truth-conducive, not truth-preserving: quasi-permanence; separable causal lines; spatio-temporal continuity; structural postulate; analogy. Enumerative induction alone is bad non-demonstrative reasoning; Goodman's new riddle is anticipated by about seven years; Wesley Salmon (1974) noticed. Vagueness (1923) as one-many meaning anticipates Unger's problem of the many (1980). The Inquiry (1940) anticipates Dummett's realism test by a quarter-century. These anticipations justify rereading late Russell as research, not popularisation.
One project thesis restated
Griffin's deepest Introduction claim: phases develop out of each other as attempts to carry forward a single philosophical project, not Broad's cartoon of a new system every few years. Science in the middle, anti-psychologism, analysis, thinner ontology when possible: the constants. Tools change (idealist dialectic, absolute realism, descriptions, types, constructions, monism, postulates); the aim does not. Politics and protest are the same energy under another description. The Companion's chapters are case studies in tool-change; the Introduction is the thesis that binds them.
Griffin: constants vs tools.
Mind, structure and the limits of empiricism
In prison in 1918 Russell rewrote his account of belief: "On Propositions" (1919) has word and image propositions sharing structure with facts, plus belief-feelings (memory, expectation, assent). It marks the turn to naturalism and neutral monism: minds and matter both built from neutral stuff, "sensations" in The Analysis of Mind (1921), "events" in The Analysis of Matter (1927). The five-minute hypothesis of 1921 Griffin finds "rather banal".
- Scepticism as a boundary problem. An observer in region A learns of region B only through what crosses the boundary; percepts give structure only. Newman showed that structure alone is nearly empty: percepts map onto any domain with enough members, anticipating Putnam's model-theoretic argument.
- Vagueness (1923). Meaning (reference) is one-many, so all language is vague; the "problem of the many" was reinvented by Unger (1980).
- Meaning and use. Russell held that one understands a word by using it correctly; Griffin suggests The Analysis of Mind may be a source of Wittgenstein's meaning-as-use (PI §43).
- Hierarchy of languages. An object language of words with "meaning in isolation" anticipated Tarski's hierarchy and answered Wittgenstein's "showing".
- Excluded middle. The Inquiry (1940) anticipates Dummett's realism test by a quarter-century, and meets the manifestation constraint at the level of words.
Russell judged enumerative induction bad non-demonstrative reasoning (HK, 1948), seven years before Goodman's "new riddle"; Wesley Salmon (1974) noticed the anticipation. After The Analysis of Matter's structural realism collapsed, Russell openly abandoned empiricism in "The Limits of Empiricism" (1936): "empiricism, though not logically refutable, is in fact believed by no one".
The narrow canon
| Narrow canon (1945-70) | Neglected work |
|---|---|
| "On Denoting" (1905), "Knowledge by Acquaintance" (1911) | Principia beyond its Introduction |
| Problems of Philosophy (1912) | Theory of Knowledge (1913, published 1983) |
| "Philosophy of Logical Atomism" (1918) | "Analytic Realism", "Philosophical Importance of Mathematical Logic" (1911) |
| Our Knowledge of the External World (1914) | Analysis of Matter, Inquiry, Human Knowledge |
Griffin's estimate: the canon, thirteen years of a fifty-year career, drew at least 70% of citations from 1945 to 1970. Pears (1967) cast Russell as heir to Hume, yet his major influences (Bradley, Kant, Leibniz, Plato, Spinoza, Frege, Wittgenstein) include no empiricist; Russell was a strict empiricist only around 1912-14. Since the Russell Archives opened in 1970, the canon has widened before 1914; after 1914, Griffin admits, even this volume treats the later work sketchily.
Mathematics in and behind Russell's Logicism, and Its Reception
Prehistory: Cauchy, Cantor, Peano
Cauchy built mathematical analysis on limits from the 1820s; Weierstrass in Berlin from the late 1850s pushed rigour further and reduced the indefinables to the cardinal numbers. Dedekind's cuts (1872) defined irrationals. Two strands then reached Russell. Georg Cantor (1845-1918), starting from limit points and derived sets, created transfinite ordinals and cardinals, kinds of order (above all well-order), set theory as the base of mathematics, and the power-class theorem: the class of all sub-classes of S has greater cardinality than S. Giuseppe Peano (1858-1932) axiomatised arithmetic in 1889 with three indefinables (initial number, successor, induction), symbolised the logic itself, ran the Rivista di matematica (1891-1906) and the Formulaire (1895-1908) with his "Peanists" Burali-Forti, Pieri and Padoa.
Grattan-Guinness's Fig. 1, condensed: the line of descent into logicism and out of it.
Paris, Friday 3 August 1900
Russell's route ran through neo-Hegelian geometry (Essay on the Foundations of Geometry, 1897), Couturat's Cantorian De l'infini mathématique, Whitehead's Universal Algebra (1898) and Moore's 1899 realism. The turn came at the International Congress of Philosophy, on a morning given to the Peanists: Peano insisted, against Schröder, on care with "the" in defining "the class such that". Russell's Autobiography says he read all of Peano at once and wrote a book by the year's end; the manuscripts show this is "absurdly wrong": the Peano papers came a month later, and writing lasted until 1902.
- Relations. The Peanists had no logic of relations, so Russell built one ("The Logic of Relations", Rivista, 1901).
- Numbers as classes. 0 is the class containing the empty class; 1 the class of classes similar to that containing 0, and so on into the transfinite, with 0, the empty class and nothing kept apart (POM, 75). Ordinals are classes of well-ordered classes.
- No dividing line. Since set-theoretic notions sit on both sides of the Peanist split between logic and mathematics, logic alone subsumes mathematics: this was logicism.
The Principles and the contradiction
The Principles of Mathematics went to Cambridge University Press in May 1902 and appeared in May 1903, in seven parts: indefinables, number, quantity, order, infinity and continuity, space, matter and motion, plus appendices on Frege and types. Russell met the denoting "sextet" (all, every, any, a, some, the) because logicism needs mathematical functions like x², and Cauchy's rule that functions be single-valued pointed him to definite descriptions. The implicational form of logicism probably came from the hypothetical status of non-Euclidean geometries.
How Russell's paradox fell out of Cantor's power-class theorem (POM ch. 10).
Applying Cantor's power-class proof to the class of "all" classes yielded the class C of classes that do not belong to themselves, which belongs to itself if and only if it does not: a double contradiction, worse than an ordinary reductio. Part VII, on dynamics, assumed that from a finite number of events one can infer others, an "elementary blunder" that G. H. Hardy spotted in his review. Russell read Frege only after sending the book off, wrote to him in June 1902, and admitted in the preface that he had been anticipated; Grattan-Guinness thinks commentators "grossly exaggerate" Frege's influence.
Collecting paradoxes, finding axioms
| Paradox | Source | Extra notions |
|---|---|---|
| Russell's class C | Russell 1901 | class, membership only |
| Greatest cardinal | Cantor (unpublished) | cardinal arithmetic |
| Greatest ordinal | named after Burali-Forti (1897); also E. H. Moore | ordinals |
| Liar | Greek | truth |
| Least undefinable number; visiting card | G. G. Berry | naming |
| Richard | Jules Richard | definability |
| Heterological | Grelling and Nelson 1908 | predicates (omitted in PM) |
From 1903 to 1907 Russell sought solutions that kept the mathematics; each either readmitted paradox or cut away legitimate theorems. Two axioms surfaced. In June 1904, reading a Whitehead draft on infinite products of cardinals, Russell saw that choosing one member from each of infinitely many classes needs an axiom, the multiplicative axiom; Zermelo published the same need as the axiom of choice that year. Cassius Keyser argued (1904) that proofs of an infinite beg the question; Russell resisted, then agreed by 1906 that an axiom of infinity was needed.
The substitutional theory (1905-6) dropped variables for substitution of entities in propositions; a paper accepted by the London Mathematical Society in April 1906 was withdrawn because of thin mathematics, "objective falsehoods", and a paradox like his own.
Principia: scope, axioms and gaps
Principia Mathematica was planned from 1906, worked out from early 1907 to late 1909, and written out entirely by Russell. Cambridge University Press needed a subsidy; the Royal Society gave £200 of the £300 asked. Volumes appeared in 1910 (750 copies), 1912 and 1913 (500 each); a fourth, on geometry, by Whitehead alone, was abandoned during the war and no manuscript survives.
- Unclear logicism. The introduction never states logicism; an inferential version seems intended, though the "pure mathematics" phrase of 1903 returns (PM 1, 90).
- Narrow scope. Superb Cantorian mathematics, real numbers and some point-set topology, but no calculus, no mechanics, no differential geometry.
- Types. The vicious circle principle: "All that contains an apparent variable must not be one of the possible values of that variable." Grattan-Guinness places orders before types, since quantification fixes orders.
- Reducibility. Types scatter integers, rationals and irrationals; the axiom of reducibility restores arithmetic but had "no philosophical basis", as the authors knew (PM 1, 55-60).
- Infinity. It needs infinitely many individuals; if these are physical, logicism becomes a posteriori (Russell 1911).
- One lonely individual. Russell's rule to assume only one individual where possible (*22.351) was forgotten in Whitehead's cardinals; the repair (a "Prefatory statement", PM 2, vii-xxxi) caused the two-year gap between volumes.
Reception, 1910s-1930s
- Students. Norbert Wiener (1913) defined ordered pairs by classes (1914); Wittgenstein and Russell devised truth tables; Jean Nicod reduced the propositional calculus to one axiom (1917).
- America. At Harvard in 1914, Sheffer (the stroke, "Boolean algebras", 1913) and C. I. Lewis, whose strict implication answered PM's conflation of implication and entailment, launching modal logic.
- Popularisers. Philip Jourdain wrote seven reviews and "The philosophy of Mr. B*rtr*nd R*ss*ll"; Russell wrote Introduction to Mathematical Philosophy (1919) during four and a half months in prison, helped by Dorothy Wrinch.
- Europe. Cassirer, Hölder, Leśniewski, Łukasiewicz, Chwistek, the Vienna Circle (Carnap's Abriss der Logistik 1929 and Aufbau 1928); Ramsey helped with PM's second edition (1925-7) and split paradoxes into mathematical and semantic.
- Odd uses. W. W. Greg used the logic of relations for textual copy-text theory (1927); J. H. Woodger axiomatised biology (1937); Keynes's Treatise on Probability (1921) followed its reductive style.
Gödel and the universality of logic
Gödel's 1931 theorem showed that in a consistent system like PM some true propositions cannot be proved, and that consistency needs a richer system: PM's hope to embrace "all those propositions which we believe to be true" (PM 1, 12-13) failed. The irony: Russell's 1922 introduction to the Tractatus had proposed a hierarchy of languages, "gold in his hands", but he never used it, ignored Gödel in the 1937 reissue of Principles, and in My Philosophical Development (1959) claimed the hierarchy disposed of Gödel's "puzzles", though it is needed to state the theorem. In 1963 he told Leon Henkin that Gödel had shown them mistaken about consistency.
Writing Principia: costs, hands, and the missing fourth volume
Grattan-Guinness reconstructs the workshop. From early 1907 to late 1909, Russell and Whitehead divided Sections; Russell wrote out the entire text. Ralph Hawtrey (1879-1975; the chapter prints 1974) read portions. Cambridge University Press had agreed to publish, but printing costs "simulated the infinite"; they applied for £300 from the Royal Society and got £200. Print runs: vol. 1 (1910) 750 copies at £1.5s.; vols. 2 (1912) and 3 (1913) 500 each. A planned fourth volume on geometry, to be written by Whitehead alone, never appeared. The title likely nods to Newton's Principia and Moore's Principia Ethica (1903). Russell's summary paper MLT went to the American Journal of Mathematics.
| PM Part | Content | Approx. pages (1st ed.) |
|---|---|---|
| I Mathematical logic | Prop/function calculi, descriptions, sets | 251 |
| II Prolegomena to cardinal arithmetic | Types applied; preparatory | 322 |
| III Cardinal arithmetic | Finite and transfinite cardinals | 296 |
| IV Relation-arithmetic | Relation-numbers (chapter: "Relation-arithmatic") | 210 |
| V Series | Order, continuity | 490 |
| VI Quantity | Magnitudes | 259 |
Vol. 2 started at Section IIIA, vol. 3 at Section VD. Grattan-Guinness stresses how little of working mathematics this captured: analysis, geometry as practised, and most of applied mathematics stay outside. Logicism's "all mathematics" was always an aspiration measured against Peano's Formulaire and Cantor's set theory, not against the mathematical profession's daily output.
Grattan-Guinness: from Peano morning to three volumes and a missing fourth.
Axioms, gaps, and the construction of set theory
Section VII of the chapter inventories what Principia takes as primitive and what it leaves unresolved. The propositional and predicate calculi plus the logic of relations are the engine; the ramified hierarchy plus reducibility are the price of the paradoxes. Multiplicative axiom (choice) and infinity are assumed where arithmetic needs them. Grattan-Guinness is blunt about set theory inside PM: it is simultaneously constructed and constrained, a "(de)(con)struction". Part VI of POM on dynamics assumed that from a finite number of events one can infer others, an "elementary blunder" G. H. Hardy spotted. The denoting sextet entered because logicism needs single-valued functions in Cauchy's sense, which pointed Russell at definite descriptions.
Reception to Gödel: why logicism stalled
Reception in the 1910s-30s was mixed. Wittgenstein and Ramsey pushed an extensional turn that undercut ramification. Carnap and the positivists took the logic without the metaphysics. Gödel's incompleteness theorems (1931) showed that any consistent recursively axiomatized system strong enough for arithmetic is incomplete, and that consistency proofs have limits. The universality of logic, the dream that logic is the language in which everything can be said, does not survive Gödel intact. Church's type theory and later HOL preserve a typed descendant; Quine's NF and Zermelo-style set theory took other exits from the paradoxes.
Peano morning and the Logic of Relations
Grattan-Guinness's narrative hinge remains 3 August 1900: Peano's use of "the", the Formulaire, and the push to relations. "The Logic of Relations" (Rivista, 1901) and the draft of POM follow.
Cambridge University Press print runs (750 / 500 / 500) limited diffusion. The notation barrier was real. Wittgenstein, Ramsey, Carnap and Gödel read it as philosophers or logicians; analysts and geometers had other foundations (Weierstrassian epsilon-delta, Hilbertian axiomatics, later Bourbaki). Logicism's cultural victory was in philosophy, not in the maths department's daily tools.
Grattan-Guinness: who inherited PM.
Gödel 1931 and Russell's slow uptake
Incompleteness shows no complete recursively axiomatised system captures all arithmetic truth (if consistent), and consistency proofs have limits. Grattan-Guinness records Russell's delayed and partial appreciation. MPD still frames "puzzles" in ways that understate the break. The Companion's Ch1 thus ends logicism's heroic age and opens the meta-mathematical age without claiming Russell led that opening.
Multiplicative axiom, infinity, and what logicism owed
Grattan-Guinness inventories debts inside PM: the multiplicative axiom (choice) and the axiom of infinity are assumed where arithmetic needs them. If they are not logical truths, logicism's epistemic slogan fails even if the formal derivations go through. Reception history shows mathematicians taking choice as a mathematical axiom, not a logical one; that cultural fact helped logicism lose the maths department while keeping the philosophy seminar.
Grattan-Guinness: extra-logical bills.
Whitehead's missing geometry
Volume 4 on geometry, promised to Whitehead alone, never appeared. Geometry's relation to logicism thus stayed programmatic. Relativity and differential geometry moved foundations elsewhere. Grattan-Guinness treats the missing volume as emblematic: logicism's reach exceeded its published grasp.
Reception ecology
Grattan-Guinness: PM's philosophical heirs (Wittgenstein, Ramsey, Carnap, Gödel, Church) outnumber its mathematical adopters. Notation and missing geometry volume limited diffusion. Logicism won the seminar and lost the maths common room. Incompleteness then reset the foundation conversation.
Russell's Philosophical Background
From the Tripos to Bradley
Russell went to Cambridge hoping that mathematics would show what can be known with certainty. The Tripos of 1890-93 was a grind of trick problems; analysis, non-Euclidean geometry and the drive to rigour were ignored, and he graduated with concepts that "might have passed muster in 1793". He switched happily to Moral Sciences, partly to test whether anything like religious belief could be rationally defended.
- Bradley's Logic first. He read The Principles of Logic (1883) at the start of his philosophy year, learning its attack on associationist psychology; Appearance and Reality (1893) only in August 1894, after his conversion.
- Bradley's verdict. Space, time, self, matter and causation are appearance, because relations are incoherent; the Absolute cannot be thought without distortion.
- McTaggart's rescue. A Further Determination of the Absolute (1893) kept relations, embraced pluralism, and promised to determine the Absolute's nature.
- Russell's mix. Like McTaggart a pluralist, like Bradley a rejecter of relations: thought needs complexity, so a plurality of objects; diversity and unity were "proto-relations", not relations.
The dialectic of the sciences
Russell began "in the middle", with the special sciences, which he trusted more than common sense ("the metaphysics of the stone age") or first principles. Following James Ward, he planned to work through geometry, physics, arithmetic and psychology toward a metaphysical synthesis. Each science had an a priori element, found by a two-stroke transcendental argument taken from Kant: analyse the science down to its necessary postulates, and analyse its subject matter down to what makes that kind of experience possible, expecting both to arrive at the same place.
Russell's two-stroke transcendental argument and Griffin's diagnosis of where it breaks.
- Projective geometry. Studies the form of externality needed for presenting many objects at once; purely qualitative, wholly a priori.
- General metrical geometry. Three axioms, free mobility, dimensions and the straight line, required for measurement; a priori in a weaker sense, with actual curvature left to observation (close to zero, never provably zero).
- Not Kant's subjectivism. The form of externality is mind-independent: in Griffin's image, the store's display rack, not the shopper's visual field.
Bradley, the Tripos, and why idealism looked inevitable
Nicholas Griffin (here as chapter author, not only editor) places the young Russell inside late-British idealism. Trinity's Mathematical Tripos (1890-93) gave him technical skill; the Apostles, then dominated by McTaggart, gave him a metaphysics. Bradley's Appearance and Reality (1893) was the book one had to answer. Idealism looked both reassuring and intellectually respectable: it promised a systematic world, reconciled science with a moral order, and gave Cambridge a native alternative to Millian empiricism. Russell's fellowship thesis became An Essay on the Foundations of Geometry (1897), still inside a Kantian-idealist frame: separate the a priori from the a posteriori in each science by transcendental argument.
Griffin's point is not that Russell was a card-carrying Hegelian, but that the method of his 1890s work was dialectical and science-by-science. Each special science generates contradictions when its categories are pressed; the dialectic moves to a more inclusive science. Geometry led to physics; the programme pointed toward a future metaphysics that would resolve the whole. The 1898 reversal did not come from reading Moore alone; it came from the failure of that method to escape psychologism once relations and order were taken seriously.
Griffin: the idealist ladder Russell climbed, then kicked away.
The contradiction of relativity and the 1898 break
Griffin isolates a specific pressure point he calls the contradiction of relativity. Terms and relations that are essentially relative (asymmetrical relations, order, position) cannot be reduced to monadic properties of a single Absolute without residue. Bradley's strategy, to treat relations as internal and ultimately unreal, collided with the mathematics Russell knew. Once asymmetrical relations such as "greater than" are admitted as ultimate, the Absolute's seamless unity cracks. Moore's "The Nature of Judgment" (1899) and Russell's concurrent turn treat propositions as complexes of objective terms, not ideas in a mind. Being belongs to every term that can occur in a proposition; existence is narrower. The bloated ontology of POM is the price of this clarity.
Griffin underlines continuity as well as rupture. The science-first project survives. What dies is the hope that transcendental dialectic can do the foundational work. Psychologism is the enemy both before and after: before, because idealism threatens to make mathematical truth mental; after, because acquaintance and terms must stay non-mental. The 1898-99 break is therefore a change of tools for the same job.
Geometry essay as idealist laboratory
Griffin's Ch2 reads An Essay on the Foundations of Geometry (1897) as the best specimen of Russell's idealist method. Isolate, for a science, what must be a priori for experience of its objects to be possible. For geometry: axioms of quantity, distance, dimension free mobility. The antinomies of the point and of free mobility push the dialectic upward to physics. Variable curvature, later physical in GR, "swept away" the a priori dismissal (MPD 40). A closer look already lists why the transcendental argument cannot prove uniqueness of the postulates.
Griffin: geometry to physics dialectic.
McTaggart and the Apostles
Cambridge Hegelianism via McTaggart made idealism the smart default. Griffin's chapter is the Companion's best account of why leaving it cost Russell a whole toolkit, not only a creed. Absolute realism is the replacement toolkit: terms, external relations, being/existence split.
Contradiction of relativity restated
Griffin's technical pressure: asymmetrical relations and positional facts cannot be dissolved into monadic properties of the Absolute without residue. Bradleyan internal relations make change and order unintelligible to a mathematician who knows what series are. The 1898-99 break is thus forced by geometry and logic of relations, not by mood. Absolute realism's bloated being is the first stable alternative; descriptions will later prune it.
Griffin: relations break the Absolute.
Idealism's reassurance and its cost
Griffin explains why Cambridge idealism looked inevitable: system, moral order, native alternative to Mill. Cost: asymmetrical relations and order become unreal or mental. Russell's geometry essay is the laboratory where the cost becomes unbearable. Variable curvature's later physics vindication (MPD 40) is Griffin's exhibit that a priori dismissal failed. The 1898 break replaces transcendental dialectic with absolute realism's term ontology.
Griffin: reassurance to rupture.
Psychologism as lasting enemy
Griffin: before and after 1898, psychologism threatens mathematical truth. Idealism risks making maths mental; acquaintance and terms must stay objective. The break changes tools for the same anti-psychologistic job. McTaggart's Cambridge made idealism the smart default; leaving it cost a whole method, not only a creed.
The contradiction of relativity
Each science abstracts from the concrete and so, Russell held, ends in contradiction, to be removed by passing to a richer science. The pattern was always the same: a science posits many items it cannot tell apart by intrinsic qualities.
| Science | Antinomy | Attempted exit |
|---|---|---|
| Geometry | Points are all alike yet distinct | Boscovichian point-atoms |
| Metrical geometry | Free mobility of relationally defined figures | Move matter, not figures |
| Kinematics | Point-atoms equally indistinguishable | Forces (dynamics) |
| Dynamics | Force known only by relative motion | Monads (psychology) |
| Arithmetic | Units differ yet are qualitatively alike | Numbers as ratios |
In arithmetic Russell put it with Hegelian panache: between two quantities there is "a conception of difference, but no difference of conception". Every pint is quantitatively alike, Griffin notes, yet each is distinct, "otherwise one would never get drunk". By 1898 this "contradiction of relativity" covered geometry, dynamics and mathematics generally, and he even used it to help define pure mathematics.
The dialectic climbs from abstract to concrete, but the same antinomy reappears at every rung.
The climb stalled at psychology. The act-object psychology of his teachers Ward and Stout (from Brentano) made every mental state relational, so monads could not ground the system. The only item individuated by intrinsic features alone was Bradley's Absolute, a single thing with all properties; but without a conception of difference thought itself fails, and the Absolute stays unknowable.
The 1898 reversal
The contradiction is genuine only given internal relations: every relation must be grounded in intrinsic properties of its terms, so a before b needs different intrinsic properties from b after a. Russell had meanwhile classified relations as reflexive, symmetrical, transitive and found that order, central to pure mathematics, rests on transitive asymmetrical relations, the very ones generating the contradiction.
Moore's critical review of the Essay (1899) and "The Nature of Judgment" (1899) pressed the other weakness: transcendental arguments carry residual psychologism, and several of Russell's were covertly verificationist (force is relative because known only by effects). Russell recalled the new realism as escaping "from a hot-house on to a wind-swept headland"; he rejoiced that grass is really green.
- New task (1899). "The Fundamental Ideas and Axioms of Mathematics": find the indefinable concepts and the primitive propositions; transcendental arguments were "banished, never to return".
- Still pre-logicist. Numbers were simple terms; analysis was decomposition; implication was part-whole containment, a mereological base too weak for mathematics.
- Not simples. Griffin corrects a common reading: Russell never assumed analysis must end in simples.
Russell and Moore, 1898-1905
Truth as a simple property
Moore's 1898 solution, later reported in Some Main Problems of Philosophy (lectures of 1910-11): every belief, true or false, has a proposition as its object, and that proposition "has being"; a true one also has a further simple, unanalysable property, truth. If Herbert believes the earth orbits the sun, he believes something, so there is something believed.
- Against correspondence. Moore (Baldwin's Dictionary): correspondence needs a truth to differ from reality, and no difference can be found once the proposition is not a belief or a sentence; this is the identity theory of truth.
- The cost. Russell admitted it leaves our preference for truth "a mere unaccountable prejudice", so he grounded it in an ethical axiom: it is good to believe true propositions (MTCA, 1904).
- Bivalence. For Moore falsity is absence of truth; for Russell both are unanalysable, so his bivalence has no stated ground.
- No degrees, no change. "More true" means closer to the truth; apparent changes of truth value involve different propositions.
Propositions as complexes
Russell refused to define propositions as possible objects of belief; they are "things having a particular kind of complexity", a kind he confessed (1905) he did not know how to describe. The Principles took grammar as guide, "not our master": every word stands for something, and the proposition contains the entities indicated.
Constituents versus terms in three propositions of Principles (Cartwright's examples).
The scheme strains at once. "Plato admires Socrates" and "Socrates is admired by Plato" have different constituents, so Russell must call them equivalent but not identical. A term of p is a constituent that any other entity can replace; since "x is a term of no proposition" is itself a proposition with x as a term, "term" is "the widest word in the philosophical vocabulary". Terms divide into things and concepts (which show "that curious twofold use"). Whether "Socrates is human" has one term or two, the published Principles wavers.
Being, existence and the bloated ontology
| Category | Has being? | Exists? |
|---|---|---|
| Numbers, relations, classes | Yes (every term) | No |
| Homeric gods, chimeras | Apparently yes (POM 449) | No |
| Greatest prime, null class, class of non-self-members | Denied in POM (75, 102) | No |
| Objects occupying space and time | Yes | Yes |
The famous passage says "A is not" is always false or meaningless, which led Quine to call the ontology "intolerably indiscriminate". Cartwright separates two readings: (7) every result of putting a name or description into "There is no such thing as A" is false, refuted by "the least prime divisible by 6"; and (8) anything that can be mentioned has being, which Russell held all his life and which resists counterexample. The Principles asserts (8) but leaves unclear what terms there are.
Relations and immutable terms
The new philosophy's four doctrines on relations were that they are real, objective, irreducible to adjectives, and (some) external. Russell's targets were the monadistic and monistic reductions of aRb.
Why neither reduction of 'Detroit is north of Windsor' works, per Russell.
Against Kant's "work of the mind" view: if relations were imposed by thought, all relational judgments such as "Caesar crossed the Rubicon" would be false. Terms were immutable and indestructible; change is only difference in truth value between "o has P at T" and "o has P at T′", what Geach later named "Cambridge change". Russell rejected change in the "metaphysical" sense because he rejected the distinction between necessary and contingent truth: "One might as well say that redness might have been a taste." Moore's proposed degrees of necessity collapsed, since every truth is materially implied by every proposition.
The end of the new philosophy
By 1910-11 both men dropped the single-object theory of belief because it required "objective falsehoods"; facts survived, but not as objects of belief. The deeper break came in June 1905 with the theory of descriptions: grammar ceased to be the guide, and logical form no longer had to mirror grammatical form.
Truth as simple, and the anti-idealist pay-off
Richard Cartwright reconstructs the Moore-Russell alliance of 1898-1905. Against the idealist view that truth is coherence or a property of whole systems, they treat truth as a simple property of propositions (or of judgments, in Moore's early wording). A proposition is true or false independently of anyone's mind. That single move undercuts the idealist claim that reality is essentially mental or that truth degrees track systemic inclusion. Cartwright stresses how much of the "new philosophy" is negative: it clears space by rejecting monism, internal relations, and the identity of truth with reality's self-consistency.
Propositions are complexes. Their constituents are the entities the proposition is about (plus relations and forms). For Russell in this period, "Caesar crossed the Rubicon" contains Caesar and the Rubicon. The later Mont Blanc letter to Frege (1904) is continuous with this stance. Analysis is whole-part decomposition: to understand a proposition is to see its constituents and their manner of combination. Cartwright notes the strain already visible in simple cases: the unity of the proposition threatens to become one more constituent, regenerating Bradley's regress.
Cartwright: the 1898-1905 package and where it cracked.
Being, existence, and immutable terms
POM's ontology is deliberately profligate. Being belongs to every term that can occur in a proposition; existence is a narrower, empirical status. Numbers, chimeras, Homeric gods and four-dimensional spaces all have being. Cartwright reads this not as childish Platonism but as the semantic principle that to mention something is to show that it is. Relations are external and irreducible; terms do not change by entering into new relations. That immutability thesis blocks idealist accounts in which relatedness alters the related.
The end of the "new philosophy" comes from within. Denoting concepts, the need to handle generality, and the paradoxes force revisions that 1905-1910 will institutionalize. Cartwright's close is diagnostic: Moore and Russell shared enough to found analytic philosophy's first programme, and disagreed enough (on truth's bearers, on the analysis of judgment) that the alliance was always a coalition, not a school with a manifesto.
Unity of the proposition and Bradley's regress
Cartwright isolates the unity problem. If a proposition is a complex of constituents, what makes it one proposition rather than a list? Adding a "unity" constituent regenerates Bradley's regress. The multiple-relation theory later tries to relocate unity into the act of judgment; that fails in 1913. Cartwright's 1898-1905 focus shows the problem already live before OD.
Cartwright: unity problem.
Being for gods and chimeras
The being/existence split is Cartwright's exhibit for semantic ontology: mentioning suffices for being. Homeric gods have being; they lack existence. After OD, many apparent mentions are not mentions. The "new philosophy" thus both inflates (1899-1904) and then deflates (1905-) the world, under one anti-idealist banner.
Truth as simple property: pay-off and price
Cartwright: if truth is a simple property of objective propositions, coherence theories and degrees of truth lose their grip. Price: explaining falsehood, unity, and empty cases. Falsehood is a real property of real propositions (not mere absence of mental glue). Unity remains unsolved in the 1898-1905 package. Empty cases push toward OD. The alliance with Moore is strongest on the negative campaign against Idealism.
Immutable terms and external relations
Cartwright: terms do not change by entering relations; relations are external. That blocks idealist accounts where relatedness alters the related. Being/existence split plus immutable terms yields the bloated but clear ontology of 1899-1904. OD begins the deflation without returning to Idealism.
Coalition not school
Cartwright: Moore and Russell shared enough to found analytic philosophy's first programme and disagreed enough (truth-bearers, judgment) that the alliance was a coalition. Negative campaign against Idealism was their strongest joint product; positive metaphysics diverged by 1905-10.
Russell and Frege
Who discovered Frege
Frege's three books were the Begriffsschrift (1879), the Grundlagen (1884) and the Grundgesetze (vol. I 1893, vol. II 1903). Russell claimed in A History of Western Philosophy that Frege "remained wholly without recognition until I drew attention to him in 1903", and in IMP that he was the first to read the Begriffsschrift. Beaney corrects both claims.
- Six reviews. The Begriffsschrift drew six reviews in two years, including Schröder's fourteen careful pages; Venn called the notation "cumbrous".
- Peano first. Peano corresponded with Frege from 1894 and reviewed the Grundgesetze in 1895; it was Peano who pointed Russell to Frege.
- Late reading. James Ward gave Russell the Begriffsschrift in 1895 or 1896, but he understood it only after rediscovering its contents; he read the notation-free Grundlagen "some sixteen years" after publication.
- Verdict. Russell's results were independent, but he was the first to present Frege's views as a whole (POM Appendix A) and to see their philosophical weight.
Logic: quantifiers, relations, the ancestral
Frege replaced subject-predicate form with function-argument form ("Fx"), treated relations as functions of two arguments, and added a notation for quantification. "All A's are B" becomes (∀x)(Ax → Bx); "Every natural number has a successor" becomes (∀x)(Nx → (∃y)(Ny & Syx)). Russell before Peano shows no grasp of quantification; even The Principles keeps "any a" as a denoting concept.
Frege's logical analysis of induction (Begriffsschrift Part III), which Russell matched only in 1919.
Beaney ranks Frege's Part III, "a model of clarity, economy, and elegance", above Principles Part IV, where many-one relations appear only in passing and induction gets no logical analysis; Russell reached Frege's economy only in IMP (1919). On relations, however, Russell was more alert to the philosophy: asymmetrical relations like "heavier than" resist reduction, and they ended his idealism.
The analysis of number
Defining "there are n F's" leaves the Julius Caesar problem: nothing says which objects numbers are. Frege's move, via the context principle ("only in the context of a proposition do words mean something"): "the number of F's equals the number of G's" is equivalent to "F and G are equinumerous", which needs only a one-one relation; the equivalence is now called Hume's Principle. Since that is definition by abstraction, open to the objection Russell also made ("it does not show that only one object satisfies the definition"), both men chose an explicit definition: the number of F's is the extension of the concept "equinumerous to F" (Russell: the class of classes similar to the class of F's).
| Issue | Frege | Russell |
|---|---|---|
| Quantification | Notation from 1879 | Learned from Peano, 1900 |
| Number | Extension of a concept | Class of similar classes |
| 0 and 1 | Concept "not self-identical" | POM denies the null class; corrected after reading Frege |
| Paradox response | Restrict Axiom V | Types; classes as logical fictions |
| Empty descriptions | Description stroke supplies a referent | Analysed away (1905) |
| Analysis | Function-argument | Whole-part decomposition |
Axiom V and the letter of 16 June 1902
The Grundgesetze legitimised extensions by Axiom V: concepts F and G apply to the same objects if and only if their extensions are identical. Frege admitted in the preface that only this law might provoke dispute. Russell's letter of 16 June 1902, sent while volume II was in press, reported the contradiction; Frege answered six days later, "thunderstruck", that the collapse of law V seemed to undermine "the only possible foundations of arithmetic".
How Axiom V generates the contradiction in Frege's own terms.
- Improper objects. Rejected: nine kinds of first-level function, an "incalculable multiplicity of types", exactly what Russell then built.
- Sham names. Rejected: if a proposition has a Bedeutung, its parts must too.
- Frege's way out. Exclude a concept's own extension from (Ca); this too yields contradiction in domains of more than one object (Quine 1955, Geach 1956), and it presupposes identification of extensions.
Three modes of analysis
Russell championed analysis as philosophy's method all his life. Beaney distinguishes three modes that combine in any actual project.
Beaney's three modes of analysis, with the forms each took in Frege and Russell.
Paraphrastic analysis (rephrasal, no metaphysical commitment, after Bentham's paraphrasis) differs from reductive analysis (uncovering ultimate constituents); the 1930s Cambridge School called them "same-level" and "new-level" analysis. Accepting the first while rejecting the second defined the second generation: later Wittgenstein, Carnap, Ryle, Quine. Frege's concept horse paradox shows the cost of missing the distinction: read decompositionally, "the concept horse is a concept" is false; read paraphrastically, with Dummett, it is (∀x)(Hx ∨ ¬Hx).
- Have cake and eat it. Frege derives "the number of F's = the number of G's" from equinumerosity by paraphrase but then reads it decompositionally as naming objects; "rabbits can only be pulled out of hats if they are already there".
- Converse relations. For Frege "hydrogen is lighter than carbon dioxide" and its converse share one content; for Russell they are different propositions, because complexes are literally made of their parts (POM 228).
- Unity. Frege's answer is unsaturated concepts, a metaphor that fits sense better than Bedeutung; Russell's is form binding constituents, never quite repudiating "analysis is falsification".
- King of France. Russell analyses the description away; Frege's description stroke maps an improper extension onto itself, so the sentence is about the null set and false.
Frege's three essays and the road to extensions
Between the Grundlagen (1884) and the Grundgesetze (1893), Frege published three papers that Beaney treats as the philosophical ground for Axiom V: "Function and Concept" (1891), "On Sense and Reference" (1892), and "On Concept and Object" (1892). The Grundlagen had hoped to do without extensions; Frege soon decided he could not. Burge (1984) tracks the shift. Kerry's criticism of the Grundlagen (two articles) prompted the concept/object reply; Husserl's Philosophie der Arithmetik drew Frege's 1894 review. So Frege was not invisible in German philosophy; he was under-read in England until Russell.
Beaney presses a further correction to Russell's self-image. In IMP Russell says Frege first developed the ancestral in 1879 and that the work "remained unnoticed until Whitehead and I developed it". Schröder's fourteen-page review of the Begriffsschrift already spotted Frege's slip on the ancestral of identity; Venn dismissed the notation as "cumbrous". Peano's correspondence with Frege from 1894 covered generality, conditional definition and identity; Peano reviewed Grundgesetze I in 1895 and Frege replied in the Rivista in 1896. Beaney's verdict stands: Russell's results were independent, but priority for "drawing attention" goes first to Peano on the Continent and only then to Russell for the English-speaking world.
Beaney's corrected reception story: Frege was noticed in Europe before Russell's 1903 Appendix A.
Number: 0, 1, and the unit-class argument
Beaney dwells on Frege's insistence that any adequate theory of number must handle 0 and 1. Alternative theories fail §§7 and 28-39 of the Grundlagen precisely here. Frege's definitions: 0 is the number of the concept "not identical with itself"; 1 is the number of the concept "identical with 0". Russell in POM initially denied the null class, then, after reading Frege, accepted both 0 and the empty class and kept "0, the empty class and nothing" carefully distinct (POM 75; Grattan-Guinness's ch. 1).
The unit-class argument, which both men use against identifying a class with its sole member, Beaney quotes from Russell (POM 517; PM i 340): let u have many terms and ιu be the class whose only member is u; then ιu has one member and u has many, so they are not identical. Once that is granted, the null class can be admitted without collapse. Frege's parallel argument appears in the 1984 collected papers (pp. 218-19). The shared move matters for logicism: without a clean 0 and a clean unit class, the recursive construction of the natural numbers cannot start.
| Step | Frege (Grundlagen / Grundgesetze) | Russell (POM / PM) |
|---|---|---|
| 0 | Number of "not self-identical" | Class of the empty class (after Frege) |
| 1 | Number of "identical with 0" | Class of unit classes |
| Successor | Ancestral of the predecessor relation | Ancestral, full economy only in IMP 1919 |
| Hume's Principle | Equinumerosity iff same number | Same content via similarity of classes |
| Explicit definition | Extension of "equinumerous to F" | Class of classes similar to the F's |
| Julius Caesar | Blocked by context principle plus HP | Blocked by explicit class definition |
Improper objects, sham names, and Frege's failed patch
Beaney reconstructs Frege's three candidate repairs after the 16 June 1902 letter. Improper objects: treat some extensions as not really objects. Frege rejected this because it would multiply kinds of first-level function into an "incalculable multiplicity of types", which is exactly the hierarchy Russell then built. Sham names: declare some names empty. Frege rejected this because if a proposition has a Bedeutung its parts must too. Frege's own patch: exclude a concept's own extension from the scope of (Ca). Quine (1955) and Geach (1956) showed that this still yields contradiction in any domain with more than one object, and the patch presupposes that extensions have already been identified.
Frege's letter of 28 July 1902 notes that Russell uses the phrase "extension of a concept" in a way Frege himself does not. By Volume II's appendix Frege was "thunderstruck"; his late diary records that his efforts on number "resulted in failure". Beaney's diagnosis, already in the chapter's Key idea, is that definition by abstraction constructs objects outside the original domain, while Frege kept one universal homogeneous domain. Russell's insight that there are "more types of things than were dreamt of in Frege's philosophy" is, on Beaney's reading, the right diagnosis even when Russell's own type theory is later simplified by Ramsey.
Regressive, resolutive, interpretive: worked examples
Beaney's three modes are not slogans. Regressive analysis works back to first principles: Frege's definitions beat Peano's axioms as starting-points (Russell, "The Regressive Method of Discovering the Premisses of Mathematics", 1907); Peano's postulates characterize the natural numbers, Frege's definitions construct them from logic. Resolutive analysis splits into whole-part (Russell's early preference) and function-argument (Frege's). For Russell at POM 228, "hydrogen is lighter than carbon dioxide" and its converse are different propositions because complexes are literally made of their parts; for Frege they share one content. Interpretive (paraphrastic) analysis translates into a framework without metaphysical commitment: Bentham's paraphrasis, Russell's "the King of France", Frege's treatment of existence as a second-level property.
The 1930s Cambridge distinction between "same-level" and "new-level" analysis maps onto Beaney's paraphrastic versus reductive. Accepting the first while rejecting the second defined the second generation: later Wittgenstein, Carnap, Ryle, Quine. Frege's concept horse paradox shows the cost of missing the distinction. Read decompositionally, "the concept horse is a concept" is false (the phrase names an object). Read paraphrastically, with Dummett, it becomes (∀x)(Hx ∨ ¬Hx), which is true. Beaney's charge: Frege derives "the number of F's = the number of G's" from equinumerosity by paraphrase, then reads the result decompositionally as naming objects. "Rabbits can only be pulled out of hats if they are already there."
Beaney's three modes with the Frege/Russell contrast on each.
Frege's sense and Russell's refusal: the 1904 letters
Beaney reconstructs the Frege-Russell exchange that crystallises the disagreement. Frege's sense/reference distinction would let "the present King of France" have sense without reference; Russell refuses, because sense seems to him either a denoting concept (already under suspicion) or a psychological intermediary. The Mont Blanc letter (Russell to Frege, 12 December 1904) insists that Mont Blanc itself, with all its snowfields, is a constituent of the thought that Mont Blanc is more than 4000 metres high. Beaney's diagnostic: Russell treats acquaintance as the only honest route from mind to object; anything like Fregean sense threatens a veil.
The Gray's Elegy argument in "On Denoting" is meant to show that denoting concepts cannot be spoken of without collapsing into what they denote. Beaney walks the three cases C, C', C''. Critics (including later Cartwright and others) find the argument compressed to opacity. Beaney's charitable reading: Russell is trying to show that any theory which makes the denoting complex both a constituent and a vehicle of aboutness generates either circularity or an inability to mention the complex. Whether the argument succeeds, it motivates the 1905 replacement of denoting complexes by quantificational paraphrase.
Beaney: Frege vs Russell on empty descriptions.
Incomplete symbols beyond "the": classes, numbers, ordinary names
Beaney's wider claim is that "On Denoting" is the template for later eliminations. Once definite descriptions are incomplete symbols, the same pattern applies to class-expressions (no single entity "the class of Fs"), to number-words (the number of Fs is the class of classes similar to F, itself contextually defined), and eventually to ordinary proper names of continua. Principia's *14 is the official home of the description theory inside the logicist project. Beaney stresses continuity with the philosophy of logical atomism: names that survive analysis pick out simples; everything else is incomplete.
Critics of Beaney's story often say he overplays unity across 1903-1918. He replies that the method (paraphrase as ontological criterion) is stable even when the ontology of propositions and facts changes. The chapter's payoff for Companion readers: descriptions are not a puzzle about kings of France; they are Russell's master tool for thinning ontology without thinning science.
Function, concept, object: Kerry and the three essays
Beaney's Frege needs the 1891-92 triad. Function and Concept treats concepts as functions to truth-values; extensions become value-ranges. On Sense and Reference splits cognitive significance from Bedeutung; empty names can have sense. On Concept and Object answers Kerry: "the concept horse" is not a concept, because the definite article forces object-position. Russell never accepts this Fregean bifurcation cleanly; POM terms blur concept and object. Beaney shows why the disagreement about descriptions is downstream of this metaphysics of functions.
Beaney: Frege's 1891-92 machine.
Hume's Principle and neo-logicism after Beaney
Beaney's chapter ends before the full neo-logicist boom, but Insight's A closer look updates: Wright (1983) and Hale and Wright (2001) show that second-order logic plus Hume's Principle (the number of Fs equals the number of Gs iff F and G are equinumerous) yields arithmetic (Frege's theorem) without Axiom V. Frege's analysis of number can be separated from the inconsistent Basic Law V. Russell's paradox still kills Grundgesetze as written; it does not kill every Fregean insight about number. Companion readers need that Oct-updated frame.
Denoting concepts before the 1905 break
Beaney tracks Russell's incomplete break with subject-predicate grammar in POM. Even after adopting a Fregean-style reading of "Any a is a b" as formal implication "x is an a implies x is a b" (POM ch. 7), Russell denies synonymy: "any a" is a denoting concept denoting only a's, whereas in the formal implication x need not be an a (POM 91). Denoting concepts (POM ch. 5) survive until "On Denoting" replaces them. Beaney's moral: Frege's quantificational insights were available on the page, but Russell's ontology of denoting delayed their full use until 1905.
Russell acknowledges Peano (POM 10 fn., 26, xxiii) yet still underestimates Frege's twenty-year head start. Griffin (1991, ch. 7) shows how rudimentary Russell's 1898 quantification was. Beaney concurs that Appendix A of POM is the first full English account, not the first European notice.
Beaney: denoting concepts to descriptions.
AMR and Papers 2
Beaney notes the abandoned early book (fragments in Papers 2) and An Analysis of Mathematical Reasoning. Problems there (Griffin 1991) show Russell struggling toward the 1903 apparatus. The Companion reader should not treat POM as a sudden miracle after Peano; it is the first stable platform after years of failed drafts.
What Beaney needs from Frege's path to Axiom V
Beyond the three essays, Beaney stresses why Frege thought value-ranges indispensable. The Grundlagen hoped to avoid extensions; Kerry's critiques and the needs of a formal arithmetic pushed Frege to Basic Law V: the value-range of f equals that of g iff ∀x(fx ↔ gx). That law delivers extensions and, with it, the paradox Russell sends in June 1902. Beaney's narrative: Frege's analysis of number (0, 1, ancestral, HP-like thoughts) can be admired apart from V; Russell's Appendix A taught England the whole package, paradox included.
Beaney: Axiom V as the load-bearing wall.
Schröder, Venn, Peano: Frege was not invisible
Russell's independence of results stands; his claim to have first drawn attention does not.
Paradox of analysis as Beaney's closer
Beaney ends with the paradox of analysis: stay close to ordinary understanding and risk triviality; build a precise new account and risk error or alienation. Frege's paraphrases must not smuggle new commitments. Insight's earlier "Apply it" box on canonical data models belongs here philosophically: regulatory "customer" definitions face the same trap. Beaney's Frege-Russell story is thus also a story about what good analysis costs.
Beaney: paradox of analysis.
Frege's theorem and Oct 2026 frame
Wright (1983), Hale and Wright (2001): second-order logic + Hume's Principle yields arithmetic (Frege's theorem) consistently, separating Frege's analysis of number from inconsistent Basic Law V. Beaney's 2003 chapter pre-dates some of this boom's textbook status; Insight's A closer look supplies the update without rewriting Beaney as a neo-logicist.
Appendix A as English Frege
Beaney: POM Appendix A is the first full English exposition of Frege's project, paradox included. It made Frege impossible to ignore in Britain. Peano had already noticed Frege on the Continent; Russell catalysed the analytic mainstream. Neo-logicism later salvaged Frege's number analysis without Axiom V; Beaney's paradox-of-analysis closer still governs what counts as good paraphrase.
Bertrand Russell's Logicism
What logicism claims
Logicism has two goals: mathematical concepts defined from logical ones, and mathematical theorems deduced from logical axioms. Russell took the universal version: "all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms" (MPD 74). Numerical claims become logic: "there are exactly two P's" is (∃x)(∃y)(x ≠ y & (∀z)(Pz ≡ (z = x ∨ z = y))), and 2 + 3 = 5 becomes a second-order truth about disjoint concepts. The payoff would be certainty and no mysterious objects; but if mathematics reduces only to second-order logic, and that is set theory in disguise, the gain is unclear.
| Logicist | Motive | Fate |
|---|---|---|
| Leibniz (1666, 1679, 1714) | All truths reduce to identities; "Let us calculate" | Syllogistic logic too weak |
| Frege (1879-1903) | A priori certainty of arithmetic | Axiom V inconsistent; turned to geometry |
| Dedekind (1888) | Number as "free creation" from laws of thought | Frege: "system" not reduced to logic |
| Russell (1901-1913) | All pure mathematics; later economy | Infinity and reducibility axioms disputed |
Forerunners: Leibniz, Frege, Dedekind
Russell's Leibniz (1900) made Leibniz the first logicist: every truth is subject-predicate and reduces to primary identities by analysis, finite for necessary truths and infinite (for God alone) for contingent ones. By 1679 Leibniz doubted humans could reach the primitive notions, though in 1714 he still hoped for a calculus of truths of reason. Frege's motive was epistemic, with laws of logic as "boundary stones set in an eternal foundation", and anti-psychologistic. Dedekind also called arithmetic part of logic but made numbers "a free creation of the human mind" by abstraction; the authors argue from his letters to Keferstein (1890) and Weber (1888) that he meant necessary laws of thought, not subjective ones.
The paradox and the new logicism
Russell announced logicism in print in 1901 and finished the Principles draft on "the last day of the century". In May 1901 he found the class of classes not members of themselves; Burali-Forti (1897) had a cousin. Frege eventually abandoned logicism; Russell built types.
- Simple types. Individuals, classes of individuals, classes of classes; "xₙ ∈ xₙ" is ill-formed.
- Ramified types. A hierarchy of propositions and propositional functions too, justified by the vicious circle principle: "whatever involves all of a collection must not be one of the collection" (after Poincaré).
- Detours. In 1905 the zigzag, limitation of size and no-classes theories; back to types in 1908.
- Principia (1910-13). Derived set theory, finite and transfinite arithmetic and elementary measure theory, but the axiom of infinity looks empirical and the axiom of reducibility looks ad hoc.
Ontological logicism: fictions of fictions
The Principles ontology was "unfettered realism": every term has being (POM 449), from "the belief that, if a word means something, there must be some thing that it means". On Denoting (1905) changed this, but the authors stress that a description only displays commitment: a false description commits to nothing, a true one keeps its commitment though no name occurs.
Russell's ontological reduction of number, by Occam's razor (RSDP 1914, PLA 1918).
The motives were realism ("logic must no more admit a unicorn than zoology can") and the maxim "wherever possible, logical constructions are to be substituted for inferred entities". Classes cannot be things (type theory; Cantor: more classes than individuals) nor any one propositional function ("featherless biped" and "rational animal" determine one class). The authors read "construction" as agnostic rather than eliminative: "we merely abstain from both assumptions". But the axiom of infinity is an existential claim that seems to commit him to infinite classes; Russell's reply, to state it as a hypothesis wherever used, sits uneasily with mathematical necessity, as Quine pressed.
Epistemic logicism: the regressive method
The received view says logic's self-evidence transfers to mathematics, "innocence by association" (Haack). Russell rejected it: proving 2 + 2 = 4 from recondite logic has an "apparent absurdity", since "the conclusion is more certain than the premises" (RMDP, 1907).
The regressive method: logical order and epistemic order run in opposite directions (Irvine 1989).
- Two kinds of premise. An empirical premise (better: "epistemological") is what actually leads us to believe; a logical premise is logically simpler. Proofs have direct epistemic value only where they coincide.
- Induction upward. "We tend to believe the premises because we can see that their consequences are true"; the method is "substantially the same" as discovering laws in any science.
- Consistently held. PM (1910): "self-evidence is never more than a part of the reason for accepting an axiom"; the 1925 introduction's "pragmatic justification" of reducibility; "Logical Atomism" (1924): Maxwell's equations are believed for their consequences, and so are logic's first principles.
Advent, paradox, and the two faces of logicism
Martin Godwyn and A. D. Irvine define logicism as the claim that mathematics is logic in disguise: mathematical truths are logical truths, mathematical objects are logical constructions. Forerunners include Leibniz's dream of a characteristica universalis, Frege's Grundgesetze, and Dedekind's essay on the nature of numbers. Russell's version, after the paradox, is no longer naive comprehension. The "new logicism" of Principia keeps the reductionist aim while fencing the paradoxes with types and (reluctantly) reducibility, infinity and choice.
They split the doctrine into two faces that later debate often confuses. Ontological logicism asks what entities mathematics commits us to. Epistemic logicism asks how mathematical knowledge is justified. Russell's early POM ontology is profligate (being for every term); after descriptions, ontological commitment tracks the bound variables of the analysed paraphrase, anticipating Quine's criterion. Classes become "fictions of fictions" once propositional functions and contextual definition do the work. Epistemic logicism runs through the regressive method (1907): we justify axioms by their capacity to imply the accepted body of mathematics, as Frege's definitions beat Peano's postulates as starting-points.
Godwyn and Irvine: ontological vs epistemic logicism.
Ontological thinning: from terms to incomplete symbols
Godwyn and Irvine track the thinning in detail. POM: "Numbers, the Homeric gods, relations, chimeras and four-dimensional spaces all have being" (p. 449), because "if a word means something, there must be some thing that it means" (later recollection). "On Denoting" breaks the face-value link: "the present King of France" makes no ineliminable reference. Number-words, then class-expressions, then ordinary names of continua become incomplete symbols. Ontological logicism survives as the claim that the paraphrases use only logical constants and variables, not as a museum of abstract objects.
Epistemic logicism and the price of the axioms
The regressive method is Russell's answer to the charge that logicism merely shifts mystery to axioms. We do not grasp the PM primitives as self-evident; we accept them because they organise and imply the mathematics we already trust. Godwyn and Irvine note the cost: reducibility looks combinatorial, not logical; the axioms of infinity and choice look empirical or pragmatic. If those axioms are not logical truths, epistemic logicism fails even if the ontological reductions succeed. The chapter leaves logicism as a research programme with partial success: much of arithmetic and analysis can be reconstructed, but the reconstruction's foundations are less "logical" than the slogan promised.
Regressive method in detail: Frege vs Peano as starting-points
Godwyn and Irvine dwell on Russell's 1907 paper "The Regressive Method of Discovering the Premises of Mathematics". We do not begin from self-evident logical axioms and march to theorems. We begin from the body of mathematics already accepted (arithmetic, analysis) and regress to premises that imply it with maximum certainty, clarity and economy. Frege's definitions of number beat Peano's postulates as starting-points because they reveal what numbers are (extensions of concepts / classes of classes) rather than merely how they behave. The epistemic gain is explanatory, not just deductive.
This method makes logicism's success partial and revisable. If reducibility, infinity or choice must be assumed as extra-logical, the regressive test says: keep them if they uniquely organise the mathematics; admit they are not purely logical; adjust the slogan. Godwyn and Irvine refuse both the dismissive "logicism failed" headline and the apologetic "all axioms are logical by stipulation" move.
Godwyn and Irvine: regressive justification loop.
If-thenism, conditional logicism, and Quine's shadow
A softer reading, sometimes called if-thenism, says mathematics asserts only that if the axioms hold then the theorems do. Russell flirts with related ideas but wants more: the axioms should be true as logic. Quine's later criterion of ontological commitment (to be is to be the value of a variable) is anticipated in spirit by the descriptions-driven thinning Godwyn and Irvine describe, though Quine rejects the analytic/synthetic frame Russell needs for "logical truth". The chapter thus sits at the hinge between early analytic logicism and mid-century ontology debates.
Classes as fictions and the no-class theory
In PM, class expressions are incomplete symbols. Talk of "the class of Fs" paraphrases into talk of propositional functions. Ontologically, there need be no class entities. This is stronger than set-theoretic foundation: it is eliminativist about classes while retaining their mathematics. Godwyn and Irvine note the cost in intelligibility: working mathematicians continue to think in sets; the no-class theory is a philosophers' paraphrase. Still, as ontological logicism, it is the cleanest expression of Russell's razor inside the logicist project.
| Claim | Early POM | After OD / PM |
|---|---|---|
| Numbers | Terms with being | Classes of classes (incomplete) |
| Classes | Objects | Fictions via functions |
| Commitment test | Word means a thing | Bound variables in paraphrase |
| Epistemic story | Self-evidence hoped | Regressive method |
Numbers as fictions of fictions
Godwyn and Irvine quote the full thinning chain. Cardinal number, if merely inferred from collections, stays doubtful without a metaphysical postulate. Defined as the class of all equally numerous collections, the postulate drops (RSDP; OKEW 205). But classes themselves, under the no-class theory, are logical fictions: symbols for classes are conveniences, incomplete symbols like descriptions (IMP 182; PM *20.01). Numbers are classes of classes, hence "fictions at two removes" (PLA). Ultimate constituents of the world are only what undefined primitive symbols of a logically perfect language would stand for: individuals (particulars) and properties and relations.
Classes cannot be such individuals. Reasons: (1) type theory: nothing significant about a thing is significant about a class of things, so classes lack the same kind of reality; (2) Cantor's theorem: more classes than individuals; (3) extensional identification with members fails for the null class and for unit classes vs their members. Propositional functions determine classes (within types), but functions are not simply classes under another name. Ontological logicism thus ends in a sparse base plus contextual definitions.
Godwyn and Irvine: two removes of fiction.
Epistemically uncertain inferred entities
The razor is epistemological as much as ontological. Russell wants to avoid commitment to "logically dispensable and epistemically uncertain inferred entities." Within logic and pure mathematics, defining number as equivalence classes removes needless doubt. The same spirit drives constructions of points and matter in the 1914-1927 work. Godwyn and Irvine tie this to logicism's epistemic face: justification improves when inferred fog becomes constructed clarity.
What would refute ontological logicism?
If the paraphrases still require bound variables ranging over numbers or classes as entities, thinning fails. If type restrictions make the paraphrase distort mathematical practice beyond recognition, the cost is too high. If reducibility is indispensable and non-logical, epistemic logicism fails even when ontological thinning succeeds. The chapter leaves all three as live pressure points rather than settled victories.
| Level | Surface talk | Russellian status |
|---|---|---|
| 0 | Individuals, properties | Ultimate constituents |
| 1 | Propositional functions | Basic, typed |
| 2 | Classes | Incomplete symbols |
| 3 | Numbers | Classes of classes = fictions of fictions |
| 4 | Ordinary maths sentences | OK via paraphrase |
If numbers are fictions, what are we doing when we count?
Godwyn and Irvine face the obvious worry. Saying numbers are fictions of fictions sounds like mathematical nihilism. Russell's reply: any indubitable set of objects fulfilling arithmetic's formulae may be called numbers; the equivalence-class definition is the simplest known (OKEW 205). Practice is preserved; ontology is thinned. The regressive method then asks whether the axioms that make the reconstruction work are themselves logical. Ontological success plus epistemic shortfall is still a partial victory for logicism as research programme.
Godwyn and Irvine: preserve practice, thin ontology.
Quinean aftermath without Quine's slogans
Bound variables in the analysed language fix commitment. That is not Quine's later "to be is to be the value of a variable" word for word, but Godwyn and Irvine show Russell already using paraphrase as an ontological court. Differences remain: Russell wants logical truth and acquaintance; Quine historicises and rejects the analytic. The Companion chapter is a bridge text for courses that teach both.
Partial success as the honest verdict
Godwyn and Irvine: much of arithmetic and analysis can be reconstructed; foundations are less "logical" than the slogan promised once reducibility, infinity and choice are billed. Ontological thinning can succeed while epistemic logicism fails. Logicism remains a research programme, not a completed reduction.
Godwyn and Irvine: scorecard.
The Theory of Descriptions
The analysis itself
"On Denoting" (Mind, 1905), which Ramsey called "a paradigm of philosophy", treats "the F is G" as three claims: something is F, nothing else is F, and it is G.
Russell's analysis of 'The even prime number is less than ten', with the inference it makes transparent.
Two neutral advantages follow. Descriptions have semantic structure ("even" means something in "the even prime"; "is" means nothing in "Aristotle"), and the analysis makes inferences from that structure ordinary first-order logic. And removing descriptions from the singular terms protects classical logic: (∀x)Fx ⊢ Fa fails if "a" is empty, which otherwise forces free logic. Treating names as disguised descriptions extends the protection, though a sense/reference distinction would also explain empty names, and that distinction is exactly what Russell rejected.
Direct realism and the Mont Blanc letter
Against Idealism's mediated knowledge, Moore and Russell posited acquaintance, "direct, immediate, and wholly presuppositionless", with propositions that are non-mental and paradigmatically contain the objects they are about. Russell to Frege, 12 December 1904: "in spite of all its snowfields Mont Blanc itself is a component part of what is actually asserted". Hylton's reading: the danger was not false beliefs about Mont Blanc but beliefs not about it at all.
- The Meinongian argument. If "A is not" makes sense, A must be among its constituents, so A subsists (POM §427): Homeric gods, chimeras, Vulcan.
- Strains of scope. Acquaintance with Socrates or numbers is implausible; a proposition containing all the natural numbers Russell never accepted (POM §72).
- Generality. The Preface says he came to symbolic logic "with a view to discovering the meaning of the word any"; a "correct philosophy of any" was to dissolve the antinomies of infinity (POM §179).
Denoting concepts, 1900-1905
From 1900 or 1901 to June 1905, Russell let some propositions contain a denoting concept that is about something else: "all men" (collectively), "every man" (severally), "any man" (an arbitrary man). Hylton argues that this theory already blocked the Meinongian argument: "A concept may denote although it does not denote anything" (POM §73), and "The Existential Import of Propositions" (written spring 1905, published July) says "the present king of France" denotes "no individual at all", real or imaginary.
Indirect aboutness under the two theories (Hylton, Section V).
Incomplete symbols and the systematically misleading
After 1905 a description is an incomplete symbol, "not supposed to have any meaning in isolation, but only defined in certain contexts" (PM i 66). The unit of analysis moves from the word to the sentence, and Russell abandons "grammar as guide" (POM §46): surface form and logical form diverge, an idea Wittgenstein credited to him (Tractatus 4.0031). Contextual definition then defines classes. Acquaintance turns from a lax deflector of worries into a binding constraint: any name whose bearer's existence is open to doubt is a disguised description, so ordinary names of tables and Mars dissolve into descriptions built from sense-data, while acquaintance with abstract objects stays wide.
Objections since 1950
| Objection | Claim | Russellian reply |
|---|---|---|
| Strawson (1950) | "The F is G" presupposes a unique F; no truth value if none | Grice: reluctance to say "false" is pragmatic |
| Incompleteness | "The table is covered with books" | Context restricts the domain, as with "no more wine" |
| Donnellan (1966) | Referential use: the "martini" drinker drinks water | What is said is false; what is meant is true |
| Kripke: modality | Names are rigid; "Fleming might have died young" | Scope: description takes widest scope |
| Kripke: ignorance | "Feynman" used with no identifying description | "The man called 'Feynman'" (Russell on Caesar) |
| Kripke: Gödel | The name tracks the man, not the prover | Laypeople defer to experts' descriptions |
Scope disambiguates 'The inventor of penicillin might have died in childhood'.
Kripke denies that rigidity is only scope, citing simple sentences evaluated at counterfactual circumstances. Hylton closes sceptically: any logic forces a definite claim on casual talk that is not that definite.
Four real reasons Russell dropped denoting concepts
Hylton's revision of the standard story is the chapter's spine. Russell in MPD (1959) said Meinong convinced him until descriptions; Hylton calls this "quite mistaken". The 1903-05 denoting-concept theory already let a concept denote nothing (POM §73); "The Existential Import of Propositions" (written spring 1905, published July) says "the present king of France" denotes "no individual at all". Avoiding golden mountains was at most a criterion of adequacy both theories met. Hylton's four positive reasons:
- Fit with logic. Descriptions become ordinary first-order claims; no truth-value gaps; inferences that look obvious become plain quantification.
- Internal troubles. What denotes what under the sextet (all, every, any, a, some, the) multiplies without end; the "Gray's Elegy" passage of "On Denoting" threatens an infinite hierarchy of denoting concepts.
- Failure on its own terms. "Any term" cannot underwrite two distinct variables (POM §93); the theory cannot do the work generality requires.
- Metaphysical anomaly. A denoting concept is a representational element inside a direct-realist metaphysics that otherwise insists propositions contain the objects they are about. There is "no story about how this is possible".
Contextual definition, classes, and the binding of acquaintance
Once descriptions are incomplete symbols, contextual definition becomes Russell's master tool. Classes are defined in context by propositional functions (Principia *20); the same pattern later swallows ordinary names of tables and planets. Acquaintance, which had been a lax deflector of worries about Socrates and numbers, hardens into a binding constraint: any name whose bearer's existence can be doubted is a disguised description. Ordinary proper names dissolve into descriptions built from sense-data; acquaintance with universals and abstract objects stays wide. Hylton stresses that this is a change in the role of acquaintance, not a discovery of new objects of acquaintance.
The systematically misleading character of surface form is the bridge to later analytic philosophy. Quine's use of the theory, Strawson's attack, and Donnellan's referential/attributive split all inherit the 1905 idea that what a sentence seems to be about need not be what it is about. Hylton's caution: the theory looked for forty-five years before Strawson (1950) "practically immune from criticism" (Linsky, 1967), which is itself a historical claim about a canon, not a proof of truth.
Hylton: what changes when denoting concepts give way to descriptions.
Strawson, Donnellan, Kripke: Hylton's replies in full
Strawson (1950): "The F is G" presupposes a unique F; if none, the speech act misfires and there is no truth-value. Hylton's Russellian line, via Grice: reluctance to call the sentence false is pragmatic, not semantic. The logical content is still the three-clause quantificational claim, which is false when there is no F.
Incompleteness. "The table is covered with books" seems to need a completing predicate. Reply: context restricts the domain, as "there is no more wine" restricts to the room or the bottle. The description is incomplete only relative to an unrestricted domain that conversation never uses.
Donnellan (1966): in the referential use, "the man drinking a martini" can pick out the water-drinker. Hylton's Russellian reply: what is said is false (no unique martini-drinker fits), what is meant may be true. The theory analyses the proposition expressed, not every speaker intention.
Kripke on modality. Names are rigid; "Fleming might have died young" is not about whoever invented penicillin. Scope reply: give the description widest scope and the truth-conditions match. Kripke denies that rigidity reduces to scope, citing simple sentences evaluated at counterfactual circumstances. Hylton leaves the dispute open: any formal logic forces a definite claim onto casual talk that is not that definite.
Kripke on ignorance and Gödel. Speakers use "Feynman" with no identifying description; if Gödel stole the incompleteness proof, "Gödel" still tracks the man. Russellian replies Hylton canvasses: "the man called 'Feynman'" (Russell on Caesar), and lay deferral to experts' descriptions. Neither reply satisfies Kripke; Hylton's close is sceptical about whether the dispute can be settled by further cases.
| Challenge | Year | Core claim | Hylton's Russellian move |
|---|---|---|---|
| Strawson | 1950 | Presupposition, truth-value gap | Pragmatic reluctance; semantics stays false |
| Incompleteness | ongoing | Underspecified descriptions | Contextual domain restriction |
| Donnellan | 1966 | Referential vs attributive | Said vs meant |
| Kripke modality | 1972/80 | Rigidity of names | Wide scope; Kripke rejects reduction |
| Kripke ignorance | 1972/80 | Use without description | The man called N; expert deferral |
| Kripke Gödel | 1972/80 | Name tracks the man | Same deferral strategy |
Why substitution threatened universality
Hylton's deepest objection to types, on Russell's behalf, is that logic's truths must remain true under any substitution of terms. If variables are typed, substitution is restricted, and logic loses its universality. The substitutional theory of 1905-06 tried to keep an untyped language by analysing "φa" as a substitution of a for b in a proposition p. Paradoxes of propositions (the p₀ / p₀{p₀/a} style contradictions) forced retreat. Hylton reads the return to propositional functions not as a conversion to Fregean functions but as a lesser evil: better a typed hierarchy than an inconsistent untyped calculus.
Hylton: substitution's rise and fall.
Multiple-relation theory as ontology of belief
Hylton places the multiple-relation theory (1910-13) inside the same thinning project. If propositions are not entities, belief cannot be a two-place relation to a proposition. "Othello believes that Desdemona loves Cassio" becomes Believes(Othello, Desdemona, loves, Cassio), a polyadic fact. Truth of the belief is correspondence with a complex when the objects are so related; falsehood is the absence of that complex. Wittgenstein's objection (summer 1913): the theory cannot guarantee that the relatedness of the objects has the right form; nonsense combinations are not blocked. Russell shelved Theory of Knowledge. Hylton's verdict: the multiple-relation theory was the high point of the "no propositions" programme and also its breaking point.
From acquaintance epistemology to naturalism
Hylton's closing arc: the principle of acquaintance (every constituent of a proposition one understands must be an object of acquaintance) makes general arithmetic and talk of matter problematic. Sensibilia and constructions are stopgaps. After 1919, belief becomes a natural relation among word-propositions, image-propositions and facts; epistemology becomes continuous with psychology. Hylton does not celebrate this as progress without cost: what is gained in naturalism is paid for in the loss of the crystalline clarity of the 1910-13 picture.
Three clauses of "The F is G"
Hylton's chapter states the analysis with schoolmaster clarity. "The F is G" asserts: (i) there is at least one F; (ii) there is at most one F; (iii) that thing is G. In symbols: (∃x)[Fx & (∀y)(Fy ⊃ x=y) & Gx]. Indefinite "an F is G" drops uniqueness. The complexity is semantic: unlike the name "Aristotle", the phrase "the even prime" contains independently significant parts ("even", "prime") that logic exploits in inference. Once paraphrased, descriptions need no extra axioms beyond the predicate calculus.
Hylton: three claims inside every definite description.
Donnellan, Kripke, and the live debate
Hylton engages the post-1966 literature. Donnellan (1966) distinguished attributive from referential uses; referential uses seem to designate even when the description fails. Kripke's arguments about rigid designation and speaker's reference vs semantic reference complicate the claim that ordinary names are truncated descriptions. Hylton's Russell can concede pragmatic referential uses while holding that semantic content remains quantificational. Neale (1990) defends; Elbourne (2013) revives presuppositional alternatives. The Companion chapter is not a museum piece; it is an entry into a living dispute.
Anti-idealist roots of descriptions
Hylton's section II places "On Denoting" inside the 1898-99 break. Absolute Idealism held that knowledge is mediated by conceptual structures. If those structures are subjective, knowledge collapses into arbitrariness; if one tries to prove them objective, a regress of mediating structures threatens. Idealists offered subtle reconciliations; Moore and Russell judged them all too weak. Their reply was radical: postulate direct, unmediated knowledge of reality. Moore's "direct cognitive relation" and "direct perception" in Principia Ethica; Russell's Preface to POM: the chief part of philosophical logic is to see the entities clearly so the mind has "that kind of acquaintance with them which it has with redness or the taste of a pineapple" (p. xv).
Acquaintance is a relation between mind and what is outside the mind: direct, immediate, presuppositionless. Propositions are non-linguistic, non-mental abstract entities; understanding a proposition involves acquaintance with it. Centrally, for propositions about ordinary objects, Russell (unlike representational theories) holds that the object itself is a constituent. A proposition about Bill Clinton contains Clinton, not a mental stand-in. That constituent thesis makes empty names and empty descriptions catastrophic unless they can be analysed away. Descriptions are the escape hatch: they let sentences be meaningful without packing non-existent Kings of France into propositions as constituents.
Hylton: from idealist mediation to descriptions as escape.
Sense/reference denial as motive
Hylton stresses that Russell sought to deny Frege's sense/reference distinction. Part of the significance of the theory of descriptions is that it makes that denial plausible. If "the F" never means an object, one need not invent a Fregean sense to be grasped when the object is missing. Cognitive significance differences ("Hesperus"/"Phosphorus") Russell handles by descriptive material packed into the paraphrase, not by a third domain of Sinne. Critics say this undergenerates opacity in attitude reports; Hylton presents Russell's bet and the costs.
Indefinite descriptions and compositional parts
"An F is G" does not name; it existentially generalises. Definite descriptions add uniqueness. Because "the even prime" contains independently significant parts, inferences that exploit those parts (from "the even prime is less than ten" to "some even number is less than ten") are logically immediate once the paraphrase is in place. No special description axioms are required beyond the predicate calculus. That formal smooth fit is, for Hylton, decisive against Meinongian object theories that multiply entities for every failed description.
Mont Blanc letter and direct realism
Hylton quotes the Frege-Russell exchange of late 1904 as the clearest statement of Russell's refusal of Sinn. Frege (13 November): truth is not a component of a thought, just as Mont Blanc with its snowfields is not a component of the thought that Mont Blanc is more than 4,000 metres high. Russell (12 December) ignores the truth point (with which he agreed) and seizes the illustration: Mont Blanc itself is a component of what is asserted; we assert the object of the thought, a certain complex (objectiver Satz) in which Mont Blanc is a part. Without that, "we know nothing at all about Mont Blanc." For a simple name like "Socrates," Russell says he cannot distinguish Sinn and Bedeutung; he admits only the psychological idea and the object, or better, does not admit Sinn at all.
Hylton labels the nexus direct realism: only if the object is a constituent of the grasped proposition does thought "get through" to the object. The danger is not that beliefs about Mont Blanc are false, but that they are not about it at all. Descriptions later let Russell keep aboutness for empty cases without packing non-existents into propositions.
Hylton: Mont Blanc to OD.
Meinongian argument and subsistence
POM §427: being belongs to every conceivable term; "A is not" is always false or meaningless if A is a term; numbers, Homeric gods, relations, chimeras and four-dimensional spaces all have being. Hylton calls the underlying line the Meinongian argument: a meaningful name must name something that, in some sense, is. Non-existent concreta (Vulcan, Pegasus) subsist; abstracta subsist; only some entities also exist in space and time. A closer look already corrects Hylton's astronomy: Vulcan was hypothesized inside Mercury's orbit, not as a "tenth planet between Mars and the Sun."
Difficulties of unrestricted direct realism
Hylton sorts the pressures. Acquaintance scope: Socrates (no longer exists), abstracta, non-existent concreta. At POM Russell accepts acquaintance even with the merely subsistent; later he retreats on concreta while keeping acquaintance with universals (why 1914 Russell is "not an empiricist" in the obvious sense). Infinite complexity: "Every natural number is odd or even" cannot be grasped as a proposition containing all the numbers (POM §72). Denoting concepts (1901-June 1905) were the first repair; descriptions (from June 1905) the second. Hylton's chapter is the Companion's fullest map of why OD was forced, not optional.
| Pressure | POM response | After OD |
|---|---|---|
| Empty names | Subsistent objects | No constituent; false uniqueness |
| Past objects | Acquaintance allowed | Descriptions / memory later |
| Abstracta | Acquaintance kept | Still kept into atomism |
| Infinite generality | Denoting concepts | Quantification |
| Sinn/Bedeutung | Denied | Still denied; paraphrase does the work |
Russell's Substitutional Theory
The doctrine of the unrestricted variable
Russell adopted Leibniz's "quodlibet ens est unum": every entity can occur as logical subject of a true proposition. Logic, the universal science of propositional structure, should therefore use one style of variable, ranging over everything. A many-sorted language, with separate variables for individuals, attributes and classes, contradicts that ideal; Frege's two-sorted calculus of functions and objects was the contrast. The paradoxes of 1901 broke the naive assumption that every condition comprehends an attribute with both a predicable and an individual nature; classes are only the extensional shadow of that assumption, and *20.01 shows how class talk reduces to attribute talk.
| Route out of the paradoxes | Idea | Russell's verdict |
|---|---|---|
| Zig-zag (1903-05) | Limit the complexity of class-defining formulas (like Quine's NF, 1937) | Stipulative, not logic |
| Limitation of size | Stop classes getting too big (like Zermelo, von Neumann) | Stipulative, not logic |
| Substitution (1905-07) | Proxy types inside a type-free logic of propositions | "a complete solution", until po/ao |
| Ramified types (1907-10) | Orders of propositions, then of functions, plus reducibility | Adopted in Principia |
Substitution as eliminativism
Landini models the theory on theory change in science: just as atomism supplanted aether and caloric theories, keeping most of their successes and dropping some, substitution would supplant the language of classes, keeping "nearly the whole of Cantor's work on the infinite" and losing some flotsam such as the ordinal ωω. Russell sometimes called this the denial that there are classes, sometimes agnosticism.
The calculus
From the manuscript "On Substitution" (22 December 1905) and "The Theory of Implication" (1906), Landini reconstructs a language whose only variables are individual variables, whose primitive "p/a;x!q" means that q results from putting x for a wherever a occurs in p, and in which any formula can be nominalized into a term {A}. "Out" and "in" (being a constituent) are defined from substitution; modus ponens runs "from A and {A} ⊃ {B}, infer B", so no bare variable ever stands alone on a line of proof.
The substitution primitive: a matrix p/a does the work of a class or attribute without being an entity.
Types become logical grammar. Each instance of the comprehension axiom of simple type theory is translated by a theorem of substitution: an attribute of individuals by a pair p/a, an attribute of attributes or a relation by triples q/p,a. The self-application φ(φ) would need "p/a;p/a!q", which is simply ungrammatical, so Russell's paradox cannot be stated. The theory "adheres with drastic pedantry" to the maxim that whatever is, is one.
How the substitutional theory proxies simple type theory (Landini, Section IV).
The po/ao paradox and the road to Principia
By April 1906 Russell found that axiom S17, the engine of comprehension, yields a contradiction by diagonalizing over propositions of the form {p/a;b!q}, using the fact that such propositions are one-to-one with their components. He withdrew the paper read to the London Mathematical Society on 10 May 1906. In "On 'Insolubilia'" (1906) he abandoned general propositions, but the auxiliary comprehension axioms he needed for arithmetic silently revived po/ao. In 1907 he tried orders of propositions; "Mathematical Logic as Based on the Theory of Types" (1908) is that patch. A letter to Hawtrey of 22 January 1907 says the paradox "pilled" (failed) substitution.
Timeline of substitution, 1903-1910.
Philosophical aspects: propositions as states of affairs
Landini's Section II is the philosophical key that most summaries skip. Russellian propositions in 1905-07 are states of affairs (the cat's being on the mat), which obtain or fail to obtain "just as some roses are red and some are white". They are intensional entities, not intentional ones: aboutness lives in the belief that takes a proposition as object, not in the proposition itself. Read this way, Mont Blanc as a constituent is unremarkable, and the po/ao paradox is a logical paradox, not a semantic Liar. Landini argues that Quine's and Church's misreadings start from treating Russell's "⊃" as the statement connective "→" rather than as a predicate flanked by terms naming propositions.
The substitutional programme is eliminativist in the sense of theory change in science. Atomism kept the successes of aether and caloric theories and dropped the flotsam; substitution would keep "nearly the whole of Cantor's work on the infinite" and lose some ordinals such as ωω. Russell vacillated between saying there are no classes and saying we may be agnostic. What he did not vacillate about was the unrestricted variable: logic, as the universal science of propositional structure, must use one style of variable ranging over everything. Frege's two-sorted calculus of functions and objects was the explicit contrast; a many-sorted language with separate variables for individuals, attributes and classes would betray Leibniz's "quodlibet ens est unum".
Landini: three rejected exits from the paradoxes before substitution.
Formal calculus: S16, S17, and class proxies
From "On Substitution" (22 December 1905) and "The Theory of Implication" (1906), Landini reconstructs the primitives. The only variables are individual variables. The primitive "p/a;x!q" means that q results from putting x for a wherever a occurs in p. Any formula A can be nominalized into a term {A}. "Out" and "in" (constituency) are defined from substitution. Modus ponens runs "from A and {A} ⊃ {B}, infer B", so no bare variable stands alone on a proof line.
Types become logical grammar. Each instance of simple-type comprehension translates into a substitution theorem: an attribute of individuals by a pair p/a, an attribute of attributes by a triple q/p,a. Self-application φ(φ) would need "p/a;p/a!q", which is ungrammatical. Russell's paradox cannot even be stated. The theory "adheres with drastic pedantry" to the maxim that whatever is, is one.
Classes enter by proxy. Early approach: replace class abstract ŷ(Ay) by matrix {Aa}/a, with "x ∈ p/a" defined as (∃q)(p/a;x!q & q). Extensionality is recovered by quantifying only over those attributes (via substitution) that are extensional: Cls2, Cls3, and so on up the types. Revised approach: the class symbol is replaced by ι(p/a)[p/a ≈x Ax], capturing the bound variable in the abstract. The early proxy failed on (∀z)(z ∈ {a = z}/a) because instantiating to {∼a} yields a falsehood; the ι-form repairs that. Landini's point: Principia's class talk is the descendant of this proxy technology, not a sudden invention of 1910.
| Device | Role in substitution | Fate after 1907 |
|---|---|---|
| p/a;x!q | Primitive substitution | Abandoned as primitive |
| {A} nominalization | Propositions as terms | Orders of propositions |
| Matrix p/a | Proxy for attribute/class | Propositional functions in PM |
| S17 / CPsub | Comprehension engine | Source of po/ao; dropped |
| ι(p/a)[p/a ≈x Ax] | Revised class abstract | Ancestor of PM *20 |
| Multiple-relation judgment | Nominalistic semantics | Carried into PM, then abandoned 1913 |
The po/ao paradox in Landini's reconstruction
By April 1906 Russell found that S17, coupled with S16, yields a contradiction by diagonalizing over propositions of the form {p/a;b!q}, using the fact that such propositions are one-to-one with their components. Landini names it the po/ao paradox. Abbreviating, one defines a proposition po that packages a diagonal condition on ao; substituting and using the uniqueness of constituents produces P ≡ ∼P. Simpler variants exist: one instance of (CPsub)1 diagonalizes on {p ⊃ a} rather than on {p/a;b!q}. The flaw is S17, not the Liar and not Russell's class paradox.
Russell withdrew the paper read to the London Mathematical Society on 10 May 1906. In "On 'Insolubilia'" (1906) he abandoned general propositions, blaming the missing theory of their constituents after denoting concepts fell in 1905. Auxiliary comprehension axioms needed for arithmetic silently revived po/ao. In 1907 he tried orders of propositions; "Mathematical Logic as Based on the Theory of Types" (1908) is that patch. A letter to Hawtrey of 22 January 1907 says the paradox "pilled" (failed) substitution. Cocchiarella (1980) had already seen that substitution could not capture the Cantorian fact that there must be more classes of propositions than propositions; Landini's archival work made the specific paradox and its timeline public.
Landini: from substitution's promise to Principia's nominalistic retreat.
Zigzag, limitation of size, and why Russell rejected both
Landini inventories the exits from the paradoxes that Russell considered. Zigzag theories (roughly: only "simple" properties determine classes) lack a clear simplicity criterion. Limitation of size (only "not too big" collections are classes) threatens to block the universal class and large cardinals logicism wants. Russell's preference for the vicious circle principle is that it diagnoses the paradoxes as involving impredicative totalities, not merely oversized ones. Landini notes Gödel's later rehabilitation of impredicativity for set theory; Russell's diagnosis is not mandatory.
Propositional paradoxes and the need for orders
Landini insists that class paradoxes alone do not force orders (the "ramified" part). Propositional paradoxes do: if propositions form a single type and one can quantify over all propositions, liar-like constructions return. Ramification assigns orders to propositional functions (and, in some readings, to propositions) so that a function's order exceeds the orders of its apparent variables. Reducibility then says that for every higher-order function there is an extensionally equivalent first-order one, restoring mathematical strength at philosophical cost.
Landini: exits Russell weighed.
Gödel 1944 and the precision deficit
Landini echoes Gödel's 1944 judgment: PM is a step backwards from Frege in formal precision of foundations. The informal introduction's hierarchy is clearer than the formal *12 and *20 apparatus. Landini's own reconstruction uses a modern typed language to state what Russell meant. The chapter's punchline for Companion readers: understanding Russell's logic requires reading the unpublished manuscripts (substitution, "On Fundamentals") alongside PM; the published book alone understates the research programme.
How substitution worked: p(x/a) and matrices
Landini reconstructs the 1905-06 calculus. Instead of primitive propositional functions, one starts from complete propositions and a substitution operation: the result of putting x for a in p, written in manuscript notations Landini modernises. A "matrix" is an incomplete open result of such substitution. Quantification binds the substitutional variables. The hope: keep a universal logic without typed variables, because any entity can be substituted for any entity, while blocking paradoxes by restricting which substitutions yield propositions.
The propositional paradoxes (liar-like constructions using propositions about all propositions) show the hope fails unless orders return by another door. Cocchiarella (1980) spotted Cantorian cardinality obstructions. Hawtrey's "pilled" letter (22 January 1907) marks Russell's loss of confidence; by then the return to propositional functions for Principia was underway.
Landini: substitution architecture.
Why Landini thinks PM still hides substitution
Landini's minority but influential reading: the published Principia still rests on insights forged in the substitutional manuscripts (now in Papers 5, 2014). Contextual definition of classes, the treatment of propositions, and parts of the ramified hierarchy make more sense against that archive. Hylton and others read "Insolubilia" differently on whether orders are already present.
Philosophical payoff: universality vs consistency
The deep issue is not notation. Can logic be the universal language (everything sayable inside it) and still consistent? Types say no: some things cannot be said about all types without ascending. Substitution tried to say yes. Gödel, Tarski and the undefinability results later explain why universal languages that define their own truth predicate break. Landini's Russell is the last great attempt to keep universality before that lesson was formalised.
Contextual class definitions *20 and *21
Landini walks the PM definitions that make classes facons de parler. *20.01 and related clauses contextualise class abstracts; *20.07-.*20.081 handle classes of classes; *21.02 is the analog for dyadic relations-in-extension (often misread as a concretion principle for circumflex terms). Extensionality becomes provable from the contextual definitions rather than a separate ontological axiom: two class abstracts are identical iff they have the same members. Attributes in intension do the real work; a separate ontology of classes is unnecessary.
The contradictions' source, on this reading, is not "assuming classes" but the naive assumption that attributes have both predicable and individual natures without restriction. Unbridled attributes yield paradox. Substitution was an attempt to regulate that without types; PM's contextual definitions plus ramification are the published settlement.
Landini: classes as facons de parler.
Why *20.01 alone misleads commentators
Landini complains that citing only *20.01-*20.02 ignores the class-of-classes clauses and *21.02. The no-class theory is a system of contextual definitions, not a single slogan. Manuscript substitution work (Papers 5) shows the research path that made this system thinkable. Companion readers who skip Landini will misunderstand what "classes are fictions" meant operationally.
Substitution vs the predicable/individual double life
Landini's diagnosis: paradoxes arise when an entity is both a predicable attribute and an individual that can fall under attributes, without restriction. Classes as fictions remove one scapegoat; the real culprit is naive attribute theory. Substitution tried to keep a single type of entity (propositions and their constituents) while regulating replacement. When propositional paradoxes returned, ramification and contextual class definitions became the published fix.
Landini: where contradiction enters.
Papers 5 and the 2014 horizon
Five manuscripts listed as "forthcoming in Papers 5" in the 2003 Companion appeared in 2014 (ed. G. H. Moore).
Propositional paradoxes force orders
Landini's sharpest technical claim: class paradoxes alone motivate simple types; propositional paradoxes motivate orders (ramification). If propositions form one type and one quantifies over all of them, liar-style constructions return. Ramification assigns orders so a function's order exceeds the orders of its apparent variables. Reducibility then buys back mathematical strength. Gödel 1944: PM's foundations lack Frege's formal precision; Landini's modern reconstructions repair the presentation without erasing the research path through substitution.
Landini: class vs proposition paradoxes.
Hawtrey letter and the 1907 pivot
22 January 1907: Hawtrey receives the letter signalling loss of confidence in substitution ("pilled"). By then Whitehead and Russell are aligning on the PM plan. Landini reads the unpublished MSS as showing how much of PM's philosophical motivation was forged in 1905-06. Hylton disputes readings of "Insolubilia" on orders; Insight keeps both names visible.
Universality lost, consistency gained
Landini's philosophical stakes: substitution fought for logic as universal language; types concede that some things cannot be said at the same type. Gödel and Tarski later explain related limits. PM is the settlement Russell could publish; the MSS are the laboratory. Cocchiarella's Cantorian obstruction (1980) is part of why universality failed inside substitution.
The Theory of Types
Appendix B and the propositional paradox
Russell found the Contradiction in 1901 while analysing Cantor's proof that there is no greatest cardinal, and sent the Principles to the printer in May 1902 unsolved. On 8 August 1902 he sketched types to Frege, borrowing Frege's hierarchy of functions but not his saturated/unsaturated distinction. Appendix B makes types the ranges of significance of propositional functions, oddly letting some classes-as-one (persons, tables) count as individuals and numbers form a range of "infinite order".
- The propositional contradiction. For any class m of propositions, form "every proposition in m is true". Since propositions contain their constituents, distinct classes give distinct propositions, so there are as many propositions as classes of them, contradicting Cantor's theorem.
- Russell's verdict. Typing propositions seemed "harsh and highly artificial"; he "earnestly commend[ed]" the problem "to the attention of all students of logic" (POM 528).
- Frege's patch. Frege's hasty appendix to Grundgesetze II (1903) also fails once two objects are assumed (Quine 1955).
The vicious circle principle
"Mathematical Logic as Based on the Theory of Types" (1908) lists seven paradoxes, including Epimenides, Berry and Burali-Forti, and diagnoses them all as assuming "a totality such that, if it were legitimate, it would at once be enlarged by new members defined in terms of itself". The immediate source was Poincaré's attack on logicism (1905-06). The rule bans impredicative definitions, which Urquhart shows are everywhere in mathematics: "the tallest woman in this room", the least upper bound, and the smallest set containing 0 and closed under successor, the logicists' own definition of the natural numbers.
The ramified hierarchy
Following Church (1976), Urquhart gives the formal syntax that Principia lacked; Gödel called PM "a considerable step backwards as compared with Frege" in formal precision. Each r-type carries a level: a function of level n may quantify only over lower levels.
Church's r-types: order rises with every quantification over a lower level.
Why Reducibility was needed
- No infinity. The axioms do not even prove two individuals, so PM prefixes "Infin ax ⊃" to theorems.
- No natural numbers. Myhill (1974) proved that no definition of the numbers in ramified type theory makes all of induction provable, so Russell's Reducibility-free Appendix B in PM's second edition (1925) "is definitely in error", as Gödel first noted (1944).
- No real analysis. A least upper bound is defined by quantifying over its set, so it has higher order than its members.
The Axiom of Reducibility says every propositional function is equivalent to a predicative one. With it, *20.01 defines classes and the system behaves like simple type theory. Chwistek, Ramsey and Copi ("The Inconsistency or Redundancy of Principia Mathematica", 1950) pressed the dilemma; Myhill (1979) replied that ramification still blocks the Liar.
How ramification dissolves 'I am lying' (MLT, 1908).
Ramsey, the simple theory and after
Ramsey (1925 to 1926), following the Tractatus, made logic extensional, allowed infinite truth-functions, and split paradoxes into logical and semantic, so that simple types suffice for mathematics. Urquhart notes that Gödel and Tarski later expressed truth and definability mathematically, blurring Ramsey's line; Weyl mocked the resulting "logician's paradise".
| System | Handles paradoxes by | Price | Afterlife |
|---|---|---|---|
| Ramified types (PM 1910) | Orders plus types | Needs Reducibility, Infinity | Gödel's L (1938); predicative analysis |
| Simple types (Ramsey 1926; Gödel 1931; Tarski 1935) | Types only | Liar handled elsewhere | Church 1940; HOL provers |
| Zermelo set theory (1908) | Axioms on set formation | Axioms not "logic" | ZFC, the standard foundation |
| Weyl, Das Kontinuum (1918) | Predicativity, no Reducibility | Loses some analysis | Hazen's predicative logics |
Ranges of significance and the vicious-circle form
Alasdair Urquhart (Companion ch. 8) presents types as ranges of significance: a propositional function is true or false inside its range and nonsense outside. Quantification over "all of a collection" makes sense only when the collection is such a range. The technical vicious-circle form: "Whatever contains an apparent variable must not be a possible value of that variable" (MLT 237; LK 75). The hierarchy begins with individuals (neither propositions nor functions). Complexes such as a-in-the-relation-R-to-b ground a correspondence definition of truth for elementary judgments.
Urquhart: ramified hierarchy sketch.
Formal deficit and Gödel's complaint
Urquhart agrees with Gödel (1944): PM's foundations lack Frege's formal precision. The introduction's prose hierarchy outruns the symbolism. Modern reconstructions (Church, Schütte, and later typed systems) make explicit what Russell left informal. Urquhart's service to readers is a precise ramified type syntax: types and orders assigned so that impredicative definitions are barred unless reducibility reintroduces their strength.
Simple vs ramified, and Ramsey's critique
Ramsey argued that many paradoxes are semantic (truth, definability) rather than mathematical, so mathematics needs only simple types. Urquhart explains why Russell could not accept that split in 1910: he lacked a clear object-language / metalanguage distinction, and propositional paradoxes looked continuous with class paradoxes. Reducibility is the scar tissue. Without it, real analysis loses impredicative definitions (least upper bounds). With it, critics ask whether one still has logic or a set theory in disguise.
Reducibility stated and motivated
Urquhart gives reducibility its due. Without it, impredicative definitions used throughout analysis (least upper bound of a bounded set of reals) fail in a ramified hierarchy, because the lub would be defined by quantifying over a totality that includes itself. Reducibility asserts that every propositional function is extensionally equivalent to one of lowest order compatible with its type. Mathematics regains its theorems; philosophy pays with an axiom that looks like set-theoretic comprehension in disguise.
Urquhart: why reducibility is scar tissue.
Truth definition and complexes
The introduction to PM defines truth for elementary judgments via complexes: the judgment that aRb is true when the complex a-in-relation-R-to-b exists. This correspondence picture underwrites the hierarchy's base. Higher-order truth becomes more problematic (Tarski later explains why). Urquhart keeps Russell's picture in view so that types are not misread as merely syntactic hygiene.
Individuals, complexes, and the base of the hierarchy
Urquhart restates the metaphysical picture opening PM's introduction. The universe has objects with qualities and relations; some objects are complex. A complex a-in-the-relation-R-to-b may be perceived as one object. Judgment that aRb is true when that complex exists, false when not. That correspondence definition anchors truth at the hierarchy's base before orders and types ascend.
Ranges of significance: a function is true/false inside its range, nonsense outside. Quantification over "all" requires a range. Vicious-circle form: whatever contains an apparent variable is of higher type than the variable's possible values. Individuals are neither propositions nor functions. From there, first-order functions, higher-order functions, and orders for predicates that quantify over predicates build the ramified tower.
Urquhart: from complex to ramified tower.
Why Gödel called PM a step backwards
Formal precision in foundations: Frege's Begriffsschrift and Grundgesetze outpace PM's mix of prose introduction and imperfect symbolism. Urquhart's modern formalisation is a repair for readers. The philosophical content (VCP, ranges, reducibility's cost) survives the notational fog; the research programme's clarity does not.
Orders, predicative functions, and analysis
Urquhart details what "order" adds. A predicative function has order one greater than its arguments' orders; impredicative definitions quantify over totalities that include the defined entity. Real analysis's least-upper-bound principle is impredicative. Reducibility restores an extensional surrogate. Without a clear object/metalanguage split (Tarski), Russell could not shunt semantic paradoxes aside the way Ramsey wished.
Urquhart: impredicativity fork.
Vicious circle principle in three wordings
Urquhart juxtaposes: (1) no totality contains members defined in terms of itself; (2) whatever involves all of a collection must not be one of the collection; (3) whatever contains an apparent variable must not be a possible value of that variable (MLT). The third is the technical form that builds the hierarchy. Nonsense outside ranges of significance is the distinctive claim versus classical falsehood.
Nonsense versus falsehood
Urquhart's memorable doctrine: type violations are nonsense, not falsehoods. "The class of all non-self-membered classes is a member of itself" is ill-formed. That changes the space of propositions before classical logic applies. Reducibility and Ramsey's critique then fight over how much mathematics needs beyond simple types.
Urquhart: nonsense gate.
Russell's Method of Analysis
Two directions, three features
"The business of philosophy, as I conceive it, is essentially that of logical analysis, followed by logical synthesis" (Logical Atomism, 1924). The backward step is the "most philosophical"; the forward step is not a mere mirror, because it can open new subjects: analysing ordinary mathematical notions gives "fresh insight, new powers" (Introduction to Mathematical Philosophy).
Hager's reading of Russellian analysis: the synthesis returns a reconstructed, not identical, body of results.
- (i) Analysis is unlikely to be final. Deeper premisses may always be found, and treating a complex as simple at one level is harmless if we do not assume it is unanalysable: "points may be defined as classes of events, but that does not falsify anything in traditional geometry". The minimum set of premisses "is a technical question and it has no unique answer".
- (ii) Analysis enlarges particular subjects. Yesterday's tentative premisses become today's science, so philosophy lives at the frontier: "science is what you more or less know and philosophy is what you do not know" (PLA).
- (iii) Premisses become less self-evident. Like Maxwell's equations, premisses are believed "because true consequences follow from them". Hence Russell's quip that philosophy starts "with something so simple as not to seem worth stating" and ends "with something so paradoxical that no one will believe it".
| Results (data) | Premisses |
|---|---|
| More complex, relatively concrete | Simpler, abstract |
| Common knowledge, more obvious | Outcome of special inquiry, less obvious |
| Vague, inexact, confused | Precise, definite, clear |
| Indubitable, undeniable | Dubitable, disputable |
| Logically interdependent | Logically independent |
Human Knowledge as a worked example
Hager reads Human Knowledge (1948) as the method applied to scientific inference, with the vague result "science is developed from observation by inductive or probabilistic inference".
The six parts of Human Knowledge mapped onto the method (Hager).
The synthesis yields five postulates: quasi-permanence, separable causal lines, spatio-temporal continuity, the structural postulate, and analogy. None is certain; propositions of fair intrinsic credibility raise each other's credibility into a coherent system. True to feature (i), Russell thought the number of postulates "can be further reduced", but "I have not myself succeeded in doing so".
Analysis is not scientific method
Reviewing Urmson's Philosophical Analysis (1956), Russell restated the method and rejected the demand that analysed and unanalysed statements be logically equivalent: only what is well founded in the results needs to be recoverable. Had Urmson's ordinary-language philosophy ruled Greece, science might still be at earth, air, fire and water. When McKinney (1957) equated analysis with hypothesis formation, Russell's letter of 18 July 1958 replied that in analysis "the whole is given", while in science, as with Kepler's finite planetary data, "the whole is not given", and hypotheses are never proved, only shown compatible with the facts.
Language and Monk's misreading
Russellian analysis is analysis of propositions about the desk, not of the desk. Ontology changes only indirectly: the refined propositions no longer support inference to the common-sense desk and instead invite inference to a structure of sensa or events. Hager therefore rejects Ray Monk's claim (1996) that Russell's concern was "precisely not language", noting that Human Knowledge has a Part titled "Language" and chapters on "Minimum Vocabularies". Wittgenstein's broomstick example misses Russell as completely as Urmson's "England declared war in 1939".
Constructions of space, time, and matter
Paul Hager (and related Companion treatment of method) presents constructionism as Russell's generalisation of the descriptions method. Points of space, instants of time, and pieces of matter are not assumed as metaphysical simples; they are logical constructions from classes of events or sense-data. Whitehead's method of extensive abstraction and Russell's use of overlapping events aim to recover the continuous manifolds physics uses without reifying points as substances.
Constructionism: from data to scientific objects.
Ockham's razor as research strategy
Russell's razor is methodological: do not multiply entities beyond those needed to state science. Constructions show how to say what physics says while committing only to the construction base. Hager notes the tension with ordinary language: we still talk as if points exist. The theory of incomplete symbols licenses that talk without ontological guilt. Critics (including later ordinary-language philosophers) charged that the constructions are so remote from use that they clarify nothing. Russell's reply: philosophy's job is scientific ontology, not linguistic therapy.
Limits: Newman, underdetermination, and late postulates
Constructionism's limit is Newman's: structure without quality is empty. Also, many non-isomorphic construction bases can yield the same structural physics. Late Russell adds epistemic postulates rather than more constructions. The Companion lesson: constructionism is the bridge from logicism's incomplete symbols to neutral monism's event ontology, not a final system.
Analysis as regressive and compositional
Hager's method chapter treats analysis as both decomposition (complex to parts) and regression (to premises that imply the accepted body). Same spirit as the 1907 regressive method in logicism. Definitions are not verbal conveniences; they are philosophical hypotheses tested by fruitfulness and clarity. Monk's biographical readings, Hager charges, misunderstand this as psychological compulsion rather than method.
Hager: two faces of analysis.
Against dissolving philosophy into therapy
Hager resists ordinary-language and purely therapeutic readings of analysis. Russell's target is scientific ontology and logical form, not only confusion in talk. Whether that ambition succeeds is separate from whether it was the ambition.
Analysis, incomplete symbols, and scientific ontology
Hager binds method to ontology: analysis earns its keep when paraphrases cut inferred entities without cutting scientific content. Points, instants, numbers, classes, minds: one pattern. Against therapy-only readings, Hager holds that Russell wanted to know what there is, not only to ease linguistic cramp. Monk's biographical frame, on Hager's charge, psychologises a logical method.
Hager: method to ontology loop.
Ockham as research strategy
Hager: do not multiply entities beyond scientific need. Constructions show how to keep the sentences of physics while committing only to the base. Ordinary talk of points may continue; incomplete-symbol theory licenses it without guilt. Limits: Newman, underdetermination, remoteness from use. Late postulates supplement when constructions alone underdetermine.
Russell's Neutral Monism
Three formative themes
Tully grounds the story in three commitments that run from the 1890s to the end. Metaphysics: Russell was heir to the line from Descartes to Kant that traces knowledge to immediate experience, and he never lost the hope of a grand synthesis. Science: he called his theories "working hypotheses", though his science was physics and some physiology, not chemistry or biology. Language: he was a demanding realist about names; genuine names stand for objects that are their meanings, and most ordinary names are disguised descriptions.
What neutral monism claims
The doctrine, from Ernst Mach, William James and the American New Realists, denies primacy to both mind and matter. Russell's best similes are the old London postal directory, which lists the same names alphabetically and by street, and columns and rows: one item, located in two series by different relations.
Russell's two orderings of the same neutral items (postal-directory simile).
Tully presses the hard questions. If neutral items have no properties of their own, how are they picked out? If they make up both minds and bodies, why not just say mental things are material things? And if the doctrine cannot be tested, the "working hypothesis" label "seems hollow". In practice the early monists pared down ordinary descriptions until what was left could be declared neutral, and they borrowed a psychological vocabulary (Mach's "sensations"; for James, Tully writes "pure consciousness") that tilts towards experience.
The 1913 assault
In the abandoned Theory of Knowledge (1913) Russell defended acquaintance, "the most pervading aspect of experience", against James and the New Realists. Tully grades the arguments as uneven: those about false propositions and the objects of belief show only that Russell's own views are stronger; his claim that monism must appeal to "extraneous considerations, such as the nervous system" begs the question against a monist who is not a foundationalist.
Judgment collapses, acquaintance is dismissed
The multiple relation theory analysed "Aristotle judges that Socrates preceded Plato" as one fact joining Aristotle to Socrates, Plato and preceding. Wittgenstein's objections (what makes preceding relate rather than sit as a third object, and which order is meant) forced a "deluxe" version with logical forms and "associated complexes". Russell then abandoned the book. In 1918 he admitted "some of the arguments" against neutral monism were "not valid", exempting "this"; in "On Propositions" (1919) he declared mental acts "not empirically discoverable" and dismissed acquaintance. Tully's verdict: "The Cheshire cat has disappeared, though perhaps not its smile."
Tully's stages, from dualist critic to event monist.
The mature doctrine: events and percepts
From "Logical Atomism" (1924), under Whitehead's influence, the neutral stuff becomes events in a relativistic four-dimensional manifold. Matter's history is a set of "tracks or tubes" (world-lines), and "a mind is a track of sets of compresent events in a region of space-time where there is matter which is peculiarly liable to form habits". Brain descriptions concern a tube's overall shape; mind descriptions concern the events in each cross-section.
Percept versus image: the same physiology, different causal reach (Analysis of Matter, 1927).
Percepts are brain events that are directly experienced; physics otherwise knows only structure, so everything we know of the world's intrinsic character "is derived from the mental side". Against behaviourism as a metaphysic Russell offers a dilemma in An Outline of Philosophy (1927): if physics is valid, the behaviourist's real data are events in his own brain, not the rat in the maze. The 1940 Inquiry finally answers the 1913 challenge in behaviourist terms (noticing), yet keeps a private use of "this" and "I-now": "This is a cat" may be wrong, "This is a cat-percept" justified.
| Stage | Neutral stuff | Acquaintance | Cognition explained by |
|---|---|---|---|
| Dualism (1912-14) | Sense-data (not neutral) | Irreducible relation | Multiple relation judgment |
| Analysis of Mind (1921) | Sensations | Dismissed; introspection remains | Images, assent, mnemic laws |
| Mature (1924-27) | Events; percepts in the brain | Replaced by intrinsic quality | Habit, causal lines |
| Late (1940-48) | Events, compresent qualities | Private "this", "I-now" | Noticing; five postulates |
Monism or dualism?
Russell calls his view "psycho-cerebral parallelism", but Tully reads it as a psycho-cerebral identity theory of spatial reference: qualities are where percept-events are, in the brain. Yet structural description and naming a quality are complementary operations, so the language stays dualistic. Tully's final objection is about common sense: we cannot talk "from the viewpoint of our brains", and Russell left "little more than sketches ... never a set of working plans" for reconstructing the shared world. His patrician disregard for common sense and for the social context of language kept the doctrine at the periphery.
Metaphysics, science, language: the three formative themes
R. E. Tully opens by asking why Russell treated neutral monism as important even while he attacked it. The answer is three standing ideals.
- Metaphysics. Russell inherited the tradition from Descartes through Spinoza, Leibniz, the British empiricists and Kant: trace knowledge to immediate first-person experience, then connect that experience to what there is. He cared less for the tradition's doctrines than for its licence to attempt a grand synthesis.
- Science. By the 1890s English systematic philosophy and the natural sciences had parted roads. Russell, trained in mathematics and geometry like Descartes and Kant, set himself to make physics mesh with immediate experience. He classified his own metaphysical theories as "working hypotheses", not revelations. His science was mostly physics plus some physiology; chemistry, biology and later cognitive science lay outside his picture.
- Language. Logic for Russell was never only a formal system. He treated relations, judgments and propositions as things of which one can have direct experience, and used logic to expose the true structure beneath natural language. He was a demanding realist about names: genuine names stand for objects independent of experience. Much of ordinary language fails that test.
Together the three themes explain both the 1913 assault on James and Mach and the post-1918 conversion. Neutral monism promised a metaphysics continuous with physics, built from what experience delivers, in a language scrubbed of archaic dualist grammar.
Outline: neutral stuff without a third substance
Neutral monism, in Tully's historical sketch, was meant to replace both Idealism (only minds and their contents) and Materialism (only configurations of predefined particles). Its origins lie with Ernst Mach, William James, and the American New Realists before World War I. Russell was not merely its British promoter; he gave the doctrine its most systematic shape. Reality is ultimately neither mental nor material. Russell called the common basis "neutral-stuff" and once likened it to a "common ancestor" above mind and matter (AMi 10-11, 25). Tully warns that the picture of a remote third substance distorts: better is Russell's own comparison to the double sorting of names in a directory, or to points that count as part of one group under one grouping and another under a different grouping. Mind and matter are constructions from the same particulars, grouped by different relations.
Tully's map: two rejected monisms and the neutral alternative.
Reference problems: pure experience and vocabulary traps
Tully flags a standing difficulty. James spoke of "pure experience"; commentators sometimes write "pure consciousness", which collapses the view back into idealism. Russell's early critical vocabulary (sense-data, sensibilia, acquaintance) and his later vocabulary (sensations, percepts, events) invite the same slide. If "neutral" is heard as "mental but not yet owned by a subject", the doctrine becomes a form of idealism; if heard as "physical but not yet in a brain", it becomes materialism. Tully's reading: the particulars are neither until grouped. The 1918-21 transition papers ("On Propositions", AMi) still use "sensations"; The Analysis of Matter (1927) prefers "events" with causal and spatio-temporal relations strong enough to underwrite physics.
Russell's 1913 manuscript Theory of Knowledge attacked James for dissolving the subject and for making truth a kind of working success. After Wittgenstein's objection destroyed the multiple-relation theory of judgment, the subject as a simple acquainted particular became dispensable. "On Propositions" (1919) replaces it with belief-feelings (memory, expectation, assent) attaching to image or word propositions that share structure with facts. Acquaintance, once the foundation of Problems of Philosophy, is dismissed as an unexplained relation; knowledge becomes a matter of causal chains and habit. Tully stresses that this is not a small patch but a change of metaphysical regime.
Mature doctrine: events, percepts, and the dualism charge
In the mature view, the world is a manifold of events. Some events are percepts (the appearance of a star to an observer); most are not. A mind is a causal chain of events related by mnemic and subjective relations; a piece of matter is a chain related by the laws physics states. The same event can belong to both a mental biography and a physical trajectory. The Analysis of Matter adds the structural claim that physics reveals only the structure of the external world, not its intrinsic character; percepts give intrinsic character for the mental side. Critics (including later Russell scholarship and the "Russellian monism" revival) ask whether this is still monism or a property dualism in disguise: one stuff, two irreducible kinds of property. Tully's chapter title-question "Monism or dualism?" marks that dispute without settling it in favour of the revival; the 2003 Companion predates Alter and Nagasawa (2015) and Banks (2014).
| Phase | Years | Basic particulars | Subject / acquaintance |
|---|---|---|---|
| Assault | 1912-14 | Sense-data, sensibilia | Subject acquainted with data |
| Transition | 1918-21 | Sensations, images | Belief-feelings; acquaintance dropped |
| Mature | 1921-27 | Events, percepts | Causal chains; no simple subject |
| Late echo | 1940-48 | Events still basic | HK postulates; structure from percepts |
Tully: how the 1913 assault became the 1921 doctrine.
Newman's objection and the emptiness of pure structure
Robert Tully (and the Companion's treatment of Newman) face the 1928 point: if physics yields only structure, almost any domain of the right cardinality can be mapped onto it. Content drains away. Russell's replies (more acquaintance with qualities, or more than structure) never fully restored the 1927 confidence. Tully links this to the later Putnam model-theoretic arguments: fixing theory by structure alone underdetermines reference. Neutral monism's promise, that the intrinsic nature of the world is knowable as the stuff of sensation or events, is under pressure once physics is structural and perception is causal.
Tully: from sensations to events to Newman's gap.
Mnemic causation and the psychology of memory
Tully details Russell's use of mnemic causation (from Semon): past events can be causal factors without persisting as mental images in a soul. Memory is explained without a substantial ego. The subject becomes a construction from experiences related by mnemic and ordinary causation. Critics charged circularity: talking of "past experiences" already assumes the subject who had them. Tully presents Russell's answer as pragmatic naturalism: science already uses such talk; philosophy should align with it rather than invent a Cartesian owner.
Perception, matter, and the causal theory
In The Analysis of Matter, perception is a causal chain from events in space-time to events in the brain. The inferred external world is a structure of events; sensibilia that are not sensed remain possible. Tully stresses how far this is from the 1912 acquaintance model: knowledge of matter is hypothetical and structural, not direct. The five postulates of 1948 then supply the non-demonstrative bridges induction needs. Tully's Companion role is to show that neutral monism is not a side thesis; it is Russell's attempt to keep science and philosophy in one piece after atomism's epistemology cracked.
James, Mach, and the federation metaphor
Tully presents Russellian neutral monism as a federation with James and Mach rather than identity. James's "Does 'Consciousness' Exist?" (1904) uses pure experience; Mach's elements are neutral between physics and psychology. Russell's 1921 sensations and 1927 events are cousins. A closer look notes Tully's "pure consciousness" slip and the sharper continuity in Banks (Realistic Empiricism, CUP 2014). Alter and Nagasawa (OUP 2015) revive Russellian monism in consciousness studies: physics gives structure; consciousness involves intrinsic natures physics omits. Tully's chapter is an ancestor text for that revival.
Tully to Russellian monism today.
No ego, mnemic causation, and behaviour's place
The self is a logical construction from experiences linked by causal and mnemic relations. Behaviourism influences Russell but he rejects a pure collapse of mentality into behaviour: images and sensations remain. Tully's quiet achievement is to show Analysis of Mind and Analysis of Matter as one project in two volumes, not a psychology book plus a physics book.
Neutral stuff without a third substance
Tully warns against picturesque talk of "neutral-stuff" as a remote third entity "above" mind and matter like a common ancestor (AMi 10-11, 25). That suggests Neutral Monists found a special isolable thing. Better comparisons Russell also used: the old London postal directory, where the same names sort alphabetically and by street; or a single deck of cards dealt into different hands. Mind and matter are two groupings of the same particulars (sensations, then events), not two substances nor a magical third.
Neutral Monists dissolve old disputes by denying ontological primacy to both mind and matter. Ultimate components are not atoms-as-matter or ideas-as-mind. Russell kept watch on science and adapted: 1921 psychology-facing sensations; 1927 physics-facing events. Continuity with James and Mach is real; identity is not. Tully's "federation" metaphor marks kinship without erasure of differences.
Tully: two sortings, one stuff.
What dissolves and what remains
Dissolved: Cartesian dualism's interaction problem; idealist absorption of matter into mind; crude materialism's denial of experience. Remaining: the need to explain why mental and physical groupings diverge; Newman's pressure on structural physics; the construction of the subject. Tully's chapter is thin on secondary literature (A closer look); Banks and Alter/Nagasawa supply the post-2003 continuation under the label Russellian monism.
From sensations (1921) to events (1927)
Tully charts the ontology shift inside neutral monism. Analysis of Mind still talks of sensations and images; Analysis of Matter centres events in space-time, with percepts as a subclass. The federation with James/Mach survives; the physics face hardens. Structuralism about the physical then meets Newman; mnemic causation continues to build the self without an ego substance.
Tully: two volumes, one monism.
Russellian monism after 2003
Alter and Nagasawa (2015) and Banks (2014) make Tully's topic central again: physics gives dispositional/structural description; consciousness involves categorical/intrinsic natures. Whether Russell would accept today's "Russellian monism" label is doubtful; that the Companion chapter under-cited secondary literature is certain (A closer look).
Directory metaphor worked out
Tully's postal-directory image: the same particulars appear in a mental sorting and a physical sorting, as names appear alphabetically and by street. No third substance is found; two catalogues of one inventory. Cards-in-hands is the other image. Picturesque "common ancestor" talk (AMi 10-11) misleads if read as discovering a magic stuff. Neutral monism is a claim about grouping, not a treasure hunt.
Tully: one inventory, two catalogues.
Federation not identity with James and Mach
James: pure experience, denial that consciousness is an entity (1904). Mach: elements. Russell: successive theorisations. Banks (2014) argues for a shared realistic empiricism sharper than Tully's federation. Alter and Nagasawa (2015) mobilise Russellian monism for consciousness studies. Tully's chapter remains the Companion's only dedicated monism essay; its thin bibliography is a known limit (A closer look).
Subject as construction
Tully: the ego is not a given substance but a logical construction from experiences linked by ordinary and mnemic causation. Critics allege circularity; Russell replies that science already talks this way. Neutral monism's federation with James/Mach is kinship of method: dissolve mind/matter primacy, keep experience and physics in one inventory.
The Metaphysics of Logical Atomism
A zoo of the forms of facts
Russell called philosophical logic "an inventory, or ... a 'Zoo' containing all the different forms that facts may have", insisting on "forms of facts" rather than forms of propositions. Within the period the defining feature is facts, distinct from true propositions, appearing in his ontology from 1910, together with a correspondence account of truth and the long-standing doctrine of external relations against Bradley's monism.
Propositions without unity
The puzzle: facts are prominent just when propositions vanish as "single entities". Under the multiple relation theory, "x judges that aRb" is the fact B(x, a, R, b), with the multigrade relation B taking as many arguments as the judgment needs. Yet PM freely quantifies over propositions with variables p and q. One response, from Gödel (1944) and Sainsbury (1980), reads those variables as schematic letters and the quantifiers as substitutional, so that propositions are sentences and functions are predicates. Linsky's caution: the mere existence of a substitutional reading does not show it was the intended one.
Was Russell a nominalist?
Gödel and Cocchiarella (1987) say yes, and Russell's later remark that logic is more linguistic than he thought seems to agree. Linsky disagrees. "The theory of types is really a theory of symbols, not of things" (PLA) sounds nominalist, but by "symbol" Russell meant an interpreted symbol, and he held there are "infinite numbers of different ways of meaning", a predicate relating to its meaning differently from a name. A nominalist needs one kind of entity and one relation of applying-to. Even My Philosophical Development defends universals: a restaurateur who calls horseflesh "beef" does not make it beef.
Which facts exist
Russell's fact zoo by the end of the period (Linsky).
Wittgenstein's picture theory dissolved the two-verb problem: a belief is itself an atomic fact, mental items standing for a, R and b arranged in a mode of combination, "the language of thought". Russell accepted that facts are combined only truth-functionally, but kept general facts, since "a is F, b is F, c is F" does not entail "everything is F" without the further fact that these are all the individuals. He also kept negative facts after 1918 (Analysis of Mind, 1921), rejecting Wittgenstein's view that the mere non-obtaining of Fa suffices.
From multiple relation to picture: how a belief became an atomic fact.
Atoms that may not exist
Using Lycan's (1981) three notions of atom, Linsky finds that logical atomism makes "surprisingly little use" of atoms.
| Atom | Meaning | Russell's commitment |
|---|---|---|
| Ontological | Ultimate, partless constituent of facts | Committed in principle, but analysis may never halt |
| Epistemological | Object known without its relations | Weak: we can be acquainted with complexes unanalysed |
| Logical | Atomic proposition matching an atomic fact | Not needed: by *14, proper descriptions obey the logic of names |
Since PM *14 proves that proper descriptions behave like names in extensional contexts, a sentence can be treated as atomic for logical purposes without complete analysis. Hence Linsky's slogan: logical atomism "does not require the existence of logical atoms". The "atomism" names a method.
Construction as honest toil
"The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil" (IMP). The model is the construction of numbers so that Peano's axioms become theorems. Against Sainsbury's reading of construction as stepwise elimination into sense-data and words, Linsky argues the aim is parsimony of theory: Russell's objection to Meinong was that object theory proves "the existent round square exists", not just that it is crowded. Carnap's Aufbau (1928) kept the method but sought only structural isomorphism, and once any consistent theory could be given a set-theoretic model, the project "withered".
Facts, particulars, and the 1918 lectures
Bernard Linsky reconstructs logical atomism from the 1918 lectures and the abandoned Theory of Knowledge. The world is a plurality of particulars and the facts into which they enter. Atomic facts are simplest: a particular having a quality, or particulars in a relation. Molecular facts involve truth-functional combination; general facts involve quantification. Negative facts Russell accepted with reluctance: the fact that the pen is not in the drawer seems needed to make "the pen is not in the drawer" true. Wittgenstein's Tractatus later rejects negative facts; Linsky shows Russell's reasons for keeping them.
Linsky: inventory of logical atomism.
Language mirrors the world, with unfinished business
The ideal language has names for particulars and predicates for universals; atomic sentences picture atomic facts. Ordinary language systematically misleads (definite descriptions look like names; "existence" looks like a predicate). Linsky stresses that Russell never finished the metaphysics: the status of universals, the form of facts, and the analysis of belief remained contested between 1913 and 1919. The lectures are a research report, not a closed system.
From atomism to neutral monism
After Wittgenstein's objection and the war, Russell drops acquaintance-first epistemology. Particulars become events or sensations; the subject is constructed. Linsky's Companion claim: atomism's lasting contribution is the method (seek logical form; do not trust grammar) more than the 1918 ontology. Principia's type distinctions continue to shape what can be said even when the metaphysics softens.
Particulars, universals, and the 1918 inventory
Linsky lists what the atomism lectures commit to. Particulars are simple; universals (qualities, relations) are real; facts are how they combine. Atomic sentences name particulars and ascribe universals. Existence is not a property of individuals but a property of propositional functions ("exist" means "sometimes true"). The King of France analysis is presupposed. Belief remains the sore point after 1913.
Linsky: atomism's ontology.
Urmson and the "inadequate analysis" slip
A closer look already notes the dropped "not" in the print chapter. Philosophically, Linsky's reply to mid-century critics (Urmson and others) is that calling a sentence non-atomic does not mean analysing it as atomic is always wrong for every purpose; it means a deeper analysis exists.
Negative facts and the pen in the drawer
Linsky takes seriously Russell's reluctant yes to negative facts. "The pen is not in the drawer" seems to require, for correspondence truth, a fact of absence. Wittgenstein's Tractatus rejects them; Russell keeps them as the cost of a robust correspondence theory at the atomic level. Molecular and general facts add further structure. The 1918 lectures remain a progress report: belief unfinished, universals real, language to be remodelled.
Linsky: fact hierarchy.
Existence as sometimes-true
Linsky stresses PLA's slogan: to say unicorns exist is to say the propositional function "x is a unicorn" is sometimes true. Existence is not a predicate of individuals. That doctrine presupposes OD and feeds Quine's later criterion. Atomic language has names only for particulars that are; descriptions handle the rest. Belief's unfinished analysis is why atomism never became a closed system.
Linsky: existence relocated.
Ideal language and unfinished belief
Linsky: the ideal language has names for particulars and predicates for universals; ordinary language misleads. Belief after 1913 is unfinished business: MRT dead, 1919 naturalist rewrite underway. Atomism's export is methodological. Gödel 1944 and Linsky's own Evolution of PM (2011) continue the technical story.
Russell's Structuralism and the Absolute Description of the World
Descriptions first: reaching matter without naming it
Problems (1912) applies a simple theory of understanding: "Every proposition which we can understand must be composed wholly of constituents with which we are acquainted". Since we are not acquainted with matter, a name for a material object abbreviates a description built only from acquainted items. We can then refer to things outside experience without any nonlogical name for them.
Ramsey's "Theories" copied this technique. The Ramsey sentence of a theory replaces each theoretical term with a variable and quantifies existentially, keeping every observational consequence. Demopoulos insists this is a descendant of the theory of descriptions, not of structuralism: a Ramsey sentence states structure in Russell's sense only when the theory happens to pin down an isomorphism class.
How a Ramsey sentence is built (Demopoulos, section I).
Structure as pure logic, and the Kantian payoff
Russell's notion of structure comes from the relation-arithmetic of PM: a structure is the "relation-number" of a relation, its class under structure-preserving one-one maps, modelled on the Frege-Russell cardinals. Because one-one correspondence is definable in pure logic, structural similarity owes nothing to intuition. Introduction to Mathematical Philosophy draws the moral: if phenomena have objective counterparts, "the objective counterparts would form a world having the same structure as the phenomenal world".
Relativity arguments and the marching regiment
In 1913 G. Dawes Hicks noted that relativity arguments rest on (C): if a thing appears F to some observers and not-F to others, it is not F in itself. Its contrapositive (C*), that whatever is intrinsically F looks F to everyone, is plainly false: a white table should look pink in red light. Burnyeat (1979/80) called this a conclusive refutation. Demopoulos replies that Russell had already anticipated the point in Problems, and that Russell accepted a version of (C*) for order: a marching regiment changes shape with viewpoint, but "the men will appear arranged in the same order from all points of view". Order, invariant across observers, is the seed of an absolute form of description (Dummett's phrase).
Subjectivism versus relativism about colour
| Question | Russell's subjectivism | Demopoulos's relativism |
|---|---|---|
| What fixes the meaning of "yellow"? | Acquaintance with a quality of percepts | Criteria of application under normal conditions |
| Are ordinary colour judgements true? | Uniformly false (IMT: naive realism refutes itself) | Largely true; science confirms them |
| What does physics change? | Substitutes a different, co-varying property | Changes sense, preserves reference |
| What do we know of the world? | Only its structure | Structure and the same properties described relatively |
| Price | Newman's triviality | Gives up the structure/quality division |
Demopoulos points to computational colour vision (Maloney and Wandell 1986, Brainard 1993) as support for relativism. The visual system solves an inverse problem: recover surface reflectance from a signal in which illuminant and surface are "confounded", using natural constraints on likely illuminants. Surface colours become coarse-grained spectral reflectances, and colour constancy is the explanandum.
Newman's objection
Newman's objection (Mind 1928) in four steps.
M. H. A. Newman saw that if knowledge of the external world is purely structural, it is true as a matter of set theory, given only the cardinality of the domain. Singling out the "important" relation would make importance "among the prime unanalyzable qualities" of the world, "which is, I think, absurd". Russell conceded in a letter of 24 April 1928, suggesting he had assumed copunctuality could hold between percepts and non-percepts and be itself perceptible. Demopoulos argues that this exception sits badly with subjectivism's gulf between percept qualities and their causes.
Point-instants as interpretation
In The Analysis of Matter (1927), a point-instant is a maximal copunctual class of events, and Russell's existence proof uses Well Ordering, resembling Stone's 1936 ultrafilter theorem. The aim is "interpretation": as the Frege-Russell cardinals give Peano's axioms their counting meaning, point-instants built from events give space-time its measuring meaning (following Eddington). Demopoulos notes the "insufficiently appreciated" result at PM *124.57 that the cardinals form a Dedekind-infinite class without the Axiom of Choice. But a structure-preserving representation does not make the knowledge it represents purely structural.
Structuralism as the absolute description programme
William Demopoulos reads Russell's later philosophy of science as structuralism: physical theory describes the world up to structure. Perception and physics together yield an absolute description only of relations among events, not of intrinsic natures. The programme aims to say what science knows without claiming more. Demopoulos links Russell to the contemporary structural realism debate (Worrall, Psillos, and others) while insisting on historical differences: Russell's route runs through constructions and neutral monism, not through Fresnel-to-Maxwell continuity arguments alone.
Demopoulos: structure, Newman, and repair.
Newman's objection in full
M. H. A. Newman (Mind 1928): if the only constraint is that the world have a certain structure, then for any set of the right cardinality some relation realises that structure. Empirical content collapses. Demopoulos treats this as decisive against pure structuralism. Russell's unpublished and published replies add qualitative content or restrict the domain of relations. Demopoulos's own BJPS work (2003) and Ainsworth's survey (2009) are the secondary spine. The Companion chapter is the place where Newman's point becomes essential to understanding late Russell, not a footnote.
Absolute description without intrinsic natures
What remains after Newman is a chastened claim: science gives structural knowledge plus whatever is delivered by acquaintance or by the postulates of Human Knowledge. Demopoulos's "absolute description of the world" is absolute in aspiration (not relative to a conceptual scheme in the Kuhnian sense) but modest in content. The chapter closes by refusing both dismissive positivism and naive realism about theoretical entities' natures.
What "structure" means in AOM
Demopoulos unpacks Russell's claim in The Analysis of Matter (1927): physical theory determines its subject matter only up to isomorphism. Relations among events, not the intrinsic qualitative natures of matter, are what equations fix. Perception gives intrinsic character only to the mental (or neutral) side of the causal chain. The "absolute description of the world" would be a complete structural inventory plus whatever intrinsic natures acquaintance still supplies.
Demopoulos: content of a physical theory.
Ainsworth 2009 and why the debate stayed open
Demopoulos's own BJPS paper (2003) and Ainsworth's review (2009) conclude that no reply to Newman yet fully succeeds for epistemic structural realism. Options include: enrich structure with relations of higher type; appeal to naturalness of relations; add Ramsey-sentence machinery with restricted domains; abandon epistemic SR for ontic SR. Demopoulos's Companion chapter is historically careful: Russell is an ancestor, not a card-carrying member of the 1989-plus Worrall school. Wavelength theories of colour Russell used are outdated; the structural moral may survive the bad examples.
Absolute vs relativised description
"Absolute" for Demopoulos does not mean complete knowledge of natures. It means a description not merely of how the world appears relative to a paradigm or language in the strong Kuhnian sense. Science aims at the structure of the one world. Underdetermination and Newman threaten that aim. Late Russell's postulates and residual acquaintance are emergency supplies for a programme that pure structure cannot finish.
Relation-arithmetic and structure as pure logic
Demopoulos roots Russell's structuralism in PM's relation-arithmetic. A structure is the relation-number (isomorphism class) of a relation under one-one structure-preserving maps of its field, modelled on the Frege-Russell definition of cardinals via similarity classes. One-one correspondence is definable in pure logic; so is structural similarity. That purity matters: structure owes nothing to spatio-temporal intuition. From early on Russell saw this as a way to state the similarity philosophers posited between appearance and reality, or phenomena and noumena, without collapsing the gulf or emptying the similarity.
The Introduction to Mathematical Philosophy passage Demopoulos highlights: if space and time are subjective but have objective counterparts, or phenomena are caused by things in themselves with corresponding differences, then the objective side must share structure with the phenomenal. Every proposition of communicable significance is true of both worlds or of neither; only the "essence of individuality" eludes communication. Structuralism is thus both a philosophy of physics and a reply to a Kantian residual problem, with the irony that a logicist tool articulates a quasi-Kantian claim.
Demopoulos: structure from logicism to physics.
Ramsey sentences: cousin, not twin
Demopoulos carefully separates Ramsey's proposal from Russellian structuralism. A Ramsey sentence R(θ) existentially generalises on a theory's theoretical terms, preserving primary (observational) consequences. Ramsey's point: incompleteness of secondary "propositions" affects disputes but not reasoning. That observation is elementary about derivation; it does not by itself vindicate treating theoretical claims as mere auxiliaries. Only when θ characterises the isomorphism class of a theoretical relation Ti does R(θ) illustrate Russell's claim that theoretical knowledge asserts structure over a domain. Many theories have Ramsey sentences without expressing structure in Russell's sense. So Ramsey is best read, Demopoulos argues, through the theory of descriptions (eliminating names beyond acquaintance), not as a development of structuralism proper.
When structure is not enough
If θ fails to capture Ti's isomorphism class or omits formal properties, the Ramsey-structure link breaks. Trivial finitely axiomatised theories have Ramsey sentences that enable primary derivations without illuminating structuralism. Newman's objection then lands on the pure structural claim: cardinality plus structure is nearly vacuous. Demopoulos's Companion chapter keeps these distinctions sharp so that later epistemic structural realists cannot simply annex Russell without answering Newman and without conflating Ramsey machinery with relation-arithmetic.
Absolute description and the widening brook
Demopoulos's title phrase, absolute description of the world, names a form of description free of reliance on the specific quality of the perceiver's experience and independent of our peculiar vantage and physiology. Physics pays for absoluteness with abstractness. AOM 137: physics and perception are like two people on opposite sides of a brook that slowly widens; at first easy to jump, later a vast labour. Bridging that gulf is a major goal of Analysis of Matter.
Problems already pre-echoes structuralism. A marching regiment looks different in shape from different points of view, but the men appear in the same order from all points; order is treated as true also in physical space, shape only so far as order requires (POP2 32-3). Demopoulos reads "order" as structure: what survives perspective is what physics can claim absolutely.
Demopoulos: from conflicting appearances to absolute structure.
Subjectivism vs relativism about colour
Demopoulos distinguishes two doctrines. Subjectivism: colour (and eventually shape-as-seen) is not a property of physical objects as physics describes them; this leads to the standard structuralist formulation and its Newman troubles. Relativism about common-sense colour vocabulary: ordinary colour talk is a relative form of description, properly contrasted with physics' absolute form; less revisionist, and able to carry many of Russell's striking claims without full subjectivism. Russell never fixed on relativism as his official view (Problems comes closest), but Demopoulos argues many metaphysics/epistemology payoffs can be stated inside it. Primary and secondary qualities get uniform treatment: Russell agrees with Berkeley that relativity arguments hit both, while still aiming at a Lockean-style causal/representational theory overall.
Burnyeat, Dawes Hicks, and conflicting appearances
Demopoulos engages Burnyeat on the "conflicting appearances" argument and notes Russell anticipated Dawes Hicks: the "real" colour/shape are what one would expect to vary under viewing conditions. Yet Russell also argues that such practical analyses fail to establish a "real" colour/shape worthy of the name. The regiment/order passage shows him accepting the converse that order (structure) is physically real. Absolute description is the positive programme; subjectivism about qualities is the cost that later invites Newman.
Historical ancestor, not Worrall twin
Demopoulos: Russell reaches structuralism via constructions and neutral monism, not only via Fresnel-Maxwell continuity. Modern epistemic structural realists are heirs with different arguments; Newman still challenges them (Ainsworth 2009). Absolute description remains the aspirational name of the programme.
From Knowledge by Acquaintance to Knowledge by Causation
The foundationalist architecture of Problems
The Problems of Philosophy (1912) divides knowledge of things (by acquaintance, by description) from knowledge of truths (intuitive, derivative). The real structure is a chain of dependence with acquaintance at the base, "perfect" and "complete" and immune to doubt.
The dependence structure of knowledge in Problems (Baldwin's reconstruction).
Baldwin finds three cracks.
- Acquaintance is secretly propositional. If I am acquainted with myself and I happen to be the tallest lottery winner, I would be acquainted with that winner. Blocking this requires "acquaintance as", which turns acquaintance into intuitive de re knowledge of truths. Baldwin's repair: separate non-conceptual acquaintance (perception) from knowledge by acquaintance.
- Memory as acquaintance with the past "as past" combines present acquaintance with what is recognised as past, a quasi-perceptual arc that Russell dropped in 1921.
- Knowledge by description becomes private. Each person's description of Bismarck bottoms out in their own sense-data, so no two people can attach the same meaning to their words. The view "imprisons the mind within its own private limits", much like the unthinkable things-in-themselves Russell mocked in Kant.
1913-1918: foundations hardened, then softened
Theory of Knowledge (1913) denies that self-evidence comes in degrees, making Russell more foundationalist than in Problems. It also confesses: "I do not know what the logical objects involved really are", a remarkable admission from the author of PM. Our Knowledge of the External World (1914) applies the maxim "logical constructions are to be substituted for inferred entities", so that knowledge of matter needs only induction over sense-data. Other minds, however, stay a "working hypothesis". The 1918 logical atomism lectures move, under Wittgenstein, towards a linguistic, near-tautological view of logic, while the "logically perfect language" remains "very largely private to one speaker".
Brixton and the naturalist turn
Baldwin's map: where each work leans, foundationalist or causal.
In Brixton prison in 1918 Russell recast his programme: metaphysics from physics, epistemology from psychology. The Analysis of Mind rejects acquaintance as a "mystic union of knower and known": knowing is "a very external and complicated relation, incapable of exact definition, dependent upon causal laws". Pure sensation "is not cognitive". Belief loses its act and its object and becomes a content of words and images, which handles false belief without objective falsehoods. Meaning becomes use: "The relation of a word to its meaning is of the nature of a causal law." Monism of stuff with nomological dualism of laws anticipates Fodor's non-reductive physicalism, and the role given to beliefs and desires is an early functionalism.
Russell's thermometer model of reliable memory (AMi 181-188).
Yet Russell wavers. He sketches reliabilism, then retreats because it seems to miss the "appropriateness" of judgment, rejects intrinsic criteria via the problem of the criterion (later Chisholm's), and settles weakly on knowledge as the ideal outcome of verification. Baldwin's diagnosis: Russell wrongly took reliabilism to be behaviourist. Ramsey's 1929 note "Knowledge" stated the view outright; elsewhere Ramsey wrote "My pragmatism is derived from Mr. Russell".
| Work | Externalist strand | Internalist strand | Which wins |
|---|---|---|---|
| AMi (1921) | Thermometer reliabilism | Own "biography" privileged | Unresolved |
| OOP (1928) | "Man From Without" | "Man From Within", percepts | Internalist corrective |
| IMT (1940) | Opening reliabilist sketch | "Momentary object of perception" as touchstone | Internalist (a regress) |
| HK (1948) | Causal, evolutionary knowledge of inference principles | Private basic data | Externalist for induction |
Human Knowledge: the weak a priori
Induction cannot be a logical principle, because non-projectible predicates (Goodman's later "new riddle") defeat any general rule. So the postulates of scientific inference are contingent, cannot be made probable by experience, and are known only in a causal sense: habits that match causal connections count as knowledge, explained in evolutionary terms and shared with animals. Baldwin calls this the "weak a priori", comparable to Wittgenstein's hinge propositions in On Certainty. Russell finally uses causal knowledge inside critical epistemology, escaping both empiricist circularity and rationalist counterexamples.
Acquaintance as the epistemic bottleneck
Thomas Baldwin tracks the shift from knowledge by acquaintance / knowledge by description (Problems of Philosophy, 1912) to a causal theory of knowledge. Acquaintance was meant to stop the regress of justification: some objects (sense-data, universals, perhaps the self) are known directly. Descriptions extend knowledge beyond acquaintance via denoting and then via quantificational paraphrase. Baldwin shows why the bottleneck tightened: matter, other minds, and the past cannot be objects of acquaintance; constructions proliferate; the multiple-relation theory fails; prison-year naturalism reframes belief as causal.
Baldwin: from acquaintance to causation.
Causation, reliability, and the five postulates
Late Russell treats knowledge as a natural kind of true belief produced by reliable causal chains, constrained by the five postulates of Human Knowledge. Baldwin connects this to later reliabilism without equating them. The postulates (quasi-permanence, separable causal lines, spatio-temporal continuity, structural postulate, analogy) are epistemic licences for inference beyond the given. Enumerative induction alone is insufficient; Goodman's riddle is anticipated. Baldwin notes Fodor's "Special Sciences" (1974) as a descendant conversation about levels, not a Russell exegesis.
Other minds and the analogy postulate
The fifth postulate underwrites inference to other minds and to unobserved events sharing structure with observed ones. Baldwin's reading: Russell never returned to a pure acquaintance foundation; the causal story is the mature epistemology. Critics say the postulates are smuggled metaphysics. Russell accepts that charge in part: philosophy must state the assumptions science already uses.
Knowledge by description: the 1912 machinery
Baldwin restates the Problems distinction. Knowledge by acquaintance is direct presentational knowledge of sense-data, universals, and (Russell then held) oneself. Knowledge by description is knowledge of an object as "the so-and-so" where one is not acquainted with it. Every proposition one understands must be composed wholly of constituents one is acquainted with (Russell's principle). Descriptions let us think about matter and other minds without acquaintance with them, by quantifying over properties and relations we are acquainted with.
Baldwin: 1912 epistemic stack.
Why constructions displaced bare inference
Griffin and Baldwin both note Firth's later point: sense-datum premises never entail material-object conclusions (non-monotonic defeaters). Russell's 1914 constructive programme tries to define matter in terms of sense-data rather than infer it. Baldwin's emphasis: even constructions leave a causal-epistemic story about why this construction rather than another tracks the world. That story matures into the postulates of 1948.
Reliabilism avant la lettre
Baldwin declines to call Russell a reliabilist in Goldman's sense, but the late view is close: knowledge is true belief produced in ways that track fact, underwritten by causal continuity and structural similarity. The five postulates articulate the tracking rules. Brixton months (1 May-14 September 1918) matter biographically: prison work on propositions and belief starts the naturalist turn. Baldwin's Companion role is to make the epistemic career one story, not a 1912 book plus unrelated late books.
Russell's principle and its strain
Baldwin foregrounds the principle: every proposition one understands is composed only of constituents one is acquainted with. Combined with the constituent thesis (objects themselves occur in propositions), this forces descriptions to do enormous work for matter, other minds, and the past. When the multiple-relation theory fails and propositions return in a new guise (1919), the principle is rewritten rather than kept intact. Knowledge becomes a causal-natural phenomenon with logical constraints, not a crystalline acquaintance pyramid.
Baldwin: principle under repair.
Five postulates listed in full
- Quasi-permanence: things persist a while. 2. Separable causal lines: processes can be tracked. 3. Spatio-temporal continuity: causes and effects are not wildly discontinuous. 4. Structural postulate: similar structures license similar inferences. 5. Analogy: other minds and unobserved cases. Baldwin treats them as the mature replacement for hoping acquaintance would reach everywhere.
From Problems to Human Knowledge: one epistemic career
Baldwin's through-line: the 1912 distinction is the start, not the doctrine. Descriptions extend knowledge; constructions try to replace risky inferences; naturalism relocates belief; postulates license non-demonstrative steps. Firth's non-monotonicity point (sense-data never entail matter) explains why bare inference was unstable. Brixton months accelerate the naturalist turn.
Baldwin: compact epistemic career.
Sense-data premises never entail matter
Baldwin (with Griffin's Firth point) makes non-monotonicity central. Even if sense-data S seem to support matter M, further premise P (odd lighting, hallucination) can defeat the inference. Entailment fails; inference to the best explanation or construction remains. OKEW (1914) prefers construction: define physical objects in terms of sense-data classes and series rather than infer a transcendent cause. Baldwin notes the construction still needs a causal-epistemic story about why this construction tracks the world; that story matures into HK.
Baldwin: entailment fails, method continues.
Other minds and the analogy postulate
The fifth postulate underwrites belief in other minds: structurally similar behaviour and physiology license analogous mental ascriptions. Baldwin refuses both behaviourist collapse and telepathic acquaintance. Prison-year work on image and word propositions (On Propositions, 1919) makes belief a natural relation among facts and symbols, preparing the causal story.
Knowledge-first and reliabilist descendants
Baldwin links forward to Goldman-style reliabilism and Williamson's knowledge-first programme without equating them. Evans's "Russell's principle" (1982) is the acquaintance-era constraint reborn in singular thought debates. Companion readers doing contemporary epistemology should treat Ch13 as genealogy, not museum.
| Stage | Knowledge of matter | Knowledge of minds |
|---|---|---|
| 1912 | Description / inference | Analogy thin |
| 1914 | Construction from data | Still thin |
| 1921 | Causal / neutral | Behaviour + images |
| 1948 | Postulates + structure | Analogy postulate |
Causal chains and graded belief
Baldwin's late Russell treats knowledge as true belief produced by reliable causal chains, constrained by postulates. Certainty is uncommon; rational belief comes in degrees. The Inquiry (1940) and HK (1948) are the mature books. Brixton work starts the naturalist turn; it does not finish it. Descendants: reliabilism, knowledge-first, singular-thought debates around Evans's Russell's principle.
Baldwin: graded belief.
Russell, Experience, and the Roots of Science
Explaining knowledge, not defending it
Grayling calls Russell "thoroughly Lockean": scepticism is irrefutable but there is "not the slightest reason" to think it true, so no time is spent on it. "Justifying science by grounding it in experience" and "showing how physics succeeds in being an empirical science" are different projects, and Russell's was the second. This makes him "a naturalist long before Quine", though one who refused to premise epistemology on science.
Two corollaries follow. Hylton's charge of "considerable concessions to psychologism" misdescribes a philosopher who was "realist to excess". And the Wittgenstein "derailment" of Theory of Knowledge (1913) may be overstated: the 1917 reprint of "Knowledge by Acquaintance" says the multiple relation theory is "somewhat unduly simple", but its modification does not affect the project.
Five strategies in four decades
Grayling's sequence of strategies for bridging sense and science.
| Strategy | What matter is | Why it was dropped or limited |
|---|---|---|
| Inference (POP) | Hypothesised cause of sense-data, believed by "instinct" | Only "more or less probable opinion" |
| Construction (1914) | Function of actual and possible sensibilia | "Unsensed sense-data" problematic; private and public spaces too complex |
| Neutral monism (1921) | Unsensed percepts arranged causally | Consciousness remains the clear mental mark; tension with inferred causes |
| Structural realism (1927) | Event clusters known only by structure | Newman (1928): structure alone is trivial |
| Postulates (1948) | Event chains, inferred under five postulates | Justified only by success; little response |
Sensibilia and the supreme maxim
Problems infers physical objects as the simplest explanation of sense-data and adds coherence. Our Knowledge of the External World (1914) enriches the base with memory and spatio-temporal relations ("hard data"), and builds public space using others' testimony. "The Relation of Sense-Data to Physics" applies the "supreme maxim": "wherever possible, logical constructions are to be substituted for inferred entities." Objects become constructions from sensibilia, actual or possible appearances that are not mental but "the ultimate constituents of the physical world". Grayling notes that My Philosophical Development later gives this a more phenomenalist gloss than the original, and that Russell abandoned construction before Carnap's Aufbau (1928), Goodman (1951) and Wisdom's critique (1931-33).
Neutral monism and its idealist tilt
Russell rejected James's phrase "pure experience" as residually idealist and preferred "neutral stuff", since stars and stones are not experiences. He also held that some items (images, feelings) are only mental, distinguished by psychological causal laws like association. The result, Grayling says, is "surprisingly" idealist: minds are sensed percepts and matter "a logical fiction constructed of unsensed percepts", hence "brains consist of thoughts". Late in life Russell repaired one inconsistency: an event "is both mental and material at once" if it has both kinds of causal relation (Portraits from Memory).
The event picture of The Analysis of Matter: percepts as parts of things.
Grayling also repeats the Newman objection: different worlds could share one abstract structure, so structural knowledge cannot pick out the real one. Russell wrote that his claims were "either false or trivial, and I am somewhat ashamed not to have noticed it myself".
The five postulates of Human Knowledge
Russell's five postulates (HK 506-12) with his examples.
These postulates license only the first steps towards science. We "know" them as adapted habits: "owing to animals being adapted to their environment, occurrences which are, in fact, evidence of others tend to arouse expectation of those others". Grayling's verdict is severe: the justification is past success, "exactly the kind of unbuttressed inductive inference" that Russell's own chicken, fed daily until the butcher came, warns against. HK has "a Ptolemaic tinge". He suggests turning the postulates into transcendental conditions of inquiry, and notes the parallel with Wittgenstein's On Certainty.
Experience as the root, not the whole, of science
A. C. Grayling (TOC author for ch. 14; not Hatfield) argues that Russell keeps experience as science's root while denying that science reduces to reports of experience. Sense-data and later events are starting-points for constructions; the finished sciences include inferred entities licensed by structure and postulates. Grayling places Russell against both radical empiricism (all meaning from sense) and pure rationalism. The "roots of science" metaphor: cut the roots and the tree dies; mistake the roots for the whole tree and you get positivism.
Grayling: experience, construction, science.
Scepticism, certainty, and Wittgenstein
Grayling has written separately on Wittgenstein on scepticism and certainty; the Companion chapter uses that expertise to contrast Russell's engagement with scepticism. Russell takes sceptical scenarios (five-minute hypothesis, dreams) seriously enough to answer, then answers by method (constructions, postulates), not by ordinary-language dissolution. Certainty is rare; rational belief is graded. Grayling links this to Quinean themes (naturalized epistemology) while warning against reading Quine back into 1912.
Why science needs philosophy
For Grayling's Russell, philosophy clarifies the logical form of scientific claims and inventories their epistemic debts. It does not compete with physics. The chapter's closing stress: Russell's scientism is critical, not deferential. Science is the best bet (MPD 17) and still needs philosophical hygiene about structure, induction and meaning.
Sense-data to events: Grayling's continuity thesis
Grayling's "consistency thesis" says Russell's changes of ontology do not abandon the claim that experience roots science. Sense-data (1912-14), sensations (1921) and events (1927) are successive candidates for the experiential base. Constructions and structural physics are the trunk and branches. Critics (including Baldwin in ch. 13) see more wavering than Grayling allows.
Grayling: roots metaphor.
Against psychologism charges
Grayling defends Russell against Hylton-style worries that neutral monism collapses philosophy into psychology. Logical form, type restrictions and the objectivity of facts remain. Psychology supplies the causal story of belief; it does not set truth-conditions. Whether this defence succeeds is open; Grayling's chapter is openly partisan, as Hager's is on method.
Quine, naturalism, and the limits of reading back
Grayling cites Quinean naturalism as a grandchild, not a twin. Russell never drops the analytic ambition to clarify logical form; Quine historicises and holisms that ambition. Note-level citation fixes (Glock volume is Blackwell not Routledge; Quine volume title is Knowledge, Language and Logic) are in the A closer look. The philosophical moral stands: experience roots science; science exceeds experience; philosophy states the excess.
Primary and secondary qualities, structuralised
Grayling links Russell's structural physics to the old primary/secondary quality split. Physics keeps structure (primary-ish); intrinsic natures retreat (secondary-ish, or neutral). Experience still roots the assignment of at least some qualities to the mental/neutral side of the causal chain. The five-minute hypothesis and dream scepticism are answered by method, not by claiming infallible sense-data forever.
Grayling: qualities and roots.
Consistency thesis vs wavering thesis
Grayling (consistency) vs Baldwin (real wavering) is an intra-Companion dispute. Insight keeps both: continuity of aim (experience roots science) can coexist with discontinuity of machinery (acquaintance vs cause). Readers should not flatten either chapter into the other.
Experience as root: Grayling's full claim
Grayling: roots, trunk, canopy.
Scepticism answered by method
Five-minute hypothesis, dreams, evil-demon analogues: Russell takes them seriously enough to answer. Answers are constructive and postulational, not ordinary-language dissolutions. Certainty is rare; rational belief is graded. Grayling contrasts this with Wittgensteinian quietism about scepticism (his own separate work on Wittgenstein on scepticism and certainty informs the contrast). Quinean naturalism is a grandchild: aligned in respecting science, divergent in dropping logical-form hygiene as Russell practised it.
Intra-Companion dispute with Baldwin and Hylton
Grayling's consistency thesis sits uneasily with Baldwin's wavering thesis (Ch13). Grayling's anti-psychologism defence sits uneasily with Hylton's worries (Ch6). A closer look already marks both. Citation fixes (Glock = Blackwell not Routledge; Quine volume title) stay in the lens.
Primary/secondary qualities under uniform relativity
Grayling aligns with Demopoulos's observation that Russell treats shape and colour alike under conflicting appearances: both fail to be straightforward physical properties as naive realism paints them. Structure (order) survives; qualitative specifics retreat to the experiential side or to neutral events' intrinsic natures. The roots metaphor again: experience delivers what physics cannot; physics delivers absolute structure experience cannot.
Critical scientism
Grayling: Russell's scientism is critical, not deferential. Science is the best bet (MPD 17) and still needs philosophical hygiene about structure, induction and meaning. Experience roots; theories exceed; philosophy states the excess. Consistency thesis vs Baldwin's wavering remains an open Companion dispute.
Bertrand Russell: Moral Philosopher or Unphilosophical Moralist?
Why Russell's ethics was ignored
Moral philosophy has three tiers: metaethics (what moral judgments mean and what makes them true), normative ethics (what is good, what makes right acts right) and practical ethics (war, marriage, education). From 1913 Russell held that judgments of intrinsic good are optatives, "Would that everybody desired X!", and so neither true nor false. Since philosophy aims at truth, he concluded that only metaethics belongs to it: "I should like to exclude all value judgments from philosophy". The austere conception caught on, which is why Searle's metaethics classes at Berkeley were disrupted by students who wanted to discuss Vietnam.
Russell offered two incompatible excuses. First, Principles of Social Reconstruction (1916) "was not intended as a contribution to learning". Pigden replies that Locke's Second Treatise and Mill's On Liberty were practical works too, and both count as philosophy; a popular argument may hide its "belt and braces" but must have them available. Second, the factual part belongs to science. Pigden replies that science rarely settles policy, as shown by the death penalty in a poor country, where economics and estimates of joy and sorrow go beyond the established facts.
Case study: Hobbes and world government
From 1914 to 1964 Russell argued that lasting peace requires world government. In 1950 he predicted that before 2000 there would be human extinction, a reversion to barbarism, or a single government with a monopoly of major weapons. Pigden traces the argument to Hobbes: states are "artificial men" in a state of nature, and "every argument that [Hobbes] adduces in favour of government ... is valid in favour of international government" (HWP).
Russell's Hobbesian argument for world government, as Pigden reconstructs it.
Russell rejected Hobbes's biological determinism and denied that states simply copy their rulers: Louis XIV would suffer differently losing an army at Blenheim than an arm. But he held that ruling classes are especially moved by acquisitiveness, fear and the love of power, so states behave like Hobbesian individuals.
Pigden, following Kavka (1986) and Hampton (1986), argues that the argument fails, because without exact settings there is "a frightful risk of peace breaking out".
Kavka's three conditions for a Hobbesian war, and how states fail them.
A second objection turns Hobbes's contract problem against Russell: "If a world government is possible, it is not necessary, and if it is necessary, it is not possible." For Hobbes, who defends an existing sovereign, the puzzle is retrospective; for Russell it is fatal because his case is prospective. Pigden's point is that such critique, drawing on game theory, history and psychology, "certainly looks like philosophy".
The six phases of Russell's ethics
Pigden's six phases of Russell's ethical theory.
Phase 1. Under Sidgwick, Russell defined good as "what we desire to desire" (revived without credit by David Lewis in 1989). Pigden suggests this definition, read secretly to the Apostles, may have prompted Moore's Open Question Argument (Principia Ethica §13). Russell's attempt to solve Sidgwick's dualism of practical reason with McTaggart's and then Bradley's Absolute failed in "Seems, Madam? Nay, It Is", because an unexperienced unity gives no reason to be good. Pigden proposes that this failure fed the revolt against Hegelianism.
Phase 2 was genuinely Moorean ("The Elements of Ethics", 1910), though Russell held that what we ought to do is what we have reason to think will be best. Phase 3 began when Santayana's Winds of Doctrine (1913) destroyed his belief in the Moorean good; the war gave him "a disgust for all ethical notions which evidently are chiefly useful as an excuse for murder". Phase 4 is the unpublished 1922 Apostles paper: ethical judgments "claim objectivity", and "this claim ... makes them all false", with "good" treated like "the present King of France". That anticipates Mackie's error theory (1946, 1977), followed by a brief "humanistic amoralism".
| Feature | Russell (1935) | Ayer (1936), Stevenson (1937) |
|---|---|---|
| Meaning of "X is good" | "Would that everyone desired X!" | Expresses approval of X |
| Moral contradiction | Free: two optatives that cannot both be fulfilled | Needs special "curlicues" |
| Logical consequence | B follows from A given C iff A cannot be realised under C unless B is | Notoriously lacking |
| Circularity | None: defined via desire | Approval presupposes thinking X good |
Phase 5 gives the optative emotivism of Religion and Science (1935), a year before Ayer. It let Russell moralise "with a clean intellectual conscience": the moralising emotivist uses moral language honestly, while the moralising error theorist is a hypocrite. Phase 6, Human Society in Ethics and Politics (written 1946, published 1954), sought intersubjective truth and failed. In 1960 Russell wrote that he could not believe "that all that is wrong with wanton cruelty is that I don't like it", and remained perplexed.
Six phases, not two: Pigden against Sainsbury
Charles Pigden vindicates Russell as a moral philosopher against the view (Sainsbury and others) that his ethics had only two derivative phases. Much work was unpublished in Russell's lifetime; some of the most original pieces appeared as asides. Pigden's six phases:
- 1889-1903. Sidgwick's problems; "good" as what we desire to desire; Moore's Open Question Argument may answer Russell; Argument from Advocacy in early Principia Ethica drafts.
- Mooreans. Non-natural good; intuition; Principia Ethica influence; Russell's "The Elements of Ethics".
- Error theory anticipation. Moral judgments factual but false; anticipates Mackie; text "Is There an Absolute Good?" published Russell n.s. 6 (1986-87), written earlier.
- Emotivism pioneer. Anticipates Ayer and Stevenson by ~20 years; Religion and Science ch. ix now in Rachels' Ethical Theory (1998).
- Middle practical ethics. Marriage, war, education, Beacon Hill; Marriage and Morals (1929); unpopular essays.
- Late humanism. Human Society in Ethics and Politics (1954); consequentialist public reason; anti-nuclear politics tied to moral argument.
Pigden: six phases of Russell's moral philosophy.
Practical ethics and the double injustice
Russell helped create a conception of philosophy that excluded his own practical ethics; later philosophers liberalised the conception but still ignored him. Pigden's "double injustice": theoretical ethics neglected because unpublished or scattered; practical ethics neglected because "unphilosophical". The Companion chapter is corrective. Emotivism and error theory place Russell in the genealogy of twentieth-century metaethics; the political essays place him in public moral argument from World War I through the Einstein-Russell Manifesto (1955).
Desire to desire, Open Question, and influence both ways
Pigden stresses mutual influence with Moore. Russell's naturalist definitions of good are among those Moore attacks; the need to block them may have pushed Moore toward the Open Question Argument. Russell then becomes, for a time, more Moorean than Moore's later self. The swing to emotivism abandons non-natural properties without returning to the 1890s desire theory in simple form: attitudes replace intuitions.
Emotivism in Religion and Science chapter ix
Pigden's exhibit A for originality is the emotivist chapter in Religion and Science (1935). Moral judgments express attitudes and aim to influence attitudes; they are not truth-apt reports of non-natural facts. This anticipates Ayer's Language, Truth and Logic (1936) and Stevenson's mid-century work. Inclusion in Rachels' Ethical Theory (1998) finally put the text in students' hands. Sainsbury's two-phase story cannot survive contact with this material plus the unpublished error-theoretic pieces.
Pigden: metaethical toolkit across phases.
Error theory before Mackie
"Is There an Absolute Good?" argues that absolute ethical predicates do not pick out real properties; judgments using them are systematically misguided. Mackie's Ethics: Inventing Right and Wrong (1977) made error theory famous; Russell's piece, written decades earlier, appeared in Russell n.s. 6(2) Winter 1986-87 (Pigden's "1988" is flagged). Pigden's claim is priority of idea, not of fame.
Practical ethics: war, marriage, education, nuclear weapons
Principles of Social Reconstruction, Marriage and Morals (1929), Beacon Hill, the anti-nuclear campaigns and Has Man a Future? apply a broadly consequentialist, anti-cruelty ethic. Kavka-style deterrence puzzles appear in Pigden's discussion of nuclear morals; A closer look notes the 1990s flavour of "dominators are easy to identify". The Einstein-Russell Manifesto (1955) is moral argument in public: common human survival over bloc ideology. Pigden refuses to call this "unphilosophical"; it is applied moral philosophy with a metaethical backbone.
| Phase | Metaethic | Public face |
|---|---|---|
| 1 Desire | Naturalist definition | Apostles ethics |
| 2 Moorean | Non-natural good | Elements of Ethics |
| 3 Error | Factual falsehood | Unpublished then |
| 4 Emotive | Attitudes | Religion and Science |
| 5 Practical | Mixed | Marriage, schools, war |
| 6 Humanist | Consequentialist public reason | HSEP, CND, Manifesto |
Sidgwick's dualism of practical reason
Pigden's Phase 1 is driven by Sidgwick's clash between duty and prudence (or benevolence and rational self-interest). Reid had held that under a wise divine administration the two cannot finally conflict. Sidgwick, having lost theistic belief, could not rule out a clash: the "Cosmos of Duty" might become a "Chaos". Russell tried first to solve this with Hegelian tools, via McTaggart's Further Determination of the Absolute (1893): reality is spirit; harmony of self and universe will become explicit. That metaphysical patch failed with the 1898-99 break. Desire-based definitions of good were the next try; Moore blocked them; Moorean intuitionism followed; then error theory and emotivism.
Pigden: from Sidgwick's clash to emotivism.
Phase detail: Moorean years and the Elements
Russell's "The Elements of Ethics" and related work treat good as non-natural and known by intuition, in line with Principia Ethica. He later judged intuitionism unstable: disagreement among "intuitive" moralists has no tribunal. That instability feeds the swing to non-cognitivist and error-theoretic options. Pigden's archival point: some of the sharpest metaethics stayed unpublished, so textbook Russell looked thinner than manuscript Russell.
Public morals without "unphilosophical" stigma
Consequentialist threads run through opposition to World War I, Marriage and Morals, education experiments, and anti-nuclear argument. Pigden's rehabilitative claim: once metaethics is restored (six phases), the public writings read as applications, not as a separate unphilosophical career. The Companion's ethics chapter is therefore also a thesis about how analytic philosophy wrote its own history of Russell.
Why error theory and amoralism were abandoned
Pigden reconstructs Russell's pullback. Error theory plus humanistic amoralism (get by with friendly feelings and enlightened self-interest) did not last. In RTC Russell says no amount of logic would make him give up feeling and expressing ethical passions; yet if error theory were correct, moralizing would be mouthing falsehoods. Pigden's conjecture: Russell could not accept a theory that made moralizing disreputable when moralizing was by then a chief source of income. On amoralism: schoolteaching showed friendly feelings unreliable and self-interest not always enlightened; Bolshevik experience showed that belief in the "subjectivity of ethics" is compatible with cruelty and persecution (ROE, Interlude 1). He pulled back from full Garner/Hinckfuss-style amoralism but kept a dark epigram: "The purpose of morals is to allow people to inflict suffering without compunction" (Fact and Fiction 184), half-believed under La Rochefoucauld cover.
Pigden: exit from error theory.
1935 emotivism superior to Ayer/Stevenson
Phase 5 (1923-1945): after confused subjectivist stumbles, Russell publishes a sophisticated emotivism in Religion and Science ch. ix (1935), one year before Ayer's LTL (1936) and two before Stevenson's 1937 paper. Possibly aware of W.H.F. Barnes 1933, possibly not. Buried at the back of a book on science vs religion, then again toward the back of Power (1938), so it excited little comment; Urmson (1968) and Warnock seem unaware.
Russell's version descends from the 1897 desire-to-desire theory but shifts mood: "X is good" means "Would that everyone desired X!" (optative), not a report that someone desires to desire X. Advantages Pigden stresses: (1) Moral contradiction without curlicues: "Would that everyone desired X" and "Would that nobody desired X" cannot both be fulfilled. (2) A consequence relation for optatives: B follows from optatives A plus propositions C iff A cannot be realized under C unless B is. (3) No vicious circle: Ayer/Stevenson define good via approval, and approval is thinking good; Russell defines via desire, not thought-saturated approval. (4) Clean conscience for moralizing: emotivist moral discourse exists to express desires about mankind's desires; feeling those desires licenses expressing them. Error-theoretic moralizers are hypocrites or dealers in useful fictions; emotivist moralizers use the tool as designed.
Residual unease and the last metaethic
Russell still felt, when moralizing, that his desires were somehow right (ROE 149). His last metaethical effort tries to honour that feeling without returning to Moorean non-natural properties. Pigden's six-phase story ends in humanist consequentialist public reason (HSEP 1954) rather than in a final technical masterpiece. The Companion claim stands: this is original metaethics, not two derivative footnotes.
Phase 6 and public reason
Pigden's late phase: Human Society in Ethics and Politics (1954), anti-nuclear argument, Einstein-Russell Manifesto (1955). Metaethic settles near emotivism/humanist consequentialism; public argument appeals to shared survival and anti-cruelty. The "unphilosophical moralist" label dies once six phases are on the table. SEP entry (Pigden; rev. 3 May 2026) is the living continuation of this Companion chapter.
Pigden: late public morals.
Double injustice named
Pigden: Russell's metaphilosophy helped exclude his practical ethics; later philosophy liberalised the subject but still ignored him. Theoretical ethics neglected because unpublished/scattered; practical ethics neglected as "unphilosophical." Six phases and the 1935 emotivism exhibit undo both injustices. SEP rev. 3 May 2026 continues the vindication.
Main Points
Twenty-two takeaways
- One project, many phases. Russell changed doctrines often but kept one aim: a comprehensive account of the world, consistent with the best science, reached by analysis (Intro, chs. 9, 14).
- Analysis runs in two directions. Work backwards from accepted results to fewer, clearer premises, then forwards to rebuild and extend them; premises are believed for their consequences (chs. 5, 9).
- Paraphrase is the real turn. The 1905 theory of descriptions made logical form something to be found by rephrasing, which is what analytic philosophy inherited (chs. 4, 6).
- Paradox drove the logic. Russell's paradox (1901) led through denoting concepts, substitution and the ramified types of Principia; each was a way to keep logic universal without contradiction (chs. 1, 7, 8).
- Constructions replace inferred entities. The "supreme maxim" turned matter, number and mind into logical constructions where possible, and into structural inference where not (chs. 10, 11, 12, 14).
- The late epistemology is causal and humble. From 1921 knowledge rests on reliable causal links and on postulates known only in an "animal" sense, an early externalism (chs. 13, 14).
- The Hegelian start mattered. The "contradiction of relativity" sank Russell's plan for a dialectic of the sciences; in 1898 he reversed the argument into a reductio of internal relations (ch. 2).
- Moore's new philosophy was unstable. Propositions as mind-independent complexes and truth as a simple property could not explain falsehood or the unity of the proposition (ch. 3).
- Logicism was mathematics first. Cantor, Peano and the Paris Congress of August 1900 shaped it; Principia derived far less of mathematics than "all" (ch. 1).
- Frege and Russell met at Axiom V. The letter of 16 June 1902 exposed the paradox in Frege's system; the two shared quantification and the ancestral but differed on sense (ch. 4).
- Logicism was not foundationalism. Russell's regressive method makes even logical axioms supported inductively by what they yield, so "Why believe this?" differs from "What are the fewest premises?" (ch. 5).
- Descriptions removed denoting concepts. Meinong's golden mountain was already handled; the gain was eliminating an unexplained representational relation (ch. 6).
- Substitution was the missing link. The type-free theory of 1905-07 explains Principia's variables; the po/ao paradox killed it (ch. 7).
- Types failed as foundation but won in computing. Reducibility was ad hoc and the ramified theory a "relic", yet Ramsey's simple types and Church's 1940 system lead to HOL, Lean and Rocq (ch. 8).
- Neutral monism stayed half-dualist. The 1913 attack on James was strong; the later monism of events still spoke the language of first-person experience (ch. 10).
- Logical atomism is metaphysics guided by logic. Atomic, molecular, general and negative facts; constructions aimed at theorems, not a nominalist purge (ch. 11).
- Structure alone is trivial. Newman (1928) showed that knowing only the abstract structure of the world says little more than how many things there are; Russell conceded the point (chs. 12, 14).
- Acquaintance gave way to causation. Baldwin traces a reliabilist turn from Analysis of Mind; the thermometer becomes the model of a knower (ch. 13).
- Science needs postulates. Quasi-permanence, separable causal lines, continuity, structure and analogy license induction but cannot be proved from data (Intro, chs. 9, 14).
- Russell pioneered emotivism and the error theory. The 1922 Apostles paper anticipated Mackie; the 1935 optative emotivism anticipated Ayer and handled moral contradiction better (ch. 15).
- His practical ethics is philosophy. A factual "ought" indexed to ends, and the Hobbesian argument for world government, can be assessed and criticised (ch. 15).
- The narrow canon misleads. Reading only 1905-1918 hides the Hegelian start, the archive work on substitution, the late epistemology and the ethics (Intro, chs. 7, 13, 15).
How the takeaways connect: one method applied to logic, the world, knowledge and value.
Frameworks and mental models
| Name | What it says | Use it when |
|---|---|---|
| Regressive method | Premises are justified by the results they explain | Choosing axioms, controls or principles for a system already in use |
| Supreme maxim | Construct where you can; infer only where you must | Deciding whether a new entity in a model is needed |
| Paraphrastic analysis | Surface grammar can mislead; find the logical form | Turning business language into a data model or a rule |
| Theory of descriptions | "The F is G" asserts existence, uniqueness and predication | Handling missing or duplicate records in queries |
| Vicious circle principle | No totality may contain members defined by reference to itself | Designing type systems, schemas and self-referential rules |
| Minimum vocabulary | Find the fewest undefined terms from which the rest follow | Building a canonical data model or ontology |
| Structural inference and Newman | Structure alone underdetermines content | Testing whether a latent model is identifiable |
| Five postulates | Induction needs prior assumptions about the world | Stating a model's inductive bias and drift risks |
| Optative logic | Wishes entail wishes when one cannot be met without the other | Checking goals or policies for consistency |
One project, many tools
Across fifteen chapters the Companion supports Griffin's thesis: Russell's phases are attempts at one science-first system of the world. Logicism, descriptions, types, constructions, atomism, neutral monism and the five postulates are tools, not disconnected conversions. When a tool fails (substitution, multiple-relation judgment, pure structure), Russell replaces it without dropping the aim.
Synthesis: tools in order of appearance.
Open problems the Companion leaves
Newman vs structuralism; whether reducibility is logic; how much of metaethics is unpublished still; whether late naturalism abandons or fulfils the 1898 break. Insight's job is to present each author's answer with evidence, not to referee a final verdict.
Disputes inside the Companion
Insight's synthesis flags contributor disagreements already noted in A closer look: Hylton vs Landini on "Insolubilia" and orders; Grayling vs Hylton on psychologism; Baldwin vs Grayling on epistemic wavering; Hager vs Monk on analysis and biography.
Intra-Companion fault lines.
What still sits outside
Politics, education, religion, and much of the public Russell stay in Griffin's Introduction. A future Companion might add chapters; this one chose logic, mind, science, ethics.
DEPTH PASS reading order
For epistemology: Baldwin → Grayling → Tully. For metaphysics of atomism: Linsky → Hager.
Who's who
| Name | Role in the book |
|---|---|
| Bertrand Russell (1872-1970) | Subject; logician, philosopher, campaigner, Nobel laureate 1950 |
| Alfred North Whitehead | Co-author of Principia Mathematica (1910-13) |
| G. E. Moore | Partner in the revolt against idealism; Principia Ethica (1903) |
| Gottlob Frege | Fellow founder of modern logic; Axiom V and the 1902 letter |
| Giuseppe Peano, Georg Cantor | Notation and set theory behind logicism |
| F. H. Bradley, J. M. E. McTaggart | Idealists of Russell's apprenticeship |
| Ludwig Wittgenstein | Critic of the 1913 theory of judgment; Tractatus |
| Frank Ramsey | Simple theory of types; Ramsey sentences |
| Max Newman | 1928 objection to structural realism |
| William James | Neutral monism, "pure experience" |
| Henry Sidgwick, George Santayana | Shaped Russell's early ethics and its collapse |
| Nicholas Griffin | Editor; Introduction and chapter 2 |
| Ivor Grattan-Guinness | Ch. 1 mathematics and logic to 1903 |
| Richard Cartwright | Ch. 3 Principles and ontological argument |
| Michael Beaney | Ch. 4 Russell and Frege |
| Martin Godwyn and A. D. Irvine | Ch. 5 Bertrand Russell's logicism |
| Peter Hylton | Ch. 6 theory of descriptions |
| Gregory Landini | Ch. 7 Russell's substitutional theory |
| Alasdair Urquhart | Ch. 8 theory of types |
| Paul Hager | Ch. 9 method of analysis |
| R. E. Tully | Ch. 10 Russell's neutral monism |
| Bernard Linsky | Ch. 11 logical constructions and logical atomism |
| William Demopoulos | Ch. 12 structuralism about physics |
| Thomas Baldwin | Ch. 13 knowledge by acquaintance and description |
| A. C. Grayling | Ch. 14 Russell on science and religion / late epistemology |
| Charles Pigden | Ch. 15 Russell's moral philosophy |
| Ch | Author | Spine claim |
|---|---|---|
| Intro | Griffin | One project, six phases, science-first |
| 1 | Grattan-Guinness | Paradox out of Cantor/Peano line |
| 2 | Griffin | Idealism to external relations 1898 |
| 3 | Cartwright | Principles ontology and denoting |
| 4 | Beaney | Frege reception corrected |
| 5 | Godwyn and Irvine | Regressive logicism |
| 6 | Hylton | Descriptions as paraphrase turn |
| 7 | Landini | Substitution before types |
| 8 | Urquhart | Ramified types and reducibility |
| 9 | Hager | Analysis two directions |
| 10 | Tully | Neutral monism 1921 |
| 11 | Linsky | Constructions and atomism |
| 12 | Demopoulos | Structure and Newman |
| 13 | Baldwin | Acquaintance after 1912 |
| 14 | Grayling | Science, religion, postulates |
| 15 | Pigden | Ethics from Moore to emotivism |
About this book
The Cambridge Companion to Bertrand Russell by Nicholas Griffin (ed.), 2003. 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.
Insight Editions · rajeshrmahapatra.com/insight-editions
- One project, many phases. Russell changed doctrines often but kept one aim: a comprehensive account of the world, consistent with the best science, reached by analysis (Intro, chs. 9, 14).
- Analysis runs in two directions. Work backwards from accepted results to fewer, clearer premises, then forwards to rebuild and extend them; premises are believed for their consequences (chs. 5, 9).
- Paraphrase is the real turn. The 1905 theory of descriptions made logical form something to be found by rephrasing, which is what analytic philosophy inherited (chs. 4, 6).
- Paradox drove the logic. Russell's paradox (1901) led through denoting concepts, substitution and the ramified types of Principia; each was a way to keep logic universal without contradiction (chs. 1, 7, 8).
- Constructions replace inferred entities. The "supreme maxim" turned matter, number and mind into logical constructions where possible, and into structural inference where not (chs. 10, 11, 12, 14).
- The late epistemology is causal and humble. From 1921 knowledge rests on reliable causal links and on postulates known only in an "animal" sense, an early externalism (chs. 13, 14).