Insight Editions · No. 77
An Introduction to Ontology
Nikk Effingham
Nikk Effingham introduces ontology through debates about holes, properties, numbers, possible worlds, space, time, composite objects, and the methods philosophers use to assess what exists.
A study guide to An Introduction to Ontology by Nikk Effingham. 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 design is a double ladder. One ladder climbs through categories, from abstract (properties, numbers, worlds) to concrete (regions, past things, composites, works). The other climbs through metaontologies: Quine's criterion, metaphysical explanation, Meinongianism, fictionalism, scientific indispensability, truthmaking, Neo-Carnapianism, fundamentality. Each chapter is a worked example of method meeting subject.
The book's double ladder: categories of entity, each paired with a method.
The book's tone is part of its method. Effingham writes in the first person, jokes constantly (Mortgloriapesamplitology, the Flergbet, the Drowning-o-matic, Kevin the polyp) and repeatedly hands questions back to the reader ("I leave it to you"). Behind the jokes is a serious pedagogical claim: students learn ontology best by watching a method fail and succeed on a concrete case, then carrying it elsewhere. The chapters can be read singly, but each assumes the toolkit of Chapters 1 and 2.
| Ch | Category | Method introduced | Test case or argument |
|---|---|---|---|
| 1 | Abstracta in general | Sense versus nonsense | Russell-style refutation of permissivism |
| 2 | Holes | Theory choice; Quine's criterion | Four theories of holes (Table 2.1) |
| 3 | Properties | Metaphysical explanation | Problem of universals |
| 4 | Numbers | Reduction; indispensability; Meinongianism | Benacerraf's problem |
| 5 | Possible worlds | Fictionalism | Island universes; Brock-Rosen |
| 6 | Space | Scientific explanation | Newton's bucket; Leibniz shift |
| 7 | Past and future things | Truthmaking | Relativity of simultaneity |
| 8 | Composite objects | Neo-Carnapianism | Cultural prejudice; vagueness |
| 9 | Statues and lumps | (none new) | Paradox of constitution |
| 10 | Works of music | Fundamentality | Dodd, Levinson, Cameron |
Since 2013 three shifts matter most for readers of this book. First, grounding and fundamentality, which the book treats last, now frame much of the field. Second, philosophy of physics has moved toward emergent spacetime, changing the stakes of Chapters 6 and 7. Third, deflationary metaontologies (easy ontology, quantifier variance) have matured into full research programmes. Effingham's own later work, on time travel, presentism and constitution, extends several chapters directly.
About this book
An Introduction to Ontology by Nikk Effingham, 2013. This is a chapter-by-chapter study of the book, with summaries, maps, tables and diagrams; short quotations keep their page references. It is shared free for personal study, is not for sale, and is not endorsed by the author or publisher. Please read and buy the original.
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How This Book Works
The pairing reflects a conviction: in ontology "the methodology is itself a prime suspect" behind many problems, and it is under renewed scrutiny. Chapter 1 asks whether ontology makes sense; Chapter 2 adds Quine's theory of ontological commitment and theory choice, tested on holes. Then come properties (Ch3), numbers with Meinongianism (Ch4), possible worlds (Ch5), space (Ch6), time (Ch7), composite objects with Neo-Carnapianism (Ch8), the statue and the lump (Ch9, the one chapter without a new method) and works of music with fundamentality (Ch10).
The book's design: each chapter pairs a category with a method.
Arrangement and notation
Except for Chapter 1, each chapter takes a category and a method. Chapter 9 breaks the pattern: material constitution is already thick with methods from earlier chapters, so no new metaontology is introduced. Chapter 10 is the application chapter: musical works pull together types, abstracta, composition and fundamentality.
Effingham uses schematic letters P, Q, R for propositions and theories, and flags Terminological Alerts wherever the literature uses one word for several ideas ("abstract", "universal", "identity", "exists"). The standing advice is to fix meanings before joining a debate; many "disagreements" are verbal.
How the ten chapters are arranged after the preface.
The Basics
Ontology
The basic division: concreta in space and time, abstracta outside them.
Effingham is careful to say what kind of question ontology asks. It is not the historian's question of how many soldiers died, nor the physicist's question of which particles exist, but the general question of which categories have members at all. That is why the same events can be studied by both: the historian records the 19,420 deaths; the ontologist asks whether, in addition to the soldiers and their suffering, there are facts, numbers and properties. The book's recurring Terminological Alerts exist because these debates are notorious for verbal confusion: "abstract", "universal", "nominalism" and "exists" itself all have several uses in the literature, and many apparent disagreements dissolve once they are fixed.
Nonsense versus sense
Questions about concreta are familiar. "Is there a hippopotamus in the next room?" or "Is there a region of space a trillion light years away?" are meaningful and non-trivial: you must open the door, or physicists must investigate. Questions about abstracta look different: "Does the number 7 exist?", "Does the property red exist?", "Does the possibility of my being a pole-dancing panda exist?" Some think them defective, either meaningless like "Panda over dancing there pole?" or trivially true.
Permissivists take the trivial line. "There is a prime number between 5 and 11" is true, so there exists a number (7). "There is something you and I have in common" is true, so a property exists. After all, what does "There is a..." mean other than "There exists a..."? On this view ontology is easy, and so hardly worth a discipline (difficulty alone would not suffice either: nobody studies "Mortgloriapesamplitology", the shoe sizes of dead celebrities).
The argument against permissivism has a wider lesson that the book draws on repeatedly. The surface grammar of a true sentence need not reveal what it commits us to. "There is something that being a goat and being a man have in common" is a sentence we should assent to, and yet reading its commitments straight off its grammar produces a contradiction. So we need some theory of how true sentences relate to what exists. Chapter 2 supplies the first such theory, Quine's, and the rest of the book tests rivals. The dialetheist escape, accepting that some contradictions are true, is mentioned and set aside; most philosophers (71.4% in the 2020 PhilPapers Survey) think true contradictions impossible.
Nominalism
Nominalism, as the book uses the term, is the view that no abstracta exist; realism about abstracta (Platonism) says some do. Nominalists have four motivations:
- Naturalism. Everything is either in space and time or not; it is "just plain bizarre" for numbers to be in space and time, and odd for anything to exist outside it.
- Causal issues. Abstracta lack causal powers. The Eleatic principle says whatever exists must have causal powers.
- Epistemological issues. We know of chairs, New York (via Friends) and Chingiz Khan through causal chains. Abstracta are causally cut off, like aliens a billion light years away; so how could we know of them?
- No explanation. A physicist who adds "uselessions" to a complete theory ("They don't do anything, they're just there") offers nothing. Abstracta, with no causal role, seem equally idle.
Four motivations for nominalism, and the burden it still carries.
| View | Claim about abstracta | Main problem |
|---|---|---|
| Permissivism | True "There is" talk settles existence; easy | Russell-style contradiction |
| Nominalism | Abstracta do not exist | Must explain true-seeming abstract talk |
| Platonism (realism) | Abstracta exist | Causal and epistemic worries |
The chapter's further reading
For a general introduction to abstract entities the book recommends Chris Swoyer's article (2008). For permissivism it points to Amie Thomasson (2010) and Thomas Hofweber (2005), and to Neo-Fregeanism, a close relative of permissivism, via Matti Eklund's challenging article (2006). Attacks on ontology as a whole, it promises, return in later chapters, and they do: Meinongianism in Chapter 4, Neo-Carnapianism in Chapter 8 and fundamentality in Chapter 10 are all, in their way, challenges to the picture of ontology set up here.
Taken together, Chapter 1 does three jobs. It defines ontology and the abstract/concrete division; it defends ontology against the charges of nonsense and triviality; and it motivates nominalism while showing that nominalism carries a burden of its own. Every later chapter inherits that structure: a realist view with a commitment, an anti-realist view with a paraphrase to supply, and a method for adjudicating between them.
What ontology is not
Physicists ask whether there are black holes, Higgs bosons or superstrings; biologists ask which insects and antibodies exist; stamp collectors and UFO watchers chase their own lists. None of that is ontology in Effingham's sense. Ontologists do almost no fieldwork. They do not inventory wardrobes. They ask about categories: properties, numbers, holes, possibilities, regions, past objects, composites, works of art.
The 1 July 1916 Somme figure (19,420 British dead on the first day) shows the split cleanly. Historians ask how many died and why. Ontologists ask whether, beyond the soldiers, there are the property of being in pain, the fact that 19,420 were killed, and the number 19,420.
Permissivism spelled out
The permissivist reads natural-language "There is an X" as settling that X exists. "There is a prime between 5 and 11" yields 7; "there is something you and I have in common" yields a property; "there are ways to win elections" yields possibilities. Ontology would then be easy, and hardly worth a discipline. Difficulty alone would not save it either: nobody studies Mortgloriapesamplitology, the shoe sizes of dead celebrities, even though that too is hard.
The Russell-style trap: being a goat and being a man are both non-self-instantiating, so the permissivist gets the property being non-self-instantiating. If that property instantiates itself, it does not; if it does not, it does. Contradiction. Dialetheists who allow true contradictions can swallow this; most philosophers will not. So not every true "There is..." sentence yields an existent, and we need a theory of ontological commitment.
Permissivism reads existence off grammar and hits paradox.
Nominalism's four motives, tightened
- Naturalism. Everything is in space and time or it is not. Numbers in space and time are "just plain bizarre"; entities outside space and time are odd for a naturalist.
- Causal idleness (Eleatic). God can magic a burning bush; the number 47 never has. The Eleatic principle says existents need causal powers.
- Epistemic cut-off. We know chairs, New York (via Friends), and Chingiz Khan through causal chains. Abstracta are as cut off as aliens a billion light years away.
- No explanatory work. A physicist who adds inert "uselessions" ("They don't do anything, they're just there") explains nothing. Idle abstracta look the same.
None of this makes nominalism free. "There is a prime between 5 and 11" still seems true. Even someone who calls abstract talk nonsense must answer the awkward questions, as a Creationist who called evolution meaningless would still face the dinosaur fossils.
Abstract and concrete: handle with care
Effingham's working gloss: concreta are in space and time (and usually include events, places, and even postulated immaterial agents if they act in history); abstracta are not. He warns that philosophers redefine these terms. Some treat causal power as the mark of the concrete; some treat mind-independence differently. The book's rule is practical: say what you mean, then argue. Later chapters will put pressure on the gloss when spacetime regions, tropes and types enter.
Why difficulty is not enough
Even if ontology is hard, hardness does not make a subject. Mortgloriapesamplitology would be hard too. What licenses ontology is that its questions are open, non-trivial, and tied to theories we already care about (science, mathematics, everyday truth). Chapter 1's job is to put abstracta questions in that licensed space, not to settle Platonism.
Difficulty alone does not make a discipline.
The licensing point also blocks a common shrug: that ontology is "just semantics". Semantics matters (Chapters 2 and 8), but Chapter 1 insists the questions can be about the world. If every ontological dispute were verbal, the later methods would be idle; the book does not start there.
Dialetheism as a minority exit
The Russell-style trap on being non-self-instantiating is decisive only if true contradictions are impossible. Dialetheists accept some true contradictions and can refuse the reductio. Effingham notes the exit and moves on; the PhilPapers minority for actual true contradictions sits near 12%. For this book, classical consistency remains the default background for theory choice.
Methodology
Theory choice
We choose between theories constantly: the heliocentric model beats the geocentric one because it predicts most of what we observe. In ontology we compare the costs and benefits of rival theories on several theoretical virtues and weigh them up. The test case is holes. Intuitively a hole is "an absence, a lack of an entity": point at one in a vacuum and "There's nothing there" seems true. Yet we say "There is a hole in the bucket".
The intuition virtue is easily misread. Effingham insists that treating coherence with intuition as a virtue is not to believe that everything we intuitively believe is "true and unrevisable on philosophical grounds". Intuitions vary over time and from culture to culture, so "we'd be remarkably lucky" if ours were exactly right; and paradoxes (such as the statue and the lump in Chapter 9) force us to give some up. Intuition is one weight on the scale among several, and it can be outweighed.
Paraphrase and Quine's criterion
Many sentences have misleading surface commitments. "The Average Man lives in a house with 2.4 children" commits to no actual man; its paraphrase is about demography. "The sun has moved behind the elms" paraphrases into heliocentric facts. "There is something being a goat and being a man have in common" becomes "both fail to instantiate themselves". The original sentence is true, but means what the paraphrase says.
The Average Man example repays attention. An economist says the average man lives in London in a house with 2.4 children. Read literally, some bloke lives with two children and "the gruesome remains of a child, amassing a full 40 per cent of a whole child". Nobody infers that; the sentence's real content concerns ratios of people to houses and children. A non-native speaker who parsed it like "The prime minister lives in 10 Downing Street" would make exactly the mistake that paraphrase exists to correct. The anti-realist about holes hopes hole talk is like Average Man talk: true, useful and ontologically innocent.
Quine's criterion: commitments are read off the bound variables.
Anti-realism and ideology
The anti-realist paraphrases hole talk with predicates: "singularly perforated", "doubly perforated", "triply perforated"... This keeps explanatory power (water leaks because the bucket is singularly perforated). But a paper can have any number of holes, so the theory needs infinitely many primitive predicates, and the paraphrases grow "radically dissimilar" for complex cases.
The four theories of holes weighed in Table 2.1.
Realist options and ontological parsimony
Qualitative parsimony cuts kinds. Substance dualism posts two kinds (matter and immaterial minds); since physical explanation suffices, prefer one kind. The parsimonious move is not to deny people but to identify them with brains or bodies. Holes can be treated the same way.
- Sui generis realism (Casati and Varzi): holes are entities in a category of their own. Only "__ is a hole" is primitive, but a new category is added.
- The Ludovician theory (Argle in David and Stephanie Lewis's "Holes", 1970): holes are hole linings, physical parts of the holed object. Parsimonious, but Bargle worries holes would spin and be made of cheese, and Casati and Varzi show that enlarging an odd-shaped hole can shorten its lining, so the hole "gets smaller" (Figure 2.1).
- Holes as regions of space: a hole is the empty region r surrounded by the object's region R (Figure 2.2). It handles size, but holes can no longer move with the object.
Two kinds of ontological parsimony.
Weighing up the theories
Effingham charts the four theories (Table 2.1, redrawn below). No theory wins on every virtue, and philosophers weigh the virtues differently; usually no single virtue is decisive. "If you're looking to find a cast-iron statement of which virtue is better than which other, then the answer has not as yet been settled."
| Theory | Coherence with intuition | Explanatory power | Ideological parsimony | Ontological parsimony |
|---|---|---|---|---|
| Anti-realism | Benefit: no absences endorsed | Benefit: explains all (given the right paraphrases) | Cost: many primitives, radically dissimilar paraphrases | Benefit: no holes; most parsimonious |
| Sui generis realism | Cost: endorses things we thought did not exist | Benefit: no reason to doubt it | Benefit: only "__ is a hole" primitive (or a cost?) | Cost: a new category of entity |
| Ludovician theory | Costs: holes shrink as they grow; made of wood, cheese, denim; can spin | Benefit: no reason to doubt it (or?) | Benefit: none extra; defines holes via linings | Benefit: no extra ontology |
| Holes as regions | Cost: holes can no longer move | Benefit: no reason to doubt it (or?) | Benefit: none extra; region surrounded by a lining | Cost or benefit: parsimonious if you already believe in space |
Table 2.1 (redrawn): costs and benefits of four theories of holes.
The chapter's further reading
The book's guide: Quine's "On What There Is" (1948) for the standard statement of ontological commitment; Peter van Inwagen (1998) for "the best contemporary statement", with an argument for accepting it; introductions by Cynthia Macdonald (2005) and Bill Aune (1985); criticisms by Joseph Melia (1995) and Agustín Rayo (2007); and on the virtues, Daniel Nolan (1997), Elliott Sober (1981) and Michael Huemer (2009). On holes: the Lewises (1970), Casati and Varzi (1994), Frank Jackson (1977), Kristie Miller (2007) and Casati's shorter introduction (2009).
The chapter's method is the book's spine. Every later dispute, about properties, numbers, worlds, regions, past things, composites and works, is run as a comparison of costs and benefits, and every later metaontology is introduced as a rival to, or refinement of, the Quinean criterion set out here. Readers who master Table 2.1 have the template for the rest of the book.
Theory choice as cost-benefit
We choose theories constantly: which friend to trust about a film, heliocentrism versus geocentrism, which ontological package to accept. The shared method is a cost-benefit analysis. No ontological theory is perfect; we weigh benefits against costs and prefer the better balance.
Standard theoretical virtues in the book:
- Coherence with intuitions (folk intuitions). Good if the theory says there are chairs, that time passes, that one should save drowning children, that contradictions are not true, that solipsism is false. Bad if it denies those. Intuitions vary by culture and time, and paradoxes can force us to drop some (Chapter 9).
- Explanatory power. Prefer theories that explain more. We are not left in the dark about why the Great Train Robbery (1963, £2.6 million) happened once we have motives, opportunity and mechanism; brute truths are those we leave unexplained.
- Ontological parsimony. Prefer fewer (kinds of) entities, other things equal. Realism about holes adds a category; anti-realism may avoid it.
- Ideological parsimony. Prefer fewer primitive predicates. A paraphrase that needs infinitely many colour primitives is ideologically expensive even if ontologically thin.
- Elegance / simplicity of formulation. Prefer cleaner statements when costs are otherwise equal.
Theory-choice virtues used throughout the book.
Holes: paraphrase options in more detail
"There is a hole in the bucket" looks like it quantifies over a hole. Quinean commitment asks whether the best total theory needs holes in its domain.
- Hole linings. Identify the hole with the material that lines it. Problem: enlarge the lining (add clay to the rim) and the lining grows while the hole, intuitively, can stay the same size; shrink the lining and the lining shrinks while the hole may grow. So lining ≠ hole.
- Regions of space. Identify the hole with the empty region. That helps only if you already accept regions (Chapter 6). Also, two connected cavities can be one hole or two depending on how we count "piercings".
- Perforation paraphrase. "The bucket is perforated (in such-and-such a way)." No hole entity; cost in ideology (a family of perforation predicates) and maybe in intuitiveness.
Anti-realists owe a paraphrase that preserves truth-conditions without the entity. Realists accept holes and pay in ontology. Names and definite descriptions ("the hole in my bagel") need parallel treatment: either denote or disappear under paraphrase.
Weighing the hole theories
Realism scores on simple Quinean reading and on some explanations of shared hole-talk. Anti-realism scores on ontological parsimony and on not multiplying odd entities. Which wins depends on how heavily you weight intuition, parsimony and the success of paraphrases. Chapter 2's lasting lesson is the method, not a final verdict on holes: state commitments, offer paraphrases, score costs and benefits.
Holes illustrate the method more than they settle the world.
Names, descriptions, and ideology
Quinean paraphrase is not only for "there is a hole". Names ("Pegasus") and definite descriptions ("the hole in my bagel") need either denotations or elimination (Russell-style descriptions, paraphrase into predicates). Every eliminated entity tends to reappear as ideology: more primitive predicates. Theory choice must score ontological and ideological ledgers together; winning on one while exploding the other is not automatically rational.
Worked cost-benefit on holes
Imagine three packages only.
Package R (realism): holes exist as immaterial parasites or as regions. Benefit: "there is a hole in the bucket" is literally true; counting piercings is object-counting. Costs: odd entities; lining/region identity troubles; annihilation-count weirdness; abstract-ish holes reawaken Chapter 1 motives.
Package L (lining identity): holes = linings. Benefit: only material objects. Costs: lining size can change while hole size does not; two holes can share lining material in tangled ways.
Package P (perforation paraphrase): no holes; buckets are perforated. Benefit: ontological parsimony. Costs: ideological sprawl; some inferences about hole identity across hosts get awkward.
A rational chooser lists these rows, weights intuition, parsimony and explanatory reach, and may still disagree. That disagreement is normal in ontology; the chapter's achievement is making the ledger explicit.
Three hole packages compared on an explicit ledger.
Metaontology already on stage
Chapter 2 quietly introduces the book's standing metaontology: Quinean commitment plus theory choice. Later chapters will challenge that package (Meinong, fictionalism, Neo-Carnap, fundamentality) without discarding the habit of scoring costs and benefits. Keep the habit even when you reject Quine.
Table 2.1 in prose
Effingham's comparison (redrawn here as a table in the earlier edition layers) lines up theories of holes against virtues. Sui generis realism wins on literal reading and on some identity judgments about holes that move between hosts; it loses on ontological oddity. Lining and region reductions try to keep realism without a new category; each inherits problems from the reducing category (material boundaries; spacetime ontology). Anti-realism wins on not adding holes; it loses when perforation ideology balloons and when speakers insist they count holes as things.
Notice the pattern you will see again for properties, numbers and tables: realism simplifies semantics and complicates ontology; anti-realism simplifies ontology and complicates semantics or ideology. Chapter 2 trains you to expect that trade rather than to hunt a cost-free theory.
The standing trade: semantics versus ontology.
Cost-benefit as the book's standing decision procedure.
Explanatory power versus brute facts
A theory that leaves too much brute loses to one that explains the same phenomena with fewer primitives, other things equal. But explanation can be bought with ontology (postulate holes, universals, worlds) or with ideology (more laws and predicates). There is no master rule that ontology is always the cheaper purchase. Hole theories are the training ground for noticing when a "explanation" is only a label.
Coherence with intuition is not a veto
Folk intuition is a virtue, not a veto. Paradoxes (Chapter 9) and physics (Chapters 6-7) can force revisions. Cross-cultural variation (Chapter 8) shows intuitions are not a single timeless jury. Still, a theory that scrapes every intuition at once (no chairs, no change, no moral reasons, true contradictions, only you exist) piles costs that demand huge compensatory benefits. Ontology rarely offers such compensation.
Ontological reduction as a general tactic
Ontological reduction says Xs are nothing but Ys: holes are linings; numbers are sets; regions are sets of points. Success requires more than a mapping. The reducing entities must already be acceptable, and the identities must respect the causal and modal profile of what was reduced. Failed reductions (lining ≠ hole) teach you to demand profile match, not only formal correspondence. Successful-looking reductions still face Benacerraf-style indeterminacy when many Y-systems work equally well.
Properties
Properties and Quinean ontological commitment
The anti-realist (about properties) must paraphrase sentences that seem to quantify over properties. "The car and the sky have the same colour" might become a long disjunction: [x is blue & y is blue] v [x is red & y is red] v... But shades matter (an azure car is not the colour of an Alice blue one), and with about a thousand easily distinguished shades the paraphrase runs to a thousand pairs of disjuncts. Harder still are:
- "Blue is my favourite colour."
- "Red resembles orange more than it does blue."
- "There are three, as yet undiscovered, fundamental properties."
The anti-realist may simply deny these are true, at a cost to coherence with intuition ("Can it really be the case that blue isn't my favourite colour?").
Property talk strains anti-realist paraphrase.
The problem of universals
A second route to commitment: entities are justified if they figure in metaphysical explanations. The problem of universals has a long history and comes in many versions; Effingham takes two.
Two anti-realist answers
Trope theory posits tropes, particular instances of properties: the blackness of my jacket is distinct from the blackness of my pen; the knife's own sharpness cuts you, not sharpness in general. Tropes are a new category, but they support bundle theory (objects as bundles of tropes). Bundles of universals clash with Max Black-style cases (two exactly similar iron balls) because they need the unpopular identity of indiscernibles; trope bundles avoid that, since each ball has its own tropes.
Resemblance nominalism (here "nominalism" means anti-realism about properties, not about all abstracta) explains "a is F" by a's resembling other F things: the Taj Mahal is white because it resembles other white things; Obama is a Democrat because he resembles other Democrats. Resemblance becomes primitive ideology, and there is a threat of circularity if the job was to explain resemblance.
| Name | Explains "a is F" by | Price |
|---|---|---|
| Platonic realism | a instantiates F, outside space and time | Ch1 worries about abstracta |
| Immanent realism | a instantiates F, located where a is | Multi-location |
| Trope theory | a has an F-trope | New category; how tropes resemble |
| Resemblance nominalism | a resembles other F things | Primitive resemblance; circularity risk |
The trope theorist's bonus deserves emphasis. Bundle theory built from universals needs the identity of indiscernibles, since two bundles of the same universals would be the same bundle; built from tropes, two exactly similar iron balls are two bundles of numerically distinct tropes. So, Effingham notes, while trope theory, "like realism about properties, introduces another category, it might offer extra benefits". The resemblance nominalist, by contrast, adds no new category but pays in ideology, taking resemblance as primitive.
What are properties like?
Platonic (or transcendental) realism puts properties outside space and time (not Plato's own Forms). Immanent realism (sometimes "Aristotelianism") locates them in space and time: the property white is wholly present wherever white things are, as a time traveller might be wholly present in two places at once. David Armstrong, a leading immanent realist, holds instead that properties are in space and time but at no specific location, via states of affairs (Bill Gates having invented Windows; the Empire State Building being 381 metres tall), each composed of a property and an individual.
Platonic versus immanent realism about properties.
Two further questions: uninstantiated and negative properties.
Armstrong's view has a distinctive shape. States of affairs are "neither objects nor properties" but correspond to an individual instantiating a property: Bill Gates having invented Windows; the electron in the cloud chamber being negatively charged. Each state of affairs is composed of a property and an individual, and properties are located only derivatively, through the states of affairs they help compose. This makes Armstrong an immanent realist without saying that white is wholly present in a million places. States of affairs return in Chapter 7, as truthmakers.
Two further questions follow. (i) Are there uninstantiated properties? Being a perfect circle, being exactly 10 metres tall, being biniloctium (element 208, too unstable ever to exist). Including them gives infinitely many idle entities. (ii) Does every predicate correspond to a property? Not "__ is not a rabbit": admitting negative properties spoils the resemblance argument, since Michael J. Fox and the DeLorean would genuinely resemble each other. But restricting properties to science reopens the Quinean problem: "Blue is a colour" seems true, yet subatomic particles have no colour.
The chapter's further reading
The book recommends Alex Oliver (1996) as "an excellent introduction", with alternatives by David Armstrong (1989), Fraser MacBride (2009), Loux and Zimmerman (2003) and J. P. Moreland (2001), and the Mellor and Oliver collection (1997). For trope theory it cites Keith Campbell (1990); for resemblance nominalism, Gonzalo Rodriguez-Pereyra (2002); for location, MacBride (1998) and Parsons (2007); and for theistic approaches, Brian Leftow (2006) and Bergmann and Brower (2006).
The chapter's closing moral is that the question is never just whether there are properties. If there are, we must still decide which properties there are and what they are like, and the answers interact: restricting properties to science helps with uninstantiated properties but reopens the Quinean problem of "Blue is a colour".
One Over Many and metaphysical explanation
The problem of universals (classical "One Over Many") asks what, if anything, explains many distinct things being the same way. Electrons are charged; the Taj Mahal is white; someone is about 150,000,000 km from the sun; Brad Pitt is in love with Angelina Jolie. Do charge, white, being 150,000,000 km from, and being in love also exist?
Realists about properties say there are ontological facts that ground "a is F": a instantiates Fness. That explanation is a theoretical benefit. Anti-realists who refuse properties must either offer another explanation or treat "a is F" as brute. People who think no special explanation is required will not be moved; those who demand metaphysical explanation will score a point for realism.
Resemblance, sparse properties, tropes
Tropes are particularized properties: the charge-of-this-electron, not charge-in-general. Tropes are not repeatable universals; exact resemblance among tropes plays the role universals played. Cost: a large ontology of tropes. Benefit: avoid one entity wholly present in many places.
Four answers to the demand for metaphysical explanation.
What properties are like: location and repeatability
If properties are transcendent (Platonic), they live outside space and time and objects instantiate them. If immanent (Aristotelian), a universal is wholly present wherever it is instantiated. That second view needs a location relation story: how can one entity be wholly in many places? (The book flags "location relations" as a Terminological Alert.)
"Universal" itself is ambiguous in the literature: repeatable property; wholly multiply located entity; or irreducibly property-like entity. Effingham often avoids the word and speaks of properties to reduce confusion. Further reading points include Armstrong on universals, and debates that surface again when music types (Chapter 10) look like repeatable abstracta.
Colour predicates make the ideology-versus-ontology trade vivid.
Armstrong-style sparse realism
A common realist package (Armstrong-adjacent): only sparse properties that carve nature's joints exist as universals; abundant predicates do not each pick out a universal. That limits ontology, aids resemblance explanation, and pairs with laws as relations among universals. Anti-realists may still call even sparse universals idle. The chapter's method lesson remains: metaphysical explanation is a virtue that can tip theory choice when paraphrases grow ugly.
Josh Parsons and Max Black notes
The chapter's further discussion of location and of qualitative identity connects to puzzles like Max Black's two indiscernible spheres: if only qualitative properties exist, what distinguishes the spheres? (The book alludes without always naming Black.) Tropes and haecceities are among the responses. Flag the apostrophe error; keep the philosophical pointer.
Indiscernibility pressure on sparse property ontologies.
Natural versus abundant properties
Abundant properties correspond to almost every predicate; natural or sparse properties mark genuine resemblance and causal powers. Realists who accept only sparse universals shrink ontology and strengthen explanation. Anti-realists may still reject even sparse ones. Much of the resemblance-nominalism debate is really about whether naturalness is primitive and enough without universals.
Numbers
Ontological reductions
Sets stand in the membership relation to their members: the set of all people, all apples, or of you, a forklift truck and six hydrogen atoms from a distant star. Sets resemble collections, but set theory demands an empty set, and "a collection with nothing in it is ... well, nothing", so sets are best treated as abstract objects. With just Adam and Eve there are already {Adam}, {Eve}, {Adam, Eve}, and then sets of those sets, without end.
Two set-theoretic reductions of number, and Benacerraf's problem.
Benacerraf's problem (1965): since Zermelo and von Neumann are equally good, choosing either is arbitrary, and in general "it is bad to say that you have arrived at your conclusion by arbitrary means". Is 1 a member of 3? On von Neumann's view yes; on Zermelo's (3 = {2}) no. This threatens the whole reduction of numbers to sets (structuralism, in Shapiro and Resnik, is one response).
Constructivism deserves a word because readers often reach for it first. The idea that numbers depend on us sounds deflationary, but Effingham argues that it "does little to deflate ontological questions": if numbers depend on our minds, they exist, and we still have to ask whether they are abstract and whether they reduce to something else. If anything the view makes them weirder, since the universe "could get along quite happily without them" and they would serve a purely anthropocentric role.
Indispensability arguments for realism
Quine and Putnam argue that numbers are indispensable to science: without the gravitational constant we could not calculate forces. A naturalist, who finds out about the world by scientific means, should therefore endorse numbers. Mathematicians are also hard to overrule: as Lewis says in Parts of Classes, it would strain credulity to tell the Department of Mathematics to give up set theory on the strength of a few philosophical worries.
The book's example is Newton's law of gravitation, F = G(ma mb)/r squared, where G is approximately 6.7 × 10^-11 in newtons, kilograms and metres. Its value changes with units (the book gives about 4.3 × 10^-3 for an "interstellar" setting of parsecs and solar masses), but some number is always needed. Physics, it seems, cannot be stated without numbers, and the naturalist believes what our best physics says.
The esoteric-mathematics objection to indispensability.
Anti-realist strategies
- Constructibility (Chihara): replace talk of sets with talk of open sentences (sentences with unbound variables, like "x is a man"). Sets of particles in a galaxy nobody will ever mention have no open sentence, so Chihara appeals to what could be written: a modal move, which needs Chapter 5.
- Mathematical fictionalism (Hartry Field, 1980): "7 is a prime number" is fictionally true, like "Sherlock Holmes is a detective"; numbers are fictions like Snow White or Hansel and Gretel. Field argues science can be nominalized, rewritten without mathematical entities.
- Formalism: mathematical sentences are meaningless, neither true nor false, like a row of Sudoku digits (1 3 4 2 6 7 5 8 9 is legitimate; 1 1 2 8 3 4 5 6 7 is not). What is true is that "2 + 2 = 4" follows from the axioms: a claim about the ink blots before you.
- Constructivism: numbers depend on human activity. Effingham replies that dependence makes numbers exist, so it deflates nothing, and it makes them weirder: numbers would "pop into existence solely to service our needs".
Realist, anti-realist and Meinongian readings of number talk.
Meinongianism
The anti-realisms above accept Quine's criterion. Meinong rejects it: "There is" and "There exists" come apart. "There is a character in Lord of the Rings whose name I rarely remember" is true, and there are round squares, spherical cubes, one-dimensional hypercubes: these things are, but they do not exist. So the Meinongian can accept that there is a prime between 5 and 11 while denying that numbers exist. This is a rejection of Quine's theory rather than a full theory of commitment, since it does not yet say which of the things there are also exist.
The chapter's further reading
General introductions: Michelle Friend (2007) and Stewart Shapiro (2000). On sets, Max Black (1971), David Lewis (1990) and Penelope Maddy (1990). Benacerraf's 1965 article, with structuralist responses in Shapiro (2000) and Michael Resnik (1997). Field (1980) for fictionalism; Chihara (1990, 2007) for constructibility; Goodman and Quine (1947) for formalist rules; Maddy (1990) and John Bigelow (1988) on the realist side. For Meinong, accessible introductions by Colin McGinn (2000) and Roderick Chisholm (1973), and Priest's 2005 book.
The chapter's lasting contribution is to show that the ontology of mathematics is two questions: whether there are mathematical entities at all, and, if there are, whether numbers reduce to sets. Indispensability bears on the first; Benacerraf's problem on the second.
Psychologism, sets, and Benacerraf
Psychologism treats numbers as mental objects. It struggles with shared, infinite mathematics: the continuum is not in anyone's head, and two mathematicians seem to talk about the same 7.
Set-theoretic reductions identify numbers with sets (Frege-Russell, Zermelo, von Neumann, ...). Benacerraf's problem: if 3 is {{{∅}}} on one reduction and {∅,{∅},{{∅}}} on another, which identification is correct? Multiple adequate reductions undercut the claim that numbers are those sets. Structuralists reply that what matters is the structure, not the particular objects.
Indispensability and anti-realist exits
The Quine-Putnam indispensability argument: if our best scientific theories quantify over numbers, and we are ontologically committed to what those theories quantify over, then numbers exist. Physics writes laws with real-valued quantities; denying numbers looks like denying the science.
Anti-realist strategies in the chapter:
- Paraphrase scientific talk to avoid number quantifiers (hard for advanced physics).
- Mathematical fictionalism: number talk is true according to the fiction of mathematics, as "Hobbits live in the Shire" is true according to Lord of the Rings, without hobbits existing.
- Formalism: mathematics is manipulation of symbols by rules, not a theory of a domain of numbers.
Indispensability pushes realists; access worries push back.
Meinongianism: there is versus exists
Meinong distinguishes what there is from what exists. The golden mountain and the round square can be objects of thought (there are such objects) without existing. Quine's "On What There Is" (1948) attacks the crowded "slum of possibles" (his character Wyman) and worries about identity and non-existence. Effingham presents Meinongianism as a live metaontology: you might accept that "there is a prime between 5 and 11" without saying the prime exists in the robust sense.
Meinongianism splits being from existence.
Indispensability pressure in more detail
Science does not merely mention numbers in popularization. Measurement, differential equations, Hilbert spaces and statistical estimators quantify over mathematical objects. If Quinean commitment applies to our best total theory, nominalism about numbers looks like scientific bad faith unless a paraphrase or fictionalist embedding works inside the science, not only in the philosophy seminar.
Enhanced indispensability (explanatory) variants say that numbers earn their keep when they explain empirical patterns (periodical cicada primes are the usual toy case in the later literature). Even if you reject that enhancement, the plain Quine-Putnam form already shifts the burden onto anti-realists.
Plain and enhanced indispensability both shift the burden.
Fictionalism versus formalism
Fictionalism preserves truth-according-to-math and scientific utility while denying that numbers exist. Formalism is thinner: mathematics is rule-governed symbol play. Formalism struggles to explain applicability to the world; fictionalism struggles to explain why this fiction is so uniquely useful. Realism struggles with Benacerraf access and with set-reduction indeterminacy. Again the method is cost-benefit, not knockdown.
Reductions that relocate, not erase
Identifying numbers with sets, or sets with functions, or functions with ordered pairs, can look like ontological progress until Benacerraf asks which reduction is the fact of the matter. Often you have only shown equiconsistency or representational adequacy. Structuralism treats that as a feature: mathematics describes a pattern many systems can instantiate. Nominalists may welcome the pattern talk as a step toward paraphrase; Platonists may say the structure itself is an abstract object. The reduction strategy therefore does not by itself end the realism debate; it changes what the realist is realist about.
Benacerraf: multiple set reductions, one structure.
Psychologism's short life on stage
If numbers were private mental tokens, shared proofs would be miracles and infinite totalities impossible as completed objects of a finite mind. Effingham stages psychologism mainly to retire it quickly, clearing the field for set reductions, Platonism, fictionalism and Meinongianism. The retirement matters: many students reach for "numbers are ideas" as common sense; the chapter shows why that common sense fails as a theory.
Possible Worlds
Modal talk and possible worlds talk
Possibility comes in strengths: logical, physical, and weaker kinds (a company that "can't" repay its debts is constrained by law and economics, not by physics). Modal logic extends predicate logic with a possibility operator ◇ and a necessity operator □, interdefinable (◇φ is ¬□¬φ). But some sentences resist the operators, so we move to worlds: something is possible if it is true at some world, necessary if true at every world. At one world the Balrog does not exist but Hitler wins; at another Hitler wins by summoning the Balrog.
What a theory of worlds is supposed to deliver.
Genuine modal realism
David Lewis's Genuine Modal Realism (GMR): possible worlds are concrete spacetimes like ours, disconnected from it. There are universes with Balrogs, with your favourite soap opera played out exactly, with people who stop and sing every Lady Gaga song. "Actual" is indexical, like "here": other worlds exist but are not actual, just as other places exist but are not here.
The island universe problem
Figure 5.1 (redrawn): a series of universes culminating in an island universe. The link narrows from (a) to (c) and is gone in (d).
Imagine a normal universe (a), one with a narrow bridge between two parts (b), one joined only by a wafer-thin stretch (c; modern physics has proposed budding universes like this), and then one with that stretch removed (d). Each step seems possible, so disconnected spacetimes seem possible. But for GMR a world just is one connected spacetime, so a single world with two disconnected parts is ruled out: GMR seems to miss a genuine possibility. The links are spacetime links, not spatial: (c) might connect only through a tiny doorway on one day in 2020 AD.
Lewis's reply to the incredulous stare turns on indexicality. "Actually existing" is for Lewis like "existing here". Intuitively there are no Balrogs and no disconnected spacetimes actually, and GMR agrees: the Balrogs exist, but not in our world. Whether this reply satisfies is itself a matter of weighing intuitions against benefits, and Effingham leaves the jury, the reader, to decide.
Ersatz modal realism
When worlds were first introduced, most took them as a heuristic or as abstract entities. Ersatzism: worlds are abstract. Effingham's caricature identifies a world with a maximal consistent set of propositions. This avoids the incredulous stare but forfeits GMR's benefits:
- Literal reading: the translation of "There could have been at least one more thing than there actually is" quantifies over an object existing at another world but not at ours, a possibile, but abstract worlds have no inhabitants; the ersatzist must add more entities.
- Analysis of modality: "consistent" is a modal term, and the ersatzist needs a non-modal rule for which sets count as worlds.
- Reductions: with no possibilia, properties cannot become cross-world sets of instances.
So ersatzism avoids "the supposed madness that is GMR" but, Effingham asks, what benefit does it offer in return?
Three accounts of possible worlds.
The ersatzist caricature in the book is concrete enough to test. A world is the set of propositions true at it: at a world where Effingham is prime minister, and Two problems. Incompleteness: fictions leave matters open (Hamlet has no particular shoe size). Lewis's texts do not settle whether a particle could be both positively and negatively charged, so the fiction leaves that modal fact undetermined, but surely every modal fact is true or false. The Brock-Rosen problem (Stuart Brock, Gideon Rosen): GMR says it is true at every world that there are infinitely many disconnected spacetimes. So, by the fictionalist's own definition of necessity, "there are infinitely many disconnected spacetimes" is necessary, hence actually true: exactly what the fictionalist wanted to deny. And since the fictionalist's definitions use modal terms anyway, it is unclear what the view gains.
The chapter's further reading
The book's guide: general introductions by Joseph Melia (2003), E. J. Lowe (2002), John Divers (2002), Louis DeRosset (2009) and Rod Girle (2003); collections by Loux (1979, 2001). On GMR, Lewis's On the Plurality of Worlds (1986); on missing worlds, Philip Bricker (2001, 2006), Efird and Stoneham (2005) and Rodriguez-Pereyra (2004); on morality, Mark Heller (2003). Modal fictionalism: Gideon Rosen (1990), endorsed by Armstrong (1989), with an accessible introduction by Daniel Nolan (2011). Beyond worlds: Divers (2004, 2006), Barbara Vetter (2011) and Michael Jubien (2007).
The chapter also introduces fictionalism as a general strategy, usable well beyond worlds. The test it sets every fictionalist is the one modal fictionalism fails: say what benefit the fiction secures that the rival theories do not.
From modal talk to world talk
Modal claims ("possibly P", "necessarily Q") can be rewritten with possible worlds: P is possible iff P is true at some world; necessary iff true at all worlds. That translation is powerful for logic and philosophy of language. The ontological question is whether the rewrite commits us to worlds as entities.
Genuine modal realism (GMR)
David Lewis's GMR: other possible worlds are concrete universes, as real as ours, not overlapping ours. Benefits: a reductive analysis of modality; straightforward truth-conditions; no primitive modality. Costs: a "crazy" ontology of concrete worlds; concerns about morality and indifference (why prevent famine here if countless worlds contain every pattern?); the island universe problem (could one spacetime contain disconnected "island" regions that look like many worlds?).
Ersatz realists keep worlds but make them abstract (sets of sentences, properties, pictures). They avoid concrete pluriverses but often reintroduce primitive modality when saying which ersatz worlds are possible, so they may fail to earn GMR's main benefit. A cheap building with no doors is not a bargain.
Genuine modal realism and its two main rivals.
Fictionalism about worlds
Fictionalism: talk of worlds is true according to the fiction of possible-worlds theory, the way "According to House, ..." or "According to Lord of the Rings, ..." embeds claims without ontological commitment to the fiction's entities. Combined with Chapter 4's mathematical fictionalism, it is a general anti-realist tool: keep the utility of the idiom, drop the entities.
World talk is optional ontology, not automatic commitment.
Island universes and the figure
Figure 5.1 stages a series of universes that approach an island-universe spacetime: disconnected regions in one manifold can mimic "many worlds" without Lewis's pluriverse. Ersatz and fictionalist views avoid that geometry problem by never positing concrete worlds at all.
GMR's moral and inductive headaches
If every possibility is concretely realized somewhere, then for any famine you prevent here, there are worlds where you fail and worlds where you never tried. Lewisian replies appeal to counterpart caring and to the claim that morality concerns this world's causal sphere. Critics find the reply thin. Inductively, if every pattern exists somewhere, why think our world's regularities project? GMR answers with natural properties and eligibility; the debate is live.
Ersatz theories avoid those headaches but often help themselves to a primitive "possible" when selecting which abstract worlds count. Then they have not reduced modality; they have relocated it. Fictionalism relocates it into the fiction operator. Whether that is honesty or surrender depends on whether you think modality needs reduction at all.
Moral and inductive costs that ersatz and fictionalism avoid.
Ersatz constructions (why they under-earn)
Typical ersatz worlds: maximal consistent sets of sentences; sets of propositions; complex properties the cosmos might have; maps or pictures in a lagodonian language. To say a world is possible, you often need a prior modal notion (consistency, possible property). GMR's selling point was eliminating that prior notion by letting worlds be like ours. If ersatzism keeps modality in the background ideology, it may have a smaller ontology and a worse explanation. That is the "building with no doors" joke in systematic form: cheap to build, unfit for purpose.
Ersatz constructions often smuggle modality back in.
Fictionalism about worlds also inherits a translation task: which ordinary modal claims survive as true-according-to-the-fiction, and which were never more than loose talk? If the fiction is just classical possible-worlds semantics, most of the useful modal inferences survive. If you thin the fiction too far, you lose the benefit that made world talk attractive.
Spacetime versus many spaces
Lewisian worlds are typically entire spacetimes, not planets in one space. That blocks reading GMR as a many-planets story and matters for the island-universe worry: a single spacetime with disconnected regions is still one world in Lewis's bookkeeping, whereas GMR needs a plurality of worlds. Mixing "space" and "spacetime" language is a common slide; Effingham pauses to separate them when the figure tempts the slide.
Why talk about worlds at all?
Philosophers use worlds because they give clean truth-conditions for modality, counterfactuals, content, and supervenience claims. That utility explains the idiom's spread; it does not by itself prove realism. The chapter's sequence is deliberate: show the utility, offer GMR as the bold realist earning theory, pressure it with costs and island universes, then show ersatz and fictionalist ways to keep the idiom without the pluriverse. Methodologically, this is fictionalism's central chapter, parallel to holes for Quine.
Space
Substantivalism and relationism
Substantivalists are realists about regions of space (and spacetime); relationists are anti-realists, holding that there are only objects and the spatial relations between them. The debate is driven largely by philosophers of science, so the test is explanatory role in physics. Motion is relative: if Mr Slow rides away from you at 5 m/s and Ms Fast at 10 m/s, Ms Fast moves at only 5 m/s relative to Mr Slow. Smoke from a match rises straight up in a plane in level flight; in a closed box you cannot tell whether you are at rest or falling at constant terminal velocity. There is no absolute velocity.
The closed box thought experiment makes the point vivid. Knocked out and shut in a windowless box, you cannot tell, by any experiment inside it, whether the box is at rest or has been dropped down a bottomless pit and is falling at constant terminal velocity. Smoke on a plane in steady flight rises straight up, as on the ground. If no experiment could ever detect absolute velocity, the natural conclusion is that there is no such thing, only velocity relative to other things.
Newton's argument for absolute space
Acceleration is different: changing velocity produces inertial forces (pressed back in a plane seat, weightless in a falling lift). Newton's bucket: twist the rope and let go.
Newton's bucket and the replies to it.
Mach replied that acceleration is relative to the fixed stars, so inertial forces come from interaction with distant matter. Machians add that it is question-begging to claim to know what would happen if the stars did not exist. Einstein at first aimed to build Mach's idea into general relativity, but GR permits all the matter in the universe to rotate, which Machianism cannot allow. A third option: acceleration is absolute but brute, analysed relative to nothing, which avoids an extra entity that does no other work.
The relationist's third option deserves a fair hearing. Effingham notes that every physical theory takes some facts as brute, such as the most fundamental particles having the most fundamental properties. Treating absolute acceleration as a fundamental, unanalysed property puts it in the same class, and it avoids positing absolute space, which is otherwise "empirically undetectable" and so "a somewhat odd entity to believe in". The question, then, is whether absolute space has any other explanatory role, which the shift argument and curved geometries put to the test.
Leibniz's shift argument
Leibniz (seventeenth century) offered a reductio of substantivalism. Take a universe with two balls, Lefty at region r1 and Righty at r2, one metre to the right, with r3 empty beyond. Now imagine everything shifted one metre right: Lefty at r2, Righty at r3, r1 empty.
Figure 6.1 (redrawn): Leibniz's shift. Every object moves one metre right; for the substantivalist a distinct possibility, for the relationist no difference at all.
For the substantivalist these are two distinct possibilities, since different regions are occupied. Leibniz objects via the Principle of Sufficient Reason: God could have no reason to choose one over the other (kettles boil because switched on; Poirot assumes every death has a reason). The relationist solves it "neatly": with Lefty and Righty a metre apart, "that's the end of the matter". Effingham notes that indiscernibles and reasonless events (quantum events, the Big Bang) do seem possible, but many still find empirically undetectable differences suspect, as with absolute velocity.
Non-Euclidean geometries
The shift worry assumes Euclidean geometry, in particular the parallel postulate: through a point off a line exactly one line never crosses it.
Figure 6.2 (redrawn): the parallel postulate. Through a point off a line, exactly one line fails to cross it, true on a flat plane but false on a sphere.
On a sphere every straight line (great circle) through the point crosses the original, and triangles no longer sum to 180 degrees. Non-Euclidean geometries give space a shape, and shape can explain.
Figure 6.3 (redrawn): non-Euclidean space. Puzzling measurements (a) are explained if the curtained region is a dip in space itself (b).
Einstein's theory, confirmed many times, makes spacetime curved. Relationists can resist: Henri Poincaré proposed laws that shrink every object (including light rays and rulers) inside the zone, so distances only appear larger. Choosing between curvature and universal forces is again a matter of theoretical virtues.
Curvature versus universal forces: underdetermination.
Ontological reductions
Regions might be reduced to sets. Points get Cartesian coordinates (2 m east and 3 m north of an origin is 2,3; 2 m west and 4 m north is -2,4). Since sets ignore order, points become ordered sets <2,3>, and a region becomes the set of its points: regions turn out to be mathematical objects.
Figure 6.4 (redrawn): Cartesian coordinates. Points become ordered sets such as <2,3>; the shaded region becomes the set of all its points.
The reduction can run the other way: supersubstantivalism identifies material objects with the regions they occupy. Sider jokes that a region of spacetime "bounded out the door and barked at the postman". The gains are parsimony (one category) and explanation: an object must share its region's size and shape because it is that region. Schaffer (2009) surveys the arguments, some from quantum physics and GR.
Reducing objects to regions: the supersubstantivalist bargain.
The chapter's further reading
The book calls Barry Dainton (2010) "one of the definitive introductions", with others by Nick Huggett (1999), Tim Maudlin (2009) and E. J. Lowe (2002), and harder ones by Graham Nerlich (2003), Lawrence Sklar (1992) and John Earman (1989). Relationists include Gordon Belot (1999) and Clifford Hooker (1971); Maudlin (1993) is a substantivalist; Nerlich (1976, 1994) writes on spacetime's explanatory role; Paul Teller (1991) is "good, albeit complex"; and Jonathan Schaffer (2009) surveys supersubstantivalism.
The chapter models a naturalistic metaontology: let physics adjudicate. It also shows the limits of that method, since underdetermination (Poincaré's shrinking laws, brute acceleration) leaves the philosopher weighing virtues once more.
Regions, bucket, and what "space" sells you
Sell someone land, knock down the building, dig out the soil: they still own the region. Substantivalists treat regions (or spacetime points) as entities. Relationists say talk of space reduces to spatial relations among material objects; empty regions are not extra substances.
Newton's bucket: water climbs the walls when the bucket rotates relative to absolute space, not merely relative to the bucket. That inertial effect is Newton's case for substantival space. Mach replied that acceleration is relative to the fixed stars; without distant masses the intuition pump is question-begging. Einstein hoped GR would Machianize inertia, but GR still allows solutions where all matter rotates, which pure Machianism forbids. A third option: acceleration is absolute but brute, needing no space-substance.
Leibniz shift and geometry as physics input
Leibniz's shift: if God translated every body three metres east while preserving all relations, substantivalism counts a different world (different occupied points) while relationism counts the same world. If you accept the identity of indiscernibles for worlds, the shift is a cost for substantivalism.
Non-Euclidean geometry and GR change the cost sheet. The parallel postulate (through a point off a line, exactly one non-secant) fails on a sphere. Measured "flatland" distances that look inconsistent become consistent if space itself is curved (Figure 6.3). Metaphysics that ignores the physics of spacetime risks fighting a straw Euclidean world.
From bucket and shift to geometry as physics input.
Reductions of regions
If you dislike substantival regions, try ontological reduction: identify a region with the set of points it contains, and points with coordinate triples (Figure 6.4). That pushes ontology into sets and numbers (Chapter 4). Relationists may instead take distance relations among objects as primitive and treat region-talk as paraphrase. Either way, Chapter 6's method lesson is that scientific theory can force metaphysical options open or closed.
Reducing regions relocates commitment into sets or relations.
Crash course stages in the chapter's physics
Effingham walks through enough physics to make the metaphysics bite: inertial effects, absolute versus relative acceleration, Euclidean failure, and coordinate representations of regions. You do not need to become a physicist; you need to see why a metaphysics of space written as if Newton and Euclid were optional background is incomplete. When GR makes geometry dynamical, substantivalism and relationism both mutate (sophisticated substantivalism, spacetime relationalism). The 2013 text points at that door; later literature (Huggett and Wuthrich among others) walks further through it.
Relationism's paraphrase burden
Relationists must recover talk of empty space, of distances between where things were, and of geometry that outruns the actual material points. If only occupied relations exist, vacuum regions and uninstantiated geometries need paraphrase or denial. Substantivalists buy those truths with points or regions and then face Leibniz-shift style underdetermination. Sophisticated modern views (gauge-equivalent models, spacetime functionalism) complicate both labels; the book's Newton/Leibniz staging is the entry ticket, not the last word.
Empty space is a pressure point for relationism.
Absolute acceleration without a container
The third option after Newton and Mach: treat some accelerations as brute absolutes, not relations to space or to the stars. You avoid a substantival container and avoid Mach's distant-mass story, at the cost of taking inertial structure as primitive. Whether that is more ideological cost than substantival points is another ledger entry. GR's dynamical metric again reshapes the options by making inertial structure part of the physical field content.
Time
Realism about entities from other times
Presentism: the only material objects that exist are those that presently exist; the presentist is an anti-realist about things from other times. Eternalism: objects from all times exist, though not now, just as distant things exist though not here. "The eternalist isn't insane - he doesn't believe we can talk to the dead." The growing block theory admits past and present but not future things, so the block grows. Deviant options show the space: Futurism, Biblicism (only things from 4004 BC onwards) and "1964ism".
Theories of temporal ontology.
Singular propositions
Singular propositions are about one specific thing ("Nikk Effingham is a lecturer", Fa), unlike general ones ("All men are mortal"). "Chingiz Khan once existed" is true. Replace the name by a definite description (the x who used to be called Temujin, built Karakorum, invaded China...) and the variable ranges over a past object; by Quine's criterion Chingiz Khan exists, so presentism is false.
The constituents argument depends on a structured view of propositions, which not everyone holds. A presentist can deny that propositions have their subjects as parts, or treat
Truthmaking
Truthmaker theory: every true proposition has something that makes it true. I make
Truthmaking for the past: the presentist's options.
Special relativity
Velocity is relative, but light seemed to promise a test of absolute velocity. In 1887 the Michelson-Morley experiment found light's speed the same in every direction and season, though the earth's velocity changes as it orbits. The fix: moving clocks run slow. If Mr Slow travels at 0.5c (150,000 km/s) after Ms Fast at c, and his clock runs at half rate, he measures her at 300,000 km/s too. (The book admits the real story is more complicated and the numbers are illustrative.)
The surprise in Michelson and Morley's result is worth spelling out. If light travelled at a fixed speed relative to absolute space, the earth's motion would show up as a difference in light's measured speed, and the difference would change as the earth swung round the sun. No difference was found, in any season or direction. Either the earth is improbably at absolute rest, which the orbit rules out, or something about time and simultaneity must give.
Figure 7.1 (redrawn): simultaneity according to Einstein. Time runs upward; light paths in amber. The two observers disagree about what is simultaneous with the explosion.
In Figure 7.1(a), you beam light at an asteroid; you scratch your head at 1 s; the light returns at 2 s showing the explosion, so the explosion is simultaneous with the head scratch. In (b), your friend leaves by spaceship; his beam returns at 3 s, so for him the explosion happened at 1.5 s, simultaneous with his sneeze, which for you came after the head scratch. For the presentist, then, what exists depends on how fast you are going, and "exists for" stops being transitive: both Ms Fast and Mr Slow can exist for me while, for Mr Slow, only I exist.
Presentists reply in two ways. STR may be superseded: relativity seems to clash with quantum mechanics, so presentists hope for a theory restoring absolute simultaneity (unlike homeopaths and creationists, they "play on existing scientific concerns"). Or there is a hidden preferred frame: simultaneity is absolute but undetectable, which posits "things that are, in theory, untestable". Effingham concludes it is "not a foregone conclusion that presentism is false".
Relativity against presentism, and two replies.
The chapter's further reading
The book's guide: introductions by E. J. Lowe (2002), Thomas Crisp (2003) and Michael Rea (2003); Ned Markosian (2004) on singular propositions; Rodriguez-Pereyra (2006) and Jonathan Tallant (2011) on truthmaking; Armstrong (2004) for the monograph; Ross Cameron (2011) on distributional properties, with Parsons (2004); attacks on truthmaking by Trenton Merricks (2007) and Dean Zimmerman (2008). On relativity, Katherine Hawley (2009), William Lane Craig (2001), Sklar (1992), Craig Bourne (2006), Sider (2001), Michael Tooley (1997) on the growing block, and Bradley Monton (2011) on philosophers challenging physicists.
The chapter's method lesson is that truthmaking gives a second criterion of commitment, independent of Quine's, and that it bears on more than time: negative truths and the past both test it.
Presentism, eternalism, growing block
Presentism: only present entities exist. Dinosaurs and future Martian outposts do not (though dinosaurs did, and outposts will). Eternalism: past, present and future entities all exist; the world is a block. Growing block: past and present exist, future does not; the block grows. Deviant foils (Futurism, Biblicism, 1964ism) show how many slices one could pick; the live debate is mainly presentism versus eternalism (growing block left for the interested reader).
Singular propositions and truthmaking
"Barack Obama is president", "Nikk Effingham is a lecturer", "There is a black pigeon" can be present-tensed and easy for presentists. Harder are singular propositions about the past: that Napoleon was short, that dinosaurs existed. If the proposition's truth requires the object as a constituent, and the object no longer exists, presentists need a story (tensed properties, haecceities, surrogate propositions, or denying the singular form).
Truthmaking: truths need truthmakers, entities in virtue of which they are true. Eternalists offer past objects and events as truthmakers for past-tensed truths. Presentists must supply present truthmakers (traces, records, present tensed facts) or reject unrestricted truthmaker demands. This is a different metaontology from Quinean commitment: it asks what grounds truth, not only what bound variables range over.
Temporal ontology options under truthmaking and STR pressure.
Special relativity and simultaneity
Figure 7.1's lesson: two observers in relative motion can disagree about which distant events are simultaneous with a given event. Presentism's "only the present exists" needs a unique present; relativity makes simultaneity frame-dependent. Eternalism fits the block spacetime picture more easily. Presentists reply with preferred foliations, neo-Lorentzian interpretations, or by denying that physics settles the metaphysics. Effingham's method point: physical theory is again an input, not a sideshow.
Disagreement about simultaneity challenges a unique present.
Presentist truthmaker strategies (menu)
Typical presentist moves for past truths: (i) tensed properties of the present world ("having contained dinosaurs"); (ii) abstract surrogates (haecceities, propositions) that exist now; (iii) traces and records as truthmakers; (iv) restrict truthmaker principles so past truths need none. Each move has costs in ideology, abstracta, or explanatory thinness. Eternalists advertise past objects as straightforward truthmakers and pay in ontology and in relativity-friendly spacetime commitment.
Singular propositions: the Napoleonic test
"Napoleon was short" seems to be about Napoleon. If propositions have objects as constituents, presentism must say either that Napoleon still exists (defeating presentism), that the proposition is not singular after all, or that a present surrogate (haecceity, name-sense, tensed world-property) stands in. Eternalism takes the singular reading at face value. Truthmaking intensifies the test: even if you dodge propositional constituents, you still need something that makes the Napoleonic truth true.
Past singular truths test presentism twice over.
Growing block as a middle path
Growing block tries to honour the intuition that the past is real (history happened) while the future is open. Costs include defining the moving "edge" of reality and facing relativity's challenge to a unique edge. Effingham leaves the details to further reading, but naming the view matters so presentism and eternalism are not mistaken for the only seats at the table.
Mereology
A crash course in mereology
Take proper parthood as primitive (your hand is a proper part of you; you are an improper part of yourself). Define: x and y overlap iff some z is part of both. The xs compose y iff (i) each x is part of y, (ii) no two xs overlap, and (iii) every part of y overlaps some x. So I am composed of my atoms, and of my top and bottom halves, but not of my head and arms alone. Peter van Inwagen's special composition question (SCQ): under what conditions do some things compose a further thing?
Van Inwagen's special composition question and its answers.
Against restricted composition
Restricted answers try to fit intuition. Contact: the ys compose iff they are in contact (directly, or via chains of touching: if I shake your hand while you shake Obama's, we all compose something). But shaking hands does not intuitively create a new object ("in the vein of a hideous horror film"). Appeals to binding forces fare little better. Two arguments push toward an unrestricted answer.
The vagueness argument's key move concerns counting. "There are exactly two things" can be written using only quantifiers, variables and identity: ∃x∃y(x≠y & ∀z(z=x v z=y)). If composition were vague, a sentence like this would sometimes be indeterminate in truth value. But its only non-logical-looking parts are the quantifiers, and "the standard position is that they are not vague"; the alternative, a vague "something", seems absurd ("How could you end up not being sure" whether a thing is a table or a number?). So composition is never vague.
Universalism says any things whatever compose a further thing, so the Flergbet exists. It treats every culture as right to assert composition; the cost is wild overpopulation, softened by noting that in ordinary contexts our quantifiers are restricted ("all the beer is in the fridge" means the beer you bought, not all beer everywhere).
Universalism's defenders also point out that it is culturally neutral: on their view everyone, of every culture, is always right to assert that composition takes place and always wrong to deny it. The overpopulation it brings, objects such as the Flergbet, is softened by context. In ordinary contexts our quantifiers are restricted, so "all the beer is in the fridge" means the beer you bought, and gerrymandered objects are simply ignored; only in the unnatural context of metaphysics do we quantify over such things.
Mereological nihilism
Nihilism: composition never occurs. Only simples (things without proper parts) exist; there are no tables, only simples arranged tablewise, "treewise", "goatwise". It inherits universalism's answers to cultural prejudice (every culture equally wrong) and vagueness, plus ontological parsimony.
| Name | When do things compose? | Main cost |
|---|---|---|
| Restricted (e.g. Contact) | Sometimes, by some criterion | Arbitrary, vague, culturally biased |
| Universalism | Always | Overpopulation (the Flergbet) |
| Nihilism | Never | Underpopulation; persons; gunk |
| Eliminativism | Only for lives or minds | Ad hoc exception |
| Neo-Carnapianism | Question is verbal | Quantifier meaning; Tarskian worry |
The two extremes share benefits but fail in opposite directions.
Neo-Carnapianism
Named for Rudolf Carnap, Quine's sparring partner, Neo-Carnapianism holds that much ontology is a verbal dispute. An American says "footballs aren't round", an Englishman "footballs are round": translated, ∀x(Fx → ∼Hx) and ∀x(Gx → Hx) do not contradict, since "football" means different things.
Hirsch's quantifier variance applied to the yam.
Objections: it is hard to believe the Trobrianders really mean something different by "There is"; and the principle (here associated with Tarski) that a true singular proposition requires its referents to exist suggests the quantifiers cannot simply be swapped. Matti Eklund (2009) and John Hawthorne (2006) reply on behalf of the "heavyweight" ontologist.
The chapter's further reading
Introductions by Amie Thomasson (2010), Ned Markosian (2007, with a survey of restricted answers), Kris McDaniel (2010) and Effingham himself (2009). Restricted composition: Lynne Rudder Baker (2008), Markosian (1998), D. H. Mellor (2008), Uriah Kriegel (2008); Karen Bennett (2009) asks whether the answer is knowable. Universalism: Effingham (2011) and Daniel Korman (2010) on cultural prejudice, Sider (2001) on vagueness, Bigelow and Pargetter (2006), Rea (1998). Nihilism: Cian Dorr (2005); eliminativism: van Inwagen (1990), Merricks (2001); thinkers: Eric Olson (2007); gunk: Sider (1993). Neo-Carnapianism: Eli Hirsch, with replies by Matti Eklund (2009) and John Hawthorne (2006).
The chapter shows that even the ontology of ordinary objects is contested, and that the most intuitive position is the hardest to defend.
Special composition question
Mereology studies parthood. The Special Composition Question (SCQ): under what conditions do some things compose a further thing? Three headline answers:
- Universalism: any things compose something (even a trout-turkey).
- Nihilism: composition never occurs (or only simples exist); "tables" is talk about simples arranged tablewise.
- Restrictivism: composition sometimes occurs (organisms, artifacts, ...), under a non-trivial condition.
Argument from cultural prejudice
Different cultures carve objects differently (the book uses taytu talk among Trobriand Islanders). If our belief in tables tracks cultural habit more than joints in nature, restrictivism that privileges our folk ontology looks arbitrary. Universalists and nihilists each claim to avoid that prejudice: one accepts all fusions, the other accepts none. Conservatives (ordinary objects only) must rebut debunking arguments.
Three answers to when composition occurs.
Neo-Carnapianism and verbal disputes
Neo-Carnapians (Hirsch and others) treat many ontological disputes as verbal: quantifiers in ordinary English can be interpreted so that "there are tables" comes out true in one language and false in another without a further deep fact. Ontological questions would then be confused questions about language, not about reality. Opponents say the dispute is substantive once we speak a joint-carving language (Ontologese, Chapter 10).
Neo-Carnapianism reframes ontology as language choice.
Composition as identity and conservatism
Some hold composition as identity: the whole just is the parts, counted loosely. That shrinks the sense in which composition adds an entity. Conservatives keep ordinary objects and reject trout-turkeys without going nihilist; they need a principle (life, contact, function, ...) that is not merely "what my culture carves". Debunking arguments press them; Cotnoir/Varzi-style formal mereology clarifies the options without settling the metaphysics.
Nihilist paraphrase patterns
Mereological nihilists say there are no tables, only simples (or only particles) arranged tablewise. Ordinary true speech is reinterpreted: "There is a table here" becomes "There are simples arranged tablewise here." The pattern mirrors Chapter 2's perforation paraphrase. Costs: every ordinary object inference needs translation; organism talk and person talk get hard if organisms would be composites. Benefits: no SCQ to answer; no cultural arbitrariness about which fusions exist; extreme ontological parsimony.
Universalists pay the opposite fee: accept all fusions, including scattered and gerrymandered ones, then explain why we talk as if only some matter. Restrictivists try to sit in the middle and catch fire from both sides.
Nihilism's paraphrase pattern and its ledger.
When Neo-Carnap wins or loses
If two speakers can each speak truly in their own quantifier dialect, and nothing further settles which dialect is joint-carving, the table debate looks verbal. If there is a privileged structure (Sider-style), or if paraphrase fails to preserve assertion conditions, the debate looks substantive. Chapter 10's Ontologese is the book's later attempt to name that privileged language.
Contact, life, and other restrictivist gadgets
Restrictivists propose conditions: composition when objects contact; when they form a life; when they are fastened; when they participate in a cooperative function. Each gadget fits some folk cases and fails others (contact is too promiscuous; life ignores artifacts; fastening is vague). The cultural-prejudice argument says that whatever gadget matches our folk theory may track history, not nature. That is why nihilism and universalism advertise themselves as principle-clean, even when their conclusions sound wild.
Composition gadgets fit some cases and fail others.
Universalism's odd fusions (trout-turkeys, arbitrary sums) are a cost to intuition, not automatically a contradiction. Defenders say intuition tracks usefulness, not ontology. Nihilists say the same about tables. The symmetry is intentional: both extreme answers claim to be more honest than a restrictivism tuned to one culture's middle-sized dry goods.
Verbal disputes: diagnostic questions
Hirsch-style diagnostics ask: can each side charitably interpret the other as speaking truly in a nearby language? Do they disagree about any possible evidence? Can they agree on a translation manual that preserves assertion patterns? If yes, the dispute may be verbal. Sider-style replies insist that only some languages match structure, so charity is not the last word. Effingham leaves you with the diagnostic habit even if you reject Neo-Carnapianism overall.
Mereology beyond material objects
Though metaphysicians focus on material composition, mereology also covers days of a week, number series, and parts of structural universals. Lessons transfer: parthood is not always spatial containment; identity conditions for wholes vary by category. When music works and properties return in Chapter 10 and 3, remember that "part" talk may be analogical. The SCQ is sharpest for material objects because folk ontology is loudest there.
Material Constitution
The statue and the lump
The story: at 9 am Effingham gathers bits of clay; at 10 am he forms one huge lump (Lump); at 11 am he shapes a statue (Statue); at 1 pm he crushes it back; at 3 pm the lump goes through the chipper. Five plausible claims are jointly inconsistent:
- Lump exists from 10 am to 3 pm.
- Statue exists from 11 am to 1 pm.
- So, by the indiscernibility of identicals (if Superman is 6 ft, so is Clark Kent), Lump is not Statue.
- At noon Lump and Statue are in the same place.
- Two objects cannot be in the same place at the same time (ghosts aside).
The paradox of the statue and the lump in five premises.
The standard account
Deny (5): statues and lumps coincide without interpenetrating like walls and tables, because their atoms are shared (Wiggins, Lowe). Two objections. Double counting: asked how many objects are where Statue is, we say "one". Reply: ordinary contexts restrict the quantifiers, just as we ignore the atoms, arms and legs. The grounding objection (most popular form: Michael Burke) is harder.
Perdurantism
Perdurantism (four-dimensionalism): objects have temporal parts as well as spatial parts. A flatlander in two spatial dimensions plus time is a 3D object; we are 4D objects. An instantaneous temporal part lasts an instant. Definition: x is a temporal part of y during T iff (i) x is part of y throughout T; (ii) x exists during, and only during, T; (iii) at every instant in T, x overlaps every part of y. So Statue is a temporal part of Lump (T = 11 am to 1 pm): they do not coincide, since Lump has parts Statue lacks.
Figure 9.2 (redrawn): a perduring flatlander in three-dimensional spacetime. A cone overall, yet circular at every instant (dashed cross-sections).
The Humean supervenience argument runs through possibility. If every fact supervenes on local point-particle facts, then a world where each instantaneous stage of an object was annihilated and replaced by a qualitatively identical one (as by an evil demon) would differ in no local fact from our world. So it would be our world: our objects would be composed of instantaneous parts. The endurantist who admits the demon world is possible faces pressure to accept perdurance.
The perdurantist solution and Cameron's objection.
Other motivations for perdurantism: cultural prejudice (a culture believing in instantaneous "yamils" would commit the universalist to temporal parts); Humean supervenience (Lewis: God fixes only local point particle facts, and an evil demon could replace each instant's object); the argument from temporary intrinsics (I am short then tall; temporal parts carry the incompatible properties); special relativity; and time travel (argued, the book notes wryly, by "some earlier temporal parts of myself").
The argument from temporary intrinsics shows how persistence questions connect to logic. If I am short at one time and tall at another, am I both short and tall? Relativizing to times ("short at t1") removes the contradiction, but Lewis asks how that relativization works; perdurantism answers that distinct temporal parts bear the incompatible properties. Some, like Thomas Hofweber, deny there was ever a problem to solve.
Other endurantist theories
Anti-realism: deny Lump or Statue exists, via nihilism (simples arranged statue-wise) or by denying statues. Or identify them, in three ways:
- Crude identity: one object throughout. But then Statue persists too long, and when I melt your television in an oven it still exists as a bubbling blob.
- The Burkean approach: with sortal essentialism (objects fall under their sortals essentially), when the statue is made Lump is destroyed and replaced by Statue/Lump, as rearranged wood makes a chair from a table. Problem: which sortals count? Not "__ is a child" (no one holds a wake before your eighteenth birthday), nor a sortal for every shape.
- Relative identity: identity is relativized, to sortals (Geach) or here to times: Statue is Lump at noon but not at 10 am, as Harrison Ford and Calista Flockhart are married now but were not when Ford was married to Melissa Mathison.
| Theory | Premise denied | Key cost |
|---|---|---|
| Standard account | (5) colocation | Grounding objection |
| Perdurantism | (4), at least over time | Coincident temporal parts |
| Nihilism | Lump and Statue exist | Underpopulation |
| Crude identity | (1) or (2) | Wrong persistence |
| Burke | (1): Lump ends at 11 am | Which sortals count |
| Relative identity | Classical identity | Revises the logic of identity |
The chapter's further reading
Introductions by Laurie Paul (2010), Katherine Hawley (2010), Ryan Wasserman (2009), Matthew McGrath (2007) and Sider et al. (2007); collections by Loux (2001) and Rea (1997). The standard account: David Wiggins (2001) and E. J. Lowe (2009); the grounding objection: Michael Burke (1992), with responses by Judith Thomson (1998), Karen Bennett (2004) and Lowe (1995). Perdurantist monographs: Hawley (2001), Mark Heller (1991) and Sider (2001). Humean supervenience: Lewis (1986) and discussion by Haslanger, Wasserman, Hawthorne and Scala. Time travel: Effingham and Robson (2007). Burke's own theory: his 1994 article; relative identity: Harry Deutsch (2007), Gallois (1998), Geach (1967).
The paradox earns its place in a book about method because every answer denies something plausible, which makes it the purest exercise in weighing costs.
Numerical versus qualitative identity
English "is" and "identity" cover at least two ideas. Numerical identity: one and the same thing (Clark Kent is Superman). Qualitative identity: exact similarity (two identical billiard balls). The statue/lump puzzle is about numerical identity: is Statue one and the same object as Lump?
The standard coincidence account and grounding
Standard account: Statue and Lump are two coinciding objects sharing matter and location for a time. They differ in sortal properties and persistence conditions (Lump can survive squashing; Statue cannot). Cost: many objects where common sense sees one; the grounding problem asks what grounds their different properties if they share all microphysical parts. Benefit: keeps classical identity and Leibniz's Law.
Perdurantism's cross-time picture
Perdurantists say objects have temporal parts. Statue and Lump can share temporal parts during coincidence and differ by having different total paths in spacetime (Figure 9.1/9.2 style). A flatlander who is circular at each instant can still be a cone across time. Endurantists say objects are wholly present at each time they exist; they need another account of coincidence or must deny one of the objects.
Persistence theories reshape the coincidence puzzle.
Other endurantist and anti-realist exits
Further endurantist options in the literature the chapter surveys include dominant kind theories (only the dominant sortal object exists), constitutionalism variants, and temporary identity (controversial). Anti-realism about ordinary objects dissolves the puzzle by denying Statue, or Lump, or both, in the spirit of Chapter 8 nihilism or paraphrase. Each exit moves costs: identity law, intuition, or ontology.
Menu of standard moves against statue/lump pressure.
Why Chapter 9 adds no new method
By now the toolkit is full: identity, paraphrase, temporal parts, anti-realism about objects, grounding worries. The statue/lump case is a stress test for those tools, not a vehicle for a ninth metaontology. That is why the book's pairing scheme skips a new method here. Readers should bring Chapters 7 and 8 especially: how you think about time and composition constrains what you can say about coincidence.
Leibniz's Law as the engine
The puzzle's engine is Leibniz's Law: identicals share all properties. Lump can survive flattening; Statue cannot. If they are identical, contradiction. So either they differ (two things, or different temporal parts), or one persistence claim fails, or identity is temporary or relative (costly revisions). Keep the Law in view when evaluating clever escapes; many "solutions" quietly restrict which properties count, which is itself a theoretical cost.
Leibniz's Law turns persistence differences into non-identity.
Temporary identity and relative identity (why costly)
Some revise identity itself: objects can be identical at t1 and distinct at t2, or identical relative to a sortal. These views can block the statue/lump contradiction but rewrite logic and ordinary counting. Effingham treats them as available yet expensive. Most of the chapter's energy stays on coincidence, perdurantism and anti-realism, which keep classical identity.
Works of Music
What is a work of music?
Performances are uncontroversial; the work is what they are performances of. Three cases show the two differ:
- Bob Dylan plays Blowin' in the Wind seventeen times in one concert: seventeen performances, one work.
- Beethoven created Piano Sonata No. 14, but not every performance of it.
- Its presto agitato is so hard that many performances are poor, yet the work remains fabulous.
Works also seem unlocated (is the sonata in Belgrade or on Pluto?) and created: before 1801 there was no such sonata. This is the creatability thesis.
Realism about musical works
Julian Dodd (Works of Music, 2007) identifies works with sound-structure Types: a C sharp on a piano, or from wind in a valley, is a token of the C-sharp Type. Since Types are abstract, they are eternal, so the work existed before 1801 and composers discover rather than create. Is that a slight on genius? Dodd compares great mathematical proofs, discovered yet works of genius; Effingham notes that discovering a country is less impressive.
Five answers to what a musical work is.
Four responses to creatability: (1) Dodd's discovery; (2) locate Types where their tokens are, like immanent realism; but making the first blue object is not making the property blue, so this is closer to discovery than creation; (3) deny that abstracta cannot be created: granted, the number 7 does not trip me and
Anti-realism about musical works
The anti-realist must paraphrase, and counting is hard, as with holes. Dodd's examples: "Mozart composed more than thirty symphonies" and "Exactly one of Bruckner's symphonies is unfinished". Meinongian, Neo-Carnapian and truthmaker approaches remain open.
Anti-realism about works has resources beyond paraphrase. A fictionalist might treat work-talk as true according to a useful story (Andrew Kania has explored this), and the Meinongian, Neo-Carnapian and truthmaker approaches of earlier chapters all apply. The chapter's point is transfer: each metaontology learned on holes, numbers or composites can be run on any category whatever, from football matches to banks.
Fundamentality
Fundamentalists think what exists is obvious, and the hard question is what fundamentally exists: the bedrock that explains or entails everything else. Related terms (grounding, ontological dependence, "in virtue of") are "(probably) synonymous" and should be taken "with a pinch of salt". Derivative entities are thereby ontologically reduced to what they depend on. Sceptics doubt the notion makes sense; defenders cite Descartes, Spinoza and Aristotle.
Ontologese is meant to be a real language, not a metaphor. As physics restricts itself to natural predicates ("charged", not "schmergan"), the metaphysician stipulates the use of the most natural existential quantifier. Speaking Ontologese, one says things that sound like English but have different truth values: "There are works of music" is true in English and false in Ontologese. The disagreement with the realist about works may then be merely apparent, much like the Neo-Carnapian diagnosis of Chapter 8, though fundamentalists insist that Ontologese questions are deep, not verbal.
Schaffer's priority monism: the whole explains its parts.
| Name | What a work is | Price |
|---|---|---|
| Dodd | Eternal sound-structure Type | Works discovered, not created |
| Levinson | Indicated Type (by composer at t) | More complex Types |
| Caplan and Matheson | Fusion of performances | Needs universalism, perdurance |
| Anti-realism | Nothing | Counting paraphrases |
| Cameron | Exists, but not fundamentally | Ontologese, fundamentality |
The book closes with advice to listen to everything by The 21st Century Monads, a band of philosophers.
The chapter's further reading
The book's guide: Julian Dodd (2008) as an introduction, plus Peter Kivy (2004, ch. 4) and Jerrold Levinson (2003, ch. 8); Dodd (2007) for his own theory; Caplan and Matheson (from 2004) for musical perdurantism; Levinson (1980) for indicated types; alternatives by Andrew Kania (2006), Peter Lamarque (2002), Guy Rohrbaugh (2003), Chris Tillman (2011); fictionalism (Kania 2008) and scepticism (Aaron Ridley 2003). On fundamentality: Cian Dorr (2005) introduced "Ontologese"; Kit Fine (2001), Cameron (2010), Schaffer (2009), Sider (2009, 2011); sceptics Thomas Hofweber (2009) and Stefano Predelli (2009); Cameron (2008) on musical works.
The book ends where much current metaphysics begins: with the thought that the deepest question is not what exists, but what is fundamental and what merely depends on it.
Works versus performances
A work of music is not the same as a performance. My Body is a Cage, Blowin' in the Wind, Beethoven's Piano Sonata No. 14, Nine Inch Nails' March of the Pigs can each have many performances (or recordings). Realists treat the work as a type or other abstractum that performances token. Anti-realists paraphrase work-talk into talk of scores, instructions, historical traditions, or classes of performances.
Realism and anti-realism about musical works
Realist benefits: simple semantics for "the same work again"; explanation of correctness conditions for performance. Costs: abstracta (Chapter 1 motives return); questions about when the work comes into existence (composing? first performance?); works for recording and improvisation blur type/token. Anti-realists avoid abstracta but owe paraphrases for critical and historical practice ("all performances of the Moonlight Sonata...").
Works as types versus paraphrase into performances and scores.
Fundamentality and Ontologese
Even if tables, statues and sonatas exist, maybe only some things are fundamental. Ontologese is the joint-carving language for stating what is fundamental. Priority monism (Schaffer): the cosmos is fundamental; parts are derivative. Other views: priority pluralism (many fundamental objects); grounding structures layered by levels. Effingham applies these tools to music: a work might exist derivatively on performances, scores and practices, or on a distributional property of the universe. That last monist application is Effingham's, not Schaffer's.
Fundamentality adds a second question beyond bare existence.
Repeatable types and the abstracta bill
Musical works as types inherit Chapter 3's repeatability and Chapter 1's abstracta bill. If types are abstract, Eleatic and epistemic motives return. If works are historical individuals initiated by acts of composing (a common rival view), you gain concreteness in the historical chain but must explain shared structure across performances without a type. Improvised works and studio-constructed tracks strain both stories: sometimes the first performance or the master recording seems to be the work.
Music ontology reopens the abstracta ledger from Chapter 1.
Existence versus fundamentality as closing move
The book's last methodological turn: you might agree that musical works exist (or that tables do) and still deny they are fundamental. Grounding and priority theories let ontology become a layered map rather than a flat inventory. That does not make the earlier chapters idle; Quinean fights about existence still matter when people deny the entities altogether. Fundamentality is an extra dial, not a replacement, unless you follow Schaffer-style reframing all the way.
Bringing the toolkit to one case
For a single work such as the Moonlight Sonata, ask: Does Quinean commitment from musicology force a type? Can anti-realists paraphrase? Do performances compose a work (mereology)? Is the work an enduring individual or a perduring historical process? Is it fundamental or grounded in sounds, scores and practices? Chapter 10's point is that a mature ontologist can ask all of these without confusing them. Music is the exam; the earlier chapters are the syllabus.
One musical work, four non-identical ontological questions.
Priority monism applied with care
If the cosmos is the one fundamental object, musical works exist at best as derivative patterns in its property distribution. That can underwrite anti-realist-friendly existence without types as fundamental abstracta. It does not automatically answer performance-identity questions, and it is not Schaffer's own example. Use it as an illustration of fundamentality's flexibility, not as a citation of Schaffer on music.
Main Points
- Ontology is neither nonsense nor trivial. Permissivism collapses into a Russell-style contradiction, and the nominalist owes an account of true-seeming talk.
- Theory choice is cost-benefit analysis. Weigh coherence with intuition, explanatory power, ideological and ontological parsimony; no virtue always wins.
- Quine's criterion is the default, not the law. "To be is to be the value of a bound variable" drives paraphrase, but Meinongians, Neo-Carnapians and fundamentalists all reject it.
- Paraphrase strategies founder on counting. Holes, properties, numbers and works all generate sentences ("Exactly one of Bruckner's symphonies is unfinished") that resist paraphrase.
- Reduction is cheap but underdetermined. Benacerraf's problem shows equally good reductions make any choice arbitrary.
- Science can bear on ontology without settling it. Newton's bucket, curved spacetime and special relativity all pressure a view; brute facts, Poincaré-style forces and hidden frames keep rivals alive.
- Composition answers tend to go to extremes. Cultural prejudice and vagueness push from intuitive restriction to universalism or nihilism.
- Constitution forces a choice about identity or location. Every solution to the statue and the lump denies a plausible premise.
-
Fundamentality reframes the question. Asking what fundamentally exists lets Cameron call works of music real in English but not in Ontologese.
-
Every method has a characteristic failure. Paraphrase fails at counting; reduction at arbitrariness; fictionalism at incompleteness and Brock-Rosen; science at underdetermination; truthmaking at negative truths; Neo-Carnapianism at meaning; fundamentality at intelligibility.
- Categories interlock. Holes may be regions (Ch2, Ch6); properties may be sets across worlds (Ch3, Ch4, Ch5); works may be perduring fusions (Ch8, Ch9, Ch10). A choice in one chapter constrains others.
The single most useful habit the book teaches is to ask, of any claim that something exists or does not, what it costs: which intuitions it overrides, which primitives it needs, which entities it adds, and which metaontology it presupposes. A reader who asks those four questions can enter any of the debates surveyed here, and most of those that have arisen since 2013.
| Name | What it says | Use it when |
|---|---|---|
| Quine's criterion | Commitments are values of bound variables | Testing a paraphrase |
| Four virtues | Intuition, explanation, ideology, ontology | Comparing rival theories |
| Metaphysical explanation | Posit what explains facts | Problem of universals |
| Meinongianism | "There is" is not "exists" | Fictions, impossibilia |
| Fictionalism | True according to a story | Worlds, numbers |
| Truthmaking | Truths need existing grounds | Past, negative truths |
| Neo-Carnapianism | Disputes reflect quantifier variance | Composition debates |
| Fundamentality | Ask what is bedrock | Derivative entities |
Eight metaontologies the book introduces, from Quine's default to fundamentality.
Who's who (core)
| Name | Role in the book |
|---|---|
| W. V. O. Quine | Ontological commitment; "On What There Is" |
| David Lewis | Genuine modal realism; counterpart vibes in world talk |
| Alexius Meinong | There-is versus exists |
| Isaac Newton / G. W. Leibniz | Bucket versus shift on absolute space |
| Albert Einstein | Relativity pressure on presentism |
| Peter van Inwagen | Special composition; materialism themes |
| Eli Hirsch | Neo-Carnapian verbal disputes |
| Jonathan Schaffer | Grounding; priority monism |
| David Armstrong | Universals; truthmaking tradition |
| Nikk Effingham | Guide; later work on time, mereology, constitution |
The book ends by complicating Quine, not by discarding theory choice.
Practitioner angle (genuine links)
If you build conceptual models (enterprise ontologies, knowledge graphs, AI schemas), Effingham's lessons transfer without metaphor abuse: (1) surface vocabulary is not commitment; (2) paraphrase has ideological costs; (3) "what exists in the model" differs from "what is fundamental for explanation"; (4) verbal disputes about labels are real and need joint-carving criteria. Use the book's ledgers when a data model multiplies entity types or when a team argues whether a table "really exists" in the domain.
Transferable lessons for conceptual modelling without metaphor abuse.