20.11.02 · philosophy / metaphysics

Metaphysics — depth: grounding, persistence, and possible worlds

shipped3 tiersLean: none

Anchor (Master): Lewis 1986 On the Plurality of Worlds (Blackwell); Fine 2001 'The Question of Realism' and Fine 2012 'Guide to Ground'; Schaffer 2009 'On What Grounds What' (Monist); Kripke 1980 Naming and Necessity (Harvard)

Intuition Beginner

The foundations unit 20.11.01 asked what there is. This unit asks a sharper question: among the things that exist, which depend on which? A clay statue and the lump of clay it is made of seem to be two things in the very same place. Yet they obey different fates: the lump survives being squashed, the statue does not. How can two things share one location, and which is more fundamental?

A second cluster of questions concerns time and possibility. The tree in the garden was once an acorn and could have been a table. What is it for one and the same object to exist across time, changing its shape and size? And what is it for something to be merely possible — a way things could have gone but did not?

Metaphysics at this depth is the study of dependence, persistence, and modality: what grounds what, what survives change, and what could have been otherwise. Each question splits into opposed live answers, and the discipline consists in laying out those answers and their costs.

Visual Beginner

A table laying out the three depth questions and the live positions on each.

Depth question	Position A	Position B
What grounds what?	Priority monism: the whole grounds the parts	Priority pluralism: the parts ground the whole
How do objects persist?	Endurantism: wholly present at each time	Perdurantism: extended in time via temporal parts
What are possible worlds?	Modal realism: concrete universes like ours	Ersatz realism: abstract representations

The three panels name the load-bearing depth questions this unit develops.

Worked example Beginner

The statue and the lump. A sculptor takes a lump of clay and shapes it into a statue. The statue and the lump occupy exactly the same region and share exactly the same atoms.

Step 1. They differ in what they can survive. Squash the clay and the lump still exists (it is now a flattened lump); the statue does not (it has been destroyed).

Step 2. If the statue and the lump were one object, they would share every property. But "can survive being squashed" is a property the lump has and the statue lacks.

Step 3. So the statue and the lump are not identical, even though they share all the same matter at every moment. Two objects coincide in one place.

What this tells us: constitution is not identity. One material object can constitute another without being it, and the puzzle forces metaphysics to explain how such coincidence is possible rather than contradictory.

Check your understanding Beginner

Formal definition Intermediate+

Metaphysics at depth studies three relations that the foundations unit 20.11.01 introduced only by name. Here are their working definitions, drawn from the contemporary literature ^{[Fine 2001]}.

Grounding: an asymmetric, irreflexive, transitive, hyperintensional relation $x ⊏ y$ read " $x$ grounds $y$ " (or " $y$ obtains in virtue of $x$ "). If $x ⊏ y$ then $y$ is ontologically dependent on $x$ : $y$ holds because $x$ does. Grounding is hyperintensional, distinguishing necessarily coextensive facts that mere material implication conflates.
Mereological parthood: a two-place relation $x ⪯ y$ read " $x$ is part of $y$ ," governed by reflexivity, antisymmetry, and transitivity in classical extensional mereology. The special composition question asks under what conditions some objects $x_{i}$ have a mereological fusion $z$ with each $x_{i} ⪯ z$ .
Closest-world counterfactual: for possible worlds $w, u$ let $w ≼_{v} u$ denote " $w$ is at least as close to the actual world $v$ as $u$ is." The counterfactual $A ⇝ C$ ("if $A$ were the case, $C$ would be") is true at $v$ iff $C$ holds at every closest $A$ -world to $v$ , or at no $A$ -world if there is none ^{[Lewis 1986]}.

A grounding claim records the dependence direction explicitly. The singleton set ${S ocr a t es}$ is grounded in Socrates, written ${S ocr a t es} ⊏ S ocr a t es$ : the set exists because the man does, and not the reverse. The asymmetry is the load-bearing feature.

Two live positions on each contested question

The depth questions each admit opposed live positions; the literature has not converged.

On grounding. Priority monism (Schaffer 2009) holds that the cosmos as a whole is the sole fundamental entity and that ordinary objects are grounded in it: the whole grounds the parts ^{[Schaffer 2009]}. Priority pluralism (the more common view) holds that many fundamental entities stand on a level and jointly ground the derivative. The disagreement is not about how many things exist but about the direction of the dependence arrow.

On persistence. Endurantism holds that a persisting object is wholly present at each time at which it exists, identical across change. Perdurantism (Sider 2001) holds that a persisting object is a four-dimensional worm extended in time, with temporal parts analogous to spatial parts ^{[Sider 2001]}. Exdurantism (the stage view) holds that what exists at a time is a momentary temporal part, related to other parts by a counterpart relation.

On possible worlds. Lewisian modal realism analyses worlds as concrete, spatiotemporally isolated universes. Ersatz realism analyses worlds as abstract, maximally consistent representations (often sets of propositions). Both supply truth-conditions for modal claims; they differ on ontology, trading fidelity against parsimony.

Counterexamples to common slips

Grounding is not causation. Causation relates events across time; grounding relates facts synchronically and is independent of temporal order.
Parthood is not set-membership. A part need not be an element of a set; mereology is ontological, while set theory is one formal calculus among many.
A closest world is not an identical world. Counterfactual closeness is overall similarity, which permits small lawful differences in the closest antecedent-world.

Key argument Intermediate+

Grounding and the singleton. Consider the singleton set ${S ocr a t es}$ . It exists, but not on its own account: there is no possible situation in which Socrates fails to exist yet ${S ocr a t es}$ does. The set obtains in virtue of its member. We record this as the grounding claim

{S ocr a t es} ⊏ S ocr a t es, read: the singleton is grounded in the man .

The asymmetry is essential. Socrates does not obtain in virtue of his singleton; remove the set and the man remains, remove the man and the set vanishes. This directional dependence is what grounding adds beyond mere coexistence, and it is what a pure quantifier ontology (one that counts only what exists) cannot capture ^{[Fine 2001]}.

Closest-world counterfactual, evaluated. Take the counterfactual "if the match had been struck, it would have lit." On Lewis's analysis we evaluate it at the actual world $v$ by considering the closest worlds to $v$ in which the match is struck. Let $w^{⋆}$ be such a closest struck-world. If at $w^{⋆}$ the match lights — because the closest worlds preserve the relevant background (a dry match, oxygen present, the laws intact) — then $v ⊨ (struck ⇝ lit)$ holds. The verdict fails only if some closest struck-world is a no-light world, as it would be if the closest struck-world also contained a dampening gust.

The dependence of the verdict on the closeness ordering, not on the mere truth of the antecedent, is what distinguishes the counterfactual from the material conditional. A material conditional with a false antecedent is vacuously true; a counterfactual is not ^{[Lewis 1986]}.

Argument reconstructed. (1) A counterfactual $A ⇝ C$ is true at $v$ iff $C$ holds at every closest $A$ -world to $v$ . (2) The closest $A$ -world is the world most similar to $v$ while making $A$ true. (3) Similarity weights laws and particular facts according to their relevance to the case. (4) Therefore the truth of $A ⇝ C$ turns on the structure of modal space, not on any actual event.

Bridge. This argument builds toward 42.02.01 pending (modal logic and model theory), where the closest-world semantics for counterfactuals is formalised within Kripke frames and accessibility orderings, and appears again in 20.12.01 (philosophy of language), where rigid designation across possible worlds presupposes a worked-out modal structure. The foundational reason grounding and counterfactuals belong together is that both are dependence relations over a structured space; this is exactly the gap the foundations unit left open; putting these together, the bridge is that grounding orders reality vertically while counterfactuals order worlds horizontally, and the pattern generalises from modal metaphysics into the dependence-typed structures of proof theory and category theory.

Exercises Intermediate+

Exercise 7 (hard, symbolic).

Reconstruct the statue/lump argument as a Leibniz's-law reductio of the identity of statue and lump, and state the two main responses.

Hint

Find a property the lump has that the statue lacks, then apply the indiscernibility of identicals.

Answer

(1) The lump can survive being squashed; the statue cannot. (2) "Can survive being squashed" is a genuine property. (3) If statue = lump, then by Leibniz's law they share all properties. (4) They do not share this one. (5) Therefore statue $\neq =$ lump. Responses: (a) accept coincidence — two objects share one region, distinguished by their persistence conditions (the dominant line, Wiggins, Lowe); (b) deny there are two, treating the statue as the lump under a statue-sortal (the "constitution is identity" / eliminativist line), at the cost of explaining why we cannot then squash one without destroying the other.

Exercise 8 (hard, symbolic).

State Lewis's modal realism in one sentence, then present the ersatz-realist objection and Lewis's main reply.

Hint

What does the ersatz theorist gain by making worlds abstract, and what does Lewis say that gain costs?

Answer

Modal realism: possible worlds are concrete, spatiotemporally isolated universes of the same kind as ours, and modality is quantification over them. The ersatz objection: concrete worlds are an intolerable ontological cost, and abstract maximal representations secure the same semantics for modal and counterfactual talk without committing to a plurality of real universes. Lewis's reply: ersatz worlds cannot ground modality non-circularly, because to say which abstract representation counts as a "world" the ersatz theorist must already invoke possibility, necessity, or completeness — notions that only the concrete-worlds analysis reductively explains.

Lean formalization Intermediate+

lean_status: none. Depth metaphysics is prose-first; the correctness gate is argument reconstruction and fidelity to the primary texts of Lewis, Fine, Schaffer, and Sider. The formal apparatus relevant to this material — Kripke frames for modal logic, the closest-world ordering for counterfactuals, and classical extensional mereology — is developed in 42.02.01 pending (modal logic and model theory) and the proof-theory chapters of §42. A Lean encoding of a toy grounding relation is possible, but the contested claims here (whether the whole grounds the parts, whether temporal parts exist) are ontological and resist formal settling; the right gate is source fidelity, not type-checking.

Advanced results Master

Four contested programmes dominate contemporary depth metaphysics. Each is a live position rather than settled consensus, and on each the literature is divided.

Grounding and fundamentality. The grounding revival (Fine 2001; Schaffer 2009; Rosen 2010) analyses metaphysical dependence as a primitive, hyperintensional relation: the fact that Socrates exists grounds the fact that ${S ocr a t es}$ exists, and not the reverse ^{[Fine 2001]}. On this view the question "what is fundamental?" is not the question "what exists?" but "what grounds what?" Two positions follow. Priority monism (Schaffer 2009) holds that the cosmos is the sole fundamental entity and that parts are grounded in the whole ^{[Schaffer 2009]}. Priority pluralism holds that many fundamental entities stand on a level and jointly ground the derivative. The dispute turns on the direction of the grounding arrow, not on the inventory of what exists.

Possible worlds. Lewis's modal realism (1986) analyses modality through concrete, spatiotemporally isolated worlds: to say that unicorns are possible is to say that some concrete world contains unicorn-like creatures ^{[Lewis 1986]}. The payoff is an elegant, uniform semantics for modality, counterfactuals, and intensional content; the cost is ontological exuberance. Ersatz realism (Plantinga; Rosen) replaces concrete worlds with abstract maximal representations and keeps the semantics at a lower ontological price, paying in the loss of the modal realist's reductive ambition. Conceptualist and linguistic theories treat "possible world" talk as a useful fiction. The dispute is live because each position pays one cost to avoid another.

Persistence. Sider's Four-Dimensionalism (2001) develops perdurantism as the thesis that persisting objects are four-dimensional worms with temporal parts, dissolving the problem of temporary intrinsics: a wire is bent-at- $t_{1}$ and straight-at- $t_{2}$ by having distinct temporal parts that bear the incompatible shapes ^{[Sider 2001]}. Endurantism (Lowe; Wiggins) denies temporal parts: the object is wholly present at each time and bears its properties time-indexed. Exdurantism (Hawley; Sider's stage view) holds that what exists at a time is a momentary stage, and persistence is counterpart-theoretic. The three views disagree about whether time is relevantly like space for objects.

Mereology and constitution. The special composition question asks when some $x_{i}$ compose a $y$ . Universalism answers: always; any non-empty class has a fusion. Nihilism answers: never; only simples exist. Organicism (van Inwagen) answers: only when the $x_{i}$ are jointly caught up in a life. Each view pays a different price: universalism admits bizarre fusions, nihilism denies chairs, organicism denies tables. The statue/lump puzzle sits at the boundary: if constitution is not identity, two objects share one region, and the theorist must explain why coincidence is coherent rather than contradictory.

Synthesis. The four depth questions build toward a single layered conception of reality in which the fundamental grounds the derivative, the actual sits among many possible worlds, and objects persist by one of three competing structures; this conception appears again in 42.02.01 pending, where modal logic formalises the closest-world semantics, and in 20.13.01, where the persistence of persons reframes the endurance–perdurance debate. The foundational reason these questions belong to one subject is that grounding, modality, and persistence are three constraints on a single ontology; this is exactly why a verdict on mereology forces a verdict on persistence and constitution; the central insight is that what exists and what grounds what cannot be answered apart; the bridge is that the grounding order, the accessibility relation on worlds, and the temporal-parts structure of objects are three faces of dependence; and the pattern generalises from metaphysics into the type hierarchies of mathematics and the layered theories of the special sciences; putting these together, depth metaphysics is the systematic study of ontological dependence.

Full proof set Master

Proposition (endurantism plus Leibniz's law yields the problem of temporary intrinsics). If a persisting object is wholly present at each time at which it exists, and if it has incompatible intrinsic properties at different times, then either Leibniz's law is violated or the properties are not genuinely intrinsic.

Argument. Let $o$ be a wire that is bent at $t_{1}$ and straight at $t_{2}$ . Assume endurantism: $o$ is wholly present at $t_{1}$ and wholly present at $t_{2}$ , and the object present at $t_{1}$ is identical to the object present at $t_{2}$ (call both $o$ ). By Leibniz's law, the indiscernibility of identicals, whatever is true of $o$ at $t_{1}$ is true of $o$ at $t_{2}$ , since they are one object. But "bent" and "straight" are incompatible intrinsic shapes. So $o$ is both bent and not-bent, a contradiction. To avoid it, the endurantist must (i) deny that bentness is intrinsic and treat it as a relation to a time, (ii) index the property to the time ( $b e n t -at- t_{1}$ ), or (iii) accept temporal parts. Options (i) and (ii) are the standard endurantist replies; option (iii) is perdurantism. $□$

This is Sider's pressure point against endurantism: the problem of temporary intrinsics makes the cost of denying temporal parts visible ^{[Sider 2001]}. The perdurantist solves it by locating the incompatible properties in distinct temporal parts, so that no single thing has both. The endurantist must show that time-indexing or relationising properties preserves their intrinsic character — a substantive and contested claim, not a settled result.

Connections Master

Metaphysics — foundations 20.11.01. The depth questions here presuppose the identity, modality, and persistence notions introduced in the foundations unit; grounding and closest-world semantics are precisifications of dependence and possibility whose raw forms first appear there.
Philosophy of language 20.12.01. Kripke's rigid designation and the necessary a posteriori, developed there, supply the modal background against which the possible-world metaphysics of this unit is set; counterfactual semantics and rigid reference share the same Kripke frames.
Modal logic and model theory 42.02.01 pending. The formal semantics for necessity, possibility, and counterfactuals — Kripke frames, accessibility relations, closest-world orderings — lives here; this unit imports that apparatus and asks the further metaphysical question of what possible worlds are.
Philosophy of mind 20.13.01. The persistence of persons, central to debates about personal identity, applies the endurance–perdurance framework developed here to the self, and the constitution relation reappears as the relation between a person and their body.

Historical & philosophical context Master

The grounding revival marks a deliberate return to Aristotelian metaphysical structure. Fine's "The Question of Realism" (2001) argued that the central metaphysical question is not what exists but what grounds what — that a quantifier ontology counts what there is but misses the dependence structure that distinguishes the fundamental from the derivative ^{[Fine 2001]}. Schaffer's "On What Grounds What" (2009) pressed the point into priority monism, reviving the neo-Platonic thought that the whole might be prior to its parts ^{[Schaffer 2009]}.

Lewis's On the Plurality of Worlds (1986) gave the most systematic metaphysics of modality ever written, defending modal realism: the thesis that the worlds quantified over in modal logic are concrete universes of the same kind as ours ^{[Lewis 1986]}. The audacity of the view — and the vigour of the ersatz-realist reply — set the terms for every subsequent treatment of modality.

Sider's Four-Dimensionalism (2001) consolidated the perdurantist programme, tying temporal parts to the problem of temporary intrinsics and to the broader thesis that ontology should answer to ideology — that the world's structure is limned by our best, most joint-carving vocabulary ^{[Sider 2001]}. Kripke's Naming and Necessity (1980), already load-bearing in 20.12.01, supplies the necessary-a-posteriori framework without which the modal metaphysics of natural kinds would have no purchase ^{[Kripke 1980]}.

Bibliography Master

@book{Lewis1986,
  author = {Lewis, David},
  title = {On the Plurality of Worlds},
  publisher = {Basil Blackwell},
  year = {1986},
}

@article{Fine2001,
  author = {Fine, Kit},
  title = {The Question of Realism},
  journal = {Revue Internationale de Philosophie},
  year = {2001},
}

@incollection{Fine2012,
  author = {Fine, Kit},
  title = {Guide to Ground},
  booktitle = {Metaphysical Grounding},
  publisher = {Cambridge University Press},
  year = {2012},
}

@article{Schaffer2009,
  author = {Schaffer, Jonathan},
  title = {On What Grounds What},
  journal = {The Monist},
  year = {2009},
}

@book{Sider2001,
  author = {Sider, Theodore},
  title = {Four-Dimensionalism: An Ontology of Persistence and Time},
  publisher = {Oxford University Press},
  year = {2001},
}

@book{Kripke1980,
  author = {Kripke, Saul A.},
  title = {Naming and Necessity},
  publisher = {Harvard University Press},
  year = {1980},
}

@book{Lowe2002,
  author = {Lowe, E. J.},
  title = {A Survey of Metaphysics},
  publisher = {Oxford University Press},
  year = {2002},
}

Prerequisites

20.11.01
20.12.01

Tier anchors

beginner: Lowe 2002 A Survey of Metaphysics (Oxford) pts. on grounding, persistence, and modality in plain English; Sider 2001 Four-Dimensionalism (Oxford) intro
intermediate: Lowe 2002 A Survey of Metaphysics (Oxford) pt. III, persistence and constitution; Sider 2001 Four-Dimensionalism (Oxford University Press) chs. on temporal parts and ontological dependence
master: Lewis 1986 On the Plurality of Worlds (Blackwell); Fine 2001 'The Question of Realism' and Fine 2012 'Guide to Ground'; Schaffer 2009 'On What Grounds What' (Monist); Kripke 1980 Naming and Necessity (Harvard)

References

Lewis — On the Plurality of Worlds · 1986, Basil Blackwell; modal realism, concrete possible worlds, counterfactuals via closest worlds
Fine — The Question of Realism · 2001, Revue Internationale de Philosophie; the grounding relation and the rejection of mere quantifier ontology
Schaffer — On What Grounds What · 2009, Monist 92; priority monism, the cosmos as the one fundamental entity
Sider — Four-Dimensionalism · 2001, Oxford University Press; perdurantism, temporal parts, the stage view, the problem of temporary intrinsics
Kripke — Naming and Necessity · 1980, Harvard University Press; rigid designation and necessary identity, the modal background

Estimated time

beginner: 15m
intermediate: 45m
master: 90m