Timothy Williamson published *Modal Logic as Metaphysics* in 2013, a book whose title names exactly what the project is. His argument (developed across two decades of papers and consolidated in the book) is that modal logic is not a regional formalism for talking about possibility but the central apparatus of metaphysics itself, and that its quantifiers, when interpreted with the seriousness the formalism deserves, range over a constant domain across worlds. Necessarily everything is necessarily something. Whatever there is belongs to the same unrestricted domain at every world, including merely possible objects that are non-concrete at the actual world. Contingency concerns which objects are concrete and which properties they instantiate, rather than variation in the unrestricted domain itself. The Barcan formula and its converse, which together express the commutativity of quantification and modality, are validated. The view is the most ambitious modal-logical position currently defended, and the case for it draws on the simplicity, generality, and explanatory reach of constant-domain QML (quantified modal logic).

Williamson (*Modal Logic as Metaphysics*, 2013) is the systematic statement; "Bare Possibilia" (1998), "Necessary Existents" (2002), and "Everything" (2003) develop the apparatus. The position descends from Barcan Marcus (1946), who first proved the formula that bears her name in a quantified modal logic and pressed its philosophical consequences. Against necessitism stands a cluster of contingentist responses: Stalnaker (*Mere Possibilities*, 2012) defends the contingent-domain alternative by arguing that the apparent simplicity of constant-domain quantification is purchased at the cost of a domain populated by "merely possible" entities whose existence-status is obscure; Fine ("Necessity and Non-Existence," 2005) develops an essence-based account on which Socrates can be necessarily Socrates without necessarily existing; Adams ("Actualism and Thisness," 1981) defends actualism against the Barcan-formula consequences; Hale (*Necessary Beings*, 2013) develops a structured-necessity account on which different kinds of necessity have different scopes. Plantinga (1974b) had in effect pressed the objection in advance: such a position misreads modal claims about objects as modal claims about haecceities (individual thisnesses, the property of being *this very individual* and no other). The contemporary higher-order modal logic literature (Linnebo, *Thin Objects*, 2018; Linnebo and Rayo, "Hierarchies Ontological and Ideological," 2012) extends the dialectic into plural-quantification frameworks where the Barcan formula's status is again disputed.

The present reply sits closer to Stalnaker on the contingentist question and adds what Stalnaker alone does not specify: which contingencies the structural argument requires, and why those contingencies survive Williamson's framework whether or not constant-domain quantification is endorsed.

The challenge to the structural argument is double-edged. On one hand, necessitism would seem to make the necessary being trivially available: every possible thing exists necessarily, so the necessary being is among them. On the other hand, necessitism eliminates the explanatory work the necessary being is supposed to do. If everything exists necessarily, there is no explanatory contrast between the necessary being and contingent beings, because there are no contingent beings in the relevant sense. The argument's modal conclusion □(T∧S∧Φ) (the necessary obtaining of the three structural Conditions T, S, and Φ, borne by Time, Space, and Substance) lands trivially in a framework where everything is □ (necessary), which means it lands without distinguishing the structural conditions from anything else. Worse: the argument's premises require contingency the framework appears to deny. ◇C (the possibility of change) is defended in [§2.4](<2.4 Structural Requirement.md>) as a substantive possibility-claim grounded in Cartesian phenomenal givenness: change is possible because change is occurring in the very act of doubt. If the framework reduces all contingency to qualitative variation across a fixed domain, the question is whether ◇C remains a genuine modal possibility distinguishable from □C, and whether the contingency of which world is actual (presupposed for the proof's stretch from □◇C to □(T∧S∧Φ) being non-trivial) survives at all.

A clarification before the moves. The dispute is not with quantified modal logic, the Barcan formula's formal validity in S5, or the ambition of modal logic to do metaphysical work. The argument shares with Williamson the commitment that modal logic is not a regional formalism but the apparatus through which metaphysical structure is articulated. The dispute is with whether the move from formal validity in constant-domain QML to ontological necessitism preserves the contingencies the structural argument requires for its premises to retain their force. Three forms of the move press that question at three resolutions: the constant-domain compatibility move, the Cartesian-Certainty preservation move, and the actuality-contingency preservation move. The space they canvass is the space necessitism can occupy. Each requires the structural roles its modal apparatus presupposes, and each leaves the argument's needed contingencies intact at the level the argument needs them.

*First.* The constant-domain compatibility move presses that the structural argument's modal conclusion lands trivially in Williamson's framework. □(T∧S∧Φ) is a necessity claim, and necessitism makes every existence claim a necessity claim. The conclusion is therefore consistent with necessitism but undistinguished by it. If the structural argument's distinctive contribution is the modal status of T, S, and Φ, and necessitism flattens all modal-status differences at the existence level, the argument has not earned anything Williamson's framework does not already deliver. The conclusion would be true but not informative.

The reply distinguishes the structural sense in which T, S, and Φ are necessary from the necessitist sense in which everything is necessary. Necessitism's necessity is *constant-domain*: the same things exist in every world. The structural argument's necessity is *constitutive*: T, S, and Φ are conditions any world must instantiate to be a world. The two are compatible but not equivalent. A necessitist can accept that the things which exist in every world include configurations satisfying T, S, and Φ, and the structural argument adds that the satisfaction of T, S, and Φ is what makes the constant domain a domain of any kind rather than a featureless plurality.

The Barcan formula and its converse, applied to the structural roles, yield exactly what the argument independently derives (every world's domain instantiates T, S, and Φ), but the formula does not by itself explain *why* every world's domain instantiates them. Appendix A's *Domain Semantics and the Barcan Question* records that the structural argument quantifies over Conditions rather than individuals, so Barcan and Converse Barcan are orthogonal to the proof's load-bearing claims at the structural level.

The structural argument proposes a metaphysical explanation Williamson's formal framework does not itself provide. This should not be attributed to a general truthmaker requirement on Williamson's part: the narrower point is that constant-domain semantics alone does not establish the book's constitutive account of T, S, and Φ. The structural account answers: the constitutive role of T, S, and Φ in any configuration is what makes the constant domain a domain, and the quantifier's range across that domain is the formal shadow of the constitutive fact. The constant-domain framework is exactly the framework the structural argument's conclusion describes, with the constitutive content filled in.

*Second.* The Cartesian-Certainty preservation move presses the harder version. The argument's central premise (◇C, the possibility of change) is defended in [§2.4](<2.4 Structural Requirement.md>) as a substantive possibility-claim grounded in Cartesian phenomenal givenness: change is occurring in the very act of doubting, and that the doubting is a change cannot itself be doubted without instantiating it. ◇C must therefore mark a genuine modal possibility distinguishable from □C, the necessity of change. On Williamson's framework, the objector presses, this distinction collapses. Contingency is qualitative: which properties things have varies across worlds, but which things exist does not. If ◇C is read as the claim that change-events exist, necessitism makes that claim trivially necessary. If it is the claim that change-events possibly occur, the modal qualifier ◇ adds nothing beyond the qualitative-contingency level Williamson already permits. Either way, the Cartesian datum (that something *might* not have been changing) loses its modal weight, and the proof's first step is undermined.

The collapse does not occur, because the premise's modal weight does not sit at the level necessitism flattens. ◇C is not a claim about which entities exist in modal space; it is a claim about which configurations the structural roles permit. The Cartesian datum is not 'a change-event exists' but 'the act of thinking now occurring is itself a change', and the modal scope of the premise is over configurations of the roles, not over the existence of relata. Necessitism leaves the configuration-level intact: even if every entity exists necessarily, the question of which configuration any entity is in remains a qualitative-contingency question, and qualitative contingency is precisely what Williamson preserves. ◇C marks a substantive possibility at the level Williamson's framework licenses: configurations vary across worlds, the configuration-of-change is one configuration among others, and the Cartesian certainty is that *this* configuration is instantiated.

Stalnaker (2012) is the natural ally here; the contingentist alternative makes the same observation in a different idiom: even if necessitism is correct about domain-constancy, it is not correct that all modal weight collapses to the qualitative level, because configurations themselves carry modal scope. ◇C is a configuration-level claim and survives the necessitist reframing intact.

The explicit formalization is given at [§2.8.4](<2.8.4 Necessity Precedes Contingency.md>): □F ∧ (◇p ∧ ◇¬p) for F = T∧S∧Φ records the configuration-level contingency the Cartesian datum requires while preserving structural necessity at the F-level. Fine's (2005) essence/necessity distinction sharpens the point: that change is occurring is a Cartesian datum about the actual configuration; that change-events exist necessarily (if Williamson is right) is a Williamsonian claim about the domain. The two claims are at different levels, and the structural argument's premise is at the configuration level the necessitist framework leaves untouched.

*Third.* The actuality-contingency preservation move presses the boldest version. The structural argument's modal line decomposes into four distinct elements rather than a single stretch. *P3* is ◇C, asserted in [§2.4](<2.4 Structural Requirement.md>) as a Cartesian datum about the actual world: change is occurring in the very act of doubt. *The S5 lift* from ◇C to □◇C uses the collapse of iterated modalities: once ◇C obtains, ◇C obtains necessarily, since possibility in S5 is itself a necessary feature of modal space. *The Modal Identity* of [§2.6.6](<2.6.6 Modal Identity.md>) is the biconditional □(◇C ↔ T∧S∧Φ): the necessary equivalence between the possibility of change and the co-instantiation of the three structural roles. *The theorem-level conclusion* □(T∧S∧Φ) follows from □◇C and the biconditional by K-axiom closure under □. The objection's bite lands not on the validity of the closure but on its informativeness: if which world is actual is not itself contingent (if there are no worlds whose configuration includes change distinguishable from worlds whose configuration does not) the closure runs formally but the conclusion records no metaphysical fact the alternatives could have falsified. The argument needs it to be a substantive fact that we are in a world whose configuration includes change. On Williamson's framework, the objector presses, the contingency of which world is actual cannot be sustained. Necessitism plus constant-domain quantification reduces the difference between worlds to qualitative variation in property-distribution; the actuality of *this* configuration becomes a brute haecceitistic fact rather than a contingent one. If which world is actual is not genuinely contingent, the stretch from □◇C to □(T∧S∧Φ) loses its bite: the conclusion follows by necessity from premises that themselves cannot be modally distinguished from their alternatives.

The reply runs at the level the proof actually operates. The actuality-contingency the argument needs is qualitative-contingency, not ontological-contingency, and qualitative-contingency is exactly what Williamson preserves. The proof requires that *some* configuration is actual and that not every configuration is actual: both are qualitative-contingency claims about how the constant domain is configured, not ontological-contingency claims about what the domain contains. Necessitism endorses the first (every world has a configuration) and is silent on the second (it does not require that every configuration be instantiated). Indeed, necessitism *requires* qualitative-contingency to be substantive, since otherwise every world is qualitatively identical and the framework has no work to do. The actuality-contingency the structural argument requires is precisely the qualitative-contingency Williamson's framework treats as the residue of contingency that survives constant-domain quantification.

Adams (1981) is helpful here as a foil: Adams holds that the qualitative-contingency level is itself unstable, but his argument depends on the haecceitist reading Williamson explicitly rejects. Hale's (2013) structured-necessity account makes the cleaner point: necessities at different levels do different work, and the structural argument's necessity at the constitutive level is compatible with qualitative-contingency at the configuration level. The stretch from □◇C to □(T∧S∧Φ) is non-trivial because the configuration-of-change is one qualitative possibility among others, and the actual configuration's instantiation of that possibility is what the proof's first premise records.

Williamson's positive defense of S5 in *Modal Logic as Metaphysics* rests on considerations the structural argument finds hospitable rather than threatening. The present proof likewise derives S5 from fixed-domain semantics: real possibility ranges over one global domain W, so a witness to ◇P remains a witness at every evaluation point. This is distinct from Williamson's constant-domain thesis about **individuals**. A fixed domain of possible worlds does not by itself entail a constant domain of entities across those worlds.

The Brouwer axiom (p → □◇p) then follows for the present operator because any actual P is witnessed within W, and every world evaluates possibility over that same W. The Cartesian datum establishes ◇C through actuality; the fixed-domain account, not a prior claim that T, S, and Φ are everywhere instantiated, yields □◇C.

The disagreement is therefore precise. Williamson uses constant-domain QML to motivate ontological necessitism. The structural argument uses a fixed global domain of real possibilities to establish modal invariance, then separately derives the necessary Conditions that any world in that domain realizes. The shared formal result does not settle the ontological reading.

Fine, engaged as dialectical opponent rather than as authority, presses the structural argument from a complementary direction. Fine's "Essence and Modality" (1994) and "Necessity and Non-Existence" (2005) establish that necessity derives from essence: what makes a modal claim necessary is that it follows from the essence of some entity. Applied to the structural argument: what makes □(T∧S∧Φ) necessary? If it follows from the essence of T, S, or Φ themselves, the argument is circular: T, S, and Φ must already exist and have essences to generate the necessity, which is what the argument is trying to establish. If it follows from the essence of *change*, Fine will press that conceptual essence generates analytic necessity rather than unrestricted metaphysical necessity. The structural reply is that the argument's modality is neither essence-based in Fine's sense nor merely analytic. It is role-necessity: the necessity that anything being a *something* at all instantiates the structural conditions, grounded not in the essence of T, S, or Φ as entities but in the role-structure that any entity's being anything at all requires. The book proposes role-necessity as an alternative explanatory basis, but Fine's framework is flexible enough to ask whether structural roles themselves have essences or are grounded in more basic essences. The challenge therefore remains substantive: the book must defend the priority of role-structure rather than treating it as outside Fine's apparatus by definition.

A kindred move drops Williamson's necessitism for Linnebo's higher-order modal logic, where the Barcan formula is reformulated in plural-quantification terms and the necessitism/contingentism dialectic re-emerges at the level of pluralities rather than individuals. This is the sharpest form of the necessitist challenge, and it cannot be turned aside by the observation that the structural roles are conditions rather than individuals. The first-order replies bank configuration-level contingency as the residue necessitism leaves free, and they can do so because first-order necessitism rigidifies only the domain of individuals. Higher-order and plural necessitism rigidify properties and pluralities as well, and configurations live at exactly that level. Plural logic is not silent where configurations operate; it is a logic of that level. The reply must therefore be earned at the higher order, not asserted to lie beneath it.

It is earned by lifting Williamson's own distinction to the plural level. Necessitism already separates *being* from *concreteness*: whatever exists, exists necessarily, but what is concrete is contingent; a thing has its being in every world and is concrete in only some. Carried up to pluralities, higher-order necessitism rigidifies the *being* of configurations-as-possibilia, but its contingently-concrete machinery leaves which configurations are concretely actual contingent. That residue is precisely what the proof draws on. The first premise does not record that some plurality exists; it records that the configuration-of-change is the one concretely instantiated. The contingency at work is concreteness-contingency, and concreteness-contingency survives plural necessitism for the same reason it survives the individual case: Williamson's framework rigidifies being, not concreteness, at whichever order it is run. Linnebo's apparatus extends the dialectic to the level of pluralities; pressed there, it presses the structural conclusion no harder than at the level of individuals.

The Barcan-formula version of the modal-collapse worry, that any necessity at the existence level propagates to all existential claims, is engaged in its Spinozan-propagation form at [§4.4.9](<4.4.9 Modal Collapse Objection.md>), where the constitutive/productive distinction does the work necessitism alone cannot do.

The Lewisian concrete-plurality alternative, which gives modal claims maximal ontological weight rather than minimal, is [§4.4.1](<4.4.1 Modal Realism.md>). The modal-collapse worry, that necessary instantiation flattens all modal distinctions, is [§4.4.9](<4.4.9 Modal Collapse Objection.md>). Sider's structuralism, which deflates rather than embraces necessity, is [§4.4.7](<4.4.7 Sider's Structuralism.md>). The Whiteheadian process alternative is [§4.4.8](<4.4.8 Whiteheadian Process Metaphysics.md>). Modal fictionalism's attempt to deflate the whole apparatus is [§4.4.4](<4.4.4 Modal Fictionalism.md>).

*What converges.* Each of the three forms of the necessitist threat operates at a level the structural argument either accommodates or does not depend on. The constant-domain compatibility move is correct that □(T∧S∧Φ) is consistent with necessitism, but it underestimates what the structural argument adds: the constitutive account of what makes the constant domain a domain. The Cartesian-Certainty preservation move presses the strongest threat to the proof's first premise; ◇C survives because its modal weight sits at the configuration level Williamson preserves, not at the existence level Williamson flattens. The actuality-contingency preservation move presses the boldest version; the contingency the proof requires is qualitative-contingency at the configuration level, which is exactly what necessitism leaves intact, and which necessitism in fact requires if its framework is to do any work at all.

Williamson draws the conclusion that the formal apparatus deserves to be taken at face value: the quantifiers range over a constant domain, the Barcan formula and its converse are validated, and what looked like ontological contingency dissolves into qualitative variation across a fixed plurality. The structural argument shares the formal commitment and adds what Williamson's framework requires but does not specify: the constitutive role of T, S, and Φ in making the constant domain a domain. What Williamson left behind is not a refutation of the structural argument but a cleaner statement of the level at which the argument's necessity claim operates: not at the qualitative-contingency level where Williamson preserves modal variation, but at the constitutive level where the variation is hosted. Necessitism is the framework the structural argument's conclusion is most compatible with at the formal level, and the argument supplies the constitutive reading its formalism leaves underspecified.


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← [4.4.5 Modal Expressivism](<4.4.5 Modal Expressivism.md>) | [4.4.7 Sider's Structuralism](<4.4.7 Sider's Structuralism.md>) →
