Two-dimensional semantics promises a wedge between what a concept picks out when a world is considered as actual and what it picks out across possible worlds. If that wedge holds at the constitutive level, the Modal Identity of [§2.6.6](<2.6.6 Modal Identity.md>) (□(◇C ↔ T∧S∧Φ) read as one fact under two descriptions, connecting change-permission with co-instantiation of the three structural roles) will not survive. The challenge presses at three points: the two-intensions machinery itself, the water/H₂O analogy that most cleanly exhibits it, and the indexical anchor on which primary intensions depend. Chalmers is right on a narrower point, conceded here without qualification: the primary/secondary distinction is a genuine feature of natural-kind terms, and arguments that ignore it routinely overstate what necessary co-extension delivers. The claim contested below is narrower and specifically targeted: that the structural role definitions are natural-kind terms, and that the two-intension machinery has purchase where the appearance/nature separation it exploits does not exist.

Chalmers -(Chalmers, David J., 1996); -(Chalmers, David J., 2010) makes the cleanest contemporary case that meaning and reference come apart. Every concept, on the two-dimensional semantic framework, admits two intensions. The primary intension picks out what the concept would refer to in any world considered as actual: what 'water' would pick out if the world turned out one way rather than another. The secondary intension picks out what the concept refers to across all possible worlds, given what it in fact refers to here. For natural kind terms the two come apart cleanly: 'water' picks out the clear drinkable stuff in any world considered as actual (primary), but once we discover the actual stuff is H₂O, 'water' rigidly designates H₂O across all possible worlds (secondary). The framework tracks a gap between description and reference that rigid designation alone cannot capture.

(Jackson, Frank, 1998) develops the apparatus in parallel, pressing that conceptual analysis tracks primary intensions and that every metaphysical necessity has an *a priori* scrutable counterpart at the primary level (one derivable from a compact base by ideal *a priori* reasoning alone, without further empirical input). (Chalmers, David J., and Frank Jackson, 2001) extend the framework into a general theory of reductive explanation: wherever a target fact is not *a priori* entailed by a non-target base, a two-dimensional gap opens. (Chalmers, David J., 2012) generalizes the move still further in *Constructing the World*: all truths are *a priori* scrutable from a compact base, and wherever primary intensions fail to track secondary ones, a genuine modal gap has been uncovered. The most aggressive deployment for modal epistemology is (Chalmers, David J., 2002): ideal *a priori* conceivability, properly regimented by 2D, entails primary possibility, and primary possibility is the kind of possibility that matters for disputes over necessary existence. The framework is not a regional tool for natural-kind terms; it is a general theory of where modal weight sits and where it does not.

Applied to the present account, the framework presses a precise threat. The argument of [§2.6](<2.6 Triconditionality of Nature.md>) holds that 'causally structured succession' and Time are not merely co-extensional descriptions but the same structural fact under two modes of presentation: that the role definitions are constitutive rather than descriptive. The two-dimensionalist replies that even for terms the argument labels constitutive, primary and secondary intension can diverge: what 'causally structured succession' picks out when a given world is considered as actual may differ from what it picks out across possible worlds. The Modal Identity of [§2.6.6](<2.6.6 Modal Identity.md>) would then fail, and with it the step from □◇C to □(T∧S∧Φ): from the necessary possibility of change to the necessary co-instantiation of the three structural conditions. A wedge between description and reference at the constitutive level would leave the structural roles necessarily co-extensive with change without being identical to it: the exact space the classical theist needs to reassert an uninstantiated ground.

Three forms of the move take up the objection; the narrower point about natural-kind terms stands untouched.

*First.* The two-intensions move holds that every concept admits both intensions and that the argument illicitly collapses them at the point of constitutive terms. The pressure is sharpest in its epistemic-possibility form. The primary intension captures what the world could turn out to be for all we know *a priori*, and the Modal Identity of [§2.6.6](<2.6.6 Modal Identity.md>) is precisely the kind of *a priori* necessity claim (Chalmers, David J., 2002) has shown can fail even where the secondary intension holds necessarily. For all the argument has established analytically, the objector presses, 'causally structured succession' could turn out (in some world considered as actual) to pick out something other than Time as [§1.4.1](<1.4.1 Time.md>) defines it. Scrutability (Chalmers, David J., 2012) generalizes the pressure well beyond natural-kind terms: wherever a concept is *a priori* scrutable from a base that does not itself entail the target, a primary/secondary gap is in principle available, and the constitutive identities this argument rests on fall within that scope.

The reply runs on the analytic/*a posteriori* distinction the 2D framework itself respects, but it must keep two resolutions apart. At role-resolution, a constitutive definition fixes what counts as directed order, relational differentiation, or real bearing. At type-resolution, further arguments identify temporal succession, spatial extension, and causal persistence as the relevant realizers in genuine change. The Modal Identity does not treat the names alone as settling those richer types.

The Kripkean cases 2D was built to accommodate are *a posteriori* identities: 'water is H₂O' was discovered, not defined, and the primary/secondary gap tracks exactly what the discovery added. The structural claim is different at role-resolution: 'causally structured succession' specifies the role T plays in a genuine change-realizer rather than reporting a discovery that two independently identified things coincide. Where that role specification is analytic, primary and secondary intensions have no independent residue to track. A critic may still contest the type-level realization, but that is a dispute over the realizer arguments, not a primary/secondary gap generated by the role definition itself.

The 2D matrix is a function from possible worlds to extensions, and functions require a domain. In the present account, real possibility ranges over a fixed global domain W ([§2.1](<2.1 The S5 Pipeline.md>)). Two-dimensional semantics may offer a different representation of modal evaluation, but it must say whether its primary-intension scenarios range over that same global possibility-domain, a relative accessibility relation, or an epistemic space. Only the first is the book's alethic operator ◇. The semantic tool therefore does not by itself overturn the fixed-domain account; it either represents the same real possibilities at another level or shifts to a different relative modality. The structural roles become relevant where a scenario is offered as a genuine change-realizer, not merely because any semantic matrix is being described.

The disanalogy with the paradigm case can be stated directly. The water/H₂O primary intension is evaluable without presupposing water or H₂O: the evaluation is built from chemistry, sense-perception, and an ordinary spatiotemporal setup, none of which is the disputed term. No parallel construction is available here. Evaluating the primary intension of 'causally structured succession' requires exactly the apparatus in question: a scenario individuated from its alternatives (Space), a reasoner who carries the evaluation from premises to verdict (Time), and content held fixed long enough to be compared against the secondary intension it is meant to diverge from (Substance). The act of running the semantic tool is structured, but that act-side fact does not show that every scenario represented by the tool instantiates the loaded temporal, spatial, and causal types. The object-side question remains whether a scenario is admitted as a genuine change-world or only as an epistemic representation.

*Second.* The water/H₂O analogy sharpens the pressure. Scientists discovered water is H₂O *a posteriori*: the identity holds necessarily but was not analytically given. (Chalmers, David J., and Frank Jackson, 2001) generalize the pattern: every necessary *a posteriori* identity admits a corresponding primary-level contingency, and the identity's modal weight is confined to the secondary level. If 'causally structured succession' and Time stand in this relation, the objector presses, their identity is a discovery rather than a definition, and a discovered identity is necessary only in the restricted secondary sense that leaves the primary-level gap open, which is the gap the theist needs.

The analogy works in the natural-kind case precisely because 'water' and 'H₂O' pick out different levels of description: surface features on one side, microstructure on the other. The gap opens because a world could present the surface features while hosting different microstructure (Twin Earth's XYZ). The structural role definitions do not operate at two separable levels. 'Causally structured succession' names nothing except asymmetric directed succession among states: the full specification of what Time *is*. There is no surface/microstructure distinction to exploit, no layer of appearance hiding a different metaphysics underneath. The 2D gap requires exactly the appearance/nature separation the role definitions are designed to eliminate.

The water/H₂O example operates within an ordinary physical setting, but that does not dissolve the semantic distinction it illustrates. The relevant reply is narrower: role specifications and natural-kind identifications have different logical forms. Whether the book's type-level identifications behave more like definitions or discoveries must be defended by the realizer arguments rather than inferred from the physical setting in which semantic evaluation occurs.

*Third.* The indexical anchor presses the objection furthest. The primary intension is evaluated from the perspective of a world considered as actual, and 'considered as actual' is an indexical that depends on the evaluator's position rather than any mind-independent fact. The indexicality drives the 2D framework, the objector concludes, and if it survives so does the gap.

The indexical objection establishes an act-side point, not immediately an object-side result. Any concrete act of evaluating a primary intension unfolds through succession, distinction, and persistence. That does not by itself show that every primary-intension scenario instantiates the full temporal, spatial, and causal types. The object-side issue is semantic: whether the scenario is offered as a genuine world in W or merely as an epistemic representation. If it is a world in W, it answers to the Structural Requirement; if it is merely an epistemic representation, its variation does not by itself yield a metaphysical possibility.

Two lines support the claim rather than asserting it. First, Lewis individuates worlds as maximal fusions of spatiotemporally interrelated things (*On the Plurality of Worlds* -(Lewis, David, 1986), [§1.6](<1. Groundwork.md>)). That criterion requires the relevant worldmate relations where there are multiple inhabitants, but it does not by itself prove asymmetric temporal ordering in every sparse or changeless world. Lewisian individuation therefore supplies a concrete-world constraint without independently establishing the book's full T-type. Second, the Chalmers/Jackson scrutability framework requires an idealized *a priori* reasoner conducting the evaluation exercise from within a world considered as actual (*Constructing the World* -(Chalmers, David J., 2012), Ch. 2). That idealization strips cognitive limitations (bounded memory, finite inferential steps) while preserving the structural conditions of reasoning: directed inference from premises to conclusions (T), distinction among propositions being evaluated (S), and persistence of the cognitive state across the derivation (Φ). Any actual token of idealized reasoning is performed through an ordered cognitive practice. That establishes an act-side condition on evaluators; it does not place a temporal coordinate inside every scenario they evaluate or settle whether an abstract ideal inference is itself temporally extended.

Fine's -(Fine, Kit, 1994) distinction between essence and necessity sharpens the point. The indexical 'actual' is not merely necessarily co-present with a Time coordinate; it is essentially anchored in one. Any attempt to reconstruct an 'actuality' perspective from non-temporal resources either covertly reimports the succession from which 'now' acquires content, or fails to individuate one world from another at all. Chalmers' own framework, applied to the structural role definitions, reinstates Time in the act of trying to prise meaning loose from metaphysics.

That the indexical reading of 'actual' is itself contested only tightens the anchor rather than loosening it. Van Inwagen's 'Indexicality and Actuality' -(van Inwagen, P., 1980) argues the indexical theory is not true as Lewis states it: the words Lewis uses are ambiguous across several readings, on none of which the analysis is both genuinely indexical and correct; relatedly he presses that the indexical reading presupposes we inhabit just one possible world -(van Inwagen, P., 1980), 416. Adams's 'Theories of Actuality' -(Robert Merrihew Adams, 1974) goes further and rejects indexicality outright: actuality is a primitive, absolute property that exactly one world possesses. The structural point is indifferent to which side prevails. If 'actual' is indexical, its token use is centered on a speaker or evaluator, but indexicality need not supply a single temporal frame spanning isolated worlds. If actuality is primitive and absolute, its application need not depend on a standpoint inside the actual world. On either account, actual evaluations are structured performances; neither account alone proves that every evaluated world instantiates T, S, and Φ. The bearing is on the theory of actuality, not on the inventory of worlds: nothing here counts the plurality down to one. It shows only that whichever account of 'actual' survives presupposes the structure the argument names.

A kindred deflation drops the 2D apparatus entirely for conceptual-role semantics: (Block, Ned, 1998) and (Harman, Gilbert, 1987) argue meaning is exhausted by inferential role within a conceptual system, and the constitutive identity of [§2.6](<2.6 Triconditionality of Nature.md>) becomes one inferential-role equivalence among many rather than a modal truth. The reply is the same in thinner vocabulary. Inferential roles require reasoners whose inferences succeed each other (T), operate on distinct premises (S), and carry content across the derivation (Φ). Conceptual-role semantics has not replaced the structural roles; it has renamed them.

The adjacent modal objections are answered elsewhere. The Kripkean challenge (that necessary co-extension is not necessary identity) is handled at [§4.1.1](<4.1.1 Rigid Designation.md>), which fixes the modal-epistemic status of the necessity claim and defers to this section the two-dimensional bookkeeping that status raises. The PSR-based modal objection is handled at [§4.1.3](<4.1.3 Principle of Sufficient Reason.md>). The modal-collapse objection (that necessary instantiation flattens modal distinctions) is handled at [§4.4.9](<4.4.9 Modal Collapse Objection.md>).

*The bookkeeping, made explicit.* The modal-epistemic status of □(T∧S∧Φ) is fixed at [§4.1.1](<4.1.1 Rigid Designation.md>) as *a priori* and constitutive, and that section defers to this one the two-dimensional accounting that status invites. Here it is.

Run the derivation through the 2D matrix and the existential datum that change occurs ([§2.4](<2.4 Structural Requirement.md>)) is the clearest contingent input. The role definitions may be analytic at thin resolution, but the role-to-type bridges and the claim that they exhaust genuine change-realizers are substantive metaphysical premises, not analytic consequences of the vocabulary alone. Even that datum does not enter as ordinary empirical discovery; it is Cartesian-certain, surviving the strongest skeptical scenario ([§2.2.1](<2.2.1 Change Is Actual.md>)), so the lone contingent-primary element is not *a posteriori* in the water=H₂O way.

And the constitutive identity the datum feeds into (that the possibility of change just is the co-instantiation of the three roles, ◇C ⟷ T∧S∧Φ) has no primary/secondary gap to track. As the replies above establish, 'causally structured succession' is not scrutable from an independent base but *is* the base under another description, so its primary and secondary intensions collapse. That collapse is derived, not stipulated. A primary intension diverges from its secondary only where the concept holds some individuating feature independently of its secondary referent, a handle the primary keeps fixed while the referent varies, as 'watery stuff' keeps the surface profile while the microstructure shifts to XYZ.

The Modal Identity removes the handle. What [§2.6.6](<2.6.6 Modal Identity.md>) establishes is that between 'causally structured succession' and Time there is no individuating surplus in either direction: every specification of the one is already a specification of the other, with nothing left over. A primary intension therefore has no feature to track independently of the secondary, and no profile of its own with which to diverge. To conceive the referents coming apart, an ideal reasoner would need some feature on which they could differ; the no-surplus result is the absence of any such feature, and where there is nothing to differ on there is nothing left to conceive as different. That no-surplus result is itself fixed structurally at [§2.6.6](<2.6.6 Modal Identity.md>), established by the role definitions of §[§1.3](<1.3 Change.md>)–1.4 together with the converse at [§2.4.10](<2.4.10 Sufficiency of Structure.md>), and it holds independently of any premise about what is conceivable. Leaning on it here therefore closes the primary-intension gap without circularity. Metaphysical no-surplus is in this way epistemic no-residue, and the gap closes at the level the objection needs it open.

With the fixed-domain modal lift and the constitutive arguments at the full Condition resolution, □(T∧S∧Φ) is argued *a priori*, not read off the world as an empirical identity. The point is not that S5 alone turns an existential datum into the structural conclusion; the Structural Requirement supplies the necessary conditional, and the type-level realizer arguments supply its content. The resulting necessity is structural and constitutive in the book's intended sense. The 2D framework, asked to locate where modal weight sits, returns the verdict to the constitutive level it was built to prise open.

*What converges.* Across all three pressure points the 2D framework either requires the structure or redescribes it in more layered vocabulary. The two-intensions move needs modal space as its evaluation domain (which the roles constitute rather than presuppose) and finds no gap where the target is analytic rather than *a posteriori*. The water/H₂O analogy needs an appearance/microstructure seam the role definitions do not admit: 'causally structured succession' names nothing except what Time *is*. And the indexical anchor needs a Time coordinate from which 'actual' has determinate content at all: any attempt to reconstruct actuality from non-temporal resources either reimports the succession or fails to individuate worlds.

What stands, untouched by any of this, is Chalmers' narrower point about natural-kind terms; what falls away is the attempt to import it to the constitutive level. The role definitions of §[§1.3](<1.3 Change.md>)–1.4 are not natural-kind terms waiting to be pulled apart at a seam between surface and nature; they are role specifications whose content is exhausted by what they require. The 2D framework, turned on constitutive role definitions, does not discover a gap but reinstates the architecture it was designed to distinguish semantic layers from.


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← [4.1.1 Rigid Designation](<4.1.1 Rigid Designation.md>) | [4.1.3 Principle of Sufficient Reason](<4.1.3 Principle of Sufficient Reason.md>) →
