This line takes a different starting point from facthood, construction, or denial: it runs on the meaning of modal vocabulary alone, asking what must already be in place for the modal distinction to have content. To be *possible* is to be one way rather than another, and a determinate way just is direction + difference + bearer.

> [!argument]
> **Way Possible** · *from what "possible" means to determinate difference*
>
> 1. To be possible is to be a way things could be.
>
> 1. A determinate way has content rather than being an undifferentiated blank.
>
> 1. So possibility analytically requires the role of difference (S-role). Direction and bearer require further premises; they do not follow from the word *possible* alone.

> [!lemma]
> **Way–Possibility Convertibility** · **L7** · *Lemma toward Triconditional Necessity*
>
> A possibility must be a determinate way rather than an undifferentiated blank, so possibility presupposes difference at role-resolution. A directional reading of *way* can add the T-role; the Φ-role comes from the separate account of what bears or realizes a determinate state. The full T ∧ S ∧ Φ result therefore uses more than modal vocabulary alone.

*Why direction is not dispensable.* A way is not a qualitative profile, a set of co-instantiated properties that differs from others. A way is a directed arrangement: it converges on a determinate state by ordering its conditions jointly. Consider any journey: it requires its stages in a directed order from origin to destination (T: a directed ordering, not yet a temporal metric), distinct positions through which it runs (S: difference), and something to bear the traversal (Φ: the bearer). Remove any one condition and there is no way: not a slower or shorter way, but no way. Every possible world is a way in exactly this sense: a complete directed arrangement of how things are, not a logical listing of co-instantiated properties. A configuration stripped of direction is not a changeless world; it is an abstract collection: items enumerated without the ordering that makes them stand in any determinate relation to each other. Direction (T) is therefore not a contingent feature that some ways happen to carry; it is internal to what it is for anything to be a way at all.

*Role and type.* What the decomposition reaches, for each condition, is the *role*, not yet its specific *type*. But the three roles are not earned at one price. Difference is analytic: a way with nothing it is *rather than* is no way, so distinctness (this-rather-than-that) falls straight out of *possible*. The other two are not analytic in the same way. Direction is a directed ordering: asymmetric, influence-carrying ([§1.4](<1.4 Three Conditions.md>) Directionality), not a mere co-instantiation. Its asymmetry belongs to the role itself: a way *converges on a determinate state*, and an ordering that ran equally either way would single out no state rather than another, collapsing into a symmetric arrangement of differences (an abstract collection, not a way). The abstract-collection argument therefore excludes the symmetric-ordering case on the same ground it excludes the no-ordering case: neither is a *this-rather-than-that* directed toward its determinate state. Asymmetric ordering thus comes with the directed-arrangement reading of *way* at no further cost, as "Why direction is not dispensable" has it. What remains owed is the direction limb's *type*, not its asymmetry. Section [§3.5.3](<3. Convergence.md>) shows that the type-upgrade requires an independently secured □◇C rather than being generated by this decomposition. The bearer costs more even at the role level. The sustaining that carries one state into the next rather than leaving a bare succession of unconnected states is a being's doing, read off not the definition of *way* but the account of change in [§1.3](<1.3 Change.md>), and carried downstream at [§2.4](<2.4 Structural Requirement.md>) (the Humean reading at [§4.1.4](<4.1.4 Humean Contingency.md>)), the demotion [§1.3](<1.3 Change.md>) already marks. What the decomposition does not by itself fix is that these roles take their specific types: that the ordering is *temporal* (genuine succession, rather than an atemporal asymmetric dependence, a synchronic grounding order that carries asymmetry without before-and-after), that the distinctness is *spatial* (co-existence, a *where*, rather than bare qualitative non-identity), that the bearer is a *causal-persistent being*. The conclusion below is therefore the role-level necessity. The type-upgrades are located and weighed for the chapter at [§3.5.3](<3. Convergence.md>), where the temporal type-upgrade discharges once □◇C has been independently secured. This line does not itself supply that cross-world premise. The spatial- and bearer-type steps each close at [§3.5.3](<3. Convergence.md>) on □◇C together with a Stage-2 realizer biconditional ([§2.6.2](<2.6.2 Succession's Two Conditions.md>), [§2.6.4](<2.6.4 Persistence's Two Conditions.md>)) that transmits its necessity intact, at the same modal grade as the temporal type, though reached by a longer derivation than the temporal upgrade's □◇C alone. Specific metric richness (a temporal metric, a spatial manifold, particular fields) is contingent physical content taken up in Chapter 7.

> [!argument]
> **The Possibility Pole** · *from modal content to a partial structural result*
>
> 1. Every possibility must have determinate content, so every possible world instantiates the difference-role.
>
> 1. If a world is additionally understood as a directed arrangement, the T-role is engaged.
>
> 1. If its determinate content must be borne or realized rather than merely listed, the Φ-role is engaged.
>
> 1. Thus the full role-level T ∧ S ∧ Φ conclusion follows only with the directed-arrangement and bearer premises made explicit. Modal vocabulary alone establishes the S-role, not the whole conjunction.

*The same conclusion arrives from the opposite direction.* The second pole runs not from what a possibility requires but from what the necessity operator itself needs in order to say anything at all; it corroborates the possibility pole rather than carrying the result afresh.

> [!argument]
> **The Necessity Pole** · *from what necessity itself requires to Triconditional Necessity*
>
> 1. "□" collapses to actuality-without-alternative unless there is a non-trivial possibility-space.
>
> 1. "□" is used ineliminably and with content.
>
> 1. So a non-trivial possibility-space is corroborated, the contentful use of "□" being evidence that the space is genuine, not by itself a proof.
>
> 1. *Convertibility Lemma applied.* By the Way–Possibility Convertibility Lemma (L7) it presupposes T ∧ S ∧ Φ.
>
> 1. So □(T ∧ S ∧ Φ) is corroborated, again at role-resolution, the type-upgrades discharged or flagged at [§3.5.3](<3. Convergence.md>) as before. This pole corroborates the possibility pole rather than carrying the conclusion independently.

*Why the necessity pole holds.* The first form ran from possibility; this one runs from necessity, and turns on what "□" needs in order to say anything. A "□" with no genuine alternatives to range over collapses into a bare "so, and not otherwise, because there is no otherwise": necessity drained to vacuity. But "□" is used with content, marking some ways as ruled out and others as left open, and that contrast means something only against a real space of alternative ways. A space of genuine alternatives is a space of *ways* (each a determinate this-rather-than-that), so by the decomposition above it already carries direction, difference, and bearer. A necessity that is not empty therefore presupposes exactly what possibility did. This line turns on how "□" is *used*. And a fact about operator-usage is not yet, on its own, a fact about modal reality.

*Dependency.* At role-resolution: [§1.3](<1.3 Change.md>), [§1.4](<1.4 Three Conditions.md>) only, no S5/Euclidean axiom, no Cartesian certainty, no ¬◇∅. The role-level conclusion stands on those materials. The temporal type-upgrade at [§3.5.3](<3. Convergence.md>) is downstream of an independently secured □◇C; [§3.1](<3.1 Necessary Possibility of Change.md>) supplies strong witnesses and closures but not that full cross-world result by itself. The possibility pole supplies a partial semantic line and the necessity pole corroborates the existence of a non-trivial contrast space. Neither pole, from operator-meaning alone, proves the full triad; the direction and bearer steps depend on the additional premises identified above. The two are the possibility-pole and necessity-pole of a single fact.

*Nearest rival.* The closest contemporary program to this one grounds possibility not in the structure of a way but in the potentialities of actual objects (Vetter, Barbara, 2015); (Borghini, Andrea; Williams, Neil E., 2008); (Jacobs, Jonathan D., 2010). It shares the rejection of a plurality of worlds and differs on what does the seating: a potentiality is a property of a bearer, whereas the roles reached here are what any bearer-involving state instantiates. The present account also runs the relation as a convertibility, with neither *possibility* nor *way* reducing to the other, rather than as a reduction in one direction. The engagement is at [§4.1.12](<4.1.12 Dispositionalism about Modality.md>).

**Note (modal-predication grammar).** Natural-language predications of possibility normally arrive in tensed form, and that grammar supplies dialectical pressure toward the direction (T) limb, not an independent derivation of it. The present-indicative copula ("is possible," "obtains," "holds") is a tense form in ordinary language, not by itself a proof of metaphysical Time; even apparently tenseless complements ("2 + 2 equals 4") lean on tensed surface grammar when asserted, and Kripke semantics evaluates ◇P "at" worlds where P "holds" without thereby deriving T. The grammar is evidence, not proof: the metaphysical argument for direction stands on the decomposition above.


---

← [3.2.4 All the Way Down](<3.2.4 All the Way Down.md>) | [3.2.6 The Null-State Reductio](<3.2.6 The Null-State Reductio.md>) →
