Modal Bivalence is the exhaustiveness of the book's two primitive modal verdicts. For any contentful claim P, either P is really possible (◇P) or it is not (¬◇P). Since impossibility is defined as the negation of possibility, the formal disjunction ◇P ∨ ¬◇P is classical excluded middle applied to a modal predicate. It records the grammar of the framework; it does not by itself establish any substantive fact about which propositions are possible.

> [!premise]
> **Modal Bivalence** · **P1** · *Premise 1*
>
> For any contentful claim, either it is really possible or it is not: the two modal verdicts are exhaustive. ∀P: ◇P ∨ ¬◇P.

The grammar fixes modal status, not truth, and the two come apart. A claim can be unsettled in truth — the next coin flip will land heads — while being settled in modal status: heads and tails are both possible before either becomes actual. Vagueness raises a different problem. A string that fails to determine a proposition does not yet supply a third modal status; it fails to provide a content to which either verdict can apply.

A genuine challenge to the two-verdict framework must therefore specify what it rejects: truth-aptness, classical negation, exhaustiveness, exclusivity, or the consequence rules used downstream. These are live options, not verbal confusions. A dialetheist such as Graham Priest challenges truth-value exclusivity: some propositions may be both true and false. That alone does not yield ◇P ∧ ¬◇P, because ¬◇P states that P is impossible, not merely that P is false. To challenge modal exclusivity here, the dialetheist must extend the glut to the modal predicate itself, allowing both ◇P and its negation. The disjunction ◇P ∨ ¬◇P can remain true on that proposal while the classical claim that exactly one disjunct holds fails. Paraconsistent and intuitionist positions may instead revise what follows from a contradiction or which reductios are licensed.

This section adopts both exhaustiveness and exclusivity as part of its classical framework without pretending that the first alone supplies the second. The later replies are correspondingly limited: [§2.7.2](<2.7.2 Incoherence of ∅.md>) asks whether logical revision can furnish a model of the Null State without the defending practice instantiating the Conditions, and [§5.3.4](<5.3.4 Madhyamaka Emptiness.md>) runs the catuṣkoṭi corner by corner while preserving the wider dispute over positive metaphysical thesis-making.

The work of Modal Bivalence is limited but necessary. It fixes the two verdicts over which the later invariance theorem ranges. It does not prove that modal status is world-invariant, that both poles are occupied, or that any particular proposition is possible. Those are separate results.

*What is established.* Every contentful proposition receives one of the two modal verdicts under the classical framework used here: ◇P or ¬◇P. The result is modal exhaustiveness, not yet modal invariance.[^2-1-1-bivalence-ch4]

[^2-1-1-bivalence-ch4]: Fictionalism questions whether modal statements bear truth values at all ([§4.4.4](<4.4.4 Modal Fictionalism.md>)); dialetheist and paraconsistent revisions of classical exclusivity or consequence are met in operation, where [§2.7.2](<2.7.2 Incoherence of ∅.md>) tests the logical-revisionist rescue of the Null State and [§5.3.4](<5.3.4 Madhyamaka Emptiness.md>) runs the catuṣkoṭi, the four-cornered tetralemma, against the necessity of the Conditions; dappled-world and deflationist views question whether one global modal domain is available ([§4.1.8](<4.1.8 Metaontological Deflationism.md>)); and quietism doubts that the disjunction is meaningful ([§4.1.9](<4.1.9 Wittgensteinian Quietism.md>)).


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← [2.1 The S5 Pipeline](<2.1 The S5 Pipeline.md>) | [2.1.2 Bilateral Frame](<2.1.2 Bilateral Frame.md>) →
