Real possibility ranges over one fixed domain W, the complete set of worlds Reality permits, and that fixed domain yields a single result: modal status is invariant across every point of evaluation, a result the rest of the chapter relies on.

> [!argument]
> **Necessary Possibility** · *The Derivation*
>
> 1. *Modal Exhaustiveness.* Every contentful proposition receives one of the two classical verdicts: ◇P or ¬◇P. ([§2.1.1](<2.1.1 Modal Bivalence.md>))
>
> 1. *Global real possibility.* W is the complete domain of worlds Reality permits. ◇P means truth at some member of W; □P means truth at every member. ([§1.2.1](<1.2.1 Possibility.md>), [§2.1.3](<2.1.3 Modal Constitution.md>))
>
> 1. *Fixed-domain invariance.* Every world evaluates real possibility over that same W. A witness world for P therefore remains a witness at every point of evaluation. ([§2.1.4](<2.1.4 Modal Stability.md>))
>
> 1. *The Bottleneck.* A world-relative accessibility horizon may vary, but that is local reachability, not the global metaphysical possibility fixed by W. No variation in local reachability changes membership in W. ([§2.1.5](<2.1.5 The S5 Bottleneck.md>))
>
> 1. *Bilateral Self-Sustenance (L1).* Logical truth supplies a necessary possibility-witness and contradiction a necessary impossibility-witness, so the fixed frame is non-vacuously occupied on both sides. (□∃P: ◇P) ∧ (□∃P: ¬◇P). ([§2.1.3](<2.1.3 Modal Constitution.md>))

Together, these results close on a single conclusion:

> [!theorem]
> **Necessary Possibility** · **T1** · *Theorem 1*
>
> Whatever is possible is necessarily possible; whatever is impossible is necessarily impossible. Modal status is invariant because every point of evaluation ranges over the same total domain of real possibilities. ◇P → □◇P, with bilateral companion ¬◇P → □¬◇P.

The theorem is schematic. It does not say that anything in particular is possible; it says that once a proposition has a real modal status, changing the evaluation point does not change the global domain over which that status is evaluated. The Cartesian witness supplies the substantive input in [§2.2](<2.2 Cartesian Certainty.md>).

The derivation is also non-circular with the later Necessity of Nature. No step assumes that T, S, or Φ holds in every world. The fixed-domain theorem concerns the quantificational scope of real possibility; §[§2.4](<2.4 Structural Requirement.md>)–2.5 later identify and necessitate the structural content every world in that domain must realize.

The name is conventional. *Necessary Possibility* names the positive S5 form, while the theorem is bilateral: absence of a witness from W is as invariant as presence of one.

*What is established.* Real possibility is global, modal status is rigid across W, and both verdicts are necessarily instantiated. S5 is derived from fixed-domain metaphysical modality rather than from the Triconditions it later helps necessitate.


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← [2.1.9 What Necessity Transfer Needs](<2.1.9 What Necessity Transfer Needs.md>) | [2.2 Cartesian Certainty](<2.2 Cartesian Certainty.md>) →
