Modal Logic S4 as Theorem: τ-Modal Operators and Kripke Soundness
The modal logic S4 is derived as a theorem within Category τ — possibility and necessity operators arise from the ρ-orbit and depth structure.
Overview
VII.T13 proves that the τ-modal operators (□ = ρ-reachable, ◇ = ρ-accessible) satisfy the S4 axioms: T (□P → P), 4 (□P → □□P), and K (□(P→Q) → (□P → □Q)). Kripke soundness (VII.P07) confirms that the τ-accessibility relation generates a valid Kripke frame for S4. Modal logic is not a postulated logical system but a theorem of Category τ.
Detail
Modal logic S4 is one of the most studied normal modal logics; it adds the axiom 4 (□P → □□P, ‘if necessarily P then necessarily necessarily P’) to the minimal modal logic K. S4 is commonly interpreted as the logic of provability or knowledge, and is the basis for the Curry-Howard correspondence between intuitionistic logic and typed lambda calculus. Book VII derives S4 from Category τ. The modal operators are: □P (P holds in all ρ-reachable stages from the current stage — ‘necessarily P’) and ◇P (P holds in some ρ-accessible stage — ‘possibly P’). VII.T13 proves that these operators satisfy all S4 axioms. Axiom T (□P → P): what is necessary is actual — follows from the current stage being ρ-reachable from itself. Axiom 4 (□P → □□P): follows from the transitivity of ρ-reachability (ρ-orbit traversal is transitive). Axiom K (distribution): follows from the functoriality of ρ. Kripke soundness (VII.P07) confirms that the τ-Kripke frame (stages as worlds, ρ-reachability as accessibility) is reflexive and transitive, the characteristic frame condition for S4.
Result Statement
VII.T13: The τ-modal operators satisfy S4 axioms (T, 4, K). Kripke soundness confirmed (VII.P07). Modal logic S4 is a theorem of Category τ.