MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 75and if not xRy thenPA ⊢ σ x → Bew(¬σ y ).Any model M on this frame determines a realisation φ by puttingp φ = ∨ {σ x : M |= x p}.Then the truth conditions in M are PA-representable by the fact that for anymodal formula α,if M |= x α then PA ⊢ σ x → α φ ; whileif M ̸|= x α then PA ⊢ σ x → ¬α φ and so PA ⊢ α φ → ¬σ x .Since PA ¬σ x , the last case gives PA α φ , showing α is not PA-valid. Thereforeany PA-valid formula must be true in all models on finite strictly ordered frames,and therefore be a G-theorem.A modal formula α is called ω-valid if α φ is true for all realisations φ. Theset G* of all ω-valid formulas is a logic that includes G, but also includes ✷p →p, since Bew(σ) → σ is always true. However Gödel’s example shows thatBew(Bew(⊥ φ ) → ⊥ φ ) is not true, so G* does not contain ✷(✷p → p), andtherefore is not a normal logic. Solovay extended his analysis of G to prove thatG* can be axiomatised by taking all theorems of G and instances of ✷α → α asaxioms, and detachment as the only rule of inference.Another natural reading of ✷ in this context is “true and provable”, formalisedby modifying the definition of realisation to(✷α) φ := α φ ∧ Bew(α φ ).The fact that “provable” implies “true” might make it seem that “true and provable”has the same status as “provable”, but this is not so because of the existenceof true but unprovable sentences of PA. In general, Bew(σ) is PA-provable iffσ ∧ Bew(σ) is PA-provable, and the two are equivalent in the sense thatBew(σ) ↔ σ ∧ Bew(σ)is true, but this equivalence is not itself PA-provable unless σ is, by Löb’s theorem.The modal logic of formulas PA-valid under this modified realisation turns outto be the system S4Grz characterised by finite partial orderings (see section 5.3).This was proved in [Goldblatt, 1978] by showing that replacing ✷α by α∧✷α givesa proof-invariant translation of S4Grz into G, and then applying Solovay’s theoremfor G. 65 Since the intuitionistic propositional calculus IPC can be translated intoS4Grz (by the result of Grzegorczyk mentioned in section 5.3), these translationscan be composed to obtain a translation α ↦→ α τ of propositional formulas into65 The result was independently found by A. Kuznetsov and A. Muzavitski (Abstracts of Reportsof the Fourth All-Union Conference on Mathematical Logic, Kishiniev, 1976, p. 73, in Russian).