Proof-Theoretic Soundness and Completeness

We give a calculus for reasoning about the first-order fragment of classical logic that is adequate for giving the truth conditions of intuitionistic Kripke frames, and outline a proof-theoretic soundness and completeness proof, which we believe is conducive to automation. 1 A Semantic Calculus for Intuitionistic Kripke Models In Rothenberg (2010), we use correspondence theory (Blackburn et al., 2001) to give a cut-free calculus for reasoning about intuitionistic Kripke models (Kripke, 1965) using a fragment of first-order classical logic. Definition 1 (Partially-Shielded Formulae). We define the partially-shielded fragment (PSF) of first-order formulae: (1) ⊥; (2) P{x} iff P is an atomic propositional variable, or an atomic first-order formula with a free variable x; (3) A{x}∧B{x} and A{x}∨B{x}, iff A{x}, B{x} are in PSF; (4) Rxy, where R is a fixed atomic binary relation (5) ∀y.(Rxy∧A{y}) → B{y}, iff A{x} and B{x} are in PSF. Proposition 1. A formula in PSF is either of the form Rxy or has at most one free variable. Proof. By induction on the structure of the formula. We give the calculus G3c/PSF in Figure 1, which is useful for reasoning about sequents of formulae in PSF. A variant of it was introduced in Rothenberg (2010), based on ideas from a calculus for the guarded fragment (GF) of firstorder formulae given in Dyckhoff and Simpson (2006). Γ, P⇒P,∆ Ax Γ,⊥⇒∆ L⊥ Γ, A,B⇒∆ Γ, A ∧B⇒∆ L∧ Γ⇒A,∆ Γ⇒B,∆ Γ⇒A ∧B,∆ R∧ Γ, A⇒∆ Γ, B⇒∆ Γ, A ∨B⇒∆ L∨ Γ⇒A,B,∆ Γ⇒A ∨B,∆ R∨ Γ,Rxz, ∀y. . . .⇒A{z},∆ Γ,Rxz, ∀y. . . . , B{z}⇒∆ Γ,Rxz, ∀y.(Rxy ∧ A{y}) → B{y}⇒∆ L∀ → Γ,Rxz,A{z}⇒B{z},∆ Γ⇒∀y.(Rxy ∧ A{y}) → B{y},∆ R∀ → Figure 1: The calculus G3c/PSF for sequents of partially shielded formulae. In Figure 1, the variable y is fresh for the conclusion of the R∀ → rule, and that ∀y. . . . in the premisses of the L∀ → and R∀ → rules is an abbreviation of “∀y.(Rxy ∧ A{y}) → B{y}”. Proposition 2 (Standard Structural Rules, Rothenberg (2010)). The following rules are admissible in G3c/PSF: Γ⇒∆ Γ,Γ⇒∆,∆ W Γ,Γ,Γ⇒∆,∆,∆ Γ,Γ⇒∆,∆ C Γ⇒∆, A A,Γ⇒∆ Γ,Γ⇒∆,∆ Cut Proposition 3 (Negri (2007)). Let G3c/PSF be G3c/PSF plus the following (geometric) rules: Rxx,Γ⇒∆ Γ⇒∆ refl Rxz,Rxy,Ryz,Γ⇒∆ Rxy,Ryz,Γ⇒∆ tran Rxy, Px, Py,Γ⇒∆ Rxy, Px,Γ⇒∆ mono where Px, Py in the mono rule are atomic. Corollary 4 (Negri (2007)). The standard structural rules (Proposition 2) are admissible in G3c/PSF. Remark 1. Earlier work on geometric rules for modal logics can be found in Simpson (1994). Remark 2. The labelled sequent calculus G3I (Negri, 2007) and (Dyckhoff and Negri, 2011) can be thought of as an alternative form of G3c/PSF that hides the quantifiers and incorporates the mono rule into the axiom Ax. Definition 2 (Translation of Propositional Formulae into PSF). ⊥ =def ⊥ (A ∧B) † =def A † ∧B (A → B) =def ∀y.(Rxy ∧ A ) → B P † =def P̂x (A ∨B) † =def A † ∨B where the translation of A → B requires that the free variable of A, B is x, and y 6= x, and P̂ x uniquely corresponds to P . Recall that R-formulae occur only as strict subformulae in the translation. The extension is adapted to sequents naturally, where all formulae have the same free variable. Definition 3 (Kripke Semantics of PSF). Let M = 〈W,R, 〉 be a Kripke model, and let x̂ be a function from first-order variables into W . Then 1. M1⊥ iff M, x̂1⊥ for all x̂ ∈ W ; 2. M P{x} iff (a) M P{x} iff M, x̂ P{x} for all x̂ ∈ W , where P{x} is an atomic propositional variable; (b) M P{x} iff M, x̂ Px for some x̂ ∈ W , where P{x} is an atomic first-order formula; 3. M A ∧B iff M A and M B; 4. M A ∨B iff either M A or M B; 5. M Rxy iff (x̂, ŷ) ∈ R; 6. M (Rxy ∧A{y}) → B{y} iff M, x̂ (Rxy ∧A{y}) → B{y} iff M Rxy and either M1A{y} or M B{y}. This is extended naturally for sequents of formulae by M Γ⇒∆ iff either M1 ∧ ∧Γ or M ∨ ∨∆. Theorem 5 (Soundness and Completeness, Rothenberg (2010)). Let M = 〈W,R, 〉 be a Kripke model for Int. Then M Γ⇒∆ iff G3c/PSF ⊢ Γ⇒∆. Proof. Using Definition 3, we note the rules of G3c/PSF are sound w.r.t. the properties of M. For completeness, we show by induction of the structure of sequents (the sizes of Γ,∆ and the structure of each formula). Lemma 6 (Right Monotonicity). The rule Rxy,Γ⇒∆, Px, Py Rxy,Γ⇒∆, Py mono is admissible in G3c/PSF. Proof. Using cut. Lemma 7 (General Monotonicity). The rules Rxy,Ax,Ay,Γ⇒∆ Rxy,Ax,Γ⇒∆ Rxy,Γ⇒∆, Ax,Ay Rxy,Γ⇒∆, Ay are admissible in G3c/PSF. Proof. By induction on the derivation depth and formula size. Theorem 8. Let G be a multisuccedent sequent calculus for Int, e.g. m-G3ip (Troelstra and Schwichtenberg, 2000). Then G ⊢ Γ⇒∆ iff G3c/PSF ⊢ Γ⇒∆. Proof. By induction on the derivation height. An outline of the proof is as follows: (1) Hyperextend (Avron, 1991) G to a hypersequent calculus HG; (2) Show G ⊢ Γ⇒∆ iff HG ⊢ Γ⇒∆ | H (straightforward). (3) Extend Definition 2 so that components in hypersequents are translated with unique free variables; (4) Show HG ⊢ H iff G3c/PSF ⊢ H. (Note that instances of mono or trans can be eliminated from G3c/PSF proofs of sequents with a single free variable.) Corollary 9 (Soundness and Completeness). Let G be a multisuccedent sequent calculus for Int. Then G is sound and complete w.r.t. Int. Proof. Follows from Theorem 8.