A refined calculus for Intuitionistic Propositional Logic - DISCo

Kripke model K = 〈P, ≤, ρ, ⊩〉 where ρ is a new element (ρ ∉ ⋃ 1≤j≤n P j) andthe immediate successors of ρ are the elements ρ 1 , . . . , ρ n , formally:P = ( ⋃P j ) ∪ {ρ} ≤ = ( ⋃≤ j ) ∪ {(ρ, α) | α ∈ P }1≤j≤n1≤j≤nFinally, for every α ∈ P and every propositional variable p, α ⊩ p iff one of thefollowing conditions holds:– α ∈ P j and α ⊩ j p.– α = ρ and Tp ∈ S.We point out that, if α ∈ P j , then α ⊩ H iff α ⊩ j H; in particular, K, ρ j ✄ T jfor every 1 ≤ j ≤ n. We prove that K, ρ ✄ H for every H ∈ S.If H = Tp, by definition ρ ⊩ p. If H = Fp, then, by consistency, Tp ∉ S,hence ρ p. If H = F c p, then F c p ∈ T j for every 1 ≤ j ≤ n; it follows thatρ j ⊩ ¬p for every 1 ≤ j ≤ n, hence ρ ⊩ ¬p.Let H = T(p → B) and let α ∈ P such that α ⊩ p. Since Tp ∉ S (byPoint (2)), by definition ρ p. Let k be such that α ∈ P k . Since ρ k ⊩ p → Band ρ k ≤ α, it follows that α ⊩ B.Let H = F(A → B), then there exists k such that T k = (S c \{H})∪{TA, FB}and K, ρ k ✄ T k . It follows that ρ k ⊩ A and ρ k B, hence ρ A → B.Let H = T((A → X ∧ Y ) → C), then there exists k such thatT k= (S c \ {H}) ∪ {TA, Fp, T(X → (Y → p)), T(p → C)}and K, ρ k ✄ T k . Let α ∈ P such that α ⊩ A → X ∧ Y . Since ρ k ⊩ A andρ k X ∧ Y (otherwise, ρ k ⊩ p would follow), α ≠ ρ. Let j be such that α ∈ P j .If j = k, we have ρ k ≤ α, which implies α ⊩ C. If j ≠ k, since H ∈ T j , K, ρ j ✄T jand ρ j ≤ α, and we get α ⊩ C. The remaining cases are similar.⊓⊔By the above lemma, it follows that, if {FA} is not realizable then thereexists a closed proof table for {FA}. Since A ∈ Int implies that {FA} is notrealizable, we get:Theorem 2 (Completeness). If A ∈ Int, then there exists a closed proof tablefor {FA}.The proof of Lemma 3 implicitly defines a decision procedure for IntuitionisticLogic; indeed, starting from a finite set S of signed formulas, either a closed prooftable or a counter-model for S is built. In the following we give some insights onthe strategy we apply in the decision procedure.As usual, applying invertible rules before non-invertible ones reduces thesearch-space. In our decision procedure, cases (i)-(iv) in the definition of S correspondto the application of invertible rules. Accordingly, if there exists H ∈ Ssatisfying one of cases (i)-(iv), we firstly apply the rule related to S, H; if thesearch for a closed proof table fails, we conclude that S is not provable (there isno need to backtrack and try the application of another rule to S).12