Duplication-free tableau calculi and related cut-free sequent calculi for the interpolable propositional intermediate logics

Combining Proof-Search and Counter-Model Construction for Deciding Gödel-Dummett Logic

“…We denote by A σ the result of the application of σ to the variables in A. IL will denote the set of formulae that are provable in any intuitionistic propositional calculus (see [5]) and CL will denote the classically valid formulae. As usual an intermediate propositional logic [1] is a set of formulae L satisfying IL ⊆ L ⊆ CL and closed under the rule of modus ponens 1 and under arbitrary substitution. 2 The Gödel-Dummett logic LC is an intermediate logic: in a Hilbert axiomatic system, it is the smallest intermediate logic satisfying the axiom formula…”

“…From i ∈ I k , we derive i ∈ I k , and then Γ a , X I k X i . As X i is a variable, X i ∈ M k+1 holds thus [[X i ]] k + 1 holds by equation (1).…”

“…We obtain I 3 = {1, 2, 3} and stop. The fixpoint is [1,3]. We can derive the counter-model: from the weights we obtain X {1,2,3} {1, 2, 3, 4}, X {1,3} {2, 4}, X {3} {4} and X ∅ ∅.…”

“…It is viewed as one of the most important intermediate logics, between intuitionistic logic IL and classical logic CL, with connections with the provability logic of Heyting's Arithmetics [14] and more recently fuzzy logic [10]. Starting from proof-search in intuitionistic logic IL, the development of efficient proof-search procedures for intermediate logics like Gödel-Dummett logic has been the subject of recent studies [1,6,2].…”

“…Moreover, he showed that there is a complete proof-search strategy which is deterministic, meaning that all the logical rules become invertible. In the same time, Avellone et al [1] and Fiorino [7] investigated the ideas of the duplication-free system within the semantic tableaux approach and proposed corresponding tableaux calculi for various intermediate logics including LC. In [2], Avron claims that all these systems suffer from the serious drawback of using a rule, called [⊃ R ], with an arbitrary number of premises: this rule may introduce exponential blowup in the proof search process.…”