Cut Elimination for First Order Gödel Logic by Hyperclause Resolution

Abstract

Abstract. Efficient, automated elimination of cuts is a prerequisite for proof analysis. The method CERES, based on Skolemization and resolution has been successfully developed for classical logic for this purpose. We generalize this method to Gödel logic, an important intermediate logic, which is also one of the main formalizations of fuzzy logic.

Proof theory of witnessed Gödel logic: A negative result

“…. Q n y n P in HG ∞ can then be obtained by re-introducing the Skolemized quantifiers in the proof of ⇒ ∃xF S (x), as shown in [7] (see the result (b) above). Proof The prenex formula S := ∃x∀y(P (x) ⊃ P (y)) is valid in G w while it is not in Gödel logic.…”

“…Finally, recall that Skolemization holds for the prenex fragment of Gödel logic [9,7] and it can be easily proved for witnessed Gödel logic.…”

“…The missing step between the propositional translation sketched above and the proof of ⇒ F is the mid-hypersequent theorem of [7] (i.e. the result (a) above) which allows us to transform the proof d ′ of ⇒ F separating the quantifier and the propositional part.…”

“…As a corollary of Theorem 7 and of the undecidability of the prenex fragment of Gödel logic (Lemma 10) we also show the undecidability of the problem of determining, given a proof in RG f o ∞ , if there exists a cut-free proof of the same end sequent. To prove the key theorem below we will use the following results from [7]: Let F be a prenex formula valid in Gödel logic and let ∃xF S (x) be its Skolem form (x abbreviates x 1 , . .…”