Sequent and hypersequent calculi for abelian and łukasiewicz logics

Abstract

We present two embeddings of Lukasiewicz logic L into Meyer and Slaney's Abelian logic A, the logic of lattice-ordered Abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for L. These include hypersequent calculi, terminating hypersequent calculi, co-NP labeled sequent calculi, and unlabeled sequent calculi.

From Axioms to Analytic Rules in Nonclassical Logics

“…P 3 ) can be captured by multiple-conclusion (hyper)sequent calculi, as in the case of weak excluded middle in Gentzen LK calculus for classical logic or of Łukasiewicz axiom (see Example 7.4) in the hypersequent calculus for Łukasiewicz logic in [10], we conjecture that the expressive power of single-conclusion sequent (resp. hypersequent) structural rules is limited to N 2 (resp.…”

Expanding the Realm of Systematic Proof Theory

“…Therefore, though the calculi of [14] admit cut-elimination, their rules, which we extracted out of P 3 and N 3 axioms, are not "good" enough to ensure conservativity. This contrasts with the result for N 2 axioms (and P 3 ones, if our conjecture is true, cf.…”

“…The aim to continue the conquer further faced a serious obstacle: As shown in [3], "strong" cut-elimination for a logical system L implies that the class of algebras corresponding to L is closed under completions, whereas certain logics beyond P 3 do not admit closure under completions. Typical examples are Abelian logic AL [15,16]-the logic corresponding to compact closed categories-and infinite-valued Lukasiewicz logic L, although possessing cut-free hypersequent calculi, see [14].…”

“…Our method also works as a heuristic principle for finding appropriate rules for axioms located at levels higher than P 3 . As a case study, in Section 7 we show how to semi-automatically obtain the cut-free hypersequent calculi for AL and L, that have been discovered in [14] by trial and error.…”

“…We have finally arrived at the system HAL = HMALL − + mix + mix 0 + + split for AL introduced in [14]. A concrete cut-elimination procedure which relies on the invertibility of logical rules is contained in [13].…”