Syntactic Cut-Elimination for Intuitionistic Fuzzy Logic via Linear Nested Sequents

Abstract

This paper employs the linear nested sequent framework to design a new cut-free calculus (LNIF) for intuitionistic fuzzy logic-the first-order Gödel logic characterized by linear relational frames with constant domains. Linear nested sequents-which are nested sequents restricted to linear structures-prove to be a well-suited proof-theoretic formalism for intuitionistic fuzzy logic. We show that the calculus LNIF possesses highly desirable proof-theoretic properties such as invertibility of all rules, admissibility of structural rules, and syntactic cut-elimination.

From Display to Labelled Proofs for Tense Logics

“…Moreover, the internalized labelled calculi lend themselves nicely to uniformly proving interpolation for the class of path extensions of Kt [20]. As explained there, the labelled formalism is simpler to work with than the nested formalism when showing proof-theoretic interpolation, and thus, interpolation results are more easily obtained by making use of the internalized labelled calculi.…”

“…When translating from labelled to shallow nested, we first put our given derivation into a special form that makes use of so-called propagation rules [15,27]. Such rules allow us to eliminate certain structural rules from our labelled calculus and its derivations; this results in an internal variant of the labelled calculus that-interestingly-inherits the nice properties of the original external calculus while becoming suitable for proof-search and proving interpolation (see [20]). Furthermore, this new form of the derivation permits a stepwise translation into a derivation of a deep-nested calculus, from which, methods in [15] may be applied to further translate the proof into a proof of the corresponding shallow nested calculus.…”

Display to Labeled Proofs and Back Again for Tense Logics

“…Deep nested calculi are better suited than shallow nested calculi for proving e.g. decidability [5,19] and interpolation [26], due to the absence of the hard-to-control display rules that expand the proof-search space. Both shallow and deep nested calculi are typically internal in the sense that each sequent in a proof can be interpreted as a formula of the logic, whereas labeled calculi often appear to be external in the sense that the sequents cannot generally be interpreted as a formula of the logic (and use a language that explicitly encodes the semantics).…”

On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

“…Such calculi have been exploited to prove meaningful results; e.g. decidability [18,20], interpolation [15], and automated counter-model extraction [16,20]. We focus on the labelled and nested formalisms in this paper.…”

“…In contrast to labelled sequents, nested sequents are often given in a language as expressive as the language of the logic; thus, nested calculi have the advantage that they minimize the bureaucracy sufficient to prove theorems. The nested formalism continues to receive much attention, proving itself suitable for constructing analytic calculi [3], developing automated reasoning algorithms [13], and verifying interpolation [15], among other applications.…”