55 research outputs found
Towards a Proof Theory of G\"odel Modal Logics
Analytic proof calculi are introduced for box and diamond fragments of basic
modal fuzzy logics that combine the Kripke semantics of modal logic K with the
many-valued semantics of G\"odel logic. The calculi are used to establish
completeness and complexity results for these fragments
Bounded-analytic sequent calculi and embeddings for hypersequent logics
A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity
We present some hypersequent calculi for all systems of the classical cube
and their extensions with axioms , , , and, for every , rule
. The calculi are internal as they only employ the language of the
logic, plus additional structural connectives. We show that the calculi are
complete with respect to the corresponding axiomatisation by a syntactic proof
of cut elimination. Then we define a terminating root-first proof search
strategy based on the hypersequent calculi and show that it is optimal for
coNP-complete logics. Moreover, we obtain that from every saturated leaf of a
failed proof it is possible to define a countermodel of the root hypersequent
in the bi-neighbourhood semantics, and for regular logics also in the
relational semantics. We finish the paper by giving a translation between
hypersequent rule applications and derivations in a labelled system for the
classical cube
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