Modal Hybrid Logic

This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof 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

Refutation Systems : An Overview and Some Applications to Philosophical Logics

Refutation systems are systems of formal, syntactic derivations, designed to derive the non-valid formulas or logical consequences of a given logic. Here we provide an overview with comprehensive references on the historical development of the theory of refutation systems and discuss some of their applications to philosophical logics

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 $T$, $P$, $D$, and, for every $n\geq 1$, rule $RD^+_n$. 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

A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus

International audienceWe describe how the Davis-Putnam-Logemann-Loveland proced- ure DPLL is bisimilar to the goal-directed proof-search mechanism described by a standard but carefully chosen sequent calculus. We thus relate a procedure described as a transition system on states to the gradual completion of incomplete proof-trees. For this we use a focused sequent calculus for polarised clas- sical logic, for which we allow analytic cuts. The focusing mech- anisms, together with an appropriate management of polarities, then allows the bisimulation to hold: The class of sequent calculus proofs that are the images of the DPLL runs finishing on UNSAT, is identified with a simple criterion involving polarities. We actually provide those results for a version DPLL(T ) of the procedure that is parameterised by a background theory T for which we can decide whether conjunctions of literals are con- sistent. This procedure is used for Satisfiability Modulo Theor- ies (SMT) generalising propositional SAT. For this, we extend the standard focused sequent calculus for propositional logic in the same way DPLL(T ) extends DPLL: with the ability to call the de- cision procedure for T . DPLL(T ) is implemented as a plugin for P SYCHE, a proof- search engine for this sequent calculus, to provide a sequent- calculus based SMT-solver

Focused Proof-search in the Logic of Bunched Implications

The logic of Bunched Implications (BI) freely combines additive and multiplicative connectives, including implications; however, despite its well-studied proof theory, proof-search in BI has always been a difficult problem. The focusing principle is a restriction of the proof-search space that can capture various goal-directed proof-search procedures. In this paper, we show that focused proof-search is complete for BI by first reformulating the traditional bunched sequent calculus using the simpler data-structure of nested sequents, following with a polarised and focused variant that we show is sound and complete via a cut-elimination argument. This establishes an operational semantics for focused proof-search in the logic of Bunched Implications.Comment: 18 pages conten