Countermodel Construction via Optimal Hypersequent Calculi for Non-normal Modal Logics

International audienceWe develop semantically-oriented calculi for the cube of non-normal modal logics and some deontic extensions. The calculi manipulate hypersequents and have a simple semantic interpretation. Their main feature is that they allow for direct countermodel extraction. Moreover they provide an optimal decision procedure for the respective logics. They also enjoy standard proof-theoretical properties, such as a syntactical proof of cut-admissibility

A hilbert-style axiomatisation for equational hybrid logic

This paper introduces an axiomatisation for equational hybrid logic based on previous axiomatizations and natural deduction systems for propositional and first-order hybrid logic. Its soundness and completeness is discussed. This work is part of a broader research project on the development a general proof calculus for hybrid logics

Tableaux for free logics with descriptions

The paper provides a tableau approach to definite descriptions. We focus on several formalizations of the so-called minimal free description theory (MFD) usually formulated axiomatically in the setting of free logic. We consider five analytic tableau systems corresponding to different kinds of free logic, including the logic of definedness applied in computer science and constructive mathematics for dealing with partial functions (here called negative quasi-free logic). The tableau systems formalise MFD based on PFL (positive free logic), NFL (negative free logic), PQFL and NQFL (the quasi-free counterparts of the former ones). Also the logic NQFL− is taken into account, which is equivalent to NQFL, but whose language does not comprise the existence predicate. It is shown that all tableaux are sound and complete with respect to the semantics of these logics

Gödel’s Natural Deduction

This is a companion to a paper by the authors entitled "Godel on deduction", which examined the links between some philosophical views ascribed to Godel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Godel presented with the help of Gentzen's sequents, which amounts to JaA gt kowski's natural deduction system of 1934, and which may be found in Godel's unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Godel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Godel's Notre Dame course, and is followed by comments concerning Godel's natural deduction system