In classically based modal logic, there are three common conceptions of necessity, the universal conception, the equivalence relation conception, and the axiomatic conception. They provide distinct presentations of the modal logic S5, all of which coincide in the basic modal language. We explore these different conceptions in the context of the relevant logic R, demonstrating where they come apart. This reveals that there are many options for being an S5-ish extension of R. It further reveals a divide between the universal (...) conception of necessity on the one hand, and the axiomatic conception on the other: The latter is consistent with motivations for relevant logics while the former is not. For the committed relevant logician, necessity cannot be the truth in all possible worlds. (shrink)

Two common forms of natural deduction proof systems are found in the Gentzen–Prawitz and Jaśkowski–Fitch systems. In this paper, I provide translations between proofs in these systems, pointing out the ways in which the translations highlight the structural rules implicit in the systems. These translations work for classical, intuitionistic, and minimal logic. I then provide translations for classical S4 proofs.

A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.

The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the “logical” content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system (...) for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice and automated theorem proving of Reverse Mathematics results. (shrink)

Cut reductions are defined for a Kripke-style formulation of modal logic in terms of indexed systems of sequents. A detailed proof of the normalization (cutelimination) theorem is given. The proof is uniform for the propositional modal systems with all combinations of reflexivity, symmetry and transitivity for the accessibility relation. Some new transformations of derivations (compared to standard sequent formulations) are needed, and some additional properties are to be checked. The display formulations [1] of the systems considered can be presented as (...) encodings of Kripke-style formulations. (shrink)

RésuméLa définition de la condition suspensive, telle qu’elle nous est donnée dans l’article 1181 du Code civil français, est aujourd’hui au centre de différents projets de réforme. Si aucun projet de réforme n’a réussi à emporter l’assentiment de tous les juristes, nombre d’entre eux semblent s’accorder sur la nécessité de réformer ce texte—inchangé depuis 1804. Pourquoi un tel consensus sur ce besoin de réécriture de la définition de la condition suspensive mène à une discussion doctrinale où deux positions principales s’opposent? (...) C’est à cette question que nous tâchons de répondre en traquant l’origine de cette scission dans la lettre même du texte dont nous analysons à la fois la structure logique et la signification des termes employés pour sa rédaction. Cette analyse nous permet de mettre en évidence la sémantique ambiguë du terme “incertain”, terme pour lequel nous proposons une analyse logique de sa signification que nous comparons ensuite à celle du terme “inconnu”. (shrink)