Modal companion

In logic, a modal companion of a superintuitionistic (intermediate) logic L is a normal modal logic that interprets L by a certain canonical translation, described below. Modal companions share various properties of the original intermediate logic, which enables to study intermediate logics using tools developed for modal logic. Let A be a propositional intuitionistic formula. A modal formula T(A) is defined by induction on the complexity of A: for any propositional variable , As negation is in intuitionistic logic defined by , we also have T is called the Gödel translation or Gödel–McKinsey–Tarski translation. The translation is sometimes presented in slightly different ways: for example, one may insert before every subformula. All such variants are provably equivalent in S4. For any normal modal logic M that extends S4, we define its si-fragment ρM as The si-fragment of any normal extension of S4 is a superintuitionistic logic. A modal logic M is a modal companion of a superintuitionistic logic L if . Every superintuitionistic logic has modal companions. The smallest modal companion of L is where denotes normal closure. It can be shown that every superintuitionistic logic also has a largest modal companion, which is denoted by σL. A modal logic M is a companion of L if and only if . For example, S4 itself is the smallest modal companion of intuitionistic logic (IPC). The largest modal companion of IPC is the Grzegorczyk logic Grz, axiomatized by the axiom over K. The smallest modal companion of classical logic (CPC) is Lewis' S5, whereas its largest modal companion is the logic More examples: The set of extensions of a superintuitionistic logic L ordered by inclusion forms a complete lattice, denoted ExtL. Similarly, the set of normal extensions of a modal logic M is a complete lattice NExtM. The companion operators ρM, τL, and σL can be considered as mappings between the lattices ExtIPC and NExtS4: It is easy to see that all three are monotone, and is the identity function on ExtIPC. L. Maksimova and V.

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Concepts associés (2)

Modal companion

Intermediate logic

In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic). A superintuitionistic logic is a set L of propositional formulas in a countable set of variables pi satisfying the following properties: 1. all axioms of intuitionistic logic belong to L; 2.