Quasi-canonical systems and their semantics

Abstract

A canonical (propositional) Gentzen-type system is a system in which every rule has the subformula property, it introduces exactly one occurrence of a connective, and it imposes no restrictions on the contexts of its applications. A larger class of Gentzen-type systems which is also extensively in use is that of quasi-canonical systems. In such systems a special role is given to a unary connective $\lnot$ of the language (usually, but not necessarily, interpreted as negation). Accordingly, each application of a logical rule in such systems introduces either a formula of the form $\diamond (\psi _1,\ldots ,\psi _n)$, or of the form $\lnot \diamond (\psi _1,\ldots ,\psi _n)$, and all the active formulas of its premises belong to the set $\{\psi _1,\ldots ,\psi _n,\lnot \psi _1,\ldots ,\lnot \psi _n\}$. In this paper we investigate the whole class of quasi-canonical systems. We provide a constructive coherence criterion for such systems, and show that a quasi-canonical system is coherent iff it is analytic iff it has a characteristic non-deterministic matrix (Nmatrix) which is based on some subset of the set of the four truth-values which are used in the famous Dunn–Belnap four-valued (deterministic) matrix for information processing. In addition, we determine when a quasi-canonical system admits (strong) cut-elimination.