Finite Two-Dimensional Proof Systems for Non-finitely Axiomatizable Logics

Abstract

The characterizing properties of a proof-theoretical presentation of a given logic may hang on the choice of proof formalism, on the shape of the logical rules and of the sequents manipulated by a given proof system, on the underlying notion of consequence, and even on the expressiveness of its linguistic resources and on the logical framework into which it is embedded. Standard (one-dimensional) logics determined by (non-deterministic) logical matrices are known to be axiomatizable by analytic and possibly finite proof systems as soon as they turn out to satisfy a certain constraint of sufficient expressiveness. In this paper we introduce a recipe for cooking up a two-dimensional logical matrix (or -matrix) by the combination of two (possibly partial) non-deterministic logical matrices. We will show that such a combination may result in -matrices satisfying the property of sufficient expressiveness, even when the input matrices are not sufficiently expressive in isolation, and we will use this result to show that one-dimensional logics that are not finitely axiomatizable may inhabit finitely axiomatizable two-dimensional logics, becoming, thus, finitely axiomatizable by the addition of an extra dimension. We will illustrate the said construction using a well-known logic of formal inconsistency called mCi. We will first prove that this logic is not finitely axiomatizable by a one-dimensional (generalized) Hilbert-style system. Then, taking advantage of a known 5-valued non-deterministic logical matrix for this logic, we will combine it with another one, conveniently chosen so as to give rise to a -matrix that is axiomatized by a two-dimensional Hilbert-style system that is both finite and analytic.

Finite Hilbert Systems for Weak Kleene Logics

“…Another direction for future research worth mentioning is the study of these and other logics associated to the algebra B u (and other three-valued algebras) in the setting of different kinds of H-systems. In particular, a twodimensional version of Set-Set H-systems [21,23], whose induced logics are the so-called B-consequence relations [5], may be employed as a uniform setting for investigating pure consequence relations (like BK and PWK), their intersection (order-theoretic consequence relations) and mixed consequence relations (we use here the terminology of [16]), the latter being non-Tarskian consequence relations (lacking either reflexivity [26] or transitivity [19]).…”