Proof Theory for Positive Logic with Weak Negation

Abstract

Proof-theoretic methods are developed for subsystems of Johansson's logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and used to conclude that the considered logical systems are PSPACE-complete.Minimal propositional calculus (Minimalkalkül, denoted here as MPC) is the system obtained from the positive fragment of intuitionistic propositional calculus (equivalently, positive logic [27]) by adding a unary negation operator satisfying the so-called principle of contradiction (sometimes referred to as reductio ad absurdum, e.g., in [24]). This system was introduced in this form by Johansson in 1937 [21], and goes back to Kolmogorov's first formalization of intuitionistic logic [22], obtained by discarding ex falso quodlibet (ex falso, from now on) from the nowadays standard axioms for intuitionistic logic. A letter from Johansson to Heyting (1935Heyting ( -1936 reads [32]:[ex falso] says that once ¬a has been proved, b follows from a, even if this had not been the case before.This implies that the constructive interpretation of negation (i.e., implication) characteristic of intuitionism may give rise to doubts concerning the legitimacy of ex falso as an axiom of intuitionistic logic. More generally, by rejecting ex falso, one earns the right to study the notion of contradiction on its own and thereby, the related notion of negation.The axiomatization proposed by Johansson preserves the whole positive fragment and most of the negative fragment of Heyting's intuitionistic logic. As a matter of fact, many important properties of negation provable in Heyting's system remain provable (in some cases, in a slighlty weakened form) in minimal logic. The absence of ex falso made Johansson's system the focus of interest in the field of paraconsistency, conceived as the study of those logics which admit inconsistent non-trivial theories. From a standard paraconsistent view, minimal logic still has unfortunate features [24]. In fact, the provability of what we are going to refer to as 'negative ex falso' (a ∧ ¬a) → ¬b makes negation meaningless in inconsistent theories-since every negated formula is provable-hence preserving some of the trivial aspects distinctive of ex falso. Interestingly enough, in the setting of positive logic, negative ex falso follows already from the assumption that negation is a functional antitone operator, i.e., that it satisfies the contraposition axiom (a → b) → (¬b → ¬a) [7,8]. The latter is a theorem of Johansson's logic and one of its main

A Study of Subminimal Logics of Negation and Their Modal Companions

“…In order to keep the structure of the paper as simple as possible, we skip the broader and introductory account of the topic and refer the interested reader to [12,13]. For a proof-theoretic account, see [7].…”