Connexivity and Logics of Formal Inconsistency

Abstract

This paper discusses connexive variants of the Logics of Formal Inconsistency (LFIs). Our primary focus is on these extensions of LFIs, which are obtained by expanding their axiomatic sets to include Aristotle’s theses. We propose several non-trivial calculi, with particular emphasis on the connexive extensions of Ci. The calculi are defined over the positive fragment of classical propositional calculus. Additionally, we demonstrate that there is no non-trivial extension of Cia or Cio by the theses of Aristotle or Boethius. As a solution, we propose an intuitionistic variant of Ci that, when simultaneously extended by the so-called consistency propagation axioms and the theses of Aristotle or Boethius, remains non-trivial. A similar strategy is subsequently applied to maximal LFIs: LFI1, LFI2 and P′₁. Since the calculi based on the intuitionistic variant of Ci no longer validate the principle of gentle explosion, a key conceptual question arises: can they still be regarded as LFIs once this fundamental principle is abandoned? This motivates revising the definition of LFIs and introducing the Logics of Formal quasi-Inconsistency (LFqIs) as an alternative to LFIs.