Absurd sentences and normal deductions: A case of the logic of demodalised analytic implication with falsum

Abstract

The logic of demodalised Parry analytic implication DAI was introduced by J. M. Dunn. In its original formulation, the language of DAI consisted of classical negation, classical conjunction, and demodalised analytic implication. Later, R. L. Epstein rediscovered DAI as a content-inclusion logic, by providing semantics in terms of set-assignment models. Epstein called it the dependence logic D. He also studied DAI expanded with the constants falsum and verum in the context of algebraic analysis. More recently, A. Ledda, F. Paoli, and M. Pra Baldi investigated DAI with constants and provided an algebraic semantics in terms of implicative involutive bisemilattices. In this paper, we study DAI expressed in a language without nega- tion but with the constant falsum. First, we examine several semantic treatments of falsum, each of which gives rise to different definable negations and, consequently, to distinct logics. But we focus only on one logic, namely DAI without negation but with falsum. Second, we introduce two labelled deductive systems for the resulting logic. We prove for each deductive system the soundness and completeness theorems and establish a Prawitz-style normalisation theorem.