Algebraic Analysis of Demodalised Analytic Implication

Abstract

The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn (and independently investigated by R.D. Epstein) as a variation on a timehonoured logical system by C.I. Lewis' student W.T. Parry. The main tenet underlying this logic is that no implication can be valid unless its consequent is "analytically contained" in its antecedent. DAI has been investigated both proof-theoretically and model-theoretically, but no study so far has focussed on DAI from the viewpoint of abstract algebraic logic. We provide several different algebraic semantics for DAI, showing their equivalence with the known semantics by Dunn and Epstein. We also show that DAI is algebraisable and we identify its equivalent quasivariety semantics. This class turns out to be a linguistic and axiomatic expansion of involutive bisemilattices, a subquasivariety of which forms the algebraic counterpart of Paraconsistent Weak Kleene logic (PWK). This fact sheds further light on the relationship between containment logics and logics of nonsense.

Relating Logic and Relating Semantics. History, Philosophical Applications and Some of Technical Problems

“…In addition, both of these companions to Classical Logic in terms of appropriate matrix bundles and as semilattice-based logics are semantically described by showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. The Płonka sums of matrices are finally used to investigate the pure variable inclusion companions of an arbitrary finitary logic [see, e.g., Ledda, Paoli and Baldi, 2019;Paoli, 2007].…”

Containment Logics: Algebraic Completeness and Axiomatization

“…However, this is just one side of the coin of the logics of variable inclusion; the other side consisting of those logics verifying inferences in which the variables occurring in the conclusion are contained into the ones occurring in the premises. Consequence relations satisfying this feature are usually known as containment logics; the syntactic requirement which they share is a strengthened form of what Ferguson [24] understands as Proscriptive Principle, which also resembles the one defining logics of analytic containment [20,34].…”

Applications of Relating Semantics

“…Some interesting cases are described, for example, in[Jarmużek and Paoli, 2021;Jarmużek and Klonowski, 2021;Paoli, 2007;Ledda et al, 2019], mainly in the implicational fragment of the defined language.…”

“…The article argues that, apart from the person of R. Epstein, also D.Walton [1979a,b] is an indispensable researcher for the development of relating semantics. More on various variants of this kind of implication can be found in[Paoli et al, 2021;Ledda et al, 2019;Paoli, 2007Paoli, , 1993.…”