December 15 MON — 12.00-14.00
Sala "Enzo Paci" — Direzione del Dipartimento (Via Festa del Perdono 7, Milano)
Default Classicality, Kleene Logics and Normality Operators.
(joint work with Massimiliano Carrara)
Abstract: Many-valued logics usually come with a background picture of default classicality: deviation from classical rules of inference and valid formulas is motivated by some particular phenomena (logical paradoxes, partial information, vagueness), but as far as these phenomena are not at stake, classical logic works just fine and we can assume it. Once this picture is accepted, a question naturally arises: `How can we recapture classical logic within a many-valued logic?' That is, how can we sort out the conditions at which a classical inference/law can be legitimately drawn/asserted in a many-valued setting?
Two established approaches to this question are classical collapse by Jc Beall (2013) and minimal inconsistency by Priest (1991). The former recaptures classical inferences by strengthening the premises (if the logic is paracomplete) or weakening the conclusion (if the logic is paraconsistent), the second defines a non-monotonic relation of consequence that selects those models of the premises that are 'minimally inconsistent' (if the premises contain no contradiction, these models will be classical models). Both approaches are defined for three-valued Kleene Logics, and they do not change the expressive power of such logics.
In this paper, I propose a different solution to the problem, which in turn endows the object language of the Kleene logic of choice with the ability to express that a given sentence is not paradoxical. This solution is based on the extension of the language of Kleene logics with a normality operator ·, which is a truth-functional connective that forms true sentences out of sentences that have one of the two classical truth-values 1 or 0, and forms false sentences from sentences that have the non-classical truth value (whence the name of the operator).
The paper is divided into two parts. In the first part, I prove the two main results of the paper, showing how classical consequence can be recaptured in Kleene Logics endowed with the normality operator. This results are based on the ability to state, within the object language, that no paradox or non-classical sentence is involved.
In the second part, I explore the connections between the present approach with classical collapse and minimal inconsistency, and I discuss some conceptual virtues of the former over the latter.
Beall, Jc (2013) LP+, K3+, FDE+ and their Classical Collapse, Review of Symbolic Logic, 6/4: 742--754
Priest, G. (1991) Minimally Inconsistent LP, Studia Logica, 50/2: 321--331
