On Relevance of Entailment in a Classical Logic Context - a podcast by MCMP Team

Peter Verdée (Université catholique de Louvain) gives a talk at the MCMP Colloquium (16 April, 2015) titled "On Relevance of Entailment in a Classical Logic Context". Abstract: In this talk I present a logic that aims to determine whether implications in a classical logic context express a relevant connection between antecedent and consequent. Just like in the relevance logic tradition, the connective '->' is added to the language of classical logic as a formalisation of the relevant entailment relation in the object language (without nesting restrictions). The recursively enumerable set of theorems of the logic is defined by a set of axioms closed under a number of rules and the principle of uniform substitution. Unlike the logics in the relevance logic literature, the presented logic leaves classical negation intact in the sense that the law of non-contradiction can be used to obtain relevant implications, as long as there is a connection between antecedent and consequent. In order to realise this strengthening of traditional relevance logics, the traditional requirement that a theory of relevance should also define a new standard of deduction is given up. I present and argue for a list of requirements such a logical theory of relevant entailment in classical contexts needs to meet. I then formulate the system by presenting its axiomatisation and an algebraic semantics and demonstrate that it respects each of these requirements. This system is conceived by means of a translation into the relevant logic R and by presupposing an asymmetric treatment of symbols that occur in antecedents and symbols that occur in consequents. I moreover will give a philosophical motivation for our non-standard relevant implication and the asymmetric interpretation of antecedents and consequents. Finally I argue that there is no a priori reason to require that a successful formal theory of relevant entailment should also be a successful formal theory of (the standard of) deduction.

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