The Preservationist Approach to Paraconsistent Inference

Abstract: From a semantic point of view, an inference relation is paraconsistent iff there is at least one one inconsistent set of sentences of the underlying language and one sentence of that language that is not validly inferrable from it. On the classical understanding of validity, an argument is valid iff any model that satisfies its premisses makes its conclusion true. Thus a paraconsistent system of inference that retains the classical account of entailment must give an account of satisfaction according to which at least one inconsistent set is satisfiable. But systems called dialethic paraconsistent systems that purport to satisfy inconsistent sets give non-classical accounts of negation. It can be argued that such systems are positive, that is negation-free, systems and so not paraconsistent at all, although proof-theoretically they present useful paraconsistent sublogics of classical logic.

In the preservationist tradition, paraconsistency is achieved by variously strengthening the preservational requirements of validity. Whereas classical validity imposes only truth-preservational requirements on inference, preservationist systems require inference to preserve various mathematically well-defined measures on sets of data. An inference relation, I preserves a measure iff for every set of data, the value of the measure for the set is the value of the measure for its inferential closure. I will canvass a number of preservable measures such as incoherence-level, incoherence-dilution, and level of ambiguity, their applications and descendent research.