19/11 - Esko Turunen, PhD*

A Paraconsistent Version of Pavelka's Fuzzy Logic

In 1979 Jan Pavelka introduced a very general framework to deal with many valued logics. Pavelka's  idea was to process Zadeh's Fuzzy Sets such that theories, rules of inference, proofs as well as  tautologies may be only partial, i.e. fuzzy. Pavelka defined all his concepts in complete residuated lattices. The main issue was to study the circumstances under which the fuzzy semantic consequence operation and fuzzy syntactic operation coincide; such a property is called Pavelka style completeness.  Pavelka solved the problem in the special case that the set of truth values is the Lukasiewicz sturucure, i.e. the real unit interval equipped with standard MV-structure. The present author has recently proved that Pavelka style completeness holds if, and only if the set of truth values is a complete MV-algebra. Thus, if in particular ,the set M of truth values is a certain collection of 2x2-matrices equipped with suitable operations, then M is a complete MV-algebra. In fact, the set M extends Belnap's four valued para consistent logic. Such an approach results a complete many-valued logic that behaves consistently when looking from outside: the structure in related to Lukasiewicz logic which is a consistent logic. However, looking the logic from inside, i.e. a single 2x2-matrix, para consistency steps in. Truth and falsehood are not opposites of each other, and also contradictions and lack of knowledge is involved.

*Tampere University of Technology (Tampere, Finland)