Structural Completeness for Fuzzy Logics
by
George Metcalfe
Vanderbilt University
Coauthors: Petr Cintula

A consequence relation is structurally complete iff all of its proper extensions have new theorems. That is, if a schematic rule is admissible (preserves the set of theorems), then it is derivable in any formal system axiomatizing the consequence relation. Classical Logic has this property, but for many families of non-classical logics, intermediate, modal, and substructural, it is relatively rare, often holding only for fragments of the systems in question. We investigate structural completeness for the class of (t-norm based) fuzzy logics which includes Godel Logic, known to be structurally complete, and Lukasiewicz Logic, where the positive fragment is known to have the property, but not the full logic. Using a range of methods, we are able to obtain an almost complete picture of structural completeness for this family. In particular, we show that structural completeness for these systems is characterized by the existence of an embedding of the generators of the class of algebras for the logic into the free algebra for this class, and use this observation to obtain structural completeness for Product Logic and fragments of Basic Logic.

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Date received: May 18, 2008