18 March 2005, Colloquium on Mathematical Logic, Xavier Caicedo Ferrer

Abstract:
An axiomatic expansion E(C) of a deductive logic having a reasonable notion of equivalence defines implicitly a new connective symbol C if E(C)+E(C') proves C and C' equivalent. For some logics like classical propositional calculus, C is necessarily equivalent to a combination of classical connectives, but other calculi, like Heyting calculus or Lukasiewicz logic, allow implicitly defined connectives which are not explicitly definable in this manner, providing new concepts to the logic. We discuss the general properties of axiomatic expansions of algebraizable logics by these connectives, specially in the case of Heyting calculus and intermediate calculi, and consider the problem of characterizing those logics for which all axiomatically defined implicit connectives are explicitly definable.

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