Vitor Greati

We consider a family of non-classical three-valued logics proposed to model indicative conditionals in natural language. Among these, systems introduced by B. De Finetti, W.S. Cooper, J. Cantwell and R.J. Farrell, as well as some variants that have not appeared in the literature, but seem nevertheless to be natural objects of interest from a formal point of view. Most of these logics are not easily treatable with the standard techniques of algebraic logic. We therefore resort to non-deterministic structures and multiple-conclusion calculi to provide alternative semantical characterizations and axiomatizations. In the best cases—logics given by a finite monadic matrix—this can be done directly, in a modular way, through a procedure due to Shoesmith and Smiley. In the more involved ones—logics preserving degrees of truth—some ingenuity and more sophisticated techniques are required. We characterize these logics by a partial non-deterministic matrix, and show how to produce analytic (and effective) calculi that are complete with respect to this generalized semantics. In all cases, the calculi thus obtained can be straightforwardly converted, by a uniform procedure, into traditional single-conclusion Hilbert-style axiomatizations.