A Paraconsistent Proof Procedure Based on Classical Logic. Extended Abstract

Apparently Ex Falso Quodlibet (or Explosion) cannot be isolated within CL (Classical Logic); if Explosion has to go, then so have other inference rules, for example either Addition or Disjunctive Syllogism. This certainly holds according to the standard abstract view on logic. However, as I shall show, it does not hold if a logic is defined by a procedure—a set of instructions to obtain a proof (if there is one) of a given conclusion from a given premise set. In this paper I present a procedure pCL− that defines a logic CL−—a function assigning a consequence set to any premise set. Anything derivable by CL from a consistent premise set Γ is derivable from Γ by CL−. If Γ is (CL-)inconsistent, pCL− enables one to demonstrate this (by deriving a contradiction from Γ). The logic CL− validates applications of Disjunctive Syllogism as well as applications of Addition. Nevertheless, this logic is paraconsistent as well as (in a specific sense) relevant. pCL− derives from an intuitively attractive proof search procedure. A characteristic semantics for CL− will be presented and the central properties of the logic will be mentioned. CL− shows that (and clarifies how) adherents of CL may obtain non-trivial consequence sets for inconsistent