WinKE: A Proof Assistant for Teaching Logic

WinKE is a new interactive theorem proving assistant based on the KE calculus, a refutation system which combines features from Smullyan's analytic tableaux and Gentzen's natural deduction. It has been developed to support teaching logic and reasoning to undergraduate students. The software is supportive of and complementary to an introductory textbook on classical logic Mondadori and D'Agostino, 1997]. This paper provides a short introduction to the KE calculus and describes the main features of the WinKE system. 1 The KE Calculus In this paper we present an interactive pedagogical system for teaching classical logic based on the system KE (see D'Agostino and Mondadori, 1994] and Mondadori and D'Agostino, 1997]). This is a refutation system which, in our view, supersedes both natural deduction and Smullyan's tableau method, namely the most popular systems currently used in teaching. One drawback of the method of analytic tableaux is that its rules do not capture one of the essential semantic features of classical logic, namely the bivalent character of the underlying notion of truth (see Broda et al., 1995]). This problem could be solved by augmenting the tableau rules with the rule given below: A :A We call this rule PB, from Principle of Bivalence. The formula A which appears in the two nodes generated by an application of this rule is called PB-formula. It is well-known that in a tableau environment this rule corresponds to the cut rule of the sequent calculus. Therefore, its addition allows tableaux to represent the use of \lemmas" in proofs. However, this would be only an ad hoc adjustment. The tableau rules are complete without the PB-rule, so there is no natural way of incorporating this rule in the tableau method (or, if you wish, there is a natural way which amounts to \simulating" the system that we are going to present below). A more interesting approach consists in weakening the operational rules so that the system becomes complete only with the addition of the PB-rule. There is (under very plausible assumptions) a unique system of