Tableau-Based Theorem Proving In Normal Conditional Logics

This paper presents an extension of the semantic tableaux approach to theorem proving for the class of normal conditional logics. These logics are based on a possible worlds semantics, but contain a binary “variable conditional” operator =E= instead of the usual operator for necessity. The truth of A +B depends both on the accessibility relation between worlds, and on the proposition expressed by the antecedent A. Such logics have been shown to be appropriate for representing a wide variety of commonsense assertions, including default and prototypical properties, counterfactuals, notions of obligation, and others. The approach consists in attempting to find a truth assignment which will falsify a sentence or set of sentences. If successful, then a specific falsifying truth assignment is obtained; if not, then the sentence is valid. The approach is arguably more natural and intuitive than those based on proof-theoretic methods. The approach has been proven correct with respect to determining validity in the class of normal conditional logics. In addition, the approach has been implemented and tested on a number of different conditional logics. Various heuristics have been incorporated, and the implementation, while exponential in the worst case, is shown to be reasonably efficient for a large set of test cases.