FRANCIS J . PELLETIER Identity in Modal Logic Theorem Proving

THINKER is an automated natural deduction first-order theorem proving program. This paper reports on how it was adapted so as to prove theorems in modal logic. The method employed is an "indirect semantic method", obtained by considering the semantic conditions involved in being a valid argument in these modal logics. The method is extended from propositional modal logic to predicate modal logic, and issues concerning the domain of quantification and "existence in a world's domain" are discussed. Finally, we took at the very interesting issues involved with adding identity to the theorem prover in the realm of modal predicate logic. Various alternatives are discussed. 1o I n t r o d u c t i o n This paper is a repor t on some issues concerning the addition of identi ty to m y au toma ted theorem proving system, THINKER, in the rea lm of modal logic. Al though there is much background mater ia l which is of relevance to the overall enterprise (for some of it, see [11], [12]), for tunately not much of it is crucial for unders tanding the philosophico-logical issues involved with the addit ion of ident i ty to modal logics. In this in t roductory section, I ment ion some of this background without going into details; in the following sections we look at some deeper issues. THINKER is an automatic theorem proving system, employing a natural deduct ion format , for the full f irst-order logic with identity. While it is not impor tan t for the logic of what is to be discussed below that the systern embodies a na tura l deduction format, this perhaps explains why the emphasis below is on rules of inference rather than axioms and ra ther than on resolution-style strategies. The particular system which is implemented mirrors the Katish & Montague system [3], [4]. The system implements the full f i rs t -order predicate logic with identity (but without arbi trary function symbols). A basic distinction can be made between direct and indirect methods of (au tomated) theorem proving in general, and not just in modal logic. For example, were one interested in proofs in the simple propositional logic, there are numerous proof theories available differing axiomatic developments , different tableaux methods, different natura l deduction formulations, and also propositional resolution. A direct method of theorem proving is to Studia Logica 52: 291-308, 1993. © ].993 Kluwer Academic Publishers. Printed in the Netherlands.