Refutation systems in modal logic

Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems. Introduction: historical remarks Formal logic traditionally deals with deductive systems for inference of statements valid according to certain semantics, and is not involved in the inference of the non-valid ones. As Lukasiewicz points out in [9], out of the two intellectual acts acceptance and rejection of a statement, the latter one has been neglected in the modern formal logic. This neglect seems strange, moreover because the father of the formal logic, Aristotle, already noticed the importance of the systematic rejection of non-valid arguments. He realized that, in order to show that a syllogism was not universally valid, it was not necessary to construct a "refuting model", i.e. an example where the syllogism produces an obviously false conclusion from true premises. Instead, it was enough to infer that syllogism from others, the validity or non-validity of which had already been established, applying certain rules of inference. The typical rule used by Aristotle for that purpose was the so called "modus tollens": If A implies B, and B is rejected, then A is rejected too. Thus, he established a sort of deductive system for rejection of non-valid syllogisms. We shall call deductive systems which infer refutable statements instead of valid ones refutation systems. The statements which are inferred, Le. provably refutable in such a system will be called rejected statements. The history of refutation systems, unlike the history of the "orthodox" ones, according to the author's knowledge, is rather short and scanty. Lukasiewicz in [9], raising the general problem of the formal deduction of the nonvalid statements of a given theory, suggested a complete refutation system for the non-valid classical propositions. The system is a very natural one: the only axiom is "p is rejected" where p is a fixed propositional variable, Presented by Jan Zygmunt; Received September 30, 1992; Revised May 21, 1993 Studia Logica 53: 299-324, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.