Refutation systems in modal logic

Abstract

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 remarksFormal 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,

Gentzen-Type Refutation Systems for Three-Valued Logics with an Application to Disproving Strong Equivalence

“…Later, equivalent systems were independently considered by Goranko [19] and Bonatti [5]. We pursue this work towards similar sequential refutation systems for the above defined three-valued logics L and P.…”

“…As in the sequential calculus of Avron [20], we distinguish between two types of rules: The first type is standard in the sense that they introduce one occurrence of ∧, ∨, or ⊃ at a time. We note that the standard rules coincide with the respective introduction rules in the refutation systems for classical logic [18,19,5]. The second group contains rules that are non-standard since they introduce two occurrences of a connective at the same time, in particular this concerns negation in combination with all other connectives.…”

“…Let us note that these calculi are, in a certain sense, refutational pendants of the Gentzen-type calculi of Avron [20] for axiomatising the valid sentences of L and P, much in the same way as the complementary systems of Tiomkin [18], Goranko [19], and Bonatti [5] are pendants of standard Gentzen systems for classical logic.…”

A Generalisation of a Refutation-related Method in Paraconsistent Logics

“…5 Apart from that, however, the work on refutation systems is sporadic. The paper [8] presents an alternative rejection system for the classical propositional logic, whereas the papers [13,14] deal with rejection in modal logics. One should also note an attempt at using the notion of rejection in the framework of argumentation networks (cf., e.g., [11]).…”

Axiomatization in the Meaning Sciences

“…It is a curious question, which we will not investigate here, whether results of this kind about stronger logics can be used to address the under-generation question in a case from natural language semantics. Another curious question is whether a complete axiomatization of the invalid inferences according to a semantics, by means of "refutation axioms" and "refutation rules" as in Łukasiewicz 1957 andGoranko 1994, could shed light on whether a semantics over-generates invalidities, or equivalently, under-generates validities. 3…”