On the Proof Theory of Natural Many - valuedLogicsArnon

We claim that Proof Systems for natural many-valued logics, whether nite-valued or innnite-valued should be similar in their structure to proof systems of any other natural logic: one should not be able to tell from the structures which are used in a proof system the intended semantics. It is also preferable that standard connectives will be used, with corresponding standard rules. We demonstrate this thesis with some examples in which cut-free Gentzen-type systems, which employ either ordinary sequents or hypersequents, are used both for 3-valued logics and for innnite-valued logics. 1 The Methodological Approach In recent years there is a growing Interest in many types of nonclassical logics: modal and temporal logics, substructural logics, constructive logics, many-valued logics, paraconsistent logics, non-monotonic logics { the list is long. Obviously, there is no limit to the number of logics that logicians (and non-logicians) can produce. Some creteria are needed, therefore, to distinguish those that are \nat-ural" or \interesting" in some sense (and so deserve studying). It seems to me that the following are widely accepted virtues of a \natural" logic: Natural primitives. In other words: the primitive connectives and quantiiers of the language of the logic should intuitively correspond to concepts which are informally used outside the realm of formal logic, like: implication, negation , conjunction, necessity etc. The language might have several \conjunc-tions" (say), each corresponding to a diierent interpretation of the informal concept, but it should not include as primitives artiicial constructs, tailored for a speciic semantics (for examples: unary connectives which correspond to certain nonclassical truth values, as in 13]). The existence of a simple, illuminating semantics. On the propositional level such a semantics should provide (so I believe) a decision procedure for the consequence relation of the logic (and so, of course, also to its set of valid formulas). The existence of a nice proof system. Such a system should make it easier to nd proofs in the system, to prove results about it, and, should have the subformula property. Here again the proof system should determine not only the set of valid formulas of the logic, but also its consequence relation.