On Partial and Paraconsistent Logics

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L 4 , the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the right set-up is chosen. Our logic L 4 has a semantics that also underlies Belnap's [4] and is related to the logic of bilattices. L 4 is in focus most of the time, but it is also shown how results obtained for L 4 can be transferred to several variants

A Logical Foundation for Logic Programming II: Semantics of General Logic Programs

We suggested in [12] a general logical formalism for Logic Programming based on a four-valued inference. In this paper we give a uniform representation of various semantics for logic programs based on this formalism. The main conclusion from this representation is that the distinction between these semantics can be largely attributed to the difference in their underlying (monotonic) logical systems. Moreover, in most cases the difference can even be reduced to that of the language, that is, to the difference in the logical connectives allowed for representing derivable information. Keywords. Foundations of logic programming, negation as failure, semantics for logic programs, nonmonotonic reasoning. 1 Introduction This is a second part of the study of a general logical formalism for logic programming (see [12]). Briefly, the formalism, called biconsequence relations, involves rules (bisequents) of the form a : b fl c : d that serve as direct representations of program rules of a most ..