Algebraic Semantics for Cu lat ive ference
Zbigniew Stachniak · 1993
In this paper we propose preferential matrix semantics for nonmonotonic inference systems and show how this algebraic framework can be used in methodological studies of cumulative inference operations. The study of general properties of cumulative infer- ence systems can be based on a less general and more structured notion of a model structure. The key feature of our semantic proposal is the truth-functional inter- pretation of logical connectives, the idea well-developed in the context of logical calculi which can also be ex- ploited in the studies of nonmonotonic reasoning. We define a preferential model, or as it is called in this pa- per, a preferential matrix, as an algebra of truth-values augmented with a family V of sets of designated truth- values. We model a desired degree of nonmonotonicity by selecting an appropriate preference relation on 2). Preferential matrices have the same semantic scope as Makinson's preferential model structures. Evident simi- larities between 'classical' logical matrices and preferen- tial matrices provide an access to reach algebraic tech- niques available for methodological studies of deductive proof systems. In this context, the present paper exam- ines a list of properties of nonmonotonic inference sys- tems, starting with characterization of cumulativity and loop-cumulutivity in terms of preferential matrices. We introduce a handy notion of the monotone buse of an inference system and study the distributivity property in terms of this notion. Finally, we look at the con- sistency preservation property in the context of finding automated theorem proving methods for cumulative in- ference systems. We give a criterion for such systems to have a refutationally equivalent automated proof system based on the resolution rule. In this paper we study inference systems on propo- sitional level only. It is assumed that the reader is fa- miliar with (Gabbay 1985, Kraus, Lehmann, & Magidor 1990, Makinson 1988). The familiarity with (Brown & Shoham 1988, Makinson 1989) and the basic facts on logical matrices, as presented in (Wojcicki 1988), is an asset.