The Procedures for Belief Revision

Abstract

The idea of belief revision is strictly connected with such notions as revision and contraction given by two sets of postulates formulated by Alchourrón, Gärdenfors and Makinson in (Theoria, 48: 14–17, 1982; Journal of Symbolic Logic, 50: 510–530, 1985; Philosophical Essays Dedicated to Lennard Aqvist on His Fiftieth Birthday, pp. 88–101, 1982). In the paper expansion, contraction and revision are defined probably in the most orthodox way, i.e. by Tarski’s consequence operation (e.g. Compt. Rend. Séances Soc. Sci. Lett. Varsovie, cl.III, 23, 22–29) and Tarski-like elimination operation (see Łukowski, P., Logic and Logical Philosophy, 10: 59–78, 2002). In our approach nonmonotonicity appears as a final result of alternate using of two steps of our reasoning: “step forward” and “step backward”. Step forward extends the set of our beliefs and it is used when some new belief appears. Step backward reduces the set of our beliefs and it is used when we reject some previously accepted belief. A decision of adding or rejecting of some sentences is arbitrary and depends on our wish only. Thus, this decision cannot be logical and logic cannot justify it. In our approach logic is a tool for faultless and precise realization of extension or reduction of the set of our beliefs. But why some sentences should be added or refused depends on extralogical reasons. Such understood nonmonotonicity can be considered also on logics other than classical. A reconstruction of a given logic in its deductive-reductive form is here the basis of nonmonotonicity. It means that nonmonotonic reasoning can be formalized on the base of every logic for which its deductive-reductive form can be reconstructed. Firstly our approach is tested on the ground of the classical logic. Next it is confronted with the intuitionism represented by the Heyting-Brouwer logic. Procedures of contraction and revision are verified by using of the AGM postulates. We limit our considerations to first six conditions for contraction and revision, because of the well known relation between contraction satisfying first four conditions together with the sixth one and revision defined by this contraction and consequence operation. Satisfaction of almost every postulate is for us a good sign that our approach is reasonable. The only exception we make for the fifth postulate and for Harper identity. Analyzing the reasoning alternately extending and contracting the set of beliefs it is difficult to accept the postulate that adding previously rejected sentence we obtain again the same set of beliefs as before the contraction. This problem is deeper considered in the paper. The original AGM settles the relations between contraction and revision with, the well known Levi and Harper identities. In all cases of our approach, revision is defined on the basis of contraction which with respect to revision is prime.