EPISTEMIC ROLE OF THE LOGIC OF FALSEHOOD
The idea of belief revision is strictly connected with such notion as contraction given by the set of postulates formulated by Alchourrón, Gärdenfors and Makinson in e.g. 'On the logic of theory change: contraction functions and their associated revision functions' (Theoria 48, pp. 14-37 (1982)); 'On the logic of theory change: Partial meet contraction revision functions' (Journal of Symbolic Logic 50, pp. 510-530 (1985)). In the paper expansion and contraction are defined probably in the most orthodox way, i.e. by Tarski's consequence relation (e.g. Über einige fundamentale Begriffe der Metamathematik, Compt. Rend. Séances Soc. Sci. Lett. Varsovie, cl.III, 23, pp. 22-29) and Tarski-like elimination relation (in P. Lukowski's: 'A deductive-reductive form of logic: general theory and intuitionistic case' publ. in 'Logic and Logical Philosophy' 10, pp. 59-78 (2002)). The logic of falsehood (i.e. a logic dual in Wójcicki's sense to the given logic of truth) plays a key role for defining of the elimination relation. 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 from some previously accepted belief. A decision of adding or refusing 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 reducing of the set of our beliefs. But why some 'initial' sentences should be added or refused depends on extralogical reasons. The logic for the back-reasoning uses the class of models adequate for the logic extending the set of our beliefs. However, the class is used in a specific i.e. dual form. That is why the step forward (expansion) and the step backward (contraction) constitute the one whole. Procedure of contraction satisfies the well known AGM postulates. We limit our considerations to first six conditions for contraction. Satisfaction of almost every postulate is for us a good sing that our approach is reasonable. The only exception we make for the controversial fifth postulate.
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