Download e-book for iPad: A Logical Theory of Nonmonotonic Inference and Belief Change by Alexander Bochman

By Alexander Bochman

ISBN-10: 3642075169

ISBN-13: 9783642075162

ISBN-10: 3662045605

ISBN-13: 9783662045602

The major topic and target of this booklet are logical foundations of non­ monotonic reasoning. This bears a presumption that there's any such factor as a basic concept of non monotonic reasoning, in preference to a number of structures for one of these reasoning present within the literature. It additionally presumes that this sort of reasoning should be analyzed by way of logical instruments (broadly understood), simply as the other type of reasoning. so that it will in achieving our aim, we'll supply a typical logical foundation and semantic illustration during which other kinds of non monotonic reasoning should be interpreted and studied. The advised framework will subsume ba­ sic varieties of nonmonotonic inference, together with not just the standard skeptical one, but in addition numerous varieties of credulous (brave) and defeasible reasoning, in addition to a few new types resembling contraction inference kin that specific relative independence of items of knowledge. furthermore, a similar framework will function a foundation for a normal idea of trust swap which, between different issues, will let us unify the most techniques to trust swap present within the literature, in addition to to supply a confident view of the semantic illustration used. This ebook is a monograph instead of a textbook, with all its benefits (mainly for the writer) and shortcomings (for the reader).

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Conversely, if A W B, 'ii, there must exist a theory v included in u such that A E v and B ~ v. But u is a minimal theory containing A, and hence u = v and B ~ u. Let us denote by Fm the set of all propositions of the language. Assume now that A Wand a set u satisfies the above condition. By Reflexivity, we immediately conclude that A E u. If there exists a proposition B not belonging to u (that is, u "# Fm), then A W B, 'ii, and hence u contains some theory that also contains A. Let v be a minimal theory containing A and included in u, and assume that v "# u, that is, there is C E u \ v.

Then there exists a theory u of If- that includes a and is disjoint from b. Now, since the union of any chain of theories from If- is also a theory of If-, u is included in some maximal chain of theories Ti (by Zorn's lemma). Let If-I be a Scott consequence relation determined by Ti. Clearly, If-I is a linear consequence relation that extends If-. In addition, u is a theory of If-I, and hence a Wi b. Thus, If- coincides with the 0 intersection of linear consequence relations that extend it. As will be shown in what follows, a similar 'linear decomposition' will also hold for quite rich partially ordered structures of epistemic states.

A V -,B should belong to u by Weak Factoring, and hence Cl( u U {A}) is a world. D 28 2. Consequence Relations The next result shows that Weak Factoring is equivalent to a certain weak form of Disjunction in the Antecedent. 5. A supraclassical consequence relation is semi-classical iff it satisfies the following rule: Weak Disjunction If a, AV B f- A and a, AVe f- A, then a, A V BVe f- A. Proof. B in the above rule, we immediately obtain Weak Factoring. B f- A. e f- A. B f- A. Applying Weak Factoring to the latter and (1), we finally obtain a, A V B V f- A.

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A Logical Theory of Nonmonotonic Inference and Belief Change by Alexander Bochman

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