By Alexander Bochman

ISBN-10: 3642075169

ISBN-13: 9783642075162

The major topic and aim of this booklet are logical foundations of non­ monotonic reasoning. This bears a presumption that there's this kind of factor as a common concept of non monotonic reasoning, in place of a number of structures for any such reasoning latest within the literature. It additionally presumes that this sort of reasoning will be analyzed by way of logical instruments (broadly understood), simply as the other form of reasoning. so that it will in achieving our target, we are going to supply a typical logical foundation and semantic illustration during which other kinds of non monotonic reasoning might be interpreted and studied. The recommended framework will subsume ba­ sic kinds of nonmonotonic inference, together with not just the standard skeptical one, but additionally a variety of different types of credulous (brave) and defeasible reasoning, in addition to a few new forms reminiscent of contraction inference kinfolk that categorical relative independence of items of knowledge. furthermore, an identical framework will function a foundation for a normal thought of trust switch which, between different issues, will let us unify the most methods to trust switch present within the literature, in addition to to supply a confident view of the semantic illustration used. This e-book is a monograph instead of a textbook, with all its merits (mainly for the writer) and shortcomings (for the reader).

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Additional resources for A Logical Theory of Nonmonotonic Inference and Belief Change

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Notice that even classical consequence relations still do not coincide, in general, with the classical entailment 1=; the latter can be described as the least classical (or, equivalently, least supraclassical) consequence relation. There is, however, an intimate connection between classical entailment and classical consequence relations. Notice that any rule A f- B of a classical consequence relation is equivalent to f- A -+ B. Consequently, any classical consequence relation can be seen as a classical entailment with some set of propositions added as additional, nonlogical axioms.

2. A theory u of a Scott consequence relation If- is a least theory containing a set of propositions v if and only if u = Cnlf- (v). Proof. Clearly, Cnlf-(v) is an intersection of all theories of If- containing v. Therefore, if it is a theory by itself, it must be a least theory containing v. Assume now that u is a least theory containing v, but Cnlf- (v) is not a theory. Then there exists c such that v If- c, though no C E c is a consequence of v. The latter condition implies that, for any C E c there exists a theory Uc that contains v but does not contain C.

Proof. Let u be a theory of II-. If CI ({ A } U u) is not a theory of II-, there must exist finite sets a <;;; u, c <;;; CI( {A} U u) such that a, A II- c. If A is a basic proposition, we have a II- A --t c, and since u is a theory, there is C E c such that A --t C E u. Consequently, C E CI( {A} Uu), contrary to the assumption that c is disjoint from CI( {A} U u). lf A --t c, then there is a theory u such that a <;;; u and A --t c <;;; U. But then CI( {A} U u) is a theory of II- that includes A and a and is disjoint from c.

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