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

By Alexander Bochman

The major topic and aim of this ebook are logical foundations of non­ monotonic reasoning. This bears a presumption that there's this sort of factor as a normal conception of non monotonic reasoning, rather than a number of structures for the sort of reasoning latest within the literature. It additionally presumes that this sort of reasoning might be analyzed via logical instruments (broadly understood), simply as the other form of reasoning. with a view to in attaining our objective, we'll offer a typical logical foundation and semantic illustration within which other kinds of non monotonic reasoning might be interpreted and studied. The urged framework will subsume ba­ sic varieties 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 varieties reminiscent of contraction inference kin that specific relative independence of items of knowledge. additionally, an analogous framework will function a foundation for a common conception 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 optimistic 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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Extra resources for A Logical Theory of Nonmonotonic Inference and Belief Change

Example text

Consequently, the union UW is also a theory of If-. But it can be easily checked that this theory coincides with CI(u, v). So, CI(u,v) is also a theory of If-, and hence If- is union-closed. 0 The next result connects the notion of union-closure with that of basegeneration. 4. If a set of theories of a grounded consequence relation If- is union-closed, then II- is base-generated. Proof. Let Ll be a set of all prime propositions of If-, and A, BELl. Then Th1f-(A) and Th1f-(B) are theories of If-, and hence CI(Th1f-(A) U Th1f-(B)) is also a theory.

1. A consequence relation is strongly grounded if and only if all its theories are unions of prime theories. Let us denote by P lfo the set of prime propositions of a consequence relation If-. Then the following consequence of the above result shows that any theory of a strongly grounded consequence relation is a deductive closure of the prime propositions included in it. 2. If a consequence relation If- is strongly grounded, then any theory u of If- is equal to Thlfo(u n Plfo). Proof. If If- is strongly grounded, then each its theory u is a union of all Th(A), where A E P lfo n u, and hence u coincides with Thlfo(u n P lfo ).

10. A consequence relation is determinate iff it has a greatest theory. 1 Scott consequence relations 21 Proof. Assume that a consequence relation If- has a greatest theory u. Then A W holds if and only if A E u. Consequently, if Ai W, for any Ai E a, then a ~ u, and therefore a W. Thus, If- is determinate. For the other direction, we will show that the set u = {A I A W} is a greatest theory of a determinate consequence relation. To begin with, u is consistent. Indeed, otherwise u If-, and hence a If-, for some finite set a ~ u.

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