By Alexander Bochman
The major topic and target of this e-book are logical foundations of non monotonic reasoning. This bears a presumption that there's this kind of factor as a basic conception of non monotonic reasoning, instead of a host of platforms for this sort of reasoning latest within the literature. It additionally presumes that this type of reasoning may be analyzed by way of logical instruments (broadly understood), simply as the other type of reasoning. that allows you to in attaining our aim, we are going to offer a typical logical foundation and semantic illustration during which other kinds of non monotonic reasoning may be interpreted and studied. The instructed framework will subsume ba sic varieties of nonmonotonic inference, together with not just the standard skeptical one, but additionally quite a few kinds of credulous (brave) and defeasible reasoning, in addition to a few new varieties resembling contraction inference kinfolk that specific relative independence of items of information. additionally, a similar framework will function a foundation for a common concept of trust switch which, between different issues, will let us unify the most techniques to trust swap present within the literature, in addition to to supply a positive view of the semantic illustration used. This booklet 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
Consequence Relations Proof. Let u be a minimal theory containing A. If B ~ u, then u W 'ii implies A W B, 'ii by Monotonicity. 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.
In particular, any base-generated Tarski consequence relation will be generated by its basic propositions defined as propositions B validating the deduction theorem: if a, B f- A, then a f- B -+ A, for any a and A. Moreover, for Tarski consequence relations, any proposition is prime, so basic propositions will also be base propositions of a consequence relation. 5. If A and B are basic propositions of a Tarski consequence relation f-, then A V B is also a basic proposition of f-. As for general consequence relations, any union-closed Tarski consequence relation will be base-generated, and the reverse implication will hold only for strongly grounded consequence relations.
5 Determinate and linear consequence relations Keeping in mind our intended use of Scott consequence relations for representing epistemic states, we will describe below three classes of consequence relations that will correspond to important kinds of epistemic states. Determination. Determinate consequence relations, defined below, will correspond to determinate epistemic states described in Chapter 3. 4. A consequence relation If- will be called determinate if, for any set a, a If- implies A If-, for some A E a.