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

By Alexander Bochman

The major topic and goal of this e-book are logical foundations of non­ monotonic reasoning. This bears a presumption that there's the sort of factor as a basic conception of non monotonic reasoning, in preference to a number of structures for this kind of reasoning latest within the literature. It additionally presumes that this type of reasoning might be analyzed by way of logical instruments (broadly understood), simply as the other form of reasoning. so as to in achieving our target, we'll supply a typical logical foundation and semantic illustration within which other forms of non monotonic reasoning may be interpreted and studied. The urged framework will subsume ba­ sic sorts of nonmonotonic inference, together with not just the standard skeptical one, but in addition numerous types of credulous (brave) and defeasible reasoning, in addition to a few new forms reminiscent of contraction inference kin that specific relative independence of items of knowledge. furthermore, a similar framework will function a foundation for a basic idea of trust switch which, between different issues, will let us unify the most techniques to trust switch present within the literature, in addition to to supply a positive view of the semantic illustration used. This ebook 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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A Scott consequence relation is prime iff, for any set a, If- a holds only if If- A, for some A E a. Proof. Assume that If- has a least theory uo. Then, for any proposition A, if W A, then A does not belong to some theory of If-, and consequently it does not belong to uo. Therefore, if W Ai, for all Ai E a, then a is disjoint from uo, and hence W a holds. In the other direction, if a consequence relation satisfies the above condition, we will show first that the set u = {A 1If- A} is a theory. Indeed, otherwise we would have a If- b, for some finite a ~ u and b ~ U.

Therefore A will belong to any theory of If- that contains B, and hence B If- A will hold. Assume now that If- is not linear, and u and v are two of its incomparable theories. Then there must exist propositions A and B such that A E u \ v and B E v \ u. Consequently we have that A W Band B W A. 0 As is well-known, any partial order is an intersection of the set of linear (total) orders that extend it. Similarly, the following result shows that any Scott consequence relation is an intersection of its linear extensions.

1 Supraclassical Tarski consequence relations A Tarski consequence relation f- is called supraclassical if F ~ f-. In other words, a Tarski consequence relation is supraclassical if it subsumes classical inference. Supraclassicality is equivalent to the requirement that all theories of a consequence relation are deductively closed sets. 1. A Tarski consequence relation is supraclassical iff all its theories are deductively closed sets. Proof. Theories of a Tarski consequence relation are sets of propositions that are closed with respect to its rules.

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