By Alexander Bochman
The major topic and aim of this booklet are logical foundations of non monotonic reasoning. This bears a presumption that there's one of these factor as a normal idea of non monotonic reasoning, in preference to a number of platforms for the sort of reasoning current within the literature. It additionally presumes that this sort of reasoning may be analyzed by means of logical instruments (broadly understood), simply as the other form of reasoning. so one can in attaining our target, we'll offer a standard logical foundation and semantic illustration during which other kinds of non monotonic reasoning should be interpreted and studied. The instructed framework will subsume ba sic kinds of nonmonotonic inference, together with not just the standard skeptical one, but in addition a variety of sorts of credulous (brave) and defeasible reasoning, in addition to a few new varieties akin to contraction inference kin that specific relative independence of items of information. additionally, an analogous framework will function a foundation for a common concept of trust switch which, between different issues, will let us unify the most methods to trust switch latest within the literature, in addition to to supply a confident view of the semantic illustration used. This booklet is a monograph instead of a textbook, with all its merits (mainly for the writer) and shortcomings (for the reader).
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Extra info for A Logical Theory of Nonmonotonic Inference and Belief Change
1. 1 a set of propositions. 1 such that A E Thlf-(D). 1. 5. 1 iff it satisfies the following conditions: 1. 1 is prime; 2. 1 A , for any proposition A. Proof. 1 will be prime in If-. 1 A . 1 A by right compactness. Assume that If- satisfies the above two conditions, and let u be a minimal theory of If- containing some proposition A. 1 A . But D is a prime proposition, and hence Th1f-(D) will be a theory of If- that contains A and is included in u. Due to minimality of u, we have u = Thlf-(D). 1.
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. 6. A supra classical Tarski consequence relation is union-closed if and only if it satisfies Cn(A /\ B) = CI(Cn(A), Cn(B)) Proof. The direction from left to right is immediate. Let u and v be two theories of f- and u, v f- A. then there must exist B E u and C E v such that B, C f- A. By the above condition, this implies A E Cl(Cn(B), Cn(C)), and hence there must exist B1 E Cn(B) and C1 E Cn( C) such that C1, B1 fA.
But this is impossible due to the finiteness of Cd. Thus, u is a theory ofthis consequence relation. Moreover, assume that there is another theory Uo such that P E Uo C u, and let Pi be an atomic proposition that does not belong to uo. Then i > 1, since If- PI belongs to the consequence relation. In addition, it is easy to check that the following sequents belong to the consequence relation, for any j > 1: Consequently, Uo must contain q/, for some 1 ~ i, which is impossible, since Uo is included in u.