By Alexander Bochman
The major topic and goal of this ebook are logical foundations of non monotonic reasoning. This bears a presumption that there's one of these factor as a basic idea of non monotonic reasoning, instead of a host of structures for this kind of reasoning latest within the literature. It additionally presumes that this type of reasoning will be analyzed by means of logical instruments (broadly understood), simply as the other form of reasoning. on the way to in attaining our target, we'll offer a typical logical foundation and semantic illustration during which other kinds of non monotonic reasoning should be interpreted and studied. The prompt framework will subsume ba sic sorts of nonmonotonic inference, together with not just the standard skeptical one, but in addition quite a few varieties of credulous (brave) and defeasible reasoning, in addition to a few new forms equivalent to contraction inference family members that categorical relative independence of items of information. moreover, an analogous framework will function a foundation for a common idea 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 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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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. We will show that the latter is equal to Th1f-(A /\ B). Indeed, CI(Th1f-(A) U Th1f-(B)) is clearly included in Th1f-(A /\ B). 5 Base-generated consequence relations 39 If- (A A B) -+ C, then B -+ C E Thlf-(A), B E Thlf-(B), and hence C E Cl(Thlf-(A) U Thlf-(B)). Since Thlf-(A A B) is a theory of If-, A A B is a prime proposition of If-.
Moreover, the following result shows that adding each of these rules to a supraclassical consequence relation is sufficient for classicality. We leave the proof to the reader as an exercise. 2. A supraclassical consequence relation is classical iff it satisfies one of the following rules: Deduction If a, A f- B, then a f- A -+ B; Contmposition If a, A f- B, then a, -,B f- -,A; Disjunction If a, A f- C and a, B f- C, then a, A V B f- C. 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.
13. A Scott consequence relation is linear iff, for any propositions A and B, either A If- B, or B If- A. Proof. Assume that If- is a linear consequence relation and A W B. Then there must exist a theory u containing A but not B. In this case any theory v of If- that includes B cannot be included in u. And since the set of theories is linearly ordered with respect to inclusion, any such theory v will include u. Therefore A will belong to any theory of If- that contains B, and hence B If- A will hold.