By Alexander Bochman
The major topic and aim of this publication are logical foundations of non monotonic reasoning. This bears a presumption that there's this type of factor as a basic idea of non monotonic reasoning, in preference to a host of structures for any such reasoning current within the literature. It additionally presumes that this type of reasoning will be analyzed through logical instruments (broadly understood), simply as the other type of reasoning. which will in attaining our aim, we are going to supply a standard logical foundation and semantic illustration during which other forms of non monotonic reasoning might be interpreted and studied. The urged framework will subsume ba sic sorts of nonmonotonic inference, together with not just the standard skeptical one, but additionally quite a few types of credulous (brave) and defeasible reasoning, in addition to a few new types resembling contraction inference relatives that specific relative independence of items of information. furthermore, a similar framework will function a foundation for a common thought of trust switch which, between different issues, will let us unify the most ways to trust swap present within the literature, in addition to to supply a confident view of the semantic illustration used. This e-book 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 info for A Logical Theory of Nonmonotonic Inference and Belief Change
1. A general epistemic state E in a language C is a triple (S, l, ~), where S is a set of objects called admissible belief states, ~ is a preference relation on S, while l is a labelling function assigning a deductively closed theory in C (called an admissible belief set) to every belief state from S. If s ~ t holds, we will say that the admissible belief state t is preferred We will assume "by default" that the preference relation is a strict partial order. We can safely make such an assumption, since practically all constructions in this book will be hospitable to this understanding, and hence we will have no incentive to consider more abstract relations.
It is easy to see that a theory u is saturatable with respect to A if and only if, for any proposition B, either A ~ B or A ~ -,B belongs to u. 4. A consequence relation I- is semi-classical iff any maximal theory of I- that is consistent with some proposition A is saturatable with respect to A. Proof. If a ¥ A, there must exist a maximal theory u that is consistent with -,A and includes a. If u is saturatable with respect to -,A, either A V B or A V -,B belongs to u, and therefore either a, A V B ¥ A or a, A V -,B ¥ A.
Again, it turns out that there exists a purely internal characterization of strongly generated consequence relations. 4. A supraclassical Scott consequence relation will be called strongly grounded if all its small theories are prime. 2. A supraclassical consequence relation is strongly grounded iff it is strongly generated by some set of propositions. The following result connects the notion of strong groundedness with the general notion of right compactness. 3. A supra classical consequence relation is strongly grounded if and only if the set of its prime theories is right-compact.
A Logical Theory of Nonmonotonic Inference and Belief Change by Alexander Bochman