By Alexander Bochman
The major topic and aim 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 idea of non monotonic reasoning, rather than a host of structures for this kind of reasoning current within the literature. It additionally presumes that this sort of reasoning may be analyzed through logical instruments (broadly understood), simply as the other type of reasoning. in an effort to in attaining our aim, we'll supply a typical logical foundation and semantic illustration within which other forms of non monotonic reasoning should be interpreted and studied. The prompt framework will subsume ba sic types 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 forms resembling contraction inference family that categorical relative independence of items of knowledge. furthermore, an identical framework will function a foundation for a common thought of trust swap 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 merits (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
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. 5 Base-generated consequence relations In this section we will give a characterization of Scott consequence relations that are generated by subsets of a certain set of propositions called its base.
Since T is right-compact, U = UU and v = UV, for some subsets U and V of T. Now consider the set W of theories CI( Ui UVj), for all Ui E U and Vj E V. Since T is union-closed, W is included in T, and hence all theories are theories of If-. Moreover, the set W is directed. Indeed, any two theories CI(Ul,Vl) and CI(u2, V2) from Ware included into the theory CI(CI(Ul, U2), CI(Vl, V2)), which also belongs to W. Consequently, the union UW is also a theory of If-. But it can be easily checked that this theory coincides with CI(u, v).
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. 5 Base-generated consequence relations In this section we will give a characterization of Scott consequence relations that are generated by subsets of a certain set of propositions called its base.
A Logical Theory of Nonmonotonic Inference and Belief Change by Alexander Bochman