By Alexander Bochman
The major topic and target of this publication are logical foundations of non monotonic reasoning. This bears a presumption that there's this type of factor as a basic concept of non monotonic reasoning, in place of a number of structures for the sort of reasoning latest 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. as a way to in achieving our target, we'll offer a typical logical foundation and semantic illustration within which other kinds of non monotonic reasoning should be interpreted and studied. The steered framework will subsume ba sic types of nonmonotonic inference, together with not just the standard skeptical one, but additionally a number of kinds of credulous (brave) and defeasible reasoning, in addition to a few new forms akin to contraction inference family members that categorical relative independence of items of information. furthermore, an identical framework will function a foundation for a common concept of trust switch which, between different issues, will let us unify the most ways to trust switch latest within the literature, in addition to to supply a optimistic view of the semantic illustration used. This e-book 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
For any generating sequent ai If- Ci, if ai ~ CI(A 1\ Hu), then ai ~ u, and hence there is C E Ci that belongs to u. Therefore C E unh, and consequently C E CI(A 1\ Hu). Hence, CI(A 1\ Hu) is a theory of If-. 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.
This concludes the proof. As can be shown, the second condition above can be replaced by a weaker claim that intersection of any two theories is also a theory. It can be verified that any singular Scott consequence relation is uniquely determined by its Tarski subrelation. Also, if I- is a Tarski consequence relation, we can define the corresponding singular Scott consequence relation as follows (cf. [Gab81], Theorem 11): a I~ b iff a I- A, for some A E b. It is easy to check that I- and Ih- have actually the same theories.
The set of all admissible belief states in the above sense will be considered as the principal component of our epistemic states. It is important to note that this structure implicitly reflects dependence relations among potential beliefs. The latter, even if logically consistent, can be based on incompatible defaults and expectations, and hence cannot always be held simultaneously. Accordingly, admissible belief sets are not homogeneous; not all deletions or additions to them constitute justifiable belief sets, but only those that are supported by some admissible sets of defaults.
A Logical Theory of Nonmonotonic Inference and Belief Change by Alexander Bochman