Logics of Knowing and Believing

What is the relation of intelligence (natural or artificial) to logic? This is surely a fascinating question with many aspects. We shall not discuss what intelligence is; but we shall start by saying that logic is understood as the study of consequence and of deduction as obtaining consequences from (accepted, assumed) premises (axioms). It goes without saying that intelligence has a logical aspect (among various other aspects). Intelligent systems are said to work with various sorts of knowledge; knowledge representation and processing, and even reasoning about knowledge are considered to be important topics in AI. Knowledge may be uncertain; thus one has to work with beliefs. It has turned out that logical aspects of reasoning about knowledge and beliefs are well formalized by means of some systems of modal logic called logic of knowledge and of belief respectively. An excellent survey of propositional (Boolean) logics of knowledge and belief in relation to computational complexity (with full proofs of main results) is given by Halpern (1992). Most basic facts on these logics are surveyed in Sect. 2 of this chapter. Section 3 surveys basic facts on predicate (Boolean) logics of knowledge and belief; Hughess and Cresswell (1984) contains details. The main part is Sect. 4 where we take into consideration the fact that most knowledge in AI systems (not speaking about natural intelligence) is imprecise (vague, fuzzy), which leads us to fuzzy logic.

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