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chapter5 - CHAPTER 4 SOME EXTENSIONAL SEMANTICS Many valued...

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CHAPTER 4 SOME EXTENSIONAL SEMANTICS Many valued logics in general and 3-valued logics in particular is an old object of study which had its beginning in the work of ˆLukasiewicz (1920). He was the first to define a 3- valued semantics for a language L ¬ , , , of classical logic, and called it a three valued logic for short. He left the problem of finding a proper axiomatic proof system for it (i.e. complete with respect to his semantics) open. The same happened to all other logics presented here. They were first defined only semantically, i.e. by providing a semantics for their languages, without a proper proof systems. We can think about the process of their creation was inverse to the creation of Classical Logic, Modal Logics, the Intuitionistic Logic which existed as axiomatic systems longtime before invention of their formal semantics. Recently there is a revived interest in this topic, due to its potential applications in several areas in Computer Science, like: proving correctness of programs, knowledge bases, and Artificial Intelligence. Three valued logics, when defined semantically, enlist a third logical value, be- sides classical T, F . We denote this third value by . We also often assume that the third value is intermediate between truth and falsity, i.e. that F < < T . There has been many of proposals relating both to the intuitive interpretation of this third value and for the change to logic that follows in its wake. For obvious reasons all of presented here semantics take T as designated value, i.e. the value that defines the notion of satisfiability and tautology. If T is the only designated value, the third value corresponds to some notion of incomplete information , like undefined or unknown and is often denoted by the symbol U or I . If, on the other hand, corresponds to inconsistent information , i.e. its meaning is something like known to be both true and false then it takes both T and the third logical value as designated. We present here only five 3-valued logics semantics that belong to the first category and are named after their authors: ˆLukasiewicz, Kleene, Heyting , and Bochvar logics. Historically all these semantics were called logics, so we use the name logic for them, instead saying each time ”logic defined semantically”, or ”semantics for 1
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a given logic”. 1 ˆLukasiewicz Logic ˆL ˆL stands for ˆLukasiewicz logic semantics. This was the first 3-valued logic semantics ever to be invented. Motivation ˆLukasiewicz developed his semantics (called logic ) to deal with future contingent statements. According to him, such statements are not just neither true nor false but are indeterminate in some metaphysical sense. It is not only that we do not know their truth value but rather that they do not possess one. Intuitively, signifies that the statement cannot be assigned the value true of false; it is not simply that we do not have sufficient information to decide the truth value but rather the statement does not have one.
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