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Unformatted text preview: Chapter 3: Propositional Languages We define here a general notion of a propo sitional language. We show how to obtain, as specific cases, var ious languages for propositional classical logic and some nonclassical logics. We assume the following : All propositional languages contain a set of variables V AR , which elements are denoted by a,b,c,.... with indices, if necessary. 1 All propositional languages share the general way their sets of formulas are formed. We distinguish one propositional language from the other is the choice of its set of propo sitional connectives. We adopt a notation L CON , where CON stands for the set of connec tives. We use a notation L when the set of connectives is fixed. 2 For example, the language L {} denotes a propositional language with only one connective . The language L { , } denotes that a language with two connec tives and adopted as propositional connectives. Remember: any formal language deals with symbols only and is also called a symbolic language. 3 Symbols for connectives do have intuitive mean ing. Semantics is a formal meaning of the connec tives and is defined separately. One language can have many semantics . Different logics can share the same language. For example the language L { , , , } is used as a propositional language of classical and intuitionistic logics, some many valued logics, and is extended to the language of modal logics. 4 Several languages can share the same seman tics. The classical propositional logic is the best example of such situation. Due to functional dependency of classical log ical connectives the languages: L {} , L {} , L {} , L { , , , } , L { , , , , } , L {} , L {} all share the same semantics characteristic for classical propositional logic. The connectives have well established com mon names and readings, even if their se mantic can differ....
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 Spring '08
 Bachmair,L
 Computer Science

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