3slides - Chapter 3: Propositional Languages We define here...

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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 non-classical 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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3slides - Chapter 3: Propositional Languages We define here...

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