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Unformatted text preview: Chapter 4: Classical Propositional Semantics Language : L {¬ , ∪ , ∩ , ⇒} . Classical Semantics assumptions: TWO VALUES: there are only two logical values: truth (T) and false (F), and EXTENSIONALITY: the logical value of a formula depends only on a main connective and logical values of its subformulas. We define formally a classical semantics for L in terms of two factors: classical truth tables and a truth assignment. 1 We summarize now here the chapter 2 tables for L {¬ , ∪ , ∩ , ⇒} in one simplified table as fol lows. A B ¬ A ( A ∩ B ) ( A ∪ B ) ( A ⇒ B ) T T F T T T T F F F T F F T T F T T F F T F F T Observe that The first row of the above table reads: For any formulas A,B , if the logical value of A = T and B = T , then logical values of ¬ A = T , ( A ∩ B ) = T , ( A ∪ B ) = T and ( A ⇒ B ) = T . We read and write the other rows in a similar manner. 2 Our table indicates that the logical value of of propositional connectives depends only on the logical values of its factors; i.e. it is independent of the formulas A,B . EXTENSIONAL CONNECTIVES : The log ical value of a given connective depend only of the logical values of its factors. We write now the last table as the following equations. ¬ T = F, ¬ F = T ; ( T ∩ T ) = T, ( T ∩ F ) = F, ( F ∩ T ) = F, ( F ∩ F ) = F ; ( T ∪ T ) = T, ( T ∪ F ) = T, ( F ∪ T ) = T, ( F ∪ F ) = F ; ( T ⇒ T ) = T, ( T ⇒ F ) = F, ( F ⇒ T ) = T, ( F ⇒ F ) = T. 3 Observe now that the above equations de scribe a set of unary and binary operations (functions) defined on a set { T,F } and a set { T,F } × { T,F } , respectively. Negation ¬ is a function: ¬ : { T,F } → { T,F } , such that ¬ T = F, ¬ F = T . Conjunction ∩ is a function: ∩ : { T,F } × { T,F } → { T,F } , such that ( T ∩ T ) = T, ( T ∩ F ) = F, ( F ∩ T ) = F, ( F ∩ F ) = F. 4 Dissjunction ∪ is a function: ∪ : { T,F } × { T,F } → { T,F } , such that ( T ∪ T ) = T, ( T ∪ F ) = T, ( F ∪ T ) = T, ( F ∪ F ) = F. Implication ⇒ is a function: ⇒ : { T,F } × { T,F } → { T,F } , such that ( T ⇒ T ) = T, ( T ⇒ F ) = F, ( F ⇒ T ) = T, ( F ⇒ F ) = T....
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Bachmair,L
 Computer Science

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