4slides - Chapter 4: Classical Propositional Semantics...

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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 sub-formulas. 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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4slides - Chapter 4: Classical Propositional Semantics...

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