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# chapter4 - CHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS 1...

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CHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS 1 Language There are several propositional languages that are routinely called classical propositional logic languages. It is due to the functional dependency of classi- cal connectives discussed briefly in chapter 2. They all share the same logical meaning, called semantics. They also define the same set of universally true formulas called tautologies. We adopt here as classical propositional language the language L with the full set of connectives CON = , , , , , ⇔} i.e. the language L , , , , , ⇔} . As the choice of the set of connectives is now fixed, we will use the symbol L to denote the language L , , , , } . In previous chapters we have already established how we read and what are the natural language names of the connectives of L . For example, we read the symbol ¬ as ”not”, ”not true” and its name is negation . We read the formula ( A B ) as ” if A, then B”, ”A implies B”, or ”from the fact that A we deduce B” , and the name of the connective is implication . We refer to the established reading of propositional connectives and formulas involving them as a natural language meaning . Propositional logic defines and studies their a logical meaning called a se- mantics of the language L . We define here a classical semantics , some other semantics are defined in next chapter (Chapter 5). 1

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2 Classical Semantics, Satisfaction We based the classical logic, i.e. classical semantics on the following two assumptions. TWO VALUES: there are only two logical values. We denote them T (for true) and F (for false). Other common notations are 1 , > for true and 0 , for false. EXTENSIONALITY: the logical value of a formula depends only on a main connective and logical values of its sub-formulas. We define a classical semantics for L in terms of two factors: classical truth tables (reflexes the extensionality of connectives) and a truth assignment. In Chapter 2 we provided a motivation for the notion of classical logical connectives and introduced their informal definitions. We summarize here the truth tables for propositional connectives defined in chapter 2 in the following one table. A B ¬ A ( A B ) ( A B ) ( A B ) ( A B ) T T F T T T T T F F F T F F F T T F T T F F F T F F T T (1) The first truth values row of the above table 2 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. The table 2 indicates that the logical value of of propositional formulas de- pends on the logical values of its factors; i.e. fulfils the condition of extension- ality. Moreover, it shows 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 . It gives us the following important property of our proposi- tional connectives.
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chapter4 - CHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS 1...

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