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Chapter1p2(1) - The Foundations Logic and Proofs Chapter 1...

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The Foundations: Logic and Proofs Chapter 1, Part II: Predicate Logic
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Predicates and Quantifiers Section 1.4
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Propositional Logic Not Enough If we have: “All men are mortal.” “Socrates is a man.” Does it follow that “Socrates is mortal?” Can’t be represented in propositional logic. Need a language that talks about objects, their properties, and their relations for logic reasoning.
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Predicate Logic: The Socrates Example Introduce the propositional functions Man(x) denoting “ x is a man” and Mortal(x) denoting “ x is mortal.” Specify the domain as all people. The two premises are: Try to reach the conclusion is:
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Introducing Predicate Logic Predicate logic uses the following new features: Variables: x , y , z Predicates ( Propositional functions ): Man ( x ),… Quantifiers :  (for every),  (there exists) Propositional functions are a generalization of propositions. They contain variables and a predicate, e.g., Man ( x ) Variables can be replaced by elements from their domain .
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Propositional Functions Propositional functions become propositions and have truth values when their variables are each (i) replaced by a value from the domain or (ii) bound by a quantifier. The statement P(x) is said to be the value of the propositional function P at x . For example, let P(x) denote x > 0 and the domain be the integers. Then: P(- 3 ) is false. P( 0 ) is false. P( 3 ) is true.
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Examples of Propositional Functions Let “ x + y = z” be denoted by R ( x, y, z ) and U (for all three variables) be the integers. Find these truth values: R( ,- 2 1 , 5 ) Solution: F R( , , 3 4 7 ) Solution: T R( x , 3 , z ) Solution: Not a Proposition Now let x - y = z” be denoted by Q ( x , y , z ), with U as the integers. Find these truth values:
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Compound Expressions Connectives from propositional logic carry over to predicate logic. If P(x) denotes x > ,” 0 find these truth values: P( 3 ) ∨ (- ) P 1 Solution : T P( 3 ) ∧ (- ) P 1 Solution : F P( 3 ) → (- ) P 1 Solution : F Expressions with variables are not propositions and therefore do not have truth values. For example, P( 3 ) ∧ ( P y ) P( x ) → ( P y )
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Quantifiers We need quantifiers to express the meaning of English words including all and some : “All men are Mortal.” “Some cats do not have fur.” The two most important quantifiers are: Universal Quantifier , For all,” symbol: Existential Quantifier , “There exists,” symbol: We write as in  x P ( x ) and  x P ( x ). x P ( x ) asserts P ( x ) is true for every x in the domain . Charles Peirce (1839-1914)
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Universal Quantifier x P ( x ) is read as For all x , P( x )” or “For every x , P( x )” Examples : 1) If P(x) denotes x > 0 and the domain U for x is , the integers then x P ( x ) is false.
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