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lecture notes10 - Chapter1,PartII:PredicateLogic...

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Chapter 1, Part II: Predicate Logic With Question/Answer Animations
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Summary Predicate Logic (First‐Order Logic (FOL), Predicate Calculus) The Language of Quantifiers Logical Equivalences Nested Quantifiers Translation from Predicate Logic to English Translation from English to Predicate Logic
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Section 1.4
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Sec*on Summary Predicates Variables Quantifiers Universal Quantifier Existential Quantifier Negating Quantifiers De Morgan’s Laws for Quantifiers Translating English to Logic Logic Programming ( optional )
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Proposi*onal 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. Later we’ll see how to draw inferences.
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Introducing Predicate Logic Predicate logic uses the following new features: Variables: x , y , z Predicates: P ( x ), M ( x ) Quantifiers ( to be covered in a few slides ): Propositional functions are a generalization of propositions. They contain variables and a predicate, e.g., P ( x ) Variables can be replaced by elements from their domain .
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Proposi*onal Func*ons Propositional functions become propositions (and have truth values) when their variables are each replaced by a value from the domain (or bound by a quantifier, as we will see later). 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. Often the domain is denoted by U . So in this example U is the integers.
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Examples of Proposi*onal Func*ons 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: Q( 2,-1,3 ) Solution: T Q( 3,4,7 ) Solution: F Q( x , 3 , z ) Solution: Not a Proposition
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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 P( 3 ) P(-1) Solution : T Expressions with variables are not propositions and therefore do not have truth values. For example, P( 3 ) P( y ) P( x ) P( y ) When used with quantifiers (to be introduced next), these expressions (propositional functions) become propositions.
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Quan*fiers 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 ).
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