true if x > y Section 1.3 Quantifier, Predicates, and Validity e.g.: greater (x, y) = false otherwise A predicate, in general, has the form P(x1, x2, …, xn) true if x is a prime number prime(x) = which maps from x1, x2, …, xn to the values false otherwise true and false. where P is the name of the predicate, true if x + y = z xi are variables or parameters sum(x, y, z) = n is the degree of the predicate. false otherwise CS130 Young 1 CS130 Young 2 2 quantifiers: Proposition is simply a statement that is either Universal quantifier (for all), true or false, has no variables involved. Existential quantifier (exists) But predicates can take variables, and once quantifiers are applied to families we replace the variable by a constant, it of propositions {P(x): x }, becomes a proposition. where the nonempty set is called universe or domain. CS130 Young 3 CS130 Young 4 (x) P(x) - is true if P(x) is true for every x in e.g.: - otherwise, false. (x) P(x) - is true if P(x) is true for at least one = all the books in your local library. x in - is false if P(x) is false for every x in P(x) = the property that x has a red cover P(x) here is called a predicate. then (x)p(x) = every book in your local library has a red cover. Predicate is not completely defined unless the domain is defined. false. CS130 Young 5 CS130 Young 6 1 In general, a predicate p(x1, x2, …, xn), n > 0, is a mapping from x1, x2, …, xn to the values true and false. Where n is the degree of the predicate. The variable x in P(x) is called a free variable p is the predicate name of the predicate, but (x)P(x), where x is x s are parameters or variables. i called a bound variable(it is bound by the A predicate is a function that produces a proposition quantifier ). whenever we feed it a member of the universe(or domain). Predicate Logic is a generalization of propositional logic. CS130 Young 7 CS130 Young 8 Translate English sentences to predicates: eg: U = all reports e.g.: U = all persons A(x): x is authorized P(x): x is a philosopher T(x): x is trustworthy Q(x): x is logical F(x): x is false R(x): x is obstinate Some unauthorized reports are false. All philosophers are logical. (x)[P(x) Q(x)] (x)[A(x)F(x)] An illogical person is always obstinate. All authorized reports are trustworthy. (x)[Q(x)R(x)] (x)[A(x)T(x)] Some obstinate persons are not philosophers. Some false reports are not trustworthy. (x)[R(x) P(x)] (x)[F(x) T(x) ] CS130 Young 9 CS130 Young 10 eg: U = all birds P(x): x is a peacock T(x); x is proud of its tail S(x); x can sing In general, Some birds that are proud of their tails cannot sing. (x)S(x) (x)[S(x)] (x)[ T(x) S(x) ] (x)S(x) (x)[S(x)] No birds, except peacocks, are proud of their tails. (x)(S(x) )(x)[S(x)] (x)[ P(x)T(x) ] (x)(S(x)) (x)[S(x)] (x)[ P(x)T(x)] (x)[ T(x)P(x) ] Some peacocks cannot sing. (x)[ P(x) S(x) ] CS130 Young 11 CS130 Young 12 2 e.g. Exercise 1.3 U = the whole world B(x) is “x is a bee” F(x) is “x is a flower” Def. A wff with predicates involved is called L(x, y) is “x loves y” predicate wff. a All bees love all flowers: (x)( B(x)(y)[F(y)L(x, y)] ) or (x)( y)[( B(x)F(y) )L(x, y)] c All bees love some flowers: ( A wff with only propositional variables, (x)( B(x)(y)[F(y) L(x, y)] cannot be (x)(y)[ [B(x)F(y) ] L(x, y) ] logical operators, and the grouping symbols is called propositional wff. d Every bee hates only flowers. (x)[ B(x) (y)[L(x, y)F(y)] ] f Every bee loves only flowers. (x)[ B(x) (y)[L(x, y) F(y)] ] CS130 Young 13 CS130 Young 14 A predicate wff is valid if it is true for all interpretations. e.g. (x)P(x) (x)P(x) is valid no matter what the property P(x) describes or what U is. e.g. (x)P(x) P(a) is valid, where a is a particular member of the domain. e.g. (x)[P(x) Q(x)] (x)P(x) (x)Q(x) if both P and Q are true elements in domain then P is true elements and Q is true elements and vice versa. e.g. P(x) [Q(x) P(x)] is valid (even w/o quantifiers) if P(x) is false then the wff is true or if P(x) is true, then [Q(x) P(x)] is true and the wff is true. CS130 Young 15 3.