The Logic of Quantified Statements

Definition of Predicate Predicate is a sentence that  contains finite number of variables;  becomes a statement when specific values are substituted for the variables. Ex:  let predicate P(x,y) be “x>2 and x+y=8”  when x=5 and y=3, P(5,3) is “5>2 and 5+3=8” Domain of a predicate variable is the set of all possible values of the variable. Ex (cont.): D(x)= ; D(y)=R

Truth Set of a Predicate If P(x) is a predicate and x has domain D, then the truth set of P(x) is all xD such that P(x) is true. (denoted {xD | P(x)} ) Ex: P(x) is “5<x<9” and D(x)=Z. Then {xD | P(x)} ={6, 7, 8}

Universal Statement and Quantifier Let P(x) be “x should take Math306”; D={Math majors} be the domain of x. Then “all Math majors take Math306” is denoted xD, P(x) and is called universal statement.  is called universal quantifier; expressions for : “for all”, “for arbitrary”, “for any”, “for each”.

Truth and Falsity of Universal Statements Universal statement “xD, P(x)”  is true iff P(x) is true for every x in D;  is false iff P(x) is false for at least one x. (that x is called counterexample) Ex: 1) Let D be the set of even integers. “xD yD, x+y is even” is true. 2) Let D be the set of all NBA players. “xD, x has a college degree” is false. Counterexample: Kobe Bryant.

Existential Statement and Quantifier Let P(x) be “x(x+2)=24”; D =Z be the domain of x. Then ”there is an integer x such that x(x+2)=24” is denoted “xD, P(x)” and is called existential statement.  is called existential quantifier; expressions for  : “there exists”, “there is a”, “there is at least one”, “we can find a”.

Truth and Falsity of Existential Statements Existential statement “ xD, P(x)”  is true iff P(x) is true for at least one x in D;  is false iff P(x) is false for all x in D. Ex: 1) Let D be the set of rational numbers. “ xD, ” is true. 2) Let D = Z. “ xD, x(x-1)(x-2)(x-3)<0” is false. Why? Hint: Use proof by division into cases.

Negations of Quantified Statements The negation of universal statement “xD, P(x)” is the existential statement “xD, ~P(x)” Example: The negation of “All NBA players have college degree” is “There is a NBA player who doesn’t have college degree”.

Negations of Quantified Statements The negation of existential statement “ xD, P(x)” is the universal statement “ xD, ~P(x)” Example: The negation of “ x Z such that x(x+1)<0” is “ x Z, x(x+1) ≥ 0”.

Statements containing multiple quantifiers Ex: 1) xR, yZ such that |x-y|<1. 2) For any building x in the city there is a fire-station y such that the distance between x and y is at most 2 miles. 3) xZ such that y[3,5], x<y. 4) There is a student who solved all the problems of the exam correctly.

Truth values of multiply quantified statements Ex:  Students = {Joe, Ann, Bob, Dave}  2 groups of languages: Asian languages={Japanese,Chinese,Korean}; European languages={French, German, Italian, Spanish}.  Joe speaks Italian and French; Ann speaks German, French and Japanese; Bob speaks Spanish, Italian and Chinese; Dave speaks Japanese and Korean.

Truth values of multiply quantified statements Ex(cont.): Determine truth values of the following statements: 1)  a student S s.t.  language L, S speaks L. 2)  a student S s.t. for  language group G  L in G s.t. S speaks L. 3)  a language group G s.t. for  student S  L in G s.t. S speaks L.

Negating multiply quantified statements Example: The negation of “for xR, yR s.t. “ is logically equivalent to “xR s.t. for yR, “. Generally, the negation of x, y s.t. P(x,y) x s.t. y, ~P(x,y)

Negating multiply quantified statements Example: The negation of “ xR s.t. yZ, x>y“ is logically equivalent to “xR yZ s.t. x≤y“. Generally, the negation of x s.t. y, P(x,y) x y s.t. ~P(x,y)

The Relation among , , Λ, ν Let Q(x) be a predicate; D={x_1, x_2, …, x_n} be the domain of x. Then  xD, Q(x) is logically equivalent to Q(x_1) Λ Q(x_2) Λ … Λ Q(x_n) ;  xD, Q(x) is logically equivalent to Q(x_1) ν Q(x_2) ν … ν Q(x_n) .

Universal Conditional Statement Definition:  x, if P(x) then Q(x) . Example:  undergrad S, if S takes CS300, then S has taken CS240. Negation of universal conditional statement:  x such that P(x) and ~Q(x) Ex(cont.):  undergrad who takes CS300 but hasn’t taken CS240.

Variations of universal conditional statements Variations of xD, if P(x) then Q(x): Contrapositive: xD, if ~Q(x) then ~P(x); Converse: xD, if Q(x) then P(x); Inverse: xD, if ~P(x) then ~Q(x). The original statement is logically equivalent to its contrapositive. Converse is logically equivalent to inverse.

Necessary and Sufficient Conditions “x, P(x) is a sufficient condition for Q(x)” means “x, if P(x) then Q(x)” “x, P(x) is a necessary condition for Q(x)” means “x, if Q(x) then P(x)”

Validity of Arguments with Quantified Statements Argument form is valid means that for any substitution of the predicates, if the premises are true, then the conclusion is also true.

Valid Argument Forms: Universal Instantiation x D, P(x); aD; P(a). If some property is true for everything in a domain, then it is true for any particular thing in the domain.

Valid Argument Forms: Universal Instantiation Ex: 1) All Italians are good cooks; Tony is an Italian;  Tony is a good cook. 2) For x,y R, 74.5, 73.5 R 

Testing validity by diagrams Ex: All integers are rational numbers; 5 is an integer;  5 is a rational number. Rational numbers Integers 5

Testing validity by diagrams Ex: All logicians are mathematicians; John is not a mathematician;  John is not a logician. Mathematicians Logicians John

Testing validity by diagrams: Converse Error Ex: All Math majors are taking Math306; Bill is taking Math306;  Bill is a Math major. Math306 class Math majors Bill