Chapter 9: Syntax and Semantics II : Logic & Proofs July 23, 2009 Karin Howe

Quantified Predicate Logic Universal Quantifier: "All" –Symbolization:  –Read as: "for all" Existential Quantifier: "Some" –Symbolization:  –Read as "there exists"

Scope We talk about the scope of a quantifier - the portion of the statement that the quantifier "applies" to. For example, in the formula: (  x)(B(x)  C(x)), all of the x's are within the scope of the universal quantifier. However, in the formula: (  x)B(x) & C(x), the final x (in the C(x)) is not under the scope of the universal quantifier (and thus that x is said to be "free").

Substitution Instances A substitution instance  [t/u] of a quantified formula (  u)  or (  u)  is the formula obtained by removing the quantifier, then replacing all bound occurrences of the variable u that were bound by the removed quantifier in the formula  by occurrences of the term t. Examples: –(  x)(B(x)  C(x)) B(a)  C(a) B(b)  C(b) –(  x)(B(x) & C(x)) B(a) & C(a) –(  x)(B(x)  (  y)(C(y)) (B(a)  (  y)(C(y)

Extending Interpretations Recall our definition from chapter 8 1.If  is a 0-place predicate letter, then  is true iff I(  ) = T. 2.If  is of the form  (x 1, …, x n ) where  is a n-place predicate letter (with n > 0), and x 1, …, x n are n terms, then  is true on I iff is in I(  ) Now add the following clauses: 3.If  is of the form (  u) , then  is true on I iff for each member a of the domain of discourse  is true on I[a/u], and false otherwise. 4.If  is of the form (  u) , then  is true on I iff there is at least one member a of the domain of discourse such that  is true on I[a/u,] and false otherwise.

Determining Validity/Invalidity in Predicate Logic (  x)(P(x) & Q(x)) (P(a) & Q(a))* * Constant introduced must be new to the branch (  x)(P(x)  Q(x)) (P(a)  Q(a))  (  x)(P(x) & Q(x)) (  x)  (P(x) & Q(x))  (  x)(P(x)  Q(x)) (  x)  (P(x)  Q(x))

Practice: Truth Trees 1.(  x)(K(x)  F(x)), K(j)  F(j) 2.(  x)[(A(x) & B(x))  C(x)], (  x)(A(x) & B(x))  (  x)(B(x)   C(x)) 3.(  x)[A(x) & (B(x) & C(x))], (  x)(B(x)  D(x))  (  x)(C(x)   D(x)) 4.  (  x)(A(x)  B(x)), (  x)(A(x)  C(x))   (  x)(  B(x) & C(x)) 5.  (  x)[(A(x) & B(x))  (C(x) & D(x))],  (  x)(A(x) & C(x))   (  x)(B(x) & D(x)) 6.(  x)[(A(x) & B(x))  (C(x) & D(x))], (  x)(A(x)  B(x)),  (  x)(A(x)  C(x)),   (  x)(A(x)  B(x))

Practice: Symbolization 1.All who love are blind. ~bumper sticker Dictionary: L(x) = x loves; B(x) = x is blind 2.HE WHO DIES WITH THE MOST TOYS IS DEAD ~bumper sticker Dictionary: D(x) = x is dead; T(x) = x dies with the most toys 3.What's good for M & M Enterprises is good for the country. ~Catch-22 Dictionary: E(x) = x is good for M & M Enterprises; C(x) = x is good for the country

1.The man who dies rich dies disgraced. ~Andrew Carnegie Dictionary: D(x) = x is dead; R(x) = x is rich; G(x) = x is disgraced 2.Those who deny freedom to others deserve it not for themselves. ~Abraham Lincoln Dictionary: O(x) = x denies freedom to others; T(x) = x deserves to be free 3.All that glitters is not gold. ~proverb Dictionary: L(x) = x glitters; G(x) = x is gold

1.Not all prostitutes are junkies ~newspaper Dictionary: P(x) = x is a prostitute; J(x) = x is a junkie 2.There are old pilots and there are bold pilots–but there are no old bold pilots. ~sign in an Air Force ready room Dictionary: O(x) = x is old; B(x) = x is bold; P(x) = x is a pilot 3. Vulcans never bluff ~Spock Dictionary: V(x) = x is a Vulcan B(x) = x is bluffing

1.He who hesitates is lost. ~proverb Dictionary: H(x) = x hesitates; L(x) = x is lost 2.With a name like Smuckers, it has to be good. ~advertisement Dictionary: S(x) = x is a product made by Smucker's; G(x) = x is a product that has to be good 3.There is an even prime. ~Mathematics text Dictionary: E(x) = x is even; P(x) = x is prime

Practice: Proofs Practice CPL Problems Lab #6