Introduction to Logic
 Tools
for
Thought 		 
Exercise 7.3 - Relational Evaluation

Consider a language with object constants a and b and relation constants p and q where p has arity 1 and q has arity 2. The following is a truth assignment for this language.

p(a)i = 1
p(b)i = 0
q(a,a)i = 0
q(a,b)i = 1
q(b,a)i = 1
q(b,b)i = 0

Say whether each of the following sentences is true or false with this truth assignment.

a. ∀x.(p(x) ⇒ q(x,x))
b. ∀x.∃y.q(x,y)
c. ∃y.∀x.q(x,y)
d. ∀x.(p(x) ⇒ ∃y.q(x,y))
e. ∀x.p(x) ⇒ ∃y.q(y,y)