For each wff find an interpretation in which it is true and

For each wff, find an interpretation in which it is true and one in which it is false.

Solution

a)

(x) ( [A(x)B(x) ][A(x)B(x)]’)

Domain = all dinosaurs

True:A(x) = dinosaur had two legs

       B(x) = dinosaur had four legs

All dinosaurs had two legs or four legs, and no dinosaurs had both four legs and two legs.

False:A(x) = dinosaur eats meat

B(x) = dinosaur eats plants

All dinosaurs ate meat or ate plants, and no dinosaurs ate both meat and plants.

(Ornitholestes was believed to be an omnivore.)

b) (x) (y) [p(x,y) --> p(y,x)]

    x is a boy.   y is a girl

True:

   p(x,y) = boy x is in a relationship with girl y

all boys x is in a relationship with all girls y also implies all girls y is in a relationship with all boys x.

False:

p(x,y) = boy x admires girl y.

all boys x admires girls y but all girls y may not admire boys x.

c)(x)[P(x)-->(y)Q(x,y) ]

True:

The domain is all positive integers.

P(x) =“x > 0”

Q(x,y) =“y / x is a positive integer”

False:

The domain is all positive integers.

P(x) =“x = 0”

Q(x,y) =y / x is a positive integer

d)

(x)[A(x)(y)B(x,y) ]

True:

The domain is all positive even integers.

A(x) =“x < 3”

B(x,y) =“y / x is a positive integer”

False:

The domain is all positive even integers.

A(x) =“x > 3”

B(x,y) =y / x is a positive integer