1 10 points each Let Px Qx and Rx be the statements x is a c

1) (10 points each)

Let P(x), Q(x), and R(x) be the statements x is a clear explanation, x is satisfactory, and x is an excuse, respectively. Suppose that the domain for x consists of all English text.

Express each of these statements using quantifiers, logical connectives, and

P(x), Q(x), and R(x).

(a)All clear explanations are satisfactory.

(b)Some excuses are unsatisfactory.

2) (10 points each)

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers.

(a)?x(|x| > 0)

(b)?x?y(x = 1/y)

1) (10 points each)

What rule of inference is used in each of these arguments? See Rosen 7^th ed, page 72.

(a) Kangaroos live in Australia and are marsupials. Therefore, kangaroos are marsupials. (b) It is hotter than 100 degrees today or the pollution is dangerous. It is less than 100 degrees outside today. Therefore, the pollution is dangerous.

(c)Steve will work at a computer company this summer. Therefore, this summer Steve will work at a computer company or he will be a beach bum.

(d)If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, then I will understand the material.

1) (10 points each)

(a)For the following set of premises, what relevant conclusion(s) can be drawn? Explain the rules of inference used to obtain each conclusion from the premises:

I am either dreaming or hallucinating. I am not dreaming. If I am hallucinating, I see elephants running down the road.

(b) Use rules of inference to show the following (state all rules used),

if ?x(P(x) ? (Q(x) ? S(x))) and ?x(P(x) ? R(x)) are true, then ?x(R(x) ? S(x)) is true.

Solution

P(x) : x is a clear explanation

Q(x): x is satisfactory

R(x): x is an excuse

For all x, THERE EXISTS a satisfactory explanation which can be an excuse