Question

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. c) Some excuses are not clear explanations. d) Does (c) follow from (a) and (b)?

Answer 1

Answer:

a) $\forall x,P(x) \to Q(x)$

b) $\exists x:R(x) \land(\sim Q(x))$

c)$\exists x:R(x) \land(\sim P(x))$

d) Yes, it does.

Step-by-step explanation:

P(x) = “x is a clear explanation”

Q(x) = “x is satisfactory”

R(x) = “ x is a excuse”

We will be denoting “such that” as “:” and “not” as “~”

a) All clear explanations are satisfactory.

$\forall x,P(x) \to Q(x)$

b) Some excuses are unsatisfactory.  

$\exists x:R(x) \land(\sim Q(x))$

c) Some excuses are not clear explanations.

$\exists x:R(x) \land(\sim P(x))$

d) Does (c) follow from (a) and (b)?

Yes, it does.

(a) can be expressed as “If x is a clear explanation then x is satisfactory”

Recall that  

$P\rightarrow Q$ is equivalent to $\sim Q \rightarrow \sim P$

So (a) can be paraphrased as

“If x is not satisfactory then x is not a clear explanation”

Joining b) and a) we get

“Some excuses are not satisfactory and if a excuse is not satisfactory then the excuse is not a clear explanation”

From here we deduce that,

“Some excuses are not clear explanations”