Let $P(x),Q(x),R(x)$ be the statements $x$ is a clear explanation,$x$ is satisfactory,$x$ is an excuse,respectively. Suppose that the domain for $x$ consists of all the English text. Express each of these statements using quantifiers, logical connectives and $P(x),Q(x),R(x)$.
a. All clear explanations are satisfactory.
b. Some excuses are unsatisfactory.
c. Some excuses are not clear explanations
Please Corret Me
a. $\forall x ~ (P(x) \to Q(x))$
b. $\exists x ~ (R(x) \to \neg Q(x))$
c. $\exists x ~ (R(x) \to \neg P(x))$
I am not completely sure why I have used implication instead of a conjunction, I am wondering can someone explain,
Best Answer
(b) and (c) are wrong. Restricted existentials need conjunctions. 'Some $A$s are $B$s" says that something which is an $A$ is also a $B$, so $\exists x(Ax \land Bx)$.
$\exists x(Ax \to Bx)$ is true if there is something which satisfies the condition $Ax \to Bx$, and anything that doesn't satisfy $A$ will make the antecedent of the condition false and hence make the condition true. So $\exists x(Ax \to Bx)$ is true if something isn't $A$, which isn't what you want at all.
Any elementary logic text will explain how to express restricted quantifications and stop you making this very elementary mistake in translation -- e.g. look for Paul Teller's excellent Primer, now freely available online.