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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,

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    @AndréNicolas , For my answers in b,c (above) I tried to change implication to the disjunction and changing 'existence' to 'for all' and I find it very obvious that I am wrong , As Always,Thanks for the help. – 2012-12-13

3 Answers 3

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(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.

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Assuming that by Vx you mean $\forall x$ (for all x) you are correct on the first one. The reason you use implication is that you are transcribing "all A are B" into "A(x) implies B(x)" which is an implication.

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It should be $\wedge$ in both b and c. It might be more intuitive to write (for example, b) $\exists x~ \neg (R(x) \to Q(x))$.