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Consider the following first order logic statement

$I)\forall x\forall yP\left ( x,y \right )$

$II)\forall x\exists yP\left ( x,y \right )$

$III)\exists x\exists yP\left ( x,y \right )$

$III)\exists x\forall yP\left ( x,y \right )$

Which one of the following is true about $I,II,III,IV?$

$A)$ If $II)$ is true , then $III),IV)$ is true

$B)$  If $IV)$ is true , then $II),III)$ is true

$C)$ None of these

If I put logic then answer that I am getting $C)$ , but according to given answer $II)$ and $IV)$ has same power, So answer will be $I)$ Which one is correct?

雨が好きな人
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Srestha
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1 Answers1

2

Assuming that your second III) was supposed to be IV)...

(A) If II is true, then III and IV are true.

II says that for all $x$, there exists a $y$ such that $P(x,y)$. If this is true, then III is true because III states that there exists an $x$ and $y$ such that $P(x,y)$. Well, since II says that for all $x$ there is a certain $y$ that makes this true, there certainly does exist a pair of $x$ and $y$ such that $P(x,y)$. In fact, there are as many pairs as there are $x$s. So III is true in this case.

The truth of IV does not depend on the truth of II because IV states that there exists an $x$ such that for all $y$ $P(x,y)$. To see that II and IV are different types of statements, consider the following: ‘for all people, there is a day of the year that is their birthday’ and ‘there exists a person such that all the days of the year are their birthday’. Clearly, these are totally different statements. So IV is not necessarily true in this case.

(B) If IV is true, then II and III are true.

IV says that there exists an $x$ such that for all $y$, $P(x,y)$. The relationship between IV and II is the same as before, so II is not necessarily true in this case.

Once again, for III to be true we just need a single pairing of $x$ and $y$ that satisfy $P(x,y)$. So III is true because IV says that there is a single $x$ that can pair up with all $y$.

So, since (A) and (B) are both not true, the answer is (C)—none of these.

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