Logic - Logically implies question

Question

$\forall x(A(x) \rightarrow B(x))$ logically implies $\exists x(A(x) \land B(x))$

Is the above statement true or false? I have no clue on how to start figuring this out. Can someone help me please?

No. If $A(x)$ is false, then $A(x)\rightarrow B$ for any $B$. — Thomas Andrews, Jan 26 '15 at 03:30
Then I suggest you review propositional logic. If $p$ is false, then $p\rightarrow q$ is always true, for any $q$. — Thomas Andrews, Jan 26 '15 at 03:32
Yes, i understand that, but what i am asking is why you said my original question is false because of the fact that it is always true if A(x) is false — Torched90, Jan 26 '15 at 03:36
Pick some $A(x)$ so that $\forall x(\lnot A(x))$. Then $\forall x(A(x)\rightarrow B(x))$ but it is not true that $\exists x(A(x)\land B(x))$. — Thomas Andrews, Jan 26 '15 at 03:38

score 1 · Answer 1 · answered Jan 26 '15 at 03:33

I feel like vacuous truths would disprove this. For example, I could let $A(x)$ be "$x\in \emptyset$" and $B(x)$ be "$x$ is a seven-headed purple fire-breathing dragon." The first statement of yours ($\forall x(A(x)\to B(x))$ is true (since there are no members of the empty set). However, the second statement is false since there does not exist $x$ such that $A(x)$ is true.

score 0 · Answer 2 · answered Jan 26 '15 at 04:15

Torched90, you seem to be asking a lot of questions concerning quantifiers (your question history suggests this anyway) and their properties concerning relations such as $\to, \lor$, and $\land$. I would like to give you six claims that you can explore for yourself concerning their truthfulness (truth or falseness will appear in an enumerated list below the properties).

$\forall x(P(x)\to Q(x))$ implies $\exists x(P(x)\land Q(x))$
$\forall x(P(x)\to Q(x))$ and $\forall xP(x)\to \forall xQ(x)$ have the same truth value.
$\forall x(P(x)\land Q(x))$ and $\forall xP(x)\land \forall xQ(x)$ have the same truth value.
$\exists x(P(x)\lor Q(x))$ and $\exists xP(x)\lor \exists xQ(x)$ have the same truth value.
$\forall xP(x)\lor \forall xQ(x)$ and $\forall x(P(x)\lor Q(x))$ are logically equivalent.
$\exists xP(x)\land \exists xQ(x)$ and $\exists x(P(x)\land Q(x))$ are logically equivalent.

Are these claims true or false? Try to verify for yourself before looking at the list of answers below (or before asking another very similar question on MSE).

False
False
True
True
False
False

If you are still stuck on understanding why these claims are true or false, then I would suggest reviewing or reading up on some basic propositional logic. The book here has a gentle introduction to logic if you do not know of another book.

Logic - Logically implies question

2 Answers2