$\forall x(Px \rightarrow Qx)\stackrel{?}{\equiv}\neg\exists x(Px \rightarrow \neg Qx)\stackrel{?}{\equiv}\neg\exists x(Px \wedge\neg Qx)$

Question

(1) The statement $\forall x(P(x) \rightarrow Q(x))$ reads "For all $x$, if $P(x)$, then $Q(x)$".

(2) The statement $\neg\exists x(P(x) \rightarrow \neg Q(x))$ reads "There does not exist an $x$ such that if $P(x)$, then not $Q(x)$".

(3) The statement $\neg\exists x(P(x) \wedge \neg Q(x))$ reads "There does not exist an $x$ such that $P(x)$ and not $Q(x)$".

I know from De Morgan's Law that statements $1$ and $3$ are logically equivalent, and know that statement $2$ is inequivalent. However, all three statements seem to me to be saying the saying thing!

Could someone explain the difference in meaning between statement $2$ and the other statements, perhaps with an example?

Answer 1

We can work with the different equivalent transformations of the formulas.

We agree that 1 and 3 are equivalent and 3 says that "there is no object $a$ such that $P(a)$ and $\lnot Q(a)$".

But this is consistent with: "for some $b$, $\lnot P(b)$ and $Q(b)$".

Consider 2 now: it is equivalent to $¬∃x(¬P(x) \lor ¬Q(x))$ and to $∀x(P(x) ∧ Q(x))$

But this is not consistent with "for some $b$, $\lnot P(b)$ and $Q(b)$".

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Example: 1 may be "for every x, if x is Male, than it is Human" that is the same as "there is no x that is Male and not Human", which is True.

Thus, 2 will be "there is no x such that if x is Male, than x is not Human", which is False because it amounts to "for every x, x is a Male and it is Human" (there are Women).

Answer 2

$\forall x(P(x) \rightarrow Q(x))\tag1$

Every Penguin is Queer.

$\neg\exists x(P(x) \wedge \neg Q(x))\tag3$

No non-Queer Penguin exists.

$$\neg\exists x(P(x) \rightarrow \neg Q(x))\tag2$$ reads "There does not exist an $x$ such that if $P(x)$, then not $Q(x)$".

Since existentially quantified conditionals are easy to misread and confusing to accurately parse, we transform what you wrote to the following equivalent statements:

• Such a thing doesn't exist: it is either not a Penguin or not Queer
• This is false of everything: it is either not a Penguin or not Queer
• Everything is a Queer Penguin.

This shows that sentence (2) is logically stronger than sentence (1), which is equivalent to sentence (3).