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I was reading about existential quantifier on Wikipedia whereby I came across this explanation for it's negation.

Generally, then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically,

¬∃x∈X P(x) = ∀x∈X ¬P(x)

A common error is stating "all persons are not married" (i.e., "there exists no person who is married"), when "not all persons are married" (i.e., "there exists a person who is not married") is intended.

¬∃x∈X P(x) = ∀x∈X ¬P(x) ≠ ¬∀x∈X P(x) = ∃x∈X ¬P(x)

Now, let X = {all persons}, p = x is married and e = ∃x∈X, p is true

The above example "not all persons are married" seems logically equivalent to "some persons are married", i.e., "e". Here, the negation of e seems to imply e itself. But that doesn't make any sense. How can e be logically equivalent to the negation of e?

Maybe I'm missing something or maybe I am not able to think of quantifiers and logical expression in a formal or rigorous manner. What is the correct way to think logically (no pun intended)?

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    $\begingroup$ You have to be careful about the word “some” in formal logic, which doesn’t carry the same implications we often associate with everyday talk. “Not all persons are married” is equivalent to “some persons are NOT married.” But the latter is not equivalent in formal logic to “some persons ARE married.” In other words “some X are Y” is not equivalent in formal logic to “some X are not Y,” even though in ordinary language we often draw that inference. $\endgroup$ Jun 8 at 18:51
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    $\begingroup$ "not all persons are married" is equivalent to "some persons are not married" and this does not seem equivalent to "some persons are married". Mathematical discourse precisely takes care of the ambiguities that could exist in informal speech. $\endgroup$
    – user65203
    Jun 8 at 18:53
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    $\begingroup$ In your case: "Not all people are married" is equivalent to $ ¬∀x∈X P(x) = ∃x∈X ¬P(x)$: Not all people are married is equivalent to saying "Some people are not married." I understand what you are suggesting, but logic focuses on only that which is logically implied, not what we know by common sense. $\endgroup$
    – amWhy
    Jun 8 at 18:53
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    $\begingroup$ The main problem is that $\lnot \forall x Q(x)$, which is true if we let the domain of x be "people", and $Q(x)$ mean "people with four eyes and a mouth on top of their head, and no nose. It is true that $\lnot \forall x, Q(x)$. But we can not logically say, therefore $\exists x Q(x)$. That is false, because there are no people with four eyes and a mouth on top of their head and no nose. $\endgroup$
    – amWhy
    Jun 8 at 19:02
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You've incorrectly negated the existential statement: "not all people are married" is equivalent to "some people are not married," not "some people are married." I suspect your confusion comes from two assumptions you're making which are unjustified, one about the term "some" in general and one about the specific nature of marriage. Specifically:

  • That "some" means "some but not all."

  • That there are some married people in the first place.

The first reflects a possible discrepancy between the natural-language meaning of "some" and its meaning as a translation of the precise quantifier "$\exists$." This is just something you have to move past (or avoid using the word "some" in this context - but recognize that others will use it). The second reflects the danger of implicitly allowing "outside knowledge" to creep into the logical analysis of a statement; this is something we always have to avoid doing.


Actually there's a second issue re: "some" as a translation for "$\exists$," which doesn't play a role in your specific question but is still worth noting: namely, that the latter doesn't specify whether only one or more than one example exists. So both "some $X$ is $Y$" and "some $X$ are $Y$" may feel a bit off. Again, this is just something we have to learn to work with: that insofar as we use "some" as a translation for "$\exists$," there will be discrepancies between its usage in the context of logic and its natural-language connotations.

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  • $\begingroup$ I think I was a little lost between the logical and colloquial or "natural" meaning of "∃". I do understand "some" implies "one" or "all" in formal logic. But as you stated in your second point, I got a little imprecise while using ∃ with a real world natural sounding sentence and that's how the error crept in. Thank you for clearing it up. $\endgroup$
    – J.S
    Jun 9 at 20:11
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“Not all persons are married” is logically equivalent (in standard logic) to “there exists a person who is not married”. On the other hand, “Some persons are married” is logically equivalent to “there exists a person who is married”. So these two statements are not negations of one another. Indeed, as long as there are two people in the universe, one of whom is married and one of whom isn’t, then both statements are simultaneously true.

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Rather than using the word "some ... are ..." for $\exists$, think about it with the language "there exists ... such that ...".

You see that:

$$\neg \exists x \in X: P(x) \iff \forall x \in X: \neg P (x)$$

can be expressed as:

"It is not the case that there exists an $x$ in $X$ that has $P$" means the same thing as "Every $x$ in $X$ does not have $P$."

That is:

"No $x$ have $P$" means the same as "All $x$ do not have $P$".

Now:

$$\neg \forall x \in X: P(x) \iff \exists x \in X: \neg P(x)$$

can be expressed as:

"It is not the case that all $x$ in $X$ have $P$" means the same as "There exists (at least one) $x$ in $X$ which does not have $P$"

That is:

"Not all $x$ have $P$" means the same as "At least one $x$ does not have $P$."

But if you say "It is not the case that all $x$ have $P$" is completely consistent with the statement "No $x$ have $P$"

If I were to say: "All people are over $4$ metres tall", you could instantly disprove it by pointing to yourself, and say, "No they are not all over $3$ metres tall, look at me, I'm under $3$ metres tall." (Probably).

But in fact "all people are not over $4$ metres tall."

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  • $\begingroup$ "There exists" sounds indeed more precise in defining logical statements than "for some" which seems to arise confusion. Maybe I'll have to get more familiar with mathematical logic and then it won't be much of a problem understanding these terms. Thanks, for it was a really detailed explanation. $\endgroup$
    – J.S
    Jun 9 at 20:24
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In ordinary language, some usually implies some not, otherwise it would be pointless to mention the fact. Nobody would say "some giants are tall".

That restriction does not exist in logics. Some can be all (but not none).

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  • $\begingroup$ There's also the issue that in natural language "some" often means "more than one." Of course that's not relevant to the specific error in the OP, but it's worth noting to drive the point home. $\endgroup$ Jun 8 at 19:06

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