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)?
Best Answer
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.