I've really confused myself here: $$ P(x) = ``\text{x has a tail}" $$ How would I write: $$ ``\text{Not everything has a tail}" $$
would it: $$ ¬∀x P(x) $$
be correct?
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It isn't the same thing. The negation is there is something that doesn't have a tail. – André Nicolas Jul 10 '15 at 05:51
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I do not want to negate "Not everything has a tail" I'm confused as to how to write it out. – Feath Jul 10 '15 at 05:57
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1That should be $\lnot\forall$, not $\lnot\exists$. (Remember that $\exists$ means "exists.") – Akiva Weinberger Jul 10 '15 at 06:04
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"Everything doesn't have a tail" is not logically equivalent to ""Not everything has a tail."
Use instead $\lnot \forall xP(x)$, or equivalently $\exists x \lnot P(x)$, there is something that doesn't have a tail.
André Nicolas
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so this is the way to write out "Not everything has a tail" in proper notation? – Feath Jul 10 '15 at 06:03
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I'm confused as to how you know you have the negate the entire thing and not just the "everything" – Feath Jul 10 '15 at 06:06
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I modified the answer to give you two alternatives. The simplest is $\lnot\forall xP(x)$. Then I gave an equivalent assertion using the existential quantifier. I suspect that one, for no good reason, may be preferred. – André Nicolas Jul 10 '15 at 06:06
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"Not everything" has a certain property means there is something that doesn't have the property. – André Nicolas Jul 10 '15 at 06:08