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?

  • $\begingroup$ It isn't the same thing. The negation is there is something that doesn't have a tail. $\endgroup$ – André Nicolas Jul 10 '15 at 5:51
  • $\begingroup$ I do not want to negate "Not everything has a tail" I'm confused as to how to write it out. $\endgroup$ – Feath Jul 10 '15 at 5:57
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    $\begingroup$ That should be $\lnot\forall$, not $\lnot\exists$. (Remember that $\exists$ means "exists.") $\endgroup$ – Akiva Weinberger Jul 10 '15 at 6:04

"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.

  • $\begingroup$ so this is the way to write out "Not everything has a tail" in proper notation? $\endgroup$ – Feath Jul 10 '15 at 6:03
  • $\begingroup$ I'm confused as to how you know you have the negate the entire thing and not just the "everything" $\endgroup$ – Feath Jul 10 '15 at 6:06
  • $\begingroup$ 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. $\endgroup$ – André Nicolas Jul 10 '15 at 6:06
  • $\begingroup$ "Not everything" has a certain property means there is something that doesn't have the property. $\endgroup$ – André Nicolas Jul 10 '15 at 6:08

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