The negation of "Everything is a goat" is not "Nothing is a goat". Asentence and its negation must have opposite truth values; that is, ifone is true, the other is false. A sentence and its negation cannotboth be true and they cannot both be false. But, as I think yourealize, "Everything is a goat" and "Nothing is a goat" canboth be false: if there are some things that aren't goats and somethings that are, then the two claims will be false. So this shows that"Nothing is a goat" is not the negation of "Everything is a goat".Might the negation of "Everything is a goat" be "Something is a goat"?No, for both these claims could be true: imagine that there is at least onegoat and furthermore that everything is a goat.
(All these errors are facilitated by the false assumption that nouns like "nothing" and "everyone" function like "Harry" or "the animal in the shed" do. For more on this error, see Question 49.)
What then isthe negation of "Everything is a goat"? It's "Something is not a goat."If this claim is true then "Everything is a goat" is false. And if it'sfalse — false that there's at least one thing that isn't a goat — thenthat must be because everything is a goat. Finally, what is the negation of"Nothing is a goat"? It's "Something is a goat", for these sentencesalways have different truth values, that is, if one is true the otheris false.