Suppose a statement $P$ is true for all values of $n$. Would the converse of this be
I am inclined to believe that the former is the correct answer, but that would mean that "no" falls under the category of "some", which does not seem right. On the other hand, if the latter were the correct answer, then "some" is completely ignored. Hence I am at a loss!
*Note: because within predicate logic with quantifiers, the term "converse" is very ambiguous, I will assume, unless told otherwise by Trogdor, that what seems to be meant is "negation".
The negation of "$\forall n, P(n)$" is "$\exists n$ such that $\lnot P(n).$"
In other words, the negation of your statement is "There exist(s) $n$ such that it is not the case that $P(n)$ holds" or "The statement is not true for one or more values of n."
As the other answer already said, "converse" is a nebulous term when one is not talking about a sentence of the form "$A \to B$". However, one sees in mathematical writings that some extend the notion of "converse" in the natural way to sentences of the form "$\forall x \in S\ ( A(x) \to B(x) )$", where the converse is usually understood as "$\forall x \in S\ ( B(x) \to A(x) )$". It is in any case better to spell out what one means instead of just using the word "converse", so as to avoid any misunderstanding.
