Here's a question I got wrong on a HW assignment recently, which asked students to negate the given statement and assign that negation a truth value.
Q: There are exactly 3 points on every line. my answer: Some lines have more than 3 points. (true)
explanation: I took this to be a universally quantified statement:
for every line l in the set of all lines L, if p(l) gives the number of points incident with l, p(l) = 3
I was marked wrong on the negation because I didn't also state that lines could have less than 3 points. Assuming that's correct, in this case of universally quantified negation it is not sufficient to state that p(l) does not equal 3 for at least one line l.
here's what my teacher wrote when I asked for a clarification:
Yes, your negation is half of the correct negation. But there are two parts to it. Recall in class there are two statements for the negation of Playfair's version of the parallel postulate. You need them both there to have a complete negation of the original statement.
This partly makes sense to me because the negation of my answer doesn't yield the original question. But it also doesn't because my answer did (I think) negate the statement, and did so in a way that is consistent with most everything I have read about how to negate logic statements: to negate ~(P -> Q) it is enough to claim at least one case where P -> ~Q.
Any clarification would be much appreciated!