I asked this question, caused by a confusion that I was able to crystallize in the comment section of ryang's answer.
What is negation? One could define it like this: $P \oplus \neg P$
That is not sufficient, however. There are many things that are of opposite truth-values. Take the proposition "all men are mortal". The proposition "bears have wings" has the opposite truth-value, but it isn't its negation. If this was all a negation was, the flipping of a truth value, then $\neg P$ for all true propositions could be the same, untrue proposition, and vice versa.
So, what else goes into a negation? They have to use the same terms, perhaps? Given that the aformentioned examples use different terms (man $\neq$ bear, mortal $\neq$ have wings), they cannot be negations of each other. So, it makes sense that a negation of a proposition uses all of the same terms, though the logical symbols will differ. However, I can construct many different propositions that contain all of the same terms from the negated propositon, that are of opposite truth-values from the proposition. So, what else is necessary for a negation to be valid?
In the linked-to answer, it appears that an attempted negation must not just flip the truth-value in that case, but it must be of a form that always flips the truth-value.
Take the sentence "all line are straight". It's negation is "there exists a line that is not straight", because it is of an opposite truth-value, AND it's of a form that always produces an opposite truth value: $\exists x \in L, \neg S(x)$.
An attempted negation could be "all lines are not straight", which is also of the opposite truth-value, but of a form that does not always produce an opposite truth-value: $\forall x \in L, \neg S(x)$
Another attempted negation could be "all non-lines are straight", which would also have the opposite truth value in this scenario, but its form does not always produce opposite truth values: $\forall x \not \in L, S(x)$
I'm asking if this is correct:
A proposition and its negation are of a form such that all semantic interpretations of them yield opposite truth-values for them, and the terms inside the proposition and the negation are always the same (with respect to each other) within any interpretation.
If so:
To say the negation of any arbitrary, true proposition is the universally false proposition $P \land \neg P$ is not okay, because although it will always yield an opposite truth value, a valid negation also requires the preservation of terms. And to say the negation of e.g. $\neg(\forall x \in L, S(x)) \iff \forall x \in L, \neg S(x)$ is not true, because although the terms are preserved and although many semantic interpretations may yield opposite truth-values to the LHS and RHS, it isn't true that ANY semantic interpretation would do so.
Or, have I misunderstood what a negation is?
Best Answer
Typically, the negation of a sentence $\phi$ is considered the sentence $\neg \phi$. That is, negating a sentence is seen as a purely syntactical operation: the negation of $\phi$ is just that very sentence $\phi$ but with a negation sign in front of it. This is why we can talk about 'the' negation of a sentences rather than just 'a' negation.
Of course, many sentences are logically equivalent to $\neg \phi$, and sometimes these are also considered 'negations' of $\phi$, and I think that this is really the concept you are after. Personally, I think a better term to use here is that of contradictories, and this is really about semantics . That is, $\phi$ and $\psi$ are contradictory if and only if they always (i.e. under any interpretation) have opposite truth-values ... which is the same as saying that $\psi$ is logically equivalent to $\neg \phi$.
But yes, the key here is that two sentence are contradictory (or, if you want, that one sentence is a contradictory sentence of/to another sentence) if they always have opposite truth-values, and not any two sentences that, under some interpretation, just merely happen to have opposite truth-values. In the latter case we would never talk about the one being a negation of the other.
So yes, you are right about this, and the quotes in your post also make this very same point.
Finally, you write:
I am not sure what you mean here. Do you mean that $\neg$ is defined as $P \oplus \neg P$? That is surely not correct. Negation is an operator, not a sentence. You can talk about the (or a) negation of a sentence, and that would be some other sentence, but negation by itself is not a sentence.
Maybe you meant to say that the negation of $P$ is $P \oplus \neg P$? That is not correct either. $P \oplus \neg P$ is (at least under classical logic) a sentence that is always true (a tautology). This is not the negation of $P$. If it is a negation of anything, then it would be a negation of a contradiction such as $P \leftrightarrow \neg P$.
I think what you tried to get at is simply that $\neg P$ is a negation of $P$. And yes, that's right. And like I said above, I would consider $\neg P$ the negation of $P$, while $P$ is 'merely' a contradictory of $\neg P$ ... I would consider 'the' negation of $\neg P$ to be $\neg \neg P$.