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?
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$\begingroup$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:
What is negation? One could define it like this: $P \oplus \neg P$
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 $\neg \neg P$.
$\endgroup$ 6 $\begingroup$The negation just flips the truth value, but don't get confused between a statement and the truth value of a statement.
We can express the negation of "All men are mortal" in words as "Not all men are mortal" if we're lazy. This has the opposite truth value of "All men are mortal", and it has the same truth value as "Bears have wings", but it is not the same statement as "Bears have wings".
Is "Not all men are mortal" the same statement as "It is not the case that all men are mortal"? Is it the same statement as "There exists a man who is not mortal"? These questions are awkward to answer when we're talking about English sentences: if the first two sentences are different, then what is the 'real' negation? So we switch to more formal notation.
The negation of $\forall x \in L, S(x)$ is just $\neg \forall x\in L, S(x)$. I am happiest saying that it is not the same statement as $\exists x\in L, \neg S(x)$, but it is an equivalent statement: they have the same truth value.
Of course, if we know that all lines are straight, then $\forall \in L, S(x)$ is equivalent to any other true statement, if our definition of 'equivalent' is 'has the same truth value'. But the point of the rules of negating quantifiers is that simplifying $\neg \forall x\in L, S(x)$ to $\exists x\in L, \neg S(x)$ is guaranteed to produce an equivalent statement, whether we know anything about lines or not.
This is closest to your discussion of
a form such that all semantic interpretations of them yield opposite truth-values for them
but I think it is not the definition of negation - that's just a description of what we want from a rule that lets us rewrite the negation as an equivalent statement, if the rule is supposed to be useful in all cases.
$\endgroup$ $\begingroup$In classical logic, a negation, by definition, flips a sentence’s truth value.
What is negation? I asked this question in the comment section of ryang's answer.
To be clearer: negation in classical logic logically flips a formula's truth.
- Thus, two (atomic/compound) sentences with opposite truth values (in our universe) aren't necessarily negations of each other!
\begin{align} A(x)\,&:=\;\text{$x$ is alive}\\A\,&:=\;∀x\;A(x),\text{ i.e., All men are alive}\\ I(x)\,&:=\;\text{$x$ is invisible}\\ I\,&:=\;∀x\;I(x),\text{ i.e., All men are invisible}\\N(x)\,&:=\;\text{$x$ lives in Norway}\end{align}
In some alternate universe (and thus interpretation), the first four formulae above are all true, so that $A$ and $I$ here have the same truth value. However, $A$ and $¬A$ certainly have opposite truth values, as does $N(x)$ and $¬N(x),$ even as $N(x)$ in this universe has no fixed truth value.
- Thus, ceteris paribus, negating a (atomic/compound) formula necessarily flips its “current” truth value.
Now, if two formulae $A$ and $I$ consistently have the same value in the specified interpretation, then they are equivalent formulae. However, this does not mean that $¬A$ negates $I,$ since in another interpretation $¬A$ could be equivalent to $I.$
On the other hand, two logically equivalent formulae, that is, two formulae that share a truth value regardless of interpretation, do have negations that must again be logically equivalent.
Thus, two (atomic/compound) formulae can be equivalent but their negations not logically equivalent.
However, two (atomic/compound) formulae logically having opposite truth values must effectively be negations of each other.
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.
If by “any arbitrary, true proposition” you just mean a proposition that is true in a particular interpretation, then yes its negation isn't necessarily a contradiction.
But if by “any arbitrary, true proposition” you mean a logically true proposition, that is, a validity, then its negation is indeed logically equivalent to $\:P \land \neg P.$
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 andalthough many semantic interpretations may yield opposite truth-values to the LHS and RHS, it isn't true that ANY semantic interpretation would do so.
Ignoring the part that I struck out, that's exactly right: \begin{gather}∀x{∈}L\;S(x)\tag1\\ ∃x{∈}L\;¬S(x)\tag{¬1}\\ ∀x{∈}L\;¬S(x)\tag2\end{gather} $(1)$ certainly has a different truth value from $(¬1),$ but may have have the same truth value as $(2).$
P.S. Side point: I removed the postfix comma because in a longer sentence it may be understood as a delimiter that you may not actually intend.
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