What is negation?

Question

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?

Answer 1

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 $\neg \neg P$.

Answer 2

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.

Answer 3

What is negation?

1. In classical logic, to negate a formula (including an open formula that has no definite truth value) means to logically flip its truth value.

2. Therefore, these statements are equivalent:

   • $P$ and $Q$ are negations of each other;
   • regardless of interpretation, $P$ and $Q$ have opposite truth values;
   • $P↔Q$ is unsatisfiable;

   (A more pedantic definition of ‘negation’ than #1 considers $(¬A∨¬B)$ to be merely logically equivalent to the negation $¬(A∧B)$ of $(A∧B);$ in this case, change “$P$ and $Q$” in bullet 2 to “the truth-functional form of $P$ and $Q$”, and change “unsatisfiable” to “a contradiction”.)

3. On the other hand, these statements are equivalent:

   • $P$ and $Q$ are logically equivalent;
   • regardless of interpretation, $P$ and $Q$ have the same truth value;
   • $P↔Q$ is valid, i.e., $P\equiv Q;$

4. Two formulae with opposite truth values (in a given interpretation) do not necessarily negate each other.

   For example, $(\forall x\;x^2\le0)$ and $(x=x)$ have opposite truth values in the universe $\mathbb R,$ but the same truth value in the universe of all imaginary numbers (that is, in this interpretation, the two formula are equivalent); so, they aren't mutual negations.

   the negation of $(\neg\forall x {\in} L\;S(x))$ is not $(\forall x {\in} L\;\neg S(x)),$ 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.

   Yes.

5. Notice that negation preserves logical equivalence.

6. By the definition in #1:

   • Negating a valid formula gives an unsatisfiable formula, and vice versa.

   • Negating an invalid formula gives a satisfiable formula, and vice versa.

     Observe that both $\big(\forall x\,P(x)\big)$ and its negation $\big(\exists x\,¬P(x)\big)$ are satisfiable and invalid.

   To say the negation of any arbitrary, true proposition is the 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.

   Yes.

   $(\forall x\;x^2\ge0)$ is true in the universe $\mathbb R,$ and its negation $(\exists x\;x^2<0)$ is true in the universe of all imaginary numbers so is not a propositional-logic contradiction.

   $(x=x)$ is true in every interpretation (i.e., valid), and its negation $(x\ne x),$ although false in every interpretation (i.e., unsatisfiable), is not a propositional-logic contradiction.