How do we know that the negation of a statement is unique? (Mathematical Logic by Chiswell and Hodges)

Question

I am reading Mathematical Logic by Ian Chiswell and Wilfrid Hodges and have been trying to solidify my understanding of pages 24-26 (Section 2.6 - Arguments using 'not'). In this section, the authors start off by defining negation with the following sentence: "If $\phi$ is a statement, we write $\lnot \phi$ for the statement expressing that $\phi$ is not true."

Based on this, how do we know that the negation of a statement is unique? I hope the below examples will illustrate my point:

Cat Example 1: If $\phi$ is the statement "The cat is black", then $\lnot \phi$ is "It is not true that the cat is black"

Cat Example 2: If $\phi$ is the statement "The cat is black", then $\lnot \phi$ is "The cat is blue"

Inequality Example 1: If $\phi$ is the statement "$5 < 7$", then $\lnot \phi$ is "It is not true that $5 < 7$"

Inequality Example 2: If $\phi$ is the statement "$5 < 7$", then $\lnot \phi$ is "$5 \ge 7$"

Of these four examples/claims, I would agree (based on the above definition of negation) with Cat Example 1 and Inequality Example 1. I would not agree with Cat Example 2 because the cat not being black does not imply that it has to be blue (there are other colors the cat could be).

Similarly (and this is where I get confused because I think most people would disagree with me here and say that Inequality Example 2 is correct), I would also not agree with Inequality Example 2 because it implicitly assumes that if "$5 < 7$", then "$5 \ge 7$" is the only other, unique option (just like how Cat Example 2 assumed the if the cat was not black, then the cat being blue was the only other option).

Of course, the difference between Inequality Example 2 and Cat Example 2 is that it IS actually true that "$5 \ge 7$" is the only other option, but this requires prior knowledge/the assumption of the trichotomy property of natural numbers. This makes me think that I should remedy Inequality Example 2 by thinking of "$5 \ge 7$" not as the negation of $\phi$, but as a completely new statement which has to be derived. So, perhaps it should be the conclusion of the following derivation using the $(\to E)$ rule?:

$\lnot \phi$, $(\lnot \phi \to \psi)$

$\therefore \psi$

where $\phi$ is "$5 < 7$", $\lnot \phi$ is "It is not true that $5 < 7$", and $\psi$ is "$5 \ge 7$".

Lastly, and I'm hoping I have written this post to be clear enough on its own, but for added context, this post is a follow-up to one I posted here the other day.

Answer 1

The negation is unique. "The cat is not black iff (the cat is red or the cat is white or the cat is any non-black color or no color at all)". The negation of a statement $\phi$ is all statements which, if they are true, mean that $\phi$ is not true. It's essentially a bunch of statements joined by an "Or". A statement made up of a composition of ors is true if any one of the statements is true.

The cat being blue therefor implies the veracity of the negation of "the cat is black". The negation is true if the cat is green, but "the cat is blue" is not true if the cat is green. The negation can be true without "the cat is blue" being true, so the statements aren't equivalent. The multiple ors are essential to forming the negation.

It's a good rule of thumb to think of logical negation as set complements, e.g. union of ways a cat can be non-black. Generally, interpret the negation as broadly as possible.

Answer 2

Summary :

Negation of Statement $P$ is always $\lnot P$ in very general terms.
In Context , Negation might be Equivalent to something else , but Context is Critical.

Elaboration :

CAT Example 1 is Correct , When we are given that "Cat is Black" , the negation is "It is not true that Cat is Black"

CAT Example 2 is not Correct , When it is not true that "Cat is Black" , we can not claim that "Cat is Blue" , because we do not know what other Possibilities can be there.

In Chess , we might Point to a Queen to claim "That Queen is Black" , with negation "That Queen is White" , because we have 2 Possibilities.
Even here , we have to be careful to know the Context : Are we talking about that Queen Piece ? If the Piece itself is variable , the negation will not be valid , because that Piece might change too Eg we might have bren Pointing to a King to claim "That Queen is Black" & the negation reply might be "No , That King is Black" in general.
More-over , that Entity itself might change : We might have been Ppint to a Square to claim "That Queen is Black" with the negation reply "No , That Square is Black"
We should know the Domain over which the variable ranges , to give the Equivalent negated Statement.

We can avoid all that by making the negation "It is not true that that Queen is Black" , when we are not aware of the Context.

Claiming "It is true that $5<7$" with negation "It is not true that $5<7$" is Correct.

Claiming "It is true that $5<7$" with negation "It is true that $5 \ge 7$" is not really valid , unless we know the Domain & incorporate the trichotomy. With trichotomy it is valid.

Eg Consider Complex Domain : "It is true that $5 < 7 +2i$" has the negation "It is not true that $5 < 7 +2i$"
"It is true that $5 < 7 +2i$" will not have the negation "It is true that $5 \ge 7 +2i$" , because the trichotomy is not valid in the Context here.

Matrix Example : "It is true that Matrix $P$ is Positive Definite" , negation is "It is not true that Matrix $P$ is Positive Definite" , which is very general & always true.
We can not make the negation like "It is true that Matrix $P$ is Negative Definite"

Answer 3

Inequality Example 2: If $\phi$ is the statement "$5 < 7$", then $\lnot \phi$ is "$5 \ge 7$"

I would not agree with Inequality Example 2 because it implicitly assumes that if "$5 < 7$", then "$5 \ge 7$" is the only other, unique option (just like how Cat Example 2 assumed the if the cat was not black, then the cat being blue was the only other option).

How do we know that the negation of a statement is unique?

To be very clear, you're not actually discussing the uniqueness of the negation: not in the sense of being one of a kind nor whether, for example, the sentences ¬(A→B) and A ∧ ¬B are multiple negations of the sentence A→B.

What you are actually doubting is that 5 < 7 comprehensively and exhaustively ‘opposes’ 5 ≥ 7 in all possible hypothetical scenarios. Since to negate a statement means to logically flip its truth value, that is, to flip its truth value regardless of interpretation, I agree with you that those two sentences are not technically negations of each other: take an interpretation where the universe is $\mathbb C$ and where any inequality involving any imaginary part is defined as false.

Here's another example to bolster your point. In the universe $\mathbb C,$ there are two ways to define an irrational number: either as $\mathbb C{\setminus}\mathbb Q$ or as $\mathbb R{\setminus}\mathbb Q.$ As the sentences i is rational and i is irrational are both false in the second interpretation, they are not technically negations of each other.

In practice (outside of formal logic) though, I suspect that few will bat an eyelid if you declare that 5 ≥ 7 and i is irrational are the negations of 5 < 7 and that i is rational, respectively.