Capturing the "antonym phrase" concept in Formal Logic

Question

Let's start with an example.

True is the antonym of False and Positive is the antonym of Negative. This can be represented in Logic as $\neg$False$(x)$ $\equiv$ True$(x)$ and $\neg$Negative$(x)$ $\equiv$ Positive$(x)$.

However the phrase "False Negative" has an antonym "True Negative", but not "False Positive" or "True Positive". Similarly the phrase "Good Person" has an antonym "Bad Person" but not "Bad Non-Person" or "Good Non-Person".

If we represent the concept of antonym phrase in logic it would become $\neg($False$(x)\land $Negative$(x))$ $\equiv$ True$(x)\land$Negative$(x)$, which is obviously not a valid logic sentence, as it does not stay consistent with the definition $\neg$Negative$(x)$ $\equiv$ Positive$(x)$.

I am wondering is there a way to represent the antonym phrase concept correctly using formal language. I think this would require two things: $(1)$ We need to somehow narrow down the domain of objects, and $(2)$ We need to somehow capture the order of the elements of the phrase, however I am not sure how to achieve these two requirements.

Answer 1

• $False(Negative(x))$
• $False(x)\land Negative(x)$

These are not the same.

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When people say "false negative" in english, they refer to the first one.

The test can be negative, and the fact that it is positive or negative can be false. But the test itself cannot be false as you wrote in the second one.