# The antonymical fallacy

I think there's an informal fallacy that doesn't have a name of its own. I tried looking and couldn't find it, so I named it the antonymical fallacy. Maybe it does already exist. If it does, please tell me! It is closely related to the False dilemma informal fallacy.

So the idea is this: There is a difference between the negation and the antonym (or antonym-like word) of something. Some antonyms are indeed the logical negation of the original word, but not all!

not-tall does not equal small (it can me medium sized), not-legal does not equal illegal (it can be alegal), not-deterministic does not mean random [1], not-perfect does not mean imperfect (It can be of average quality).

In general, ¬P≠antonym(P).

The example of the fallacy is often used as follows:

- Either P or ¬P (Law of the Excluded Middle)
- P->Q
- antonym(P)->Q
- (P∨antonym(P))->Q (From 2,3)
- ∴Q

This is true if premise 3 were ¬P->Q, but it's not in the case of the fallacy. Or this is true if you also prove that ¬(P∨antonym(P))->Q, but that's overcomplicating things!

Examples of the fallacy:

- (Either everyone becomes a doctor or it isn't the case that everyone becomes a doctor)
- If everyone is becomes a doctor, we die (from famines, no one farms)
- If no one becomes a doctor, we die (from illnesses)
- If everyone or no one becomes a doctor, we will die
- Therefore, we will die (!)

Not-everyone ≠ None.

This ignores that between the doctor and not-doctor world, there are world with some doctors in which we live.

- (Either a number is positive or it is not positive)
- If a number is positive, it has magnitude
- If a number is negative, it has magnitude
- If a number is positive or negative, it has magnitude
- Therefore, every number has magnitude

Not positive ≠ Negative.

This ignores the fact that zero has no magnitude, and it is neither positive or negative.

- (Either the water is in part A of the box or it is not in part A)
- If the water is in part A, the water is in a section of the box
- If the water is in part B (B is the rest of the box), the water is in a section of the box
- Therefore, the water is in a section of the box

Not in part A ≠In part B

This ignores the possibility that the box may be full of water, and thus both A and B contain water, therefore the water is in both sections, not only one.

[1] See this, section 'Examples of the Standard Argument', this for one excluded logical possibility. Logical possibility does not entails correctness, obviously.