# If something is neither true nor false, what is it?

I saw the other day something similar to the following:

1. one of the following is true.
2. the above is false.
3. $1 + 1 = 5$

You can probably see the problem with this. I can clearly state that $3$ is false, but what would I call $1$ and $2$?

To clarify, I really meant if there were some state between true and false that could make these consistent.

• Presumably, you mean "exactly one of the following is true" Dec 7 '16 at 16:10
• A similar "paradox" is attained with 1) statement 2 is false, 2) statement 1) is true. Dec 7 '16 at 16:11
• @Omnomnomnom you could have at least one of the following is true, but I don't see the affect either would have. Dec 7 '16 at 16:12
• I'd say that it lacks a definite truth value. Dec 7 '16 at 16:16
• @SimpleArt are you saying that you don't understand the problem with my setup? If 1 is true, then 2 is false. But since 2 is false, 1 can't be true. Dec 7 '16 at 16:29

These statements taken together are called inconsistent. That means that they cannot all be simultaneously true. But the first and second are neither true nor false without broader context. Using the language of first order logic, they might be said to be formulas with "free variables." Here's an example of a formula with free variables $$4x+3y=9$$ This is neither true nor false because I haven't told you what $x$ or $y$ are. If I use quantifiers to get rid of all the free variables, then I have a sentence which may be true or false: $$\forall x\forall y (4x+3y=9)$$ $$\forall x\exists y (4x+3y=9)$$ $$\exists x\exists y (4x+3y=9)$$ The first statement is false, while the second two are true.

• I think that technically your statements can't be made into first order formulas or sentences because in first order logic, "you can only quantify over variables, but not sets." The idea of inconsistency is still what you are looking for. Two relevant (and totally awesome) results related to this are Godel's completeness and incompleteness theorems. Dec 7 '16 at 16:29
• Thank you for this, I've tried to clear up my question. Dec 7 '16 at 17:08

In the 3 statements you've shown, statement #3 is definitely false, as you have already mentioned.

To clarify, I really meant if there were some state between true and false that could make these consistent.

No, statements #1 and #2 are inconsistent statements, they contradict each other.

Actually, since there's no discrepancy that statement #3 is false, it is a bit of a red herring, irrelevant. Alternately, statement #3 can be omitted and statement #1 and #2 can be re-written as:

1) Statement 2 is true.
2) Statement 1 is false.

or

The next statement is true.
The previous statement is false.

If you are looking for a word or phrase to describe the relationship of the statements, I would say the statements are "paradoxical".

You can also use the term: non sequitur.

noun: non sequitur; plural noun: non sequiturs; noun: nonsequitur; plural noun: nonsequiturs

a conclusion or statement that does not logically follow from the previous argument or statement.

Too long to repeat the whole Wikipedia article.

What you are looking for are the Liar paradox and its variants. Here are the possible resolution.

• This doesn't answer the question, namely: what truth value do you assign to such a statement, or is there a word for the inability to assign such a truth value. Dec 7 '16 at 16:19

This may not satisfy anyone reading this... but here goes...

In the software world, something that is neither true or false is likely not yet to have been determined. In which case it is null (or unset).

• That's just called three-valued logic and has no context to this question (which is using law of excluded middle) Dec 7 '16 at 19:03
• @Ephemeral I'm not sure about that - the question is asking what sort of truth value ("state") 1 and 2 may have, and to me that sounds like many-valued logic. Dec 7 '16 at 19:06
• Haha, I know what null is! Though I'm not sure how it fits in mathematically. Dec 7 '16 at 21:12
• I am not sure this is true; I am playing with reinventing if/else in terms of ifTrue (Coproduct, fThen), and ifFalse(Coproduct, fThen), so that (Coproduct ifTrue fThen1) ifFalse fThen2. I want to make a random Coproduct generator(Left ()) or (Right ()) depending on if it rolls 0 or 1. I want to make a variable name to describe a value which is always either Left () or Right () but I never know what it is at time of call; So it is not null or unset, it is an actual function, but it is undecidable at time of getting it. Dec 19 '16 at 20:23
• I am currently curious if there is a word for "something that is undecidable". Dec 19 '16 at 20:25

The answer you are looking for is "undecidable". What is neither true nor false is called an undecidable proposition. This is about the liar paradox and Gödel's incompleteness theorems.

Every attempt to establish the truth of the first proposition leads to a contradiction in the second. And you would never want inconsistency, ever. Not in mathematics and logic.

So, you decide it as "undecidable" to avoid contradiction.