If something is neither true nor false, what is it? I saw the other day something similar to the following:


  
  one of the following is true.
  the above is false.
  $ 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.
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.
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.