True or false or not-defined statements Is it correct to say that for a statement to be either true or false it has to be well defined?
For example: the statement 
$$\frac{1}{0} = 1$$
is neither true nor false because the expression on the left simply isn't defined.
Or the statement: 
sdfjinrivodinvr

is not true or false because it doesn't make sense.
Or are these "expressions" even statements if they are not well-defined?
 A: One way to make precise the distinction you're trying to make is the notion of a well-formed formula in logic. Roughly speaking this is a formula which is built up from other formulas in a meaningful way, so it can be assigned some kind of meaning and it is meaningful to talk about whether or not it is true. A formula which is not well-formed does not in any meaningful sense have a truth value. 
In a suitable formal system for talking about arithmetic operations, the expression $\frac{1}{0}$ is already not well-formed; division $\frac{a}{b}$ should only be well-formed if $b \neq 0$. 
A: If a statement does not make sense, it is neither false nor true. As Pauli said, it's not even wrong. 
A: Mathematics is not the study of bits of ink on paper (or pixels on screens, indeed), it is the study of concepts and abstract ideas. Hence, when you look at some ink on a piece of paper, you have to first decide "does this correspond to an abstract idea?" before asking "what mathematical meaning does that idea contain?". Before you ask if $\frac{1}{0}=1$ is true or false, you need to ask what those symbols mean. Well, usually you don't need to ask, because it's obvious, but when you're unsure you ought to remember that just because you wrote down a thing, doesn't mean there's anything in it.
Hence I would argue that (unless you give meaning to it, and there is no "obvious" meaning in this case) $\frac{1}{0}=1$ is neither true nor false, because truth or falsity is a property of abstract mathematical concepts, and this pattern of pixels does not map to any such thing.
In programming terminology, I would describe it as a compile error, or a parse failure :)
A: The statement $1/0=1$ could reasonably be construed as meaning that the expression to the left of "$=$" is defined, and its numerical value is $1$.  And that is certainly false.
While in high school Jubal Harshaw won a debate by citing the British Colonial Shipping Board as the authority supporting some factual statement.  But the British Colonial Shipping Board never existed; he made it up.  Is his assertion false, or just meaningless?
(Some may know that Jubal Harshaw himself is a character in a novel that has a legal notice in its front matter, saying all persons in this story are fictitious.  So one might wonder whether my assertion about what Jubal Harshaw did is true or false.)
A: I believe the original question of the author remains still unanswered - are there such mathematical statements that are well-formed and understandable, but still are not true or false? We have been focusing too much on the examples.
First of all, there are such statements which have not been proven true or false, but could, given enough time and a smart brain. For instance prime number theory - whether given large number is a prime or not, can not be proven true or false at a moment (without testing all the candidates) but possibly could be in the future.
Then there are statements that can not be proven, and the fact that they can not be proven has been proved. For instance:
http://www.edge.org/q2005/q05_9.html#dysonf
So the answer is YES.
