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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:


is not true or false because it doesn't make sense.

Or are these "expressions" even statements if they are not well-defined?

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Your first expression is false, because the right-hand side is defined, and so if it were true then the left-hand side would also be defined. Your second statement is not a mathematical problem. –  Clive Newstead Sep 22 '12 at 23:16
The question reminds me of Pauli's remark: en.wikipedia.org/wiki/Not_even_wrong –  Trevor Wilson Sep 22 '12 at 23:18
@Thomas: I guess that would depend what you meant by each side of the equation. On a more formal level, if you have an interpretation of a logical system then you can only declare some formula $\phi$ in that system to be true or false if $\phi$ is actually a valid formula. So if $\phi$ is not then the truth or falsity (or otherwise) of $\phi$ is meaningless. –  Clive Newstead Sep 22 '12 at 23:20
How come all the debate is focusing on the first question? I want to know if sdfjinrivodinvr is true! –  Trevor Wilson Sep 22 '12 at 23:30
The division "operation" in unpleasant to try to accommodate in a formal system. For when we are defining term, we cannot say that if $a$ and $b$ are terms, then $a\div b$ is a term. There are workarounds, but nothing direct. –  André Nicolas Sep 22 '12 at 23:37
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5 Answers 5

up vote 4 down vote accepted

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 :)

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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 $b \neq 0$.

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Thanks for the answer. So just to make sure that I understand, you would say that the first statement isn't true or false because it is not a well-formed formula, right? –  Thomas Sep 22 '12 at 23:38
@Thomas: yes, that's one interpretation. It is possible to give other interpretations to the symbol $\frac{1}{0}$ (for example in projective geometry) such that it is well-defined, and then it is just not equal to $1$. –  Qiaochu Yuan Sep 23 '12 at 8:00
I don't think this can be described as a well-formedness condition, because $\frac 1x$ certainly is a well-formed term, and we're then going to need some convention about what it means when $x$ is $0$. In principle one could decide never to allow the term $\frac 1x$ except in contexts where it can be proved that $x\ne 0$, but that would be very far from the syntactic character I would expect of the word "well-formed". (...) –  Henning Makholm Sep 23 '12 at 12:09
(...) Letting provability intrude on well-formedness in this way would also mean that we could never speak about the meaning of formulas themselves without reference to a proof system, which would make it difficult even to formulate soundness and completeness properties for a proof system. –  Henning Makholm Sep 23 '12 at 12:10
@HenningMakholm: in the formalism I know, division isn't an operation, it's just shorthand for multiplication by inverse, and it is a theorem that (in a field) an inverse always exists except if you are zero. There's then a subtlety as to how you translate formulae involving division into something valid... I might edit my answer with my thoughts on this. –  Ben Millwood Sep 23 '12 at 14:42
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If a statement does not make sense, it is neither false nor true. As Pauli said, it's not even wrong.

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Strange though it may seem, have an upvote! –  The Chaz 2.0 Sep 24 '12 at 0:41
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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.)

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Interesting. I had actually thought of an analogy to my question along those lines. It is kind of like asking questions that do not have answers because they assume things that are not true. If one asks for the reason why the sky is green, then that question isn't well-defined because it assumes something that is false. –  Thomas Sep 23 '12 at 0:50
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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:


So the answer is YES.

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Any statement can be proven, if we take a sufficiently strong set of axioms - particularly if we take the statement itself as an axiom. There are results of the form "statement $X$ is not provable in the specific system $Y$" but these cannot be summarized as "$X$ cannot be proven". –  Carl Mummert Sep 24 '12 at 1:42
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