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In the appendix of Terence Tao's book Analysis, he writes:

Thus well-formed statements can be either true or false; ill-formed statements are considered to be neither true nor false.

A more subtle example of an ill-formed statement is $ \ 0/0 = 1$; division by zero is undefined, and so the above statement is ill-formed.

So we have:

  • (i) $ \ $ ill-formed statements are neither true nor false.
  • (ii)$ \ $ The statement "$0/0=1$" is neither true nor false.

Consider the following statements:

(1) $ \ $The statement "$0/0=1$" is true.
(2) $ \ $The statement "$0/0=1$" is not true.
(3) $ \ \forall x \in \mathbf{R}, x/x=1$

For statement (1)：Since (ii), "0/0=1" is not true, and hence statement (1) is false. But on the other hand, as statement (1) contains ill-formed symbols, by (i) it is neither true nor false.

Similarly, because of (ii) statement (2) is true, but it is neither true nor false by (i). Again we get a contradiction.

As for statement (3), what's the truth value of it? False, or neither true nor false? I don't know. Maybe it depends on the outcome of our discussion about statements (1) and (2).

I'm so confused. How do we understand the logical state of "neither true nor false"? How to solve the "paradoxes" above?

Any insights are much appreciated.

Sky subO
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    "True" and "false" with regard to mathematical statements is not a philosophically absolute statement, but a statement about what can be proved. Of course, you could have a mathematical system where $0/0$ is defined axiomatically, but that is not the result of a proof. Thus, a "not true" statement is not necessarily false; it is simply without a proof of its truth. Similarly, "not false" statements have no proof of their falsity, but that does not mean they are true. That is the source of your paradox. – Keith Backman Sep 23 '24 at 20:36
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    You may not state "Since (ii), "0/0=1" is not true, and hence statement (1) is false", because it is known that (1) is not false, which you pretend to ignore. –  Sep 23 '24 at 20:51
  • Statement (3) is neither true nor false, because division is defined as a function $\mathbb R\times\mathbb R\setminus{0}\to\mathbb R$. – Lavender Sep 23 '24 at 20:52
  • When considering ill-formed statements you no longer have that not true implies false and not false implies true. So you can have (1) be false and (2) be true without contradiction. – CyclotomicField Sep 24 '24 at 00:02
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    The quotation marks around "$0/0=1$" are intended to indicate that we're not talking about dividing zero by zero but rather about a string of five symbols $0,/,0,=,1$. This situation is often expressed by saying that putting quotation marks around a string of symbols indicates that the string is being mentioned, not used. See Use–mention_distinction in Wikipedia (or ask Google) for more information. What makes a statement ill-formed is that it contains nonsense (like $0/0$), not that it mentions nonsense. In particular, statements (1) and (2) in your question are well-formed. – Andreas Blass Sep 24 '24 at 00:09
  • @KeithBackman You're right, but I approach these statements with great care and haven't made such mistakes ( not true implies false or not false implies true）. – Sky subO Sep 24 '24 at 14:43
  • @YvesDaoust There is a subtle difference between statement (1) and "$0/0=1$" itself. It is "$0/0=1$"that is not false. I think (1) is false because "$0/0=1$" is not true and (1) claims that it is, – Sky subO Sep 24 '24 at 15:20
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    @SkysubO: this is a sophism. "Supercalifragilisticexpialidocious" is not false, hence it is true. –  Sep 24 '24 at 15:23
  • @YvesDaoust I don't understand what you are saying, but here I know that not false doesn't imply true. – Sky subO Sep 24 '24 at 18:04
  • @AndreasBlass Thanks for the comment. It's very helpful! – Sky subO Sep 24 '24 at 18:17
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    "But on the other hand, as statement (1) contains ill-formed symbols" What is an "ill-formed symbol"? Statement 1) refers an ill-formed statement but statement 1) is not an ill-formed statement. "Iskabibble" is an ill form statement but "My sister believes 'ishkabibble'" or "'17 is prime' has the same truth value as 'ishkabibble'" are not ill-formed statements. – fleablood Sep 24 '24 at 23:51
  • "The statement X is true" and "X" are no longer equivalent. (They were never the same. They were always different statements. "The statement X is true" was a statement about X and whether it was true. "X" was an independent statement.) If X is a well-formed statement (Such as "Peter Pan is an invisible red dragon") then "The statement X is true" and "X" are equivalent meaning they are both true or both false but they are not the same statment. But with ill-formed sentences and not true not being equivalent to false. "X is true" and "X" are NOT equivalent. to be cont.... – fleablood Sep 25 '24 at 00:00
  • @fleablood I think you've hit the nail on the head. Thanks a lot! – Sky subO Sep 25 '24 at 15:53
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    @ryang Your answer helped me a lot. Thank you so much! – Sky subO Sep 27 '24 at 13:18

3 Answers3

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  • (i) $ \ $ ill-formed statements are neither true nor false.
  • (ii)$ \ $ The statement "$0/0=1$" is neither true nor false.

Consider the following statements:

(1) $ \ $The statement "$0/0=1$" is true.
(2) $ \ $The statement "$0/0=1$" is not true.
(3) $ \ \forall x {\in} \mathbf{R}\; x/x=1$

For statement (1), by (ii), "0/0=1" is not true, and hence statement (1) is false. But on the other hand, as statement (1) contains ill-formed symbols, by (i) it is neither true nor false.

The ill-defined mathematical statement 0/0 = 1 is neither true nor false, thus the well-formed statement "The statement 0/0 = 1 is true" is false.

Similarly, because of (ii), statement (2) is true, but it is neither true nor false by (i). Again we get a contradiction.

"The statement 0/0=1 is not true" isn't ill-formed, and is in fact and indeed a true statement.

As for statement (3), what's the truth value of it?

∀x∈R x/x = 1 is technically an ill-defined statement in usual mathematical contexts, where it has no truth value.

ryang
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  • Your assertion that $\forall x \in R,x/x = 1$ is technically ill-formed is a reference to one of your own posts which doesn't justify the terminology. If you are claiming that this is a documented resolution of the question, then please give a proper reference. – Rob Arthan Sep 24 '24 at 22:09
  • @RobArthan The linked post indicates that an ill-formed mathematical statement can be a well-formed formula (specifically: a well-formed sentence). Indeed, ∀x∈R x/x = 1 is such an example, and calling it an ill-formed statement is with reference to mathematical analysis, which the OP's question and the book they're quoting is with reference to. In other words, this discussion is premised on what Terence Tao, in his book Analysis, means by an "ill-formed statement". – ryang Sep 25 '24 at 04:32
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    Well all I can say is that you (and Tao) are using bad terminology. You are conflating form with function and making that even worse by offering "ill-formed" as something that is not the complement of "well-formed", whereas one always expects "ill-X" to be the complement of "well-X". You (and Tao) should be saying "ill-defined" rather than "ill-formed' to make it clear that you are making a statement about the meaning of a construct and not is form. – Rob Arthan Sep 26 '24 at 21:16
  • @RobArthan "You are conflating form with function" Thank you for pinpointing that it's more helpful and accurate to call something like 0/0 = 1 an 'ill-defined mathematical statement' rather than an 'ill-formed mathematical statement', and I've since edited both posts to relabel them. – ryang Sep 27 '24 at 03:53
  • @RobArthan (To be clear though, my original posts did consider well-formed statements and ill-formed statements to be complementary sets, likewise well-formed sentences (WFF's) and ill-formed sentences (IFF's), and called 0/0 = 1 a well-formed sentence that's an ill-formed mathematical statement. Gosh, esoteric. But this is moot now, since we've retired that label.) – ryang Sep 27 '24 at 03:59
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The answer to this question depends on whether you are working in mathematical logic or in mainstream mathematics. Tao's comment is from the mainstream mathematics perspective, where we think of expressions like $0/0$ as being undefined and treat formulas involving them as having no meaning. From this point of view, "for all real numbers $x$, $x/x = 1$" would be meaningless. I think Tao is being rather unhelpful in his use of the term "ill-formed": there is nothing wrong with the form of the expression $0/0$: it is an expression whose form I can comprehend, but that is a little problematic when we ask what it denotes, if anything. In general, it is undecidable whether an expression is undefined, whereas we expect to be able to recognise well-formed mathematical expressions (precisely so that we can ask questions like "is $x/y$ defined?" for such and such a value of $x$ and $y$).

In mathematical logic (the world you're likely in if you're writing it completely symbolically as $\forall x \in \Bbb{R}\,x/x=1$), things are a bit different. In the usual semantics for first-order logic, all functions are total. It is standard in first-order axiomatisations of fields to define $x/0$ to be $0$. Alternatively, you can formulate your axioms so that they don't assign a value to $x/0$, but this is less common as it makes the metatheory more complicated (however, it is quite a popular approach when logic is applied in computer science). Many systems that build on first-order logic follow the approach that functions are total, but many other systems model the notion of a partial function in the system and their rules for reasoning about function application require you to prove that an argument of function lie in its domain.

Rob Arthan
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How do we understand the logical state of "neither true nor false"?

Consider the function $f: N^+ \to N$ such that $f(x)=1$ for all $x\in N^+$.

$\forall x:[x \in N^+ \implies f(x)=1]$

From this definition, we see that $f(1)=1$ is true and $f(1)=0$ is false.

What about $f(0)=1$? "Ill-formed" does not quite capture it. Rather, it is of indeterminate truth value, i.e. the above definition of $f$ cannot be used to determine a truth value of $f(0)=1$ since $0$ is not in the domain of $f$ as defined here. Informally, we can say that $f(0)$ is undefined.

As for the definition of division on the real numbers, the $\div$ operator can be defined in terms of multiplication as follows:

$\forall x,y,z\in R:[y\neq 0 \implies [x\div y =z \iff x=z\times y]]$

Informally, based on this definition of the $\div$ operator, we can say that $0\div 0=1$ is of indeterminate truth value since this definition cannot be used to assign a truth value to this expression. Informally, we can also say that $0\div 0$ is undefined.

Dan Christensen
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