Is the pseudomenon a statement?

Question

I'm asking this because I'm teaching a class on paradoxes for kids, and I realized I have no idea what the answer to this question is. It is a research question in the pedagogical sense, I suppose.

I'm referring to the famous "liar's paradox," which in one form asserts: "This sentence is false." According to the logic definition of the word "statement," does it qualify as a statement?

The easy answer would be that it does not because it does not have a distinct truth value. But the reason why it does not have a distinct truth value is quite different from the reason one would give in most examples of non-statements. Moreover, if it is not a statement, then would we say that the continuum hypothesis is also not a statement? And taking that a step further, can we confidently call any conjecture a statement unless it has been established as not being independent of the axiomatic structure in use? Lastly, does the status as statement depend on the axiomatic structure? For example, is the axiom of choice a non-statement in ZF, but a statement in ZFC?

I do not work in logic so please forgive my naivete. As I said I'm asking for the benefit of some young students who I'd like to have a better answer for.

Answer 1

I will add a point which sound important for this discussion.

There is only one strict definition of "statement" - the definition in mathematical logic, where a statement is defined in terms of syntax of a formal language (say, of 1st order predicate logic). This definition is correlated with that formal language, and cannot help answer your question about English which is another language. What would help is a definition in semantic terms. Logicians use "interpretations" which are functions in their "semantics", and this permit to talk about semantics strictly. Here is a "definition" (or "criterium") of "statement" (or, of "qualifying as statement") in natural languages:

A statement is a sentence $S$, for which there exists at least one interpretation such, that the question "Is $S$ true?" can be answered "Yes" or "No" in this interpretation.

A paradox has no interpretations as explained by Asaf Karagila. Therefore, it is not a statement. The continuum hypothesis has at least one such interpretation. Therefore, it is a statement.

To fully comply with the "definition", I should have said above "the continuum hypothesis formulated (phrazed) in English, or any other natural language". Probably, I could improve the wording of the "definition", so that it is "works" both for natural languages and mathematical formulas, but at this time, this is the best short wording which I could figure.

Answer 2

You have made the well known confusion between true and provable. (And throughout this answer I will assume we work in classical first-order logic.)

Truth value is given in a particular structure. The continuum hypothesis has a truth value, it just can be different in different models of $\sf ZFC$. And in every model of $\sf ZF$ the axiom of choice has a truth value, but in some models it's true and in other it is false.

The fact that a statement is not provable from a particular theory does not preclude it from having a truth value, in a particular model for the theory. Two simpler examples are these:

1. $\forall x\forall y(x\cdot y=y\cdot x)$. Is this true in the language of groups?

   Well, first of all the question should be if this is provable, not true. The answer is that it is not provable, since some groups are abelian and others are not.

2. $\exists x\forall y(x\cdot x\cdot x=2\land y\cdot y\cdot y=2\rightarrow x=y)$, namely there is a unique cubic root to $2$. Is this true in the language of fields?

   Again, the question should be if this is provable from the theory of fields, and again the answer would be negative. In $\Bbb Q$ and $\Bbb C$ there are no such $x$, and three such $x$ respectively; and in $\Bbb R$ there is just one.

Let me digress for a moment, and analyze this mistake. I see it coming mainly from two possible sources:

1. Bad terminology. We sometimes use "true in $T$" as a synonym for "provable from $T$", because the completeness theorem tells us that $T$ proves something if and only if it is true in all the models of $T$. So sometimes we say things like "$\aleph_1\leq2^{\aleph_0}$ is true in $\sf ZFC$" when we really mean to say that it is provable.

2. Misunderstanding incompleteness, in particular in foundational theories. Sometimes it is convenient to think that set theory has only one model, the universe of mathematics, rather than considering it equal to other theories like group theory or fields, and then we think that if $\sf ZFC$ can't prove something, then that something has no truth value.

   But this is just misunderstanding that within a given structure (or a universe of mathematics, if you will), a statement has a truth value.

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So what about the liar paradox? If you look at the definition of a well-formed formula, which is really, "a legal statement" you will see that we construct them by induction from the constant symbols and free variables in our language, through the relations and so on.

But this also means that we can draw a tree and deconstruct a statement to its primitive elements. In the liar paradox, "This statement is false", this means that the statement $\varphi$ is in fact $\lnot\varphi$. So when drawing the tree we have just one cycle, which means that this is not a well-formed formula. So it cannot be assigned a truth value in any structure.

But wait, you might say, what about Godel's incompleteness theorem where he wrote a statement "This statement is not provable"? Well, this is not quite accurate. He did something a bit more complicated than that, but this answer is long enough as it is, so I'll stop here and let you digest what I have already written.

Answer 3

There's a big difference between:

1. a liar sentence
2. the continuum hypthesis (i.e. the sentence "the continuum hypothesis is true")

The continuum hypothesis is a statement, and a perfectly good statement at that. Its kind of like the statement "for all $x$ and $y$, we have $xy=yx$" in group theory. The axioms of group theory cannot prove this statement (because not every group is Abelian) nor can they refute it (because some groups are Abelian), nonetheless it is a perfectly good statement in the language of group theory. Similarly, the usual axioms of set theory (called $\mathrm{ZFC}$) neither prove the continuum hypothesis nor refute it, which is equivalent to saying that the continuum hypothesis is true in some models of $\mathrm{ZFC}$ and false in others. But that doesn't mean its an ill-formed statement, just the like inability to prove or refute "for all $x$ and $y$, we have $xy=yx$" does not make this an ill-formed statement of group theory. For this reasons, I don't think the continuum hypothesis really belongs in a class about paradox. Strictly speaking, there's nothing paradoxical about the continuum hypothesis; its just surprising that such a natural-sounding statement is independent of the usual axioms of set theory.

The concept of a liar sentence, on the other hand, is very different. Liar sentences are amazing because if they're true then they're false, and if they're false then they're true. This means that the English language can express sentences that cannot be assigned a classical truthvale (i.e. $\mathbf{true}$ or $\mathbf{false}$) even in principle. A consequence of this is that formal languages must be carefully designed so that they cannot express liar sentences, otherwise thins tend to explode the moment you try to assigned truthvalues to the sentences in a consistent way. That is surely surprising and paradoxical.