Halbach & Zhang’s ‘Yablo without Gödel’
In an article entitled ‘Yablo Without Gödel’ (Footnote: Volker Halbach, and Shuoying Zhang, PDF Yablo without Gödel. Analysis, 77(1), 53-59, 2016.) in the journal Analysis, (Footnote: Analysis is published quarterly by the Oxford University Press on behalf of the Analysis Trust, see also the Wikipedia entry.) Halbach & Zhang claim that they can generate the Yablo paradox (Footnote: Stephen Yablo. PDF Paradox Without Self-Reference, Analysis 53 (4): pages 251-252. 1993.) without using any circularity or self-reference, and without using methods such as the diagonal lemma or Gödel numbering.
However, they make a ridiculous assumption that renders their claim of a proof utterly laughable. It is also rather amusing that the journal Analysis proudly proclaims on their website that it is the “most established and esteemed journal for short papers in philosophy. We are happy to publish excellent short papers ….” Apparently they are also very happy to publish complete crap.
I sent a short paper to Analysis pointing out the error in Halbach & Zhang’s article, but it was rejected on the basis of a referee’s report. That report demonstrated an alarming lack of any in-depth knowledge by the referee of what he was talking about. Below is a summary of the error in Halbach & Zhang’s article; you can see a copy of the paper that I sent to Analysis and a copy of the referee’s report at A referee’s comments: laughable or depressing?
The assumption of an enumeration within a Language
In defining a formal language, Halbach & Zhang state:
We work in a language [ T ] of predicate logic. Apart from the identity symbol, the language contains a binary predicate symbol <, and a ternary predicate symbol Sat(x, y, z). For each formula φ in the language, there is a closed term φ in the language. (Footnote: Here the closed term should appear as a φ with a bar over it. If no bar is showing, your browser is not displaying the content as intended. Perhaps you have the CSS styling turned off in your browser.) This can be achieved by adding countably many new constants, c1, c2, … to the language and then fixing some 1-1-mapping between the set of constants and the set of formulae in the language with all constants. This mapping doesn’t play any role in the proof of the paradox; it only helps to motivate our version TS of the T-sentences below. No assumptions about the structure of sentences are used in our proof. We write φ for the constant for φ.
Halbach and Zhang specifically refer to one single language which includes all the terms that they refer to, including the postulated enumeration function φ (where φ is the free variable of the function φ ) within that same language. But although Halbach and Zhang assert that all these terms belong to the same language, they do not provide any proof that it is possible that the function φ can be in that same language. In fact, as shown below, it is trivially easy to show that it is impossible.
Halbach & Zhang assert that whenever the free variable φ in the function φ is substituted by a valid substitution value, the result is a closed term, that is, the term can have no free variables. But if the substitution value is a formula with one or more free variables, then, since all terms belong to the same language, (Footnote: In Halbach & Zhang’s account, only one language is involved and there is no mention of a meta-language.) the resultant term is necessarily a term that contains one or more free variables that are not subject to a quantifier, and yet Halbach & Zhang assert that the term is closed and has no free variables. For example, if the substituted value is ∀x, x > y, then ∀x, x > y has one free variable y, and hence this term is not a closed term. (Footnote: The rather unusual terminology for the mapping function tends to obfuscate. As a conventional example, given the function f(x) = x + 2, and if the x is substituted by y3 − 3y, we obtain the new function h(y) = y3 − 3y + 2. The free variable x of the function f has been substituted by a function with a free variable, and the result is evidently a function with a free variable.)
It is worth pointing out here that, besides the obvious claim implicit in the title of their article, Halbach & Zhang specifically assert that their claim relies neither on Gödel numbering nor the diagonal lemma, and moreover, make no mention of a meta-language (Gödel numbering is a function in a meta-language to the language being coded).
A contradiction arises, what a surprise - not !
Halbach & Zhang discover that their system results in a contradiction. That’s not a surprise to me. But the astonishing thing is that Halbach & Zhang think that the contradiction shows something deep and profound, whereas all it shows is their lack of in-depth knowledge of their subject matter. Halbach, along with Albert Visser, has also published two papers (Footnote: Volker Halbach & Albert Visser, PDF Self-reference in Arithmetic 1, PDF Self-reference in Arithmetic 2, Review of Symbolic Logic, 7(4), Part 1: 671-691, Part 2: 692-712 (2014).) in the Journal of Symbolic Logic that amount to forty-two pages, and which are supposed to be an authoritative review of self-reference. But throughout the papers Halbach and Visser simply assume that a formal language can self-reference - and even though the papers are meant to be a review of self-reference, there is no attempt at all to prove that such a self-reference can occur in a fully formal language. Unfortunately this is completely typical of the current approach of academic people who like to call themselves “logicians”, where an unprovable assumption has been elevated to a dogmatic mantra that no-one is allowed to question.
Enumeration within a Language?
It is elementary to demonstrate that an enumeration of the formulas of a given language L can be achieved in a meta-language to the language L. But it is also easily shown that there cannot be such an enumeration within the language itself, as follows:
We assume that there is an enumeration of the formulas of a given language L within L itself (this is what Halbach & Zhang assert). Then there must be some function f(x) that defines that enumeration. Since the enumeration function is itself a formula to be enumerated, it follows that for one specific numerical value of the free variable x in f(x), we must have a result such as:
f(123…) = f(x)
where 123… is some specific numerical value. This, of course, is absurd, since now we have a non-variable term on the left side and a variable term on the right side. It follows that Halbach & Zhang’s claim that there can be a formula f in the same language L and which enumerates the formulas of that same language is an assumption that has no logical basis. (Footnote: For a more formal proof, see PDF The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System.)
This conclusion can also be adduced by considering the inverse of Halbach & Zhang’s purported mapping function, as follows:
If it is possible for the predicate Sat to evaluate from the values given to its variables, then it must be able to access the information contained in any constant that is purportedly given by the mapping φ. In other words it must be able to access the inverse of the purported mapping function φ. If we designate that inverse by Map−1(x), then for example, we have, for the formula ∀x, x > y, that ∀x, x > y = c where c is (according to Halbach & Zhang) a constant, and so the inverse would be Map−1(c) = ∀x, x > y. This, of course, is absurd, since we now have a constant term on the left-hand side of the equality, but a free variable term y on the right-hand side of the equality.
It is obvious that it is the assumption of an enumeration function within the language itself, and the resultant conflation of constant and variable terms, is the source of the contradiction that Halbach & Zhang remark upon in their article. It isn’t at all surprising that a conflation of constant and variable terms results in a contradiction, since the system that Halbach & Zhang have described is a system that is lacking the fundamental requirements of logical systems.
Halbach & Zhang say in their article that they “hope to further our understanding of what the source of the paradox is”, and indicate their total bafflement at what is patently obvious. As demonstrated on the page Yablo’s paradox, it is child’s play to see what the source is. It is no surprise that the source is the same as in many similar paradoxes - the source is a self-reference engendered by vagueness of definition.
What is very surprising is that, while one can presume that referees for the journal Analysis examined Halbach & Zhang’s article, they appear to have simply accepted Halbach & Zhang’s assumptions without subjecting them to even the most basic sort of critical appraisal. But when there is an assumption to the effect that, although something is utilized within a proof it is not actually essential to the proof (such as Halbach & Zhang’s hypothetical function), that assumption should always be viewed with more than a modicum of suspicion.
Note that you can see a copy of the paper that I sent to Analysis and a copy of the referee’s report at A referee’s comments: laughable or depressing?
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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