Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com
I sent a short paper to the journal Analysis (Footnote: Analysis is published quarterly by the Oxford University Press on behalf of the Analysis Trust, see also the Wikipedia entry), pointing out an error in an article (Footnote: ‘Yablo Without Gödel’, Volker Halbach, and Shuoying Zhang, Yablo without Gödel (PDF). Analysis, 77(1), 53-59, 2016) that had been recently published in their journal. 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. I wrote to Analysis to point out the referee’s errors, but that was ignored.
Below is a copy of the original text that I sent, and a copy of the referee’s report.
Halbach & Zhang have published an article Yablo Without Gödel (2016) that claims that they can generate the Yablo paradox without using methods such as the diagonal lemma or the recursion theorem, and avoiding any explicit circularity. However, they make an assumption that conceals the fact that their system lacks the fundamental requirements of a logical system.
Halbach & Zhang (2016) have published an article that claims that they can generate the Yablo paradox (Yablo 1993) without using methods such as the diagonal lemma or the recursion theorem, and avoiding any explicit circularity. However, it is readily demonstrated that their method relies on the use of an assumption that is logically untenable.
2: Conflation of constant and variable terms
Halbach & Zhang (hereinafter referred to as HZ) state:
We work in a language [T] of predicate logic. Apart from the identity symbol, the language contains the identity symbol, 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 & Zhang, 2016: 54)
Now, whenever the free variable in the function φ is substituted by a valid substitution value, HZ state that 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, the resultant term is necessarily a term that contains one or more free variables that are not subject to a quantifier, and yet HZ 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 (Note: 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(x) = 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).
Clearly, the conflation of constant and variable terms is the source of the contradiction that HZ remark upon. It is not at all surprising that a conflation of constant and variable terms results in a contradiction; the system that HZ have described is a system that is lacking a fundamental requirement of a logical system.
This conclusion can also be adduced by considering the inverse of the one-to-one 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 function φ; that is, 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 HZ) 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.
When there is an assumption to the effect that, although something such as a function is utilized within a proof, it is not actually essential to the proof (such as HZ’s hypothetical function), that assumption should always be viewed with more than a modicum of suspicion.
Halbach, V. and Zhang, S., 2016. Yablo without Gödel, Analysis 77(1):53-59.
Yablo, Stephen, 1993. Paradox without self-reference, Analysis, 53(4): 251–252.
The referee’s response was:
The author claims to have identified an error in Halbach & Zhang’s “Yablo without Gödel”. In particular, Halbach & Zhang claim that φ will be a closed term; but, according to the author, when φ is open, φ must also be open.
Unfortunately, the author is wrong. Here are two ways to show this.
Example 1. In standard Gödelisation, φ will be the numeral associated with the formula φ. Whether φ is open or closed, φ is a numeral; hence, φ is closed.
Example 2. Start with a countable basic language, L. Add countably many new constant symbols c1, cn, … ; call the resulting language L*. There are countably many L*- formulas. Enumerate these φ1, φm…. Now just define φi = ci .
Note: Example 2 is a variant on Halbach & Zhang’s own approach; see the passage quoted by the author.
Here the referee demonstrates quite clearly that he has completely missed the entire point of my paper - the crucial point that should be obvious to any discerning reviewer is that Halbach and Zhang assume that an enumeration of the functions of a language can be achieved in the same language as the functions being enumerated.
Regarding the Example 1 provided by the referee, well, of course it is the case that φ is closed in standard Gödelisation - the referee is stating the outstandingly obvious. But in Halbach & Zhang’s case it is also obvious that the function φ is not in the same language as φ since it is in a language that is a meta-language to the language of φ. Since Halbach and Zhang claim that they are working within a single language with no mention of meta-language, the referee’s comment is not merely irrelevant, it shows his complete lack of grasp of the subject matter.
Halbach and Zhang make no mention whatsoever of Gödelisation, Gödel numbering, or meta-language in relation to their claim, except to specifically assert that:
And in fact, Halbach and Zhang specifically refer to one single language which includes all the terms that they refer to, including the postulated enumeration function φ (and where φ is the free variable of the function φ), and they state:
We work in a language of predicate logic…For each formula in the language, there is a closed term φ in the language.
Halbach and Zhang’s assertion that φ is in the same language as the formulas of the language could hardly be any clearer. But that passes right over the referee’s head.
Regarding the Example 2 provided by the referee, the referee refers to a language L that has countably many formulas and countably many constants. The referee then asserts that the formulas can be enumerated– but I hadn’t claimed that they couldn’t - of course they can be enumerated, that is patently obvious.
But, as is the case for his example 1, the referee completely misses the point and fails to explain how that enumeration might be possible within the language L. He simply asserts:
“just define φi = ci”
and simply assumes that such a definition is possible within the language itself, and provides no proof whatsoever of this assertion.
It is easily shown that there cannot be such an enumeration within the language itself, as shown fairly informally at Enumeration within a Language? and for a formal proof, see The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System (PDF).
In summary, the referee’s response demonstrates an unfortunate fact - that referees all too often don’t actually apply any logical thought to what they do, but instead simply rehash conventional dogma - a process that is almost guaranteed to stifle any dissenting voice. On the one hand, the referee’s ignorance is laughable; but it is also depressing to think that people who are purportedly well-educated can be so woefully inadequate at what they do.
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There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.
There is now a new page on a contradiction in Lebesgue measure theory.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
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