Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2018 www.jamesrmeyer.com

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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.

**Abstract **

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.

**1: Introduction**

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

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

For example, if the substituted value is **∀ x, x > y**, then

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

**3: Conclusion**

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.

[1]

Halbach, V. and Zhang, S., 2016. *Yablo without Gödel*, Analysis 77(1):53-59.

[2]

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 will be a closed term; but, according to the author, when

Unfortunately, the author is wrong. Here are two ways to show this.

**Example 1.** In standard Gödelisation, will be the numeral associated with the formula

**Example 2.** Start with a countable basic language, ** L**. Add countably many new constant symbols

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

Halbach and Zhang make no mention whatsoever of Gödelisation, Gödel numbering, or meta-language in relation to their claim, except to specifically assert that:

- their claim relies neither on Gödel coding nor the Gödel diagonal lemma, and
- they are working within a single language.

And in fact, *Halbach and Zhang* specifically refer to one * single* language which includes

*We work in a language of predicate logic…For each formula in the language, there is a closed term in the language.*

Halbach and Zhang’s assertion that 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

“just define ** _{i}** =

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.

Footnotes:

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

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A review of Buldt’s *The Scope of Gödel’s First Incompleteness Theorem*

There is now a new page on a contradiction in Lebesgue measure theory.

There is now a new page Halbach and Zhang’s *Yablo without Gödel* which analyzes the illogical assumptions used by Halbach and Zhang.

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.

13 Jan 2017 Ned Block’s Blockhead

8 Apr 2016 Are we alone in the Universe?

13 May 2015 Good Math, Bad Math?

31 Mar 2015 Cranks and Crackpots

16th Mar 2015 Bishops Dancing with Pixies?

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

– a page with general links:

– and a page relating specifically to the Gödel mind-machine debate:

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Copyright © James R Meyer 2012 - 2018

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