Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com
The Liar Paradox comes in many guises. A simple example is “This sentence is false”. The paradox is that if the sentence is true, then it would be saying that it itself is false. And if the sentence is false, then, since it states that it is false, it appears to be true. Hence the paradox.
In 1993, Stephen Yablo claimed that he had devised a Liar type paradox that does not involve any circular self-reference whatsoever. (Footnote: Stephen Yablo. Paradox Without Self-Reference, Analysis 53 (4): pages 251-252. 1993. It can be viewed Paradox Without Self-Reference: here.) It goes like this:
Imagine an infinite sequence of sentences S1, S2, S3, …, each to the effect that every subsequent sentence is false:
The sentence S1 is the sentence: “For all k that are greater than 1, Sk is false”
The sentence S2 is the sentence: “For all k that are greater than 2, Sk is false”
The sentence S3 is the sentence: “For all k that are greater than 3, Sk is false”
and so on.
Now, suppose that some sentence Sn is true. This sentence Sn states that every sentence above it is false. So the sentence immediately above Sn (which is Sn+1) must be false. But Sn+1 states that every sentence above it is false. So, on the one hand, Sn states that Sn+1 and Sn+2 must be false, but on the other hand, Sn+1 actually states that Sn+2 must be false - so therefore Sn+1 is true - contrary to what Sn states.
So, our supposition that some sentence Sn is true is incorrect, and no sentence Sn can be true, and so all the sentences must be false. But each sentence actually states that all sentences above it are false, so they are stating the truth.
This is the paradox. You will see that it is just like the Liar Paradox. Stephen Yablo, who came up with the paradox, claims that it is a Liar-like paradox that is not in any way circular, and so that it proves that self-reference is neither necessary nor sufficient for producing a Liar-type paradox - because none of the sentences actually refers to itself, but only to the subsequent sentences.
But Yablo’s claim that there is no circular self-reference is erroneous. The paradox depends on the way the paradox is presented, where the vague phrase ‘and so on’ is used. This simply conceals the fact that a complete and logical definition is simply:
For all n, the sentence Sn is: “For all k greater than n, Sk is false”
When the paradox is clearly defined like this, instead of relying on the vagueness of ‘and so on’, or similar euphemisms, we can see quite clearly that the definition does of course include a self reference. We have, in the definition, an object ‘S’. So what is this object S? The only information we have on the thing that is ‘S’ is that it is defined in terms of S - which of course is a circular self-reference.
Compare the difference with a similar definition that does not include any self-reference:
The sentence Sn is: “The circumference of a circle is more than n times its diameter.”
The expression, “The circumference of a circle is more than n times its diameter” has one variable term n, and so it is not a proposition, and so the concept of correct or incorrect does not apply to it. But once we substitute a value for the n, we get a proposition which can be either correct or incorrect. For example:
The first sentence S1 is: “The circumference of a circle is more than 1 times its diameter.”
The second sentence S2 is: “The circumference of a circle is more than 2 times its diameter.”
And, crucially, these propositions are completely independent of how they were generated - and so is any decision that is made regarding whether it is correct or incorrect. In other words, anyone could make a decision on the proposition without knowing anything about how it was generated.
But with Yablo’s definition:
For all n, the sentence Sn is: “For all k greater than n, Sk is false”
when we substitute a value for n, for example 3, we get that the third sentence is:
“For all k greater than 3, Sk is false”.
Clearly this is not independent of how it was generated. Taken on its own, it is utterly meaningless, because S is not defined. But if you try to determine what S is, you find that it is defined in terms of itself, and so it is defined as a circular self-reference, contrary to Yablo’s claim.
Note that Yablo’s claims of non-circular self-reference have been disputed elsewhere. (Footnote: Beall, J.C. Is Yablo’s paradox non-circular? Analysis 61(3): pages 176-187, (2001) Beall, Is Yablo’s paradox non-circular?: Details It can be viewed Beall, Is Yablo’s paradox non-circular?: here PDF.) (Footnote: Graham Priest. Yablo’s paradox. Analysis 57: pages 236-242, 1997. Priest, Yablo’s paradox: Details It can be viewed Yablo’s paradox - Priest: here (PDF).)
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There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.
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.
There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
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