Yablo’s Paradox - A Liar Paradox without Self-Reference?

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.PDF Paradox Without Self-Reference, Analysis 53 (4): pages 251-252. 1993. It can also be viewed at PDF Paradox Without Self-Reference.) It goes like this:

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, S_n = “For all k greater than n, S_k 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, a thing ‘S’ with a subscript - in fact this is a function with one free variable, here denoted by ‘n’. When this free variable of the function is substituted by some definitive number ‘k’, it must evaluate as some definitive non-variable object. So what is this function S? And what might it evaluate as for numerical values of its free variable? The only information we have on the function ‘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:

For all n, S_n = “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.

For all n, S_n = “For all k greater than n, S_k 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, S_k 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. PDF Is Yablo’s paradox non-circular? Analysis 61(3): pages 176-187, (2001) Beall, Is Yablo’s paradox non-circular? Details. ▪ Graham Priest. Yablo’s paradox. Analysis 57: pages 236-242, 1997, DOI:10.1093/analys/57.4.236. Priest, Yablo’s paradox: Details. ▪ Cook, Roy T. There Are Non-circular Paradoxes (But Yablo’s Isn’t One of Them!) The Monist 89, no. 1 (2006), doi:10.5840/monist200689137, pages 118-149.)

Also, for fun, there is a quite hilarious article that was written about the Yablo paradox by Volker Halbach and Shuoying Zhang. I have written a webpage about it, see Halbach & Zhang’s ‘Yablo without Gödel’ it is well worth reading for the entertainment value of the laughable inanity of the authors’ proclamations.

Other paradoxes