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27. Enough Already with Self-Reference:

Yablo’s Paradox

The Question:

Imagine a school with an infinite number of pupils. Some of them played a prank and Ms. Perplex, the head-mistress, interrogates them, one by one in alphabetical order by their names: Albert, Bertha, Cecily, David,…, Xavier, Yolanda, Zacchary, Zara, Zbulun, Zelda…..

Albert says “all the kids behind me are lying.” Bertha says, “all the kids behind me are lying.” Cecily says “all the kids behind me are lying.” And so on, and so on.

Exasperated, Ms. Perplex wonders “who is telling the truth?”

 

The Paradox:

If Bertha, Cecily and all those following them lie then, indeed, Albert told the truth. But if Cecily, David and all those following them lie, then Bertha actually told the truth; hence, Albert lied. On the other hand, if David and those following him lie, then Cecily had told the truth about them. So Bertha – who had maintained that Cecily et al. were lying – had been lying after all.

Ms. Perplex is at a loss. She cannot believe any of the pupils. None can be unambiguously said to speak the truth…but none can be unambiguously accused of speaking untruth.

 

Background:

In 1985, Stephen Yablo, a doctoral student in philosophy at the University of California at Berkeley, published a paper entitled “Truth and Reflection” in which he practically tripped over a paradox that would be named after him. Towards the end of his paper, he discussed some versions of the Liar Paradox (see Chapter ###) and then gave an example, nearly as an afterthought, of a paradox that was not based on self-reference. As it turned out, it was this example, not his paper, that would make Yablo a household name among aficionados of paradoxes.

Several years after the publication of “Truth and Reflection”, buoyed by the reception that his by-the-way example had received, Yablo published a two-page paper entitled “Paradox without Self-Reference”. In it he, first of all, pointed out that self-reference is not sufficient for a paradox. For example, the Christian hymn “So dear Lord to Thee we raise, this our hymn of grateful praise” refers to itself but is not paradoxical.

But, second of all, he asked the question whether self-reference is necessary to produce paradoxes. After all, if paradoxes could be avoided by eschewing self-references why not, in Yablo’s words, impose an absolute ban on all self-reference? “This may be using a cannon against a fly, but at least it stops the fly,” Yablo cited his critics as saying. Except, as he put it, it does not.

 

Dénouement:

When we talked about the Liar’s paradox, we said that //// enhanced. So, when Albert said his piece, he was actually saying in ‘enhanced mode’, “I am telling the truth when I say that everybody behind me lies.” Well, since not everybody behind him lies – Bertha tells the truth when she says that everybody behind her lies – the second part of Albert’s ‘enhanced’ statement is actually a lie. So, his statement contains both a truth and an untruth. And that’s the basis for the paradox.

Ms. Perplex’s interrogation is a version of Yablo’s example. None of the pupil’s statements refer to their own statements; they each refer to the ones down the line. Nevertheless, they are paradoxical since their veracities cannot be determined. So Yablo made his case: self-reference is not necessary to produce paradoxes.

 

Technical supplement:

Did Yablo really make his case? He was not seeking simply to present yet another paradox, he was after a paradox that did not depend on self-reference. By presenting the paradoxical assertions as a string of statements, each referring to the later ones, Yablo maintained that there is no self-reference.

Critics dispute that. Some of them maintain that by only quoting the first few staements and then ending the description of the paradox with ‘and so on, and so on’, as I did at the beginning of the chapter, is illegitimate.

More convincingly, others maintain that self-reference is not absent in Yablo’sd paradox, but only hidden, as was the case of the ‘loop liars’ inb Chapter ###:

Alice: “Bob’s statement is true.”

Bob: “Alice’s statement is not true”.

In this dialogue, the loop liars do not refer to their own, but to each other’s statements. But the dialogue’s statements do refer to the circle of which they are parts; hence they refer to themselves.

According to Yablo, his string of statements is not similar to a circle of loop-liars, essentially because his statements are arranged in a chain, not in a circle. Nevertheless, critics say, self-reference does exist, though hidden, in Yablos’ paradox in the following sense: by asserting that all statements behind this one, are false, each pupil uses his or her own location in the string as a reference point in order to specify which statements are not true.

So, once again, we have self-reference.

Comments, corrections, observations: