A Yabloesque variant of the Bernardete Paradox

Here I want to present a novel version of a paradox first formulated by José Bernardete in the 1960s – one that makes its connections to the Yablo paradox explicit by building in the latter puzzle as a ‘part’. This is not the first time connections between Yablo’s and Bernardete’s puzzles have been noted (in fact, Yablo himself has discussed such links). But the version given below makes these connections particularly explicit.

First, we should look at Bernardete’s original. Imagine that Alice is walking towards a point – call it A – and will continue walking past A unless something prevents her from progressing further. There is also an infinite series of gods, which we shall call G1, G2, G3, and so on. Each god in the series intends to erect a magical barrier preventing Alice from progressing further if Alice reaches a certain point (and each god will do nothing otherwise):

(1) G1 will erect a barrier at exactly ½ meter past A if Alice reaches that point.

(2) G2 will erect a barrier at exactly ¼ meter past A if Alice reaches that point.

(3) G3 will erect a barrier at exactly 1/8 meter past A if Alice reaches that point.

And so on.

Note that the possible barriers get arbitrarily close to A. Now, what happens when Alice approaches A?

Alice’s forward progress will be mysteriously halted at A, but no barriers will have been erected by any of the gods, and so there is no explanation for Alice’s inability to move forward (other than the un-acted-on intentions of the gods, which isn’t much of an explanation). Proof: Imagine that Alice did travel past A. Then she would have had to go some finite distance past A. But, for any such distance, there is a god far enough along in the list who would have thrown up a barrier before Alice reached that point. So Alice can’t reach that point after all. Thus, Alice has to halt at A. But since Alice doesn’t travel past A, none of the gods actually do anything.

Now let’s change the puzzle a bit. Imagine that Alice is an expert logician enjoying her morning walk (which, as usual, passes through point A). Alice will continue walking unless she hears someone utter a paradoxical sentence or set of sentences. Hearing a paradox is paralyzing to Alice, however. Upon hearing such a thing, she will instantly stop in her tracks (and she is able to detect paradoxes instantaneously, the minute they are uttered). Finally, Alice walks in total silence, never uttering a word.

Abstract Light Painting, by Alexander Nie. CC-BY-SA-2.0 via Flickr.

As before, we also have an infinite series of gods G1, G2, G3, … and each god intends to act in a particular way if Alice reaches a certain point on the path past A. But now they are not erecting barriers, but are instead merely making utterances:

(1)       G1 will say:

“Everything the other gods have said so far is false.”

if Alice makes it ½ meter past A.

(2)       G2 will say:

“Everything the other gods have said so far is false.”

if Alice makes it ¼ meter past A.

(3)       G3 will say:

“Everything the other gods have said so far is false.”

if Alice makes it 1/8 meter past A.

And so on.

In short, each god in the series will accuse all of the other gods who have already spoken of being liars, if Alice makes it far enough. Now, what happens when Alice approaches A?

Again, Alice’s forward progress will be halted at A: Imagine that Alice did travel past A. Then she would have had to go some finite distance past A. But, for any such distance, there is a god far enough along in the list (in fact, infinitely many of them) who would have said:

“Everything the other gods have said so far is false.”

before Alice reached that point. Let Gm be any one of the gods whose point Alice has passed. Notice that if Alice passed god Gm, then she also passed all of Gm+1, Gm+2, Gm+3… Now, Gm uttered:

“Everything the other gods have said so far is false.”

when Alice passed the appropriate point (that is, when Alice has reached 1/(2m) meters past A). But before that each of the gods whose number is greater than m (i.e. Gm+1, Gm+2, Gm+3,…) will have already said the same thing about the gods who spoke before them. As a result, Gm’s utterance can be neither true nor false.

Assume that Gm’s utterance is true. Gm’s utterance amounts to his saying that each of Gm+1, Gm+2, Gm+3,… was lying when they made their respective utterances. So each of Gm+1, Gm+2, Gm+3,… must in fact be lying. But then each of Gm+2, Gm+3, Gm+4,… must be lying. But Gm+1’s assertion that:

“Everything the other gods have said so far is false.”

is equivalent to saying that each of Gm+2, Gm+3, Gm+4,… is lying. So Gm+1 is telling the truth. Contradiction, so Gm cannot be telling the truth.

Thus, Gm utterance must be false. But we can run the same argument given in the previous paragraph on Gm+1, Gm+2, Gm+3,… just as easily as on Gm (after all, if Alice passed the point at which Gm makes his utterance, then she also passed all the points corresponding to Gm+1, Gm+2, Gm+3,…). Thus, all of Gm+1, Gm+2, Gm+3,… are lying as well. But then Gm’s assertion that:

“Everything the other gods have said so far is false.”

is true after all. Contradiction again.

Note: The reader who finds the previous two paragraphs difficult may want to consult my previous discussion of the Yablo paradox here.

Thus, Alice cannot walk any distance past A, no matter how short, since doing so would mean she would have to pass a point at which a paradox had already been uttered. So she halts when she reaches A. But, since she doesn’t pass A, no one (neither Alice nor any of the gods) has said anything. So what, exactly, stopped Alice?

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