Mixed Yablo Paradoxes

The Yablo Paradox (Yablo, Stephen 1993) is an infinite sequence of sentences of the form:

S₁:        For all m > 1, S_m is false.

S₂:        For all m > 2, S_m is false.

S₃:        For all m > 3, S_m is false.

:           :           :           :

S_n:        For all m > n, S_m is false.

:           :           :           :

Loosely put, each sentence in the Yablo sequence ‘says’ that all of the sentences ‘below’ it on the list are false. The Yablo sequence is paradoxical – there is no coherent assignment of truth and falsity to the sentences in the list – as is shown by the following informal argument:

Proof of paradoxicality: Assume that S_k true, for arbitrary k. Then, for every m > k, S_m is false. It follows that S_k+1 is false. In addition, it follows that, for every m > k + 1, S_m is false. But given what S_k+1 says (that every sentence ‘below’ it is false), it follows that S_k+1 is true. Contradiction. Thus, S_k cannot be true, and must be false. Since k was arbitrary, it follows that, for any n, S_n is false. So S₁ is false. In addition, for all n > 1, S_n must be false. But given what S₁ says (that every sentence ‘below’ it is false), it follows that S₁ is true. Contradiction.

Not too long after the discovery of the Yablo Paradox, Roy Sorensen published a paper that included a variant of the Yablo Paradox called the Dual of the Yablo Paradox, which we obtain by replacing the universal quantifications (the “for all”s) with existential quantifiers (“there exists”s). Thus:

S₁:        There exists an m > 1 such that S_m is false.

S₂:        There exists an m > 2 such that S_m is false.

S₃:        There exists an m > 3 such that S_m is false.

:           :           :           :

S_n:        There exists an m > n such that S_m is false.

:           :           :           :

Loosely put, each sentence in the Dual Yablo sequence ‘says’ that at least one of the sentences ‘below’ it on the list is false. The Dual of the Yablo Paradox is also, as its name suggests, paradoxical. The proof is left to the reader (hint: the reasoning is a sort of ‘mirror image’ of the reasoning for the Yablo paradox: begin with “Assume that S_k is false, for arbitrary k…)

So there are (at least) two sorts of sentences that might occur in infinite Yabloesque sequences – sentences of the form:

S_n:        There exists an m > n such that S_m is false.

which we shall call Y-exists sentences, and sentences of the form:

S_n:        For all m > n, S_m is false.

which we shall call Y-all sentences (having grown up in the South of the United States, I quite enjoyed writing that!) One obvious question to ask is what happens when we consider infinite sequences of sentences where some of the sentences are Y-exists sentences, and some of the sentences are Y-all sentences – we can call such a sequence a mixed Yablo sequence. The answer is relatively straightforward:

Theorem 1: Any mixed Yablo sequence containing only finitely many Y-all sentences is paradoxical.

Proof: If there are only finitely many Y-all sentences in the list, then there is a k such that all sentences below S_k are Y-exists sentences. Assume, for any j > k, that S_j is false. It follows that S_j+1 is true. In addition, it follows that, for every m > j + 1, S_m is true. But given what S_j+1 says (that some sentence ‘below’ it is false), it follows that S_j+1 is false. Contradiction. Thus, S_j cannot be false, and must be true. Since j was any arbitrary number greater than k, it follows that, for any n > k, S_n is true. So S_k+1 is true. In addition, for all n > k+1, S_n must be true. But given what S_k+1 says (that some sentence ‘below’ it is false), it follows that S_k+1 is false. Contradiction.

Theorem 2: Any mixed Yablo sequence containing only finitely many Y-exists sentences is paradoxical.

Proof: A dual, ‘mirror-image’ of the reasoning in Theorem 1, left to the reader.

Theorem 3: Any mixed Yablo sequence containing infinitely many Y-all sentences and infinitely many Y-exist sentences is not paradoxical.

Proof: If there are infinitely many Y-all sentences in the list, and infinitely many Y-exists sentences in the list, then there is a Y-all sentence ‘below’ any Y-exists sentence, and a Y-exists sentence ‘below’ any Y-all sentence. Assign truth to each Y-exists sentence and falsity to each Y-all sentence. Then any Y-exists sentence will indeed be true, since there will be at least one false Y-all sentence ‘below’ it, and any Y-all sentence will be false since there will be at least one true Y-exists sentence ‘below’ it.

These technical results have the following, rather interesting upshot: When we move from consideration of the Yablo Paradox and its Dual in isolation to a consideration of mixed Yabloesque sequences more generally, it turns out that most of these sequences are not paradoxical. In technical jargon, both the collection of sequences that contain only finitely many Y-all sentences, and the collection of sequences that contain only finitely many Y-exists sentences, are countably infinite – they are infinitely many such sequences, but this collection of sequences is the smallest size of infinite collection. The collection of infinite Yabloesque sequences that contain both infinitely many Y-all sentence and infinitely many Y-exists sentences, however, is a much larger collection. It is what is called continuum-sized, and a collection of this size is not only infinite, but strictly larger than any countably infinite collection. Thus, although the simplest cases of Yabloesque sequence – the Yablo Paradox itself and its Dual – are paradoxical, the vast majority of mixed Yabloesque sequences are not!

Featured image credit: ‘Wormhole’, by Paco CT. CC BY-NC-SA 2.0 via Flickr