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

S 1 : For all m > 1, S m is false.

S 2 : For all m > 2, S m is false.

S 3 : 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 S 1 is false. In addition, for all n > 1, S n must be false. But given what S 1 says (that every sentence ‘below’ it is false), it follows that S 1 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:

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

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

S 3 : 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