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“Too Many” Yabloesque Paradoxes

The Yablo Paradox (due to Stephen Yablo and Albert Visser) consists of an infinite sequence of sentences of the following form:

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

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

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

: : :

Sn: For all m > n, Sm is false.

Sn+1: For all m > n+1, Sm is false.

Hence, the nth sentence in the list ‘says’ that all of the sentences below it are false. The sequence is genuinely paradoxical – there is no way to assign truth and falsity to each of the sentences in this list so that a sentence is true if and only if what it says is the case and a sentence is false if not. For some background on the Yablo paradox and variations on it, see my previous discussion here.

There are numerous variations on the Yablo Paradox. Many of these proceed by varying the quantifier used at the beginning of each of the sentences. For example, we obtain the Dual of the Yablo Paradox by considering an infinite sequence of sentences of the form:

Sn: There exists an m > n such that Sm is false.

In other words, each sentence in the Dual of the Yablo Paradox ‘says’ that at least one of the sentences below it is false. We obtain the Schlenker Unwinding (named after Philippe Schlenker) by considering an infinite sequence of sentences of the form:

Sn: For infinitely many m > n, Sm is false.

In other words, each sentence in the Schlenker Unwinding ‘says’ that infinitely many (but not necessarily all) of the sentences below it are false. And we obtain the Yablo Unwinding (named after C) by considering an infinite sequence of sentences of the form:

Sn: For co-infinitely many m > n, Sm is false.

In other words, each sentence in the Yablo Unwinding ‘says’ that all but finitely many of the sentences below it is false.

Here I want to explore another construction that results from substituting a common quantifier into the original Yablo construction. Consider what happens when we replace “for all” with “for too many”. We obtain the following sequence:

S1: For too many of the m > 1, Sm is false.

S2: For too many of the m > 2, Sm is false.

S3: For too many of the m > 3, Sm is false.

 : : :

Sn: For too many of the m > n, Sm is false.

Sn+1: For too many of the m > n+1, Sm is false.

Hence, each sentence in the list ‘says’ that too many of the sentences below it are false. The first step in analyzing this construction is to consider what, exactly, we mean by saying that too many sentences are false. In the present context – an analysis of semantic paradoxes – the following seems like a natural reading:

“Too many of the sentences are false.”

Is equivalent to:

“The number of sentences that are false is (somehow) too large to be compatible with an acceptable assignment of truth and falsity to all sentences in the list.

In short, if too many sentences are false, then that means the list would be paradoxical.

We can now analyze this list to see if there is an acceptable assignment of truth and falsity to each sentence in the list. The first step in doing so is to note that no sentence in the list can be true:

If any sentence in the list is true, then given what it says, too many of the sentences below it would be false – that is, the collection of sentences below it that will be assigned falsity is too large to allow for an acceptable assignment of truth and falsity. So if there is an acceptable assignment of truth and falsity to the sentences in the list, then no sentence is true on that assignment.

But if no sentence on the list is true, then it follows that every sentence on the list must be false. Given what each sentence says, it also must be the case that all sentences in the list being false is, nevertheless, not too many.

So far, this seems okay. We have shown that all sentences on the list are false, and it turns out that even if all of the sentences on the list are false, this isn’t too many. But now consider the bit of reasoning in the offset passage above. The reasoning amounts to a proof of the following claim:

If one or more of the sentences in the list is true, then there are too many false sentences below it.

A new, sort of meta-level puzzle now arises. Combining this claim with our overall conclusion (i.e. all of the sentences are false, but this turns out not to be too many) we arrive at the following conflicting claims:

  1. If all of the sentences in the list are false, then there are not too many false sentences in the list.
  2. If only some of the sentences in the list are false (and hence at least one is true), then there are too many false sentences in the list.

This seems to violate the following, I think rather obvious, principle:

Let X be a proper subset of Y (i.e. Y contains every thing contained in X, and also contains at least one thing not contained in X). Then, if there are too many things in X for some condition C to hold, then there are too many things in Y for condition C to hold.

Thus, despite the appearance of an apparently acceptable assignment of truth and falsity to each sentence in the list, the “too many” variant of the Yablo paradox seems paradoxical (or, at the very least, very puzzling) after all!

Featured image: “Infinity” by MariCarmennd9. CC0 via Pixabay

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