Monday, February 07, 2011

Reasoning badly from Yablo’s paradox

Paradoxes can be hard to resolve, so it can be hard to reason well from them. A nice example is a recent argument that the past is finite, by Laureano Luna (2011: ‘Reasoning from paradox’, The Reasoner 5(2), 22–23). I shall vary the details. Suppose there’s a place where, once a year, every year, someone says “No previous utterance here was true,” with nothing else ever having been said there. Details can be varied, so long as we would, were the past infinite, have an infinite sequence of similar utterances. Indeed, it’s because we can vary the details that the following contradiction seems to follow from supposing the past to be infinite (rather than from, say, supposing that language need not begin with evident truths).
......Each utterance in that place concerned the past, so it seems that each utterance should be either true—were none of the previous utterances there true—or else false—were at least one of them true. But none of them can be either, on pain of Yablo’s paradox: Were no utterance there true then, via what each said, each would be true (and not true); but were any of them true, then since none of the earlier ones would have been true, its immediate predecessor would also, via what it said, have been true (and not true).
......But even if such sequences of utterances are impossible, the past might be infinite. One possibility is that simply infinite sequences, e.g. the natural numbers, are indefinitely extensible (are Potential Infinite) in the sense that while there’s always a next element, e.g. a bigger number, there’s no complete collection of them all. Standard mathematics assumes that such isn’t the case, but we’ve yet to discover that it isn’t. And if it is, then although we naturally think of past years as stretching back in time forever, the past couldn’t be the whole of such an infinite sequence, and so our infinite sequence of utterances would’ve been impossible too. And yet the past might, even so, be infinite. E.g. there might have been, before the Big Bang, some infinitely slow process, which took an infinite time to complete (and before which there might have been something else, possibly with no beginning); such a process has an infinite duration in the sense that we might go any natural number of years back into it and not reach its beginning, and also in the sense that were it the unit of time, all the time since the Big Bang would be relatively infinitesimal.
......Another possibility is that truth, or descriptive accuracy, is essentially a matter of degree. We might take an utterance of “No previous utterance here was true” to be asserting that none of those previous utterances described the past well enough for it to be classed as true. Yablo’s paradox would then be ruling out every possibility except the possibility that all those utterances described the past only vaguely, that they were all vaguely true. (Does it seem that they would then have been failing to describe the past well enough to be classed as true? If so, note that the suggestion is that either classification—true or not—would be less accurate than that of vaguely true.) So, our contradiction may well have been due to our having used, in effect, a rather artificial language. So we seem to have shown only that either the language of those utterances doesn’t allow such sequences of sentences, or something else (e.g. maybe the past must be finite, or maybe the natural numbers are indefinitely extensible).

1 comment:

Sam Alexander said...

It's just a matter of truth not being definable. Like most liars paradoxes, Yablo's paradox assumes a truth predicate with certain properties which aren't actually mutually possible.

If it's consistent that the past goes back for at least N days, for every natural number N, then by the compactness theorem, it's consistent that the past goes back infinitely far (granted, people living in the universe constructed by the compactness theorem might have access to nonstandard naturals and thus might *think* their past was finite; that's an interesting thing to ponder on its own)