Logic

Are all paradoxes false? That is, when philosophers talk about paradoxes, is it always assumed that there's actually a solution out there which will resolve the problem?

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In his paper, 'The Ways of Paradox', W. vO. Quine draws up a useful classificatory scheme for paradoxes, dividing them up into three groups. 'Veridical paradoxes' are conclusions that seem profoundly counter-intuitive, and yet are actually perfectly sound. An example might be the paradox of the ravens: on the face of it, it doesn't seem right to claim that an observation of a red pencil can lend any support at all to the hypothesis that all ravens are black, but there's a good argument to suggest that actually it does lend a (very) small amount. (Basically, the hypothesis that all ravens are black is logically equivalent to the hypothesis that all non-black things are non-ravens, and the appropriate method of confirmation for the hypothesis in that latter formulation would seem to be to take a non-black thing and to check whether it's not a raven).

'Falsidical paradoxes' are ones where an absurd conclusion seems, on the face of it, to be supported by a good argument, but where further scrutiny can reveal that there was a hidden fallacy lurking in the argument after all. Zeno's paradoxes of motion would be examples of this. Zeno argued that there could be no such thing as motion on the grounds that any such motion would have to involve the crossing of an infinite number of finite distances, which could never possibly be achieved in a finite time. But a better understanding of the mathematics of the infinite, and the realisation that it's actually perfectly possible for an infinite sequence to add to a finite sum, revealed the flaw in the argument.

But then the really scary paradoxes are the 'antinomies'. An example (perhaps not the best, but the simplest) would be the liar paradox. A little reflection should reveal that there can be no consistent assignment of truth or falsity to the proposition, "This sentence is false". If it's true, then it's false; but, if it's false, then it's true. Moreover, if we decide that it's actually neither true nor false, then it turns out that it's false, in which case it's true. Most perplexing! Now, Quine makes an interesting suggestion at this juncture. "One man's antinomy", he writes, "is another man's falsidical paradox, give or take a couple of thousand years." His attitude seems to be that there has to be some hidden fallacy lurking in the line of thought that leads to the recognition of paradox in these antinomies. The universe of logic just can't be inherently self-contradictory. If it seems to us that it is, then we have to work on the assumption that what is faulty is not logic itself, but simply our understanding of it. The difference between antinomies and falsidical paradoxes, on this view, will not rest on whether there is a fallacy there, but on whether we have managed to find it yet. We might smugly pat ourselves on the backs for successfully identifying the fallacy in Zeno's paradoxes, and relegating those from the category of antinomies into the category of falsidical paradoxes: but the fact that we continue to regard several other paradoxes as antinomies just goes to show that there's still a lot of work left for us to do. And I suspect that most philosophers probably would share this attitude, that there must be something going wrong in an antinomy. The challenge -- and it is, in some cases, a very formidable challenge indeed -- is to figure out precisely what.

In trying to understand why Quine or others would not countenance antinomies, or real paradoxes, perhaps it would help to add that the conception of paradox in play here is that of an argument, a collection of premises that entails a conclusion. The arguments that appear to be paradoxes are ones whose premises seem obviously true, whose reasoning seems impeccable, and whose conclusion seems obviously false. So, on this conception what would a real paradox be? An argument that leads from true premises via correct reasoning to a false conclusion.

Now, why be confident that there couldn't be such a thing? Because what we mean by "correct reasoning" is just reasoning that leads from true premises to a true conclusion. To judge something to be a real paradox would then be akin to judging something both to be an apple and not to be an apple. That is not a possible judgment. So, the judgment that there are no real paradoxes doesn't stem from an optimistic confidence in the powers of the human mind to unravel mysteries. We know there are no real paradoxes, and that judgment isn't any more exciting than our judgment that no number is both even and not even.

As Jasper Reid says, however, sometimes we are confronted by arguments that leave us speechless -- we will reject that they are real paradoxes, but for the life of us we cannot say where the error lies.

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