Monday, April 04, 2011

Russell, Cantor, Curry, and Simmons’ paradox

This post is the fourth part of Vaguely True Liars.

The paradoxes of infinity being tangential here, I’ll just touch upon Russell’s paradox, which concerns collections (although it was originally formulated in terms of predicates). The obvious collections – e.g. the class of all chairs – don’t include themselves, as members (as one of the collected things), but some might, e.g. the collection of all the non-chairs would not itself be a chair. So, let the collection of all the non-self-membered collections be called ‘R’. One member of R is the class of all chairs; but is R a member of itself, or not? If it is then it’s self-membered, so it shouldn’t be; but if it isn’t then it’s eligible to be, so it ought to be. That’s Russell’s paradox. The obvious resolution – akin to my first suggestion for Grelling’s paradox – is to consider, not R, but the collection of all the other non-self-membered collections.

If I’m right about Grelling’s paradox, predicates won’t always be associated with classes, so it would be reasonable to resolve these two paradoxes differently. But of course, things are not quite that simple. We should, for example, replace ‘non-self-membered’ with ‘well-founded’ if we want to rule out such possibilities as a collection V that contains U and all the other non-self-membered collections, where U contains V and all those others. And then there’s Cantor’s paradox: If there was a (well-founded) collection of all the other (well-founded) collections, then there would be more sub-collections than there are collections, via Cantor’s diagonal argument, which shows that any collection – even an infinitely big one – has more sub-collections than it has members (at least if it’s a well-founded and non-variable collection) [i]. However, a sub-collection (of some collection) is just another collection (of some of the original collection’s members), and so Cantor’s argument is that any collection of all the others would be bigger than itself. For such reasons, Russell’s paradox may well belong with the paradoxes of infinity [ii].

So let’s look next at a paradox due, in its essentials, to Haskell Curry. Suppose I say ‘if what I’m now saying is true, then pigs fly’. That’s clearly making the same type of assertion as my earlier Liar statement. But if we generalise ‘pigs fly’ to ‘the Pope’s next assertion is true’, we get the following argument for papal infallibility. Suppose I say ‘if what I’m now saying is true, then the Pope’s next assertion is true’. And suppose, just suppose, that what I said was true. What I said was that if what I said was true, then so was whatever the Pope then said. So we are supposing that if we suppose that what I said was true – as we are doing – then we are also supposing that what the Pope said next was true. So we are, in effect, supposing (just supposing) that what the Pope said was true. To recap, we supposed that what I said was true, and thereby supposed that what the Pope said was true. But that means that what I said – that if what I said was true, then so was what the Pope said – was true, and hence that what the Pope said next was true.

But of course, the Pope might, for all we know, have said that pigs fly, so the argument was certainly fallacious. It would be interesting to consider why it was; but what’s most apposite here is how unlike the Liar paradox it was. The most obvious difference is that the word ‘not’ did not appear in it (although ‘not’ could be introduced via the material conditional) [iii]. Furthermore, by deriving fact from mere possibility, Curry’s paradox can seem more like the Ontological Argument, than the Liar paradox. Now, I suppose that how we resolve my particular example should depend upon how we treat future contingents, as well as how we treat self-reference. Regarding the former, note that what I said would have been like my Liar statement (or possibly true), had the Pope next said something false (respectively true).

But in view of the latter, Simmons’ paradox (which I recently considered in the post that that link links to) is more apposite, because that paradox can be resolved by noting that reference is in general a matter of degree. So to sum up, it may well be that a natural kind of paradox arises from our tendency to ignore – to see past – the imprecision of description. As we use language, we naturally focus upon apposite elements of truth and falsity – much as we see the picture, not the pixels – and so we tend to overlook such possibilities as Liars being vaguely true. An obvious danger is that, by considering too few possibilities, we misconstrue the evidence for our favourite logic. Indeed, since standard set theory was developed in response to such paradoxes of infinity as Cantor’s, there’s some danger of the language of modern science having been built on shaky foundations [iv].

[i] For the mathematical details, see any introduction to set theory. For a mathematician’s view of the philosophical details, see Peter Fletcher, ‘Infinity’, in Dale Jacquette (ed.), Philosophy of Logic (Amsterdam: Elsevier, 2007), 523–585.

[ii] For more on the kind of paradox that includes Russell’s and Cantor’s (and the Burali-Forti), see Stewart Shapiro and Crispin Wright, ‘All Things Indefinitely Extensible’, in Agustin Rayo and Gabriel Uzquiano (eds.), Absolute Generality (Oxford: Clarendon Press, 2006), 255–304.

[iii] For some discussion, see Graham Priest, Beyond the Limits of Thought, second edition (Oxford: Clarendon Press, 2002), 168–9, 278.

[iv] For historical details, see Ivor Grattan-Guinness, The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel (Princeton University Press, 2000).

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