Monday, July 26, 2010

Russell's Paradox and Logical Conservativism (The Russell's Paradox Series, Part I of IV)

Graham Priest often treats Russell's Paradox as the basis of a positive argument for dialetheism. He has even said that in certain respects it is a more invulnerable dialetheist argument than the argument from the semantic paradoxes, since there's no issue of meaningfulness--since it concerns sets rather than sentences, one can't get around it by denying that some sentence is meaningful.

This is, of course, not the only dialetheist position on the paradox. JC Beall, if I understand him correctly, follows Hartry Field in separating the paradox into two versions, the traditional set-theoretic one, and a paradox about the property 'does not apply to itself'--Beall accepts classical orthodoxy about the former and handles the latter dialethetically, just as Field accepts classical orthodoxy about the former and handles the latter by denying the relevant instances of the Excluded Middle. To me, the Field/Beall position on the taxonomy of the paradoxes seems bizarre. Russell's Paradox is about sets, and the 'does not apply to itself' paradox is about properties. They have similar structures, but what of it? Lots of paradoxes have Russell's-Paradox-ish structures, and saying that this means that the 'does not apply to itself' paradox is a version of Russell's Paradox strikes me as making no more sense than describing the Liar as 'the sentence version of' Russell's Paradox. (In fact, I'm inclined to think that, in so far as "applies to" can be paraphrased as something like "is said of itself in a true sentence", the "does not apply to itself" paradox has a lot more in common with the Liar than it does with Russell's Paradox.) In both cases, one can talk that way if one wishes, but it doesn't strike me as shedding much light on anything. I think that things are kept clearer by regarding the 'does not apply to itself' paradox as an interesting puzzle in its own right that doesn't have much of anything to do with Russell's Paradox.

In any case, a third possible dialetheist position on Russell's Paradox might be to take naive set-theoretic realism and the associated contradictions as something that one might as well accept once one has accepted dialetheism, but something which the non-dialetheist has no particular reason to be bothered about unless they are confronted with some separate compelling argument for dialetheism. Someone who held this position might think that, e.g. the Liar Paradox formed the basis for a compelling argument for dialetheism, and consistent solutions to the Liar are severely wanting, so, given the existence of Liar sentences, we should all be dialetheists (at which point we would have no particularly good remaining reason to reject the unrestricted comprehension axiom of naive set theory), but that, in the absence of other reasons to accept true contradictions, there's nothing particularly problematic about orthodox, consistent views about set theory.

Now, I'm not a dialetheist of any sort, and I think that the Liar Paradox can be plausibly solved within a consistent framework. (In fact, once I'm done with the Russell's Paradox series, I've been thinking about starting up a series on my preferred solution to the Liar.) That said, in this and in follow-up posts, I want to argue that dialetheists should accept Dialetheist Option #3.

To see why, let's start with what I'd like to think of as something like the Principle of Logical Conservativism (PLC), which is a special case of a general rule of ontological caution. Here's how I describe it in my dissertation:


Any time we expand our ontology, the following seems like an eminently reasonable principle of caution:

• Until we have concrete evidence to the contrary, we should take whatever principles we previously took to apply to absolutely everything whatsoever to apply to the newly discovered objects as well.

By analogy, when we discover the existence of a distant galaxy that we have not been yet able to send probes to photograph, or SETI messages to search for alien life in, the fact that we can imagine or describe perpetual motion machines gives us no reason to think that aliens living on one of the planets of the newly discovered galaxy have developed such a machine. As far as we know, such machines are impossible. If, however, when we send the probes and the SETI messages, we make contact with aliens living in that galaxy, and they show us what seem to be perpetual motion machines, after careful consideration of their evidence, we should be prepared to revise our old ideas about physical possibility. Just so for sets and logical possibility.


To expand on the thought a bit:

At one point in In Contradiction, Graham Priest says that his view that the cumulative hierarchy of sets postulated by ZFC and similarly orthodox set-theoretic options is an interesting mathematical structure that mathematicians may have their own reasons to choose to concentrate on, but that the whole universe of naive set theory--including sets with contradictory properties, such as the Russell Set and the set of all ordinal numbers--also exists, is "quite compatible" with a view often expressed by more conservative theorists who are agnostic about the existence of any sets outside the cumulative hierarchy. To me, this exactly misses the point--assuming that one has a reason to believe that at least the sets in the cumulative hierarchy exist, a reasonable stance might be (i) confidence that no sets with inconsistent properties, like the Russell Set and the set of all ordinal numbers don't exist, grounded in a general confidence that no objects have inconsistent properties, combined with (ii) agnosticism about the existence of sets that are outside of the cumulative hierarchy but whose existence would be compatible with what we know about the logical structure of reality in general.

The realm of sets, if it exists, is notoriously epistemically inaccessible to us. (In fact, as Benaceraff famously pointed out, if we assume that it exists, it still seems to be the case that, if every set in that realm disappeared tomorrow, we'd never know.) Even if we assume that there's a compelling case for set-theoretic realism--i.e. for the conclusion that at least some sets exist--the question of which sets exist is still very much open.

Different set theories have different comprehension axioms that give us different results about that question, and we don't have any direct evidence that settles the question of which one gets us the right result. The unrestricted comprehension axiom of naive set theory gives us a particularly simple and clear-cut formula for deciding which sets exist, but one which, as far as we know, can't be the right one, since, as far as we know, there are no true contradictions. If we had evidence of the existence of sets with inconsistent properties, then unrestricted comprehension might be the best option, but that's not the situation that we're in.

Now, a fair-minded critic might start to worry at this point that I'm throwing in this conditional--and the "concrete evidence to the contrary" clause in the principle of caution, above--to make my stance on all this sound more fallibilistic and open-minded than it really is. After all, given the Benacerraf-style worries just mentioned, what would ever count as concrete evidence of the existence of inconsistent sets?

All I can say is that, if you share this concern, you should stay tuned for the next installment!

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