Monday, January 11, 2010

A Question About Truth And Sets

The Liar Paradox is sometimes seen as a problem for "naive truth theory" roughly parallel to the problem for naive set theory posed by Russell's Paradox. In the current literature, however, there seems to be an interesting difference. When it comes to semantic paradoxes, it's not hard to find arguments for truth-value gaps, truth-value gluts or even (as in, for example, J.C. Beall's case) a combination of both solutions. When it comes to the set-theoretic paradoxes, though, lots of paraconsistentists argue for re-capturing the inferential power and intuitive appeal of naive set theory by simply embracing set-membership gluts when it comes to cases like the Russell Set. Other than Penelope Maddy's "Proper Classes," though--an article that's high on technical detail and low on philosophical motivation--it seems pretty hard to find anyone who proposes that we just rehabilitate naive set theory by embracing set-membership *gaps* when it comes to cases like the Russell Set or the set of all ordinal numbers, etc.*

Am I wrong about this? Are there legions of gappy naive set theorists out there I haven't heard of?

If not, does anyone have any thoughts about *why* there don't seem to be any (or almost any), given the level of structural symmetry between the two types of paradoxes?

*And, really, that doesn't seem to be *exactly* what Maddy is up to in that article, although she's at least in the vicinity of that position.


Colin said...

Hartry Field might count, given his views on what he calls "conceptual property theory" expressed in 'Saving Truth'. I think his conception of conceptual properties is roughly the same thing as a gappy naive set theory.

Ben said...

Yeah....interesting point. I agree that Field's paracomplete approach to paradoxes about properties in seem to be closely structurally parallel what gappy naive set theory would probably have to look like, but the issue is sort of complicated by the fact that he seems to be (somewhat strangely, to my way of thinking) content with ZFC-orthodoxy as a solution to the set-theoretic paradoxes, saying e.g on p. 384 of "Saving Truth From Paradox" that “the possibility that we should take a dialetheic attitude toward set theory itself” is “a possibility that Priest takes seriously but I suspect few of his readers will, any case I am unwilling to..."

Granted, to the extent that the sort of properties Field believes in are supposed to have categories of objects that have them and categories of objects that don't, and that this is the source of the problem, there's probably room for doubt about how much of the distinction he's appealing to is terminological--what we reserve the term 'set' for--and how much of it is philosophically substative.