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.