Last weekend, I read Jon Barwise and John Etchemendy's book "The Liar: An Essay on Truth and Circularity." The post is in two parts, divided up in a way that I hope will be intuitive.
The book is, as the name suggests, an attempt to get around the Liar Paradox. I'm sure anyone likely to glance at this book already knows plenty about it, but just for easy reference, here's the Liar Sentence again.
LS: This sentence is false.
Now, in all fairness to them, the book came out in the same year as "In Contradiction," so you can't blame them for not arguing convincingly against dialetheism, which was after all not much on the scene in 1987. On the other hand, to the extent that they are trying to "solve" the semantic paradoxes, though, they are at least trying to show that e.g. the Liar Sentence does not have its prima facie feature of being true if it is false and false if it is true. In practice, the set of arguments that would beg that question is necessarily co-extensive with the set of arguments that would beg the question against the dialetheist's position that the Liar Sentence really *is* both true and false.
Barwisee and Etchemendy argue that sentences like the LS, or, more precisely, the propositions expressed by those sentences, are false, but that, remarkably, they are not also true. And this despite the fact that they assume bivalance--every proposition is either true or false, and there are no truth value gaps.
How do they accomplish this minor miracle?
First things first, they model propositions as set-theoretic objects. This doesn't sound possible for self-referential sentences, given that standard Zermello-Frankel set theory forbids self-membership. As such, they go with Peter Aczel's "universe of hypersets," an exciting-sounding phrase for an alternate set theory (which contains the Zermello-Frankel universe of well-founded sets) in which circularity is permitted. (The sets in the orthodox hierarchy exist in Aczel's conception of the universe of sets, they just don't exhaust it. "Hypersets" are just sets outside of the hierarchy.) On the face of it, the adoption of Aczel's set theory to shed light on the semantic paradoxes sounds like it would let set-theoretic paradoxes through the back door, since these are kept out of orthodox set theory by the cumulative hierarchy of sets where sets can only have sets beneath them in the hierarchy as members. Not so, Barwise and Etchemendy claim, since Aczel has shown that we don't need the cumulative hierarchy to ban things like the Russell Set (the set of sets that are not members of themselves), we just need the set/class distinction. The Russell Set is not as set at all but a class, and there's no class of sets that are not members of themselves.
(If this sounds dangerously arbitrary, dear reader, I'm with you. In fact, here's a question I haven't been able to get a good answer to yet. If anyone wants to help me out in the comments, please jump in, since I suspect that I'm missing something important here. If we invented a new term that was neutral between sets and classes, like "grouping," where a grouping can be either a set or a class, then can't we re-instate the paradox by reference to a Russell Grouping? If not, why not?)
In any case, using Azcel hypersets, they establish models of propositions for both the "Russellian" and "Austinian" conceptions of propositions. (It's not clear how much of a historical/exegetical claim they are making about either Russell's or Austin's actual views in using these terms.) True Russellian propositions are made true by the world as a whole, and true Austinian propositions are made true by particular situations. (For example, the Austinian proposition "Jill has the 3 of Hearts," about a particular game, is false if the speaker has mistaken Lucy for Jill, but by coincidence Jill has the 3 of Hearts in another game across town.) Barwise and Etchemendy prefer the Austinian conception, although they first model Russellian propositions not just because problems in that model set up the Austinian model, but because (like the relationship of Zermello-Frankel sets and Aczel's hypersets) the Russellian sets are ultimately contained in the universe of Austinian sets.
Now, once we start the formal modelling--and this is true for both the modelling of Russellian propositions and later for the model of Austinian propositions--we get to my first and most important problem with the book.
(1) They blatantly beg the question, but making one of the rules of the model that no set-theoretic proposition-object and its "dual" will be included. Translated out of the set-theoretic context, this means that they set it up as a rule in advance that no proposition will be true and false. "We included a formal rule to ban inconsistency and, amazingly, no inconsistencies were generated by he model!"
This rule has, as you might expect, some funny consequences once the whole thing gets going. In the Russellian case, the result is that, since the world does not make the LS true, it is false, but the fact that its false is not included in "the world." Yes, you read that right. The world does not include the fact that it is false. Why not? Because, if it did, then the LS would be true as well as false, since it claims that it is false and the world which makes it true or false would include the fact that it is indeed false.
Now, they trumpet the fact that the fact that the LS is false gets to be included in "the world" as a great advantage of the Austinian alternative. On the face of it, it would be....except...well....we'll get to that next post.