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.

Yep.

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.

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## 12 comments:

(above comment had typos, they are fixed here.)

Interesting post. I haven't read the book, but it sounds interesting. (I'm pretty sceptical of non-existence solutions to the Liar, but anyway.)

I'm interested in your claim that Barwise and Etchemendy beg the question. Against who? Dialetheists?

If that's what you're suggesting, I don't think you're entirely right. Dialetheists will often claim that we should avoid contradictions other-things-being-equal. Priest, for instance, thinks we should avoid admitting contradictions unless rationally mandated to do so. (It just that he thinks dialetheism is the *only* coherent option.) So it's not clear that starting out with a principle that rules out contradictions is as question-begging as you think. If the Barwise and Etchemendy project is successful, dialetheism is not rationally mandated. That would count as an argument I think.

For another thing, you seem to suggest that the B&E prject is uninteresting, given that they stipulate contradictions out in the semantics. You say:

"We included a formal rule to ban inconsistency and, amazingly, no inconsistencies were generated by he model!"

That's actually quite hard to do. Since inconsistency often crops up no matter how much you attempt to axiomatize it out. For instance classical logic includes a (semantic) rule to ban inconsistency, and yet it can't handle it's own truth predicate without generating inconsistency. So, it's not clear B&E's project is as trivial as you suggest.

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""""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."""""

Doesn't Russell's type theory also sort of prevent that from happening? Distinguishing classes and sets does not always appear that feasible.

Either way, the ZFC approach seems appropriate, and even somewhat "empirical", however challenging-- especially ZFC outfitted with a proviso denying infinity (especially as a "number"). Yet then would what Steinford logicians have to work upon, apres ZFC? Yikes--they might be required to prove something, or like stooop to economics or somethin'

On "groupings": the basic idea is that what lets the Russell Set or Russell Grouping exist is the adoption of a naive comprehension schema. ZFC goes with an axiom that restricts comprehension and forces sets to be built from already-existing sets, which effectively guarantees the Russell Set not to exist. As long as the axioms of "grouping" theory invoke the analogue of restricted comprehension, we circumvent the Russell Grouping as well. Since the only thing Aczel alters is the axiom of foundation, this essentially has no effect on the restricted comprehension already in place in ZFC.

Aaron,

Two points about begging the question.

First, I admitted that it's not entirely fair to accuse them of begging the question against dialetheism, given that "The Liar" came out in 1987, but that if the project involved in any attempted 'solution' to the Liar Paradox is surely to show that the Liar doesn't really have its apparently strange properties of being true if it is false and false if its true. When you do this by instituting a formal rule saying that *nothing* has those properties, and you are willing to bite bullets of any size whatsoever--e.g. "the Liar is false but the fact that it is false is not part of the world"--and justify it by reference to those formal rules, you are assuming precisely what needs to be proved (i.e. "begging the question," with or without an enemy representing the worst case scenario position in the conceptual space.) The whole thing at a certain level boils down to "this is not a counter-example to the LNC, and I know that, since Rule 21 states that there will be no counter-examples to the LNC!"

(2) Sure, dialetheists don't want to "mutliply contradictions beyond necessity," but Priest always regards purely formal solutions to what he regards as the obvious candidates for true-contradiction status as begging the question against dialetheism by simply assuming that the truth is consistent and working from there instead of showing it to be so. (Question-begging is always the first charge he lobs when criticizing attempted solutions to the paradoxes that motivate dialetheism.) I think he's exactly right about that, and that Barwise and Etchemendy are offering a purely formal solution par excellance.

There is also, I think, a fundamental difference between saying that the idea that true contradictions are impossible in principle and saying that, while we can make sense of them and, in fact, the best evidence suggests that there are some (the Liar Sentence is both true an false, the Russell Set both is and isn't a member of itself, etc.), we should be conservative about it and limit the number of them that we posit. Given the former position, an arbitrary formal rule against contradictions might be OK in certain contexts (although still radically illegitimate, I think, in contexts where the LNC itself is being defended), but given the latter, it would pretty much always be the height of irrationality to ban real (if, in any given domain, improbable) possibilities before the investigation begins.

J.,

How and in what sense is ZFC "empirical?"

I don't get it.

Colin,

Sure, if you instituted the right sort of formal rules regulating what kinds of groupings existed, exactly parallel to those regulating sets, you could prevent a Russell Grouping, but my question was whether, just used informally as a term that encompassed both sets and classes, one would be making some sort of category mistake or incoherent claim by asking whether the Russell Grouping was a member of itself? If not, it seems like the set/class distinction may be (while mathematically useful for delineating sets as a well-behaved universe for a certain kind of study) beside the point philosophically for the debate about whether there are true contradictions, since, on the face of it, a Russell Grouping is just as dangerous to the LNC as a Russell Set.

On ZFC vs. Aczel, I'd say that my worry pretty much applies to both, but that the issue is called attention to in Aczel, since by allowing "hypersets," we've admitted that the notion of a set outside of the cumulative hierarchy--e.g. one that's a member of itself--is a perfectly coherent idea, so all that's left to prevent the Russell Set is the set/class distinction.

(Of course, I've tended to think that the only reason some people who can see Tarski's hierarchy of artificial languages as a totally unworkably arbitrary "solution" to the semantic paradoxe don't think the same about ZFC and the set-theoretic paradoxes is that we all speak natural languages involving the truth predicate, whereas we don't actually speak set-theoretic jargon in ordinary life, so the artificiality of the solution doesn't jump out as vividly.)

Ben, I dunno I guess I don't have any intuitions about groupings or whether we have some natural analogue of set theoretic concepts. But on Aczel, you've got something wrong. You say, "all that's left to prevent the Russell Set is the set/class distinction." It really doesn't depend on any distinction

betweensets and classes, it just requires that the conception of sets be one according to which there is no Russell Set. This is guaranteed by axioms of set theory which are preserved even in Non-Well-Founded set theory and so there is no Russell Set. This is entirely independent of what one thinks about classes or their putative relation to sets.The liar paradox/epimenides is semantic; Russ. paradox (and Cantor's) was logical (or set theoretic). The two are often conflated. It might be recalled Cantor's set theory was hardly accepted in toto by the mathematicians (or brighter logicians, presumably) people of the day. Infinity (not to say multiple infinities) is considered absurdity by some "finitists" and/or constructivists.

ZFC doesn't really posit any sets, universal or otherwise (tho' he does include an infinity axiom, right). We don't pretend to be set theorists, but ZFC's "separation" axiom limits predication to a subset, right. So the problem (if there really is a big problem) with self-referentiality has been prevented. Yet the hierarchy suggested in the separation schema seems somewhat akin to Russ. theory of types; or for that matter Tarski's metalanguage, used to make truth statements about object language (or observation language, according to Quine).

If that restricted subset in ZFC is construed to be akin to an "object language" the hierarchy (while more formalized) seems rather Tarskian. The real question then (at least for filosophers, instead of mathematicians) concerns the status of mathematical objects (not to say predicates): which is to say (however crudely) calculus has a lot more to do with the real world than does set theory (and did things pretty well even before set theory came about).

Colin,

Thanks for that. I would point out, though, that as far as the set/class distinction being what remained in the non-well-founded universe to prevent paradox, I'm pretty sure that the mistake traces back to Barwise and Etchemendy's presentation. I'll look up the quotes when I get home tonight.

J,

ZFC postulates the existence of sets in exactly the same way that the old naive set theory did. Just not as many.

Now, you don't have to be a mathematical realist about the literal truth about ZFC--although I'd take a Quinean line on this, and say if you think there are true statements about the ZFC set universe and you can't paraphrase the sets away, you are a realist about sets--but this is surely a quite separate question from what ZFC per se postulates.

Colin,

OK, I found the quote. It's in Chapter 3, on p. 45 of my edition of the Barwise and Etechemendy book, at the beginning of the section entitled "The consistency of ZFC/AFA":

"...On Zermelo's conception the Russell 'set' is actually the universe of all sets. And since this is a proper class, not a set at all, the familiar reasoning that derives a contradiction from the definition of

zis blocked But the set/class distinction is the key here, not the banning of self-membership."On Aczel's conception, we still have the set/class distinction, only now there is a proper class of sets that

docontain themselves, as well as a proper class thatdo not. (See Exercise 14.) In both cases there is no Russell set, only a Russell class...."So I would argue that if I'm confused about the importance of the set/class distinction in preventing paradoxical sets once self-membership is allowed, so are they.

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