Monday, February 18, 2008

Negation and Denial

I want to post on more recent reading soon, but meanwhile I have one last footnote to the discussion of Barwise and Etchemendy. I didn't include this in the main line of discussion, both because it might be of more general interest and because it can be intelligibly separated from the rest of their argument.

Remember that the oddest upshot of their modelling of propositions as hyper-sets is that (a) classical bivalent logic is correct, such that everything is false if it is not true and true if it is not false, (b) the Liar Sentence expresses a truth-evaluable proposition, and (c) the Liar Sentence somehow gets to be false without also being true. Various purely formal moves nominally validated this status for both the Russellian and Austinian ways of thinking about propositions, but I think the authors end the book all-too-aware of how capricious and counter-intuitive all this sounds. Thus, they end by gesturing in the direction of the distinction between negation and denial. They claim that if they had factored in denial as well as assertion and negation, it would have been clear that the logic of their notion of propositions was entirely classical, but that it would “involve us in untangling one of the most basic conflations in the logical literature, the conflation of negation and denial” and that this “would take us far from the topic of the book.” It's hard, on a snarky level, not to think of Fermat and proofs to long to include in the margins, but never mind that right now. There's still an interesting issue about what role this distinction could contribute here.

The problem is that Barwise and Etchemendy themselves tell us very little about the distinction, or what role they think it should play. They tell us that speech-act theorists are always telling logicians to take the distinction more seriously. Fair enough. But how exactly would it be helpful here?

In one of the best turns of phrase in the book, they say that just as ignoring relativistic effects doesn't cause any problems on a trip to the corner supermarket, but knowledge of those effects becomes vitally important when approaching the speed of light, “when approaching sentences like the Liar, we risk paradox if we ignore the difference between negation and denial.”

Excellent. Sadly, we never find out what exactly the import of that distinction is here.

If anyone has any suggestions to make in the comments, I'd be glad to hear them. Tentatively, though, here's my position:

Is there a distinction? Yes. Is that distinction relevant or useful for the purposes Barwise and Etchemendy are gesturing towards? Absolutely not.

Of course, the distinction between negation and denial could be very relevant to a discussion of the semantic paradoxes if, for example, we were working within a deviant logic that denied the Law of the Excluded Middle and posited extra possibilities 'between' P and ~P. (This is precisely what Barwise and Etchemendy repeatedly deny that they are suggesting.) If, however, we are assuming classical bivalent logic, then denial and negation are still distinct, but the category of propositions we are logically or epistemically warranted in denying will, it seems to be, clearly be necessarily co-extensive with the category of propositions we are logically or epistemically warranted in asserting the negations of. As such, for someone trying to fit the square peg of granting that the Liar is truth-evaluable and that every sentence that is not true is false and vice versa into the round hole of denying dialetheism, it doesn't look like this distinction can be of any use.


Meanwhile, my friend Ryan pointed me towards another webcomic that references dialetheism , although as far as I'm concerned there's nothing in the comic itself that's quite about dialetheism.

Sunday, February 3, 2008

Barwise and Etchemendy, Pt 2 (Austinian Case)

So when we left off our story, Barwise and Etchemendy were modelling Russellian propositions with Aczelian hypersets, and the Liar Sentence came out false, but this fact wasn't included in "the world," because if it was, that would violate the "coherence condition" preventing a set-theoretic proposition-object and its "dual" from both being present in a model.

They describe this consequence as "counter-intuitive." I'd describe it as absolutely incoherent.

In any case, when they turn to Austinian propositions, one of the big advantages of the shift was supposed to be that this "counter-intuitive" consequence can be avoided. Remember, Austinian propositions include the situations they are about--so, e.g. if you are at a card game and you mistake the player holding the 3 of Hearts for Jill and claim that "Jill has the 3 of Hearts," the proposition comes out false even if by coincidence Jill actually is playing cards across town and she does happen to have the 4 of Hearts.

As such, the proposition being expressed by any given use of the LS is about some specific situation, since all propositions are on the "Austinian" model. How does this help?

Well, as in the Russellian case, the LS always comes out false. We can see why by analogy to the behavior of the Truth-Teller ("This sentence is true.") In situations that don't include any semantic facts, the Truth-Teller is false, since there is nothing to make it true. In situations that include semantic facts, the Truth-Teller is sometimes true and sometimes false, depending on what those semantic facts are...i.e. in a situation that includes the semantic fact that the Truth-Teller of that situation is true, it's true, and in a situation that includes the semantic fact that the Truth-Teller of that situation is false, it's false. Where these "semantic facts" about the Truth-Teller are supposed to come from, since they obviously don't supervene on the non-semantic facts, is a bit of a mystery to me, but whatever. Let's move on.

The Liar behaves in exactly the same way, with one major difference. The relevant semantic facts are always excluded from the situation used to determine its truth value. Why? Well, if we allowed them to included, that would give us the result that each Liar was both true and false, and that would violate the coherence rule. So it's in "the world," but not the part of it used to evaluate the truth-value of the proposition.

So the Liar of a situation that doesn't have any semantic facts in it is false--just false and not true, which is a neat trick--and the semantic fact that "the Liar of Situation 1 is false" is included not in Situation 1 but in Situation 2. Sure, Situation 2 has a Liar of its own, which is made false because there is nothing in Situation 2 to make it true (after all, the semantic fact that the Liar of Situation 1 is false is about the Liar of Situation 1, not this new Liar, so it is irrelevant), and its falsehood is a fact not of Situation 2 but of Situation 3. On and on forever.

Two comments seem to be in order about this picture.

The first is that, at the beginning of the book, Barwise and Etchemendy rejected Tarski's hierarchy-of-artificial-languages solution because it was arbitrary, didn't really solve anything on the intuitive level, etc. I agree with them on that. It is interesting, though, given that, that the situation-theoretic hierarchy they posit seems to be structurally exactly parallel to Tarski's languages model.

Secondly, and more importantly, it seems to me that the one fact being excluded from the situations is the only one that cannot be plausibly excluded from them. It is, in fact, the ONLY thing that belongs in them. If the Liar Sentence really does express a proposition, then the one and only thing its about is its own truth value. That semantic fact is the one and only fact of any kind that belongs in its Austinian "situation," so a Liar that really was 'about' a situation without semantic facts, or even without this one relevant semantic fact, is simply an incoherent non-possibility.

As such, taken as a defense of the Law of Non-Contradiction--and, as an attempt to explain why the Liar does not have its prima facie property of being true if false and false if true, that's what it amounts to, even without a dialetheist who actually asserts that it is both true and false as the conscious target target--it boils down to the following argument.

"There are no counter-examples to the LNC, therefore the Liar is not a counter-example to the LNC."

That's a perfectly valid instance of universal instantiation, but those of us who want to defend the LNC in the context of an actual argument are going to have to do better than that.

Friday, February 1, 2008

Barwise and Etchemendy, Pt 1 (Russellian Case)

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.