Sunday, December 23, 2007

Reading List

I got the preliminary reading list for my qualifying exams today. Here it is.


Jon Barwise and John Etchemendy, The Liar.

JC Beall (ed.), Liars and Heaps.

JC Beall and Brad Armour-Garb (eds.), Deflationism and Paradox.

George Boolos, Logic, Logic and Logic.

Vann McGee, Truth, Vagueness, and Paradox.

Michael Hallett, Cantorian Set Theory and Limitation of Size.

Colin McGinn, Logical Properties.

Penelope Maddy, Realism in Mathematics.

Robert Martin (ed.), Recent Essays on Truth and the Liar Paradox.

Graham Priest, In Contradiction (2nd edition).

Graham Priest, Doubt Truth to Be a Liar.

Graham Priest, Beyond the Limits of Thought (2nd edition).

Graham Priest, Towards Non-Being.

Graham Priest, JC Beall and Brad Armour-Garb (eds.), The Law of Non-Contradiction.

Saturday, December 8, 2007

Rawls and Dialetheism

Some things I've been thinking about while wrapping up my "History of Ethics" paper....

Rawls, in his "Theory of Justice," makes it clear that his primary enemy is utilitarianism. His grand project is to come up with a plausible more or less Kantishly-flavored alternative to it. Despite this, he praises utilitarianism for giving us a single consistent principle of justice, thereby ruling out the possibility of conflicting obligations. He admits that this would be best, but since he doesn't find their one principle defensible, he thinks the next best thing is to postulate a lexical ordering of obligations, such that when obligations stemming from Principle A conflict with alleged obligations rooted in Principle B, the A-obligations always win and so on down the line.

Now, to the extent that I'm a moral realist (which I try to be, or at least I do on Mondays, Wednesdays, alternate Tuesdays and maybe on Yom Kippur even when it falls on one of the other days), I take it for graned that it is never true that one is morally obliged to do P & ~P. (On Thursdays-through-Sundays, I'm even more sure of it!) So, programmatically, my sympathies are entirely with Rawls here.

What I find interesting, though, is that he makes no arguments, none whatsoever, to tell us *why* to think this should be the case. In what is, at least to me, one of the most interesting lines in the whole of the ToJ, Rawls attacks intuitionism by saying that if we don't have a knowable principle for deciding between conflicting prima facie obligations, "the means for rational discussion come to an end."

So....why? Let's put it this way. There are at least three options when dealing with a prima facie obligation to do P and a prima facie obligation to do ~P.

(1) Use some principle to decide between P & ~P, as Rawls and his utilitarian opponents both do.
(2) Admit, as Rawls castigates the inuitionist for doing, that there's no way to decide, that you just have to go with your gut on a case-by-case basis, but take it as a given that of course the conjunction of an action and its negation can't be obligatory.
(3) Say, as Graham Priest does in his chapter on Philosophy of Law in "In Contradiction," that there is no general reason to assume that contradictory prima facie obligations need always to be merely prima facie.

Now, perhaps Rawls is right that there's nothing much to discuss given (2)--it's hard not to think of Stephen Colbert's inspired White House Correspondents dinner riff on knowledge based on the gut rather than on the head--but Rawls takes it as a given that (2) being unpalatable, we must go with (1). Why?

Well, if we assume that the underlying logic of this rational discussion need be classical, it's certainly the case that in standard classical logic extended with deontic operators, you can conclude ~O(P) from O(~P), and hence [O(P) & ~O(P)] from O(P & ~P). This could be seen as problematic due to the alleged explosiveness of contradictions--that is to say, on the assumption that the underlying logic of Rawls' "rational discussion" is classical, and [O(P) & ~O(P)] is sometimes true, then any randomly chosen O(Q) would also be true. Hence, if in a lifeboat situation, you were morally obliged to save your sister and your mother from drowning, and it was impossible to do both, then it would follow that you were morally obliged to go around killing puppies. The strong moral intuition that we are not in fact so obliged might be considered, on an intuition-reliant reflective-equilibrium sort of model of moral reasoning, to be taken to be pretty good evidence that we never have inconsistent obligations.

So far, so good. By why should the means of rational discusison about morality have to be based on classical logic? It seems fairly clear that--since, as we've seen in earlier posts, the Duns Scotus proof follows from logical rules that only make sense if we assume that the Law of Non-Contradiction is universally true--if there are indeed inconsistent obligations, then the underlying logic of rational moral discussion had best be paraconsistent.

So, given that, why should we rule out (3)?

A moral philosopher could leave the refutation of dialetheism to the logician and assume that, since dialetheism is false, there are no moral dialetheias, but if inconsistent obligations are taken to be part of the motivating evidence for dialetheism, the burden is distributed the other way around. (Indeed, when Graham Priest came to Miami last spring, he told me that inconsistent obligations are the most compelling cases of true contradictions. While his primary focus in this work is on the philosophy of law, he's very clear in the chapter on this in "In Contradiction" that any normative system is likely to give rise to similar examples.) As such, the logician needs the moral philosopher to do his part for the refutation of dialetheism by giving us independent grounds for supposing moral obligations to be necessarily consistent.

The obvious move is to say that "ought implies can," and we can't engage in contradictory actions. You can't both save your mother and (by saving your sister on the other end of the lifeboat) not save your mother. Even Graham Priest, who has a detailed argument in "Doubt Truth to Be a Liar" that there are no contradictions in the "observable world," would grant this much. (As a side note, I think that his argument for the consistency of the observable world is extremely dubious, and that if the Law of Non-Contradiction is not universally and necessarily true--which of course I think it is--there are no particularly good reasons to believe the observable world to be consistent. BUT that would get us well off-track from the present discussion, so for the moment, I'm happy to grant Priest the point.) But why should we suppose that ought does indeed imply can?

After all, in contemporary work on moral responsibility, that principle is less secure than ever. Frankfurt in particular has given us some extremely compelling thought experiments that pump our intuitions in the opposite direction. In his cases, we do in fact morally judge people even when they could not to otherwise, and statistical evidence from experimental philosophers have shown that most people's pre-philosophical intuitions lie with Frankfurt.

So what's a moral realist anti-dialetheist who takes Frankfurt's examples seriously to do? There may be a way of reconstructing "ought implies can" that freely admits that "can" need not be an actual physical possibility for a given agent as required in libertarian (or even traditional compatibilist) conceptions of free will, but that is still carefully enough construed to rule out things that are never physically possible for any agent under any circumstances, but caution and independent grounding would be sorely required here to avoid making this completely ad hoc and question-begging.

Anyone have any ideas?

Friday, November 30, 2007

And The Part Of Aristotle Will Be Played By Chaospet...

A Certain Grad Student Who Writes A Web Comic just did one on dialetheism and mentioned this blog.

Whatever the philosophical merits of the particular argument in question, it made me laugh.

(The archives are worth browsing too. Sadly, most of the comics aren't about philosophy of logic, but they all make me laugh, particularly things like this, this and of course this. "Oh no, a consequentialist!")

Thursday, November 29, 2007

One Truth Or Two?

In the comments thread on the last post, Brandon made the following suggestion:

"What the dialetheist is essentially doing is suggesting that 'true' is equivocal: there is an unadulterated truth, so to speak, which is classical T; and there is another kind of truth, which is truth as found united to falsehood, which union some dialetheists represent by B."

I don't think this is entirely right, although after explaining why not, I'll circle back around to the question of why I think this might be getting at something important. I'll leave *that* as an open question for commenters, since I'm not entirely sure what to say about it.

This does seem to tie into a larger question about the relation of logical rules and the meaning of logical terms. Just because in classical logic it is taken as a given that something cannot be true or false, does it follow that "true" *means* "true, but not also false"? Quine, in his 1970 "Philosophy of Logic" book, seems to think so, suggesting (I think...I certainly don't have the book in front of me) that anyone who thinks that (P&~P) might be true is changing the *meaning* of the negation sign. Other philosophers and logicians, certainly, have seen these issues differently, as a matter of genuine disagreement between different theories about the same logical concepts, rather than a matter of the same terms being used to denote different concepts. Intuitively, I find the latter view a lot more compelling, but I'd be the first to admit that that's not an argument. In any case, for a moment at least, let's put that view to one side.

So, when a dialetheist says that a proposition is true *and* false, what do they mean by "true" and is it different from what they mean by "true" when they say that a proposition is just true?

Graham Priest, at least, in laying out his own "Logic of Paradox" (LP)--see, e.g. his brief exposition of this in his article "What is so bad about contradictions?"--is very clear on this point. Here's how he puts it. In classical logics, truth value is thought of as something like a function, where 1 (True) or 0 (False) is assigned to each proposition. In the LP, truth value is thought of as a relation, such that any given proposition can be related either to 1 (True), to 0 (False), to both 1 and 0 (True/False) or to neither.

So I think that, on this new understanding, when the dialetheist denies the Law of Non-Contradiction, claiming that some propositions are related to both 0 and 1 instead of only to one of them, they aren't really saying that there are two kinds of truth, one that excludes falsehood and one that is compatible with it, but that there's only one kind of truth, and that some propositions are true but not false and some propositions are true (in *exactly the same sense*) but also false., *that said,* everything I've said in the last couple of paragraphs has been internal to the dialetheist's description of what they are up to. I think that there *is* a legitimate question, from an outside perspective, about whether LP or any other logic featuring "truth value gluts" *really* models a situation in which some propositions are both really true and really false, or whether they are simply three-value logics where one truth value has been arbitrarily labeled "true and false," but could just as easily be thought of in some other way. This was, in fact, my initial reaction when glancing at "In Contradiction" for the first time, and, although I'm no longer convinced this is the case, at least a faint glimmering of suspicion remains in my mind on this point.

Put slightly less tendentiously, even if the third truth value really is related in some interesting way to the first two that captures some important element of paradoxical scenarios, is it, as Brandon suggests, a matter of a union between falsehood and a different thing that the word "true" can mean when altered to fit that context, or a union of what "true" always means and what "false" always means? Priest, at least, claims to be talking about the latter, but are there good reason to suspect that in practice he's really talking about the former?

I will note at least one unpleasant consequence that may hold for dialetheists if the "two truths" interpretation is correct.

"This sentence is not true" remains as puzzling as it was before the abandonment of the Law of Non-Contradiction.

Friday, November 23, 2007

Why Contradictions Don't Explode (or, "How Not To Argue Against Dialetheism," pt. 1)

If you've ever taken an intro logic class and you asked your professor how they know there are no true contradictions, you probably get some variation on the proof that anything follows from a contradiction. This is a neat little proof, and a lot of very smart people have taken it to settle the question. Sadly, it doesn't.

Sure, taken as an argument that "if there are true contradictions, using classical logic is probably not a good way to model how they work," there's nothing wrong with it--it is, after all, a perfectly valid proof--buf if it is taken as an argument against the possibility of true contradictions, it is (a) somewhat beside the point, and (b) radically question-begging.

In case anyone reads this who missed the first two posts, let me emphasize that I don't believe in the possibility of true contradictions. If it were really this easy to refute dialetheism, I'd be a happy man...although I suppose I'd also have wander off and find another topic to worry about. (I can imagine various theistic philosophers thinking the same thoughts when it occurred to them that the ontological argument might really work, that it might be *just that easy* to come up with a knock-down argument for the existence of God.) For better or for worse, it isn't this easy.

John Depoe, who I went to grad school with for a few semesters when we were getting our MAs at Western Michigan, is one of the very smart people I referred to above who apparently takes it as settling the question. I'm going to pick on him, not because he says anything unusual about it, but because he blogged about it not so long ago, so he's easily quotable and because his post is one of the clearest statements of explosion-as-an-argument-against-true-contradictions. Here's what he had to say:

"The possibility of affirming logical contradictions is strongly unintuitive, but this is not the only reason to reject the possibility of affirming logical contradictions. For those who lack this intuition (or perhaps the intuition isn’t very strong), you can consider the following proof, which shows that if you accept any logical contradiction, you can prove anything.." After giving the proof, he concludes:

" you can literally prove anything given the assumption of a logical contradiction. This is sometimes called “explosion” because once you grant a logical contradiction, you can repeat this proof over and over indefinitely, potentially proving every possible proposition is true. So, even [if] you lack the inuition that logical contradictions are impossible, this little exercise is a reminder why contradictions cannot ever be affirmed."

This sounds like a pretty devestating argument. I mean, if someone told you that if some other claim P were true, everything else would be true too, that would be a problem. We we all know that there are at least some falsehoods, right? If P entails everything, P must be false itself.

First of all, hold on. As alluded to above--point (a)--I'm worried this may be somewhat beside the point. OK, so if you can derive any Q (or ~Q) from (P&~P), then everything is both true and false. That might be embarassing for a regular, moderate dialetheist who wants at least some things to come out as just-true or just-false, but I'm not sure how much comfort it can really provide those of us who believe reality to be consistent. (This is somewhat analogous to the way that some people treat the fact that some versions of global skepticism are hard to coherently formulate as a great victory for global realism. Does that really give the rest of us any more confidence that we are justified in *our* beliefs about the external world?) Maybe all it proves is that reality really is globally inconsistent.

Perhaps (a) doesn't bother you. You say, "if I can be sure of anything on the basis of my immediate experience of the world, it's that some statements are false (and not also true.) If dialetheism generates the possibility that I'm wrong about that, then dialetheism is wrong."

Fair enough. There is, however, a much more serious objection, the one alluded to as point (b) above. This argument doesn't actually show that if there really were true contradictions, everything would follow from them. Why not? Well, let's take a look at the proof itself.

1. P & ~P
2. P (from 1 and &-elimination)
3. P v Q (from 2 and v-addition)
4. ~P (from 1 and &-elimination)
5. Q (from 3, 4 and Disjunctive Syllogism)

The problem is that step 5 begs the question. If there are any true contradictions, any at all--even if out of the infinite list of contradictory propositions, only one of them is true--then it follows that Disjunctive Syllogism is not a valid rule of inference. Remember, whether your logic is classical or paraconsistent, a disjunction is true as long as at least one of the disjuncts is true. If P is true and Q is false, or if P is false and Q is true, or if both P and Q are true, then (P v Q) is true. So far, so good. But what about the possibility that P is both true and false, while Q is just false? If that's a possibility, then the disjunction of P and Q could be true, ~P could be true and Q could still be false. This is *not* a matter of arbitrarily tweaking the rules or piling on a second absurdity to save the first. Quite the opposite. This result simply follows from the assumption that it is possible that some statements of the form (P&~P) could be true. In that scenario, plugging in Disjunctive Syllogism would at least sometimes result in generating a false (but not true) conclusion from true premises.

Now, of course, you might object that Disjunctive Syllogism *is* a valid rule of inference. After all, since it is impossible for a statement of the form (P & ~P) to be true, there is no possible scenario where (P v Q) will be true, ~P will be true and Q will be false. I agree. This is, however, to assume precisely what is supposed to be proved...that there are no true contradictions. It's not that impressive that, starting from the premise that there are no true contradictions, you get the result that absurd things would be true if there were any true contradictions.

The result could be put like this. It is false that anything would follow from a contradiction. There is, however, one (and only one) contradiction from which anything follows...the contradiction involved in assuming that the Law of Non-Contradiction is true by treating Disjunctive Syllogism as a valid rule of inference, and assuming that it is false by starting the proof with (P & ~P). This is, sadly, a terribly uninteresting result.

Wednesday, November 21, 2007

Qual Paragraph

OK to kick off, here's the little paragraph I had to put together with my topic for my qualifying exams next semester, to give some sense of the general scope of the project.


General topic: philosophy of logic, with a focus on dialetheism and the semantic and set-theoretic paradoxes

The semantic paradoxes, like the Liar and it's multi-sentence equivalents, are the best-known cases of prima facie true contradictions. The set-theoretic paradoxes seem to constitute equally good evidence for dialetheism. An artificial, purely formal solution is much more widely accepted in the set-theoretic than in the semantic case, but this sort of move seems equally arbitrary in either case. Explosion-based arguments against dialetheism are radically question-begging for familiar reasons, and at best demonstrate the existence of too many true contradictions, as opposed to none at all. In fact, this begins to look like an innate structural imbalance in the argument, since any argument either for or against true contradictions must take place either within a logical context that tolerates contradictions or within one that rules them out. A dialetheist can argue for true contradictions without begging the question by generating them within contexts in which they are supposed to be ruled out, but it is hard to see what sort of parallel move within enemy territory would be possible for those on the anti-dialetheist side. Arch-dialetheist Graham Priest also identifies candidates for the status of true contradictions in the philosophy of law (in which citizens can have real, but contradictory, legal rights and obligations) and in the metaphysics of change. Given the scope and rigor of the Priest's case and the unsatisfactory nature of many of the existing attempts to defuse the semantic and set-theoretic paradoxes which provide the best evidence for dialetheism, there should definitely be enough material here for a dissertation-length argument against true contradictions.


After Turkey-murdering-pumpkin-pie-and-whitewashing-genocide-day*, we'll start off with a quick look at how *not* to argue against the possibility of true contradictions.

*sometimes alternatively referred to as "Thanksgiving"

Tuesday, November 20, 2007

New Blog

This is, as the title indicates, about dialetheism, which is the view in the philosophy of logic that there are "true contradictions," or true statements of the form (P&~P). (This should be distinguished from "paraconsistentism," which simply denies that anything follows from a contradiction. A paraconsistentist prefers to work in a logic where this is cleared up, but does not necessarily believe that there really are any "truth value gluts." All dialetheists are paraconsistentists, but not all paraconsistentists are dialetheists.) If you've never heard of dialetheism before, I'm guessing your reaction is going to be, "that's the craziest thing I've ever heard." And it is. It is, however, also surprisingly difficult to come up with a good argument against it. If you continue to read this blog, you'll hear more about this.

(For anyone looking for the skiffy-related bits of my life, see my other blog.)

OK, so why did I set up the new blog?

I am, as most people who know me from other contexts probably know, a PhD student in Philosophy at the University of Miami, down in sunny and decadent south Florida. This is my last semester of coursework. A couple of weeks ago, I got the word that all of my course requirements have been checked off as met, and a week before that I got my dissertation subject approved. The way the system works at Miami, this means that shortly after the semester is over--i.e. in a couple of weeks--I should get my reading list for my qualifying exams. This should be about 15-20 books and a similar number of articles about my chosen subject, which I will then have five months to study. At the end of that time, during the two "reading days" between the end of classes for the spring semester and the beginning of finals week, I'll have to sit down for eight hours of examination on this topic.

In order, basically, to force myself to think out loud about all this material I'll be reading, gather my half-baked thoughts about it, etc., without boring my friends in contexts where they don't want to hear about it, I've set this up as a socially acceptable venue for that. Let the reader be warned.