Yesterday's Chaospet Comic...

...is once again worth reading.

Follow-up: "Interesting but Inconsistent?"

While looking at the discussion after my last post--for those who missed it, I made a really dumb attribution mistake that was thankfully corrected, and it was forcefully brought to my attention that my original example relied on an understanding of conditionals deeply controversial among precisely the same sort of people likely to find Impressive Scientist X examples compelling, but I argued that the same point could be made without bringing in the material conditional, and that there are simpler explanations of e.g. Bohr's failure to derive random crazy things from his inconsistent atomic theory than that he was somehow unknowingly working with an "underlying paraconsistent logic"--something closely related occurred to me.

There's a phrase that's used a lot in these discussions as if we all knew what it meant. I'm not sure that it's so clear. That's "interesting but inconsistent theory."

(The phrase "non-trivial," often appended to "interesting but inconsistent," adds bupkis. All it means is that not everything will follow from the theory, i.e. that interesting but inconsistent theories should be reasoned about paraconsistently or not at all. OK. If you believe that the moon is both made of green cheese and not made of green cheese, that theory will be "non-trivial" in precisely the same sense, but I doubt anyone would call it "interesting" in the relevant sense.)

Well, what is an 'interesting' theory? I mean, I think I know what it means before modified with "but inconsistent," but after that's there, I'm not so sure any more. Normally, when talking about consistent theories, I would take "interesting theory" (in the sense that seems to be driven at, not "interesting" as in "crazily unexpected" or anything like that) to mean "plausible theory," i.e. one that might very well turn out to be true. Or, applied to out-dated theories, one that it would have been rational to regard as quite possibly true given the evidence available at the time, even if we now understand that it is false.

Now, full-blown dialetheists banding about the phrase "interesting but inconsistent theory" might mean exactly this, since they think it's possible in principle for something to be inconsistent but true. What I'm interested in at the moment is what this "interesting but inconsistent phrase" means to people who bandy it around who are on the 2nd Grade of Paraconsistent Involvement discussed in the last post, the "I'm not a dialetheist, but..." crowd who are still holding back from the 3rd Grade where you admit that some of these theories may be true.

So, if you aren't a dialetheist, you still believe in the Law of Non-Contradiction, hence you believe that it is categorically impossible for an inconsistent theory to turn out to be true and there has never been a situation where it would have been rational on the basis of any sort of empirical evidence to believe that an inconsistent theory was true, what does "interesting but inconsistent theory" mean?

Interestingness also can't just boil down to a degree of predictive accuracy*, right? If so, interesting-but-inconsistent theories would be too easy to generate and lose the aura of respectability they gain from Bohr-type examples. After all, for any hotly disputed scientific area, where one theory predicts a bunch of effects with a certain amount of experimental support, and an obviously logically inconsistent competitor theory predicts a bunch of other effects, and there are a certain amount of experimental support for those two, if someone blandly asserted that the disputed phenomenon both existed and didn't exist, and was thus able to claim the experimental successes of both competitors for this claim (like, "if X exists, we expect to see some Y's and if it doesn't, we'd expected to see some Z's, so I expect to see both Y's and Z's..."), would that make the new conjunctive theory "interesting"?

Anyway, I'm throwing this open to the floor. On the assumption that it's never rational to believe an inconsistent theory to be actually true, what does it mean to call one 'interesting'?








*...although why we'd be interested in predictive accuracy, except as an indicator of truth, is for the most part mysterious to me.

A Thought About Underlying Logics

My apologies for the weeks between this and the last post. I'm back in Miami, slugging through my reading list, and I should be posting at a much more frequent clip from now on.

Meanwhile, I have a thought (not really a full thought, but at least the beginning of one), not about dialetheism per se but about what Graham Priest calls the "second grade of paraconsistent involvement."

Just for future reference, his "grades" are:

1st: "Gentle-strength paraconsistency" (you reject the principle that anything follows from a contradiction)

2nd: "Full-strength paraconsistency" (you think there some inconsistent but interesting, non-trivial theories)

3rd: "Industrial-strength paraconsistency" (some of those theories may be true)

4th: "Dialetheism" (some of those theories *are* true)

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Standard apologetics for the usefulness of paraconsistent logic often include historical examples of inconsistent but non-silly theories. It generally goes something like this. "Impressive Scientist X believed P and he also believed Theory Q, and he knew that Theory Q entails that not-P, but he didn't derive just any claim R, so the underlying logic he was using was clearly not classical."

Now, on the face of it the only sort of explosive logical rule that this situation would be any kind of challenge to would be a claim in the language of epistemic logic that Bxp & Bx~p entailed Bxq for any q. [Of course, (Bxp & Bx~p) is not a contradiction. Only (Bxp & ~Bxp) would be a contradiction.]

Sadly, this has nothing to do with what the historical examples are there for. The historical claim here is not that according to any sort of classical logic, it should follow from Scientist X believing P and ~P that Scientist X will believe that Q for any Q. Rather, the point is that if Scientist X believes that P and he believes that ~P, then classical logic would give him permission to draw the conclusion that Q. The fact that he never exercised this privilege is then seen as evidence that he was (albeit unconsciously, when we're talking about figures who predated the development of paraconsistent logic) operating according to paraconsistent rules of inference in which there are strict limits on what you can derive from a contradiction.

Now, I have severe doubts about the very idea that either scientific practices (or, worse yet, as is sometimes claimed, natural languages) have "underlying logics," but for the moment I'm going to put that to one side.

Instead, let's go for a simple analogy.

Einstein believed in the Special Theory of Relativity.

Einstein never drew the conclusion that "if it is not the case that it is not the case that it is not the case that it is not the case that it is not the case that the Special Theory of Relativity is true, then it is not the case that it is not the case that it is not the case that it is not the case either that Hitler won World War II or that the national flag of Australia is a pirate flag containing a jewel, that, under magnification, is revealed to be a detailed map of the surface of Mars with an X marking the spot where Elvis has high tea every afternoon at four with the Mad Hatter and a ghostly semi-physical representative of the Hegelian world-spirit."

But wait!

According to classical logic, the entire quoted claim is entailed by the truth of the Special Theory of Relativity. (Unless, of course, I slipped up while counting the number of negations.) If the STR is true then it follows that if you put an odd number of negation signs in front of the proposition that it is true, you have a false claim, and any conditional statement with a false antecedent is true, regardless of whether the consequent is true or false.

...so Einstein never drew that conclusion, or any of the infinite number of other similar conclusions classical logic would have given him permission to draw. Does it therefore follow that his "underlying logic" must have been some alternative non-classical logic, where strict rules are in place to reign in the sorts of consequents that can be put on these conditionals?

Reading List

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

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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.

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?

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!")

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.

...now, *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.