Monday, May 31, 2010

Another Revenge Paradox For Beall

In Spandrels of Truth, JC Beall abandons his previous belief in the existence of truth-value gaps, now declaring that "negation is exhaustive."

Intuitively, this means that there are now three possible statuses that statements can have on the new version of his account (unlike the old version, where there were four):

(1) (Just) true
(2) (Just) false
(3) Both true and false

In Spandrels..., Beall treads carefully around the "just" terminology, since he recognizes that for any such formulation you like, you can always devise a Liar sentence for it, e.g.

"This sentence is just false."

He does, however, sometimes express the distinction between (1)/(2) on the one hand and (3) on the other by talking about "treating" statements "classically." For example, in section 2.3.2, in his discussion of the status of non-paradoxical sentences that (like Liars) ineliminably refer to their own truth-or-falsehood status, like....

"This sentence is true."

....he says that he is open to "an asymmetric treatment of such sentences (e.g. treating gluts, some classically." (p. 15)

In section 5.4, to express his view that true contradictions only arise as by-products of the introduction of notions like "truth" into our language, and that the portion of the language that's free of such terms is also free of true contradictions, he says "our base classical." (p. 126)

And so on.

An important note about all this:

Given his rejection of gaps, it looks like "glutty" and "treated classically" are jointly exhaustive of the conceptual options. To really harp on the point, we can put this precisely as the Joint Exhaustion Principle (JEP).

JEP: Every statement is either a glut or classical.

Given Beall's account, I don't see how he could reject the JEP. He's a tpaints to explicitly reject the possibility of any sort of "paracomplete" break-downs of Excluded Middle, he goes out of his way to commit himself to a classical account of vagueness problems, etc. If someone who knows Beall's work can come up with a third option that he has room to adopt, I'd love to hear it in the comments, but right now, it sure doesn't look like he has the resources to deny the JEP.

So, given that all non-glutty statements are to be treated classically, what does that mean?

Presumably "is classical", "treated classically", etc., means at the very least that reasoning about it according to all the rules of classical logic (including those which dialetheists take to be invalid in inconsistent contexts) is appropriate when it comes to such statements.

(Note that, since Beall and I agree about the equivalence of "P is false" and "~P," I'll use them interchangeably in what follows.)

This language, however, seems to be just as prone to revenge paradoxes as the avoided language of "just false," etc. After all, what can he do with the following sentence?

# The (whole) sentence marked with the number sign is false and should be treated classically.

Like any conjunction, we have four options here:

(1) Both conjuncts are true, so the whole thing is true.
(3) The first conjunct is true, but the second conjunct is false, so the whole thing is false.
(3) The second conjunct is true, but the first conjunct is false, so the whole thing is false.
(4) Both conjuncts are false, so the whole thing is false.

(1) would entail triviality in a straightforward way. Given the assumed equivalence of "P is false" with "~P," noted above, a statement that's (a) both true and false, and (b) treated classically, is a statement from which triviality can be derived.

(2) would mean that # was not classical, which, given the JEP, would entail that # is a glut, and hence entail triviality.

(3) would entail triviality just like (1) would. If the first conjunct is false, it's also true. If the second conjunct is true, triviality is entailed by the contradiction.

(4) entails triviality in the same way. If the first conjunct is false, it's also true. If the second conjunct is false, the whole thing is (by the JEP) a glut, which makes the second conjunct true, which in turn means that we can derive triviality from the contradiction.

So....unless anyone can see a way out of the JEP on Beall's account...this looks like a pretty serious problem.

Of course, there's at least one way out:

Just as dialetheists (including Beall) standardly weaken the inferential power of their conditionals to get around Curry, Beall could weaken his even more in order to get around the Sentence # Paradox. He could claim that "P or Q", "if P, then R" and "if Q, then R" could all be true without R being true. If he made this move, however, then he would have no remaining motivation for believing in true contradictions. After all, he doesn't believe in "base language gluts," and given the sort of conditional-weakening proposal under consideration, no contradictions would follow from standard versions of any semantic paradox.

Wednesday, May 19, 2010

A Few Thoughts On Logical Fallibilism

Imagine that Aristotle had had Graham Priest's position that some (but not all) contradictions are true, and the first big treatise suggesting the impossibility of true contradictions wasn't published until 1987. Largely unquestioned academic orthodoxy held that some contradictions are true, mainstream probability theory reflected this assumption and so on. Everyone taking introductory logic classes slaved through the construction of long truth tables reflecting the three options T, F and B (for "Both"). The instructor might devote perhaps five minutes to illustrating the possibility of "Both" with a Liar sentence. Doubtless some Beginning Logic and Intro to Philosophy students would find it all a bit confusing--"how can it be *both* true and false?"--and smug TAs would sit around the bar and trade snotty jokes about the inability of undergraduates to understand basic logic.

Surely, nothing about the world described in the last paragraph is, y'know, impossible or unimaginable. (Nothing about the truth of orthodox ideas about logic makes it impossible or unimaginable for most people to be wrong about them. If you think that there are no true contradictions or truth-value gaps, etc., classical logic is the One True Logic and so on, you therefore think that in the actual world some actual people *are* mistaken about these matters. It's hard to see what possible basis there could be for arguing that it would have been impossible for things to be the mirror image of the way they are, and for the mistaken opinions to be the overwhelmingly popular ones.) Equally surely, from our perspective as residents of the actual world, we can all agree that it would be rational for denizens of the possible world in question to take that 1987 polemic against the possibility of true contradictions seriously, to (at the very least) seriously weigh the arguments and have a serious debate whether or not they should abandon dialetheist orthodoxy. After all, in the actual world, everyone with a position on the subject either thinks that (a) it's completely obvious that true contradictions are impossible, or that (b) far from being trivially obvious one way or the other, there's an interesting and important philosophical debate to be had on the matter. (No actual dialetheist that I'm aware of thinks that the existence of true contradictions is too obvious for it to be worth arguing about it.) If you have position (a), presumably you think that it would be rational for people in the possible world described above to question their belief in true contradictions. If you have position (b) in the actual world, presumably you'd advise the same position to people in the possible world where the debate was being held with a different balance of forces.

For (some) people with position (a), though, contemplation of the hypothetical might raise some sticky problems. After all, some people with position (a) brush off the very notion of taking the debate about these matters seriously by saying things like...

"You need logic to argue about logic. The very idea of arguing about it is ridiculous."


"What could be more basic than logical truths on the basis of which you could argue for them?"


"The denial of obvious logical truths is so nonsensical that I really don't care if I am begging the question."

Etc., etc., etc. I've heard variations on all of the above many times over the last couple of years of writing and thinking about dialetheism, and there are some famous and well-respected philosophers who've written sophisticated variations on the theme. For example, in a famous response to an invitation to contribute an essay to an anthology debating the Law of Non-Contradiction, David Lewis wrote:

"I’m sorry; I decline… My feeling is that since this debate instantly reaches deadlock, there’s really nothing much to say about it. To conduct a debate, one needs common ground; principles in dispute cannot of course fairly be used as common ground; and in this case, the principles not in dispute are so very much less certain than non-contradiction itself that it matters little whether or not a successful defense of non-contradiction could be based on them."

(Lewis, in other words, felt that dialetheism could be refuted with an incredulous stare.)

Now, consider the clearly possible world discussed above where dialetheism was orthodox and the camp that wanted to revise away the view that everyone else considered to be one of the most obvious and basic logical truths that there could be (that there are three possible truth-values that statements fall into, 'true', 'false' and 'both') was a tiny minority. Imagine that the David Lewis counterpart of that world (let's call him Bizarro-Lewis) responded to a similar invitation in the same way. Granted that Bizarro-Lewis' position (that the existence of the third standard truth-value was too obvious and epistemically basic to be debatable) would be *false.* That's not the issue. The question, for those who agree with the Lewissian position in the actual case, is whether Bizarro-Lewis' dogmatic refusal to even consider the opposition's arguments would be in some way irrational or unreasonable? (Rationality is, obviously, distinct from success. The history of science is littered with theories that we would now all agree are false, despite the fact that we would all agree that belief in them was, at one time, rationally justified.) If not, if the Bizarro-Lewissian position would be rational, does that mean that the philosophical community of that world would be entitled to remain in blissful ignorance of the truth of these matters, never learning better because they never seriously engaged with the debate? That seems like an....odd...position to take.

It's one that, however, it's terribly hard to see how actual-world Lewissians can avoid. If anyone can explain why we (orthodox logicians in the actual world) are justified in being dogmatically unwilling to engage in debate with dialetheists, whereas they (Bizarro-Lewissians) would *not* be justified in being similarly dogmatically unwilling to engage in debate with the dissenters of their world (who correctly deny the possibility of true contradictions), I'd like to hear it. My suspicion, however, is that any remotely satisfying answer to that question* would amount to a substantive defense of the truth of the Law of Non-Contradiction....

....or, in other words, it would amount to a contribution to the debate.

*"Any remotely satisfying answer" means, at the very least, "any answer which was such that the Bizarro-Lewissian couldn't easily offer up its exact mirror image as an answer to the mirror image question."

Monday, May 10, 2010


Sorry about the long unannounced break. I had a bit of materal stored up from before my defense, and after I burned through that, I really needed a break.

....but I'm back now. In lieu of a real Monday post, a few things:

(1) Ryan has a fun depiction of Harry Frankfurt's take on the Stone Paradox here.

(2) A few people have asked if they could read my dissertation. I'm happy to shoot a copy over to whoever...*after* I've finished various proofreading and re-formatting things I have to do for the graduate school in any case. I'll post an announcement here when its ready.

(3) Here's a schedule for the next couple of weeks:

Wednesday, May 12th: A Few Thoughts On Logical Fallibilism

Monday, May 17th: Why Theistic Libertarians Should Believe In Future Facts and Backward Causation (If They Believe in Infallible Divine Foreknowledge)

Wednesday, May 19th: Graham Priest And The Principle Of Uniform Solution