Contradictions & Explosion

Cute Comic

(Thanks to reader "M.C." for the link.)

Philosopher's Carnival: Call for Submissions

I'll be hosting the next Philosopher's Carnival, on June 28th. (The current edition is here.) If you have a blog post that you'd like to submit, you can do so here.

Also, hey, if you feel inspired to write something new with this in mind, you've got plenty of time to do so. You have something you've been burning to say about truth-value gaps, dialetheism and the Curry Paradox, right?

Or....what's that, you say? You actually don't have anything to say about that?

Well, OK, then. Posts on all topics are welcome.

(...as are philosophically-themed comics. Yes, I'm talking to you, Ryan. Write me something.)

A Revenge Paradox For Dialetheist Solutions To Curry

(Update: Never mind. See the comments.)

Dialetheists like Graham Priest and JC Beall think that some sentences are both true and false, but cleave to classical orthodoxy to the extent of continuing to insist that every sentence is either true or false. (The latter assumption is, in fact, crucial to the derivation of contradictions from the paradoxes.) Thus, Liar sentences get classified as both true and false, and the resulting contradictions are contained by rejecting classical logic in favor of paraconsistent logic. So far, so good.

For obvious reasons, this solution doesn't help with "Curry" sentences, like S1:

S1: If S1 is true, then everything is true.

If S1 gets classified as either (just) true or both true and false, triviality ensues. As such, it had best get classified as (just) false. Of course, this by itself doesn't get around the paradox, since, for familiar reasons, triviality ensues from he mere statement of S1's disquotational truth conditions. That is, however, beside the point for the purposes of this post. The important point is that, for a dialetheist who accepts that every sentence is either true or false but wishes to avoid triviality, the only option for the truth-value of S1 is that it is (just) false.

So far, so good. What, however, about S2?

S2: If S2 is either true or false, then everything is true.

Discuss.

Are Meaningless Sentences (and Bits of Burning Candle Wax) Untrue?

Solutions to the Liar Paradox according to which paradoxical sentences are meaningless face all sorts of challenges. For one thing, the partisan of such a solution needs to have a plausible error theory to explain the widespread intuition that such sentences are meaningful. For another thing, they must find a way to defuse familiar "revenge" Liars, like $, below.

$ The sentence marked with a dollar sign is either false or meaningless.

These are major obstacles, and whether or not they can be plausibly overcome is a subject for another time. What I want to focus on is an objection which I find far less initially plausible, but which I hear a surprising amount of the time.

It goes, more or less, something like this:

"Even if Liar sentences are meaningless, they're still not true, right? Meaningless claims aren't true, so that solution doesn't even help with the Strengthened Liar. ('This sentence is not true.')"

Now, in whatever sense in which we are speaking sense when we say "meaningless sentences aren't true," surely it would be exactly equally correct to say that "meaningless sentences aren't false." Meaningfulness is surely a prerequisite for falsity, just as it's a prerequisite for truth.

Forget, for a moment, about the Liar and its kindred semantic paradoxes. Let's just think about a normal case of a sentence whose meaninglessness is much less controversial, like "Green ideas sleep furiously."

Now, given the two claims we just endorsed:

(1) Meaningless sentences aren't true.

&

(2) Meaningless sentences aren't false.

Given these two claims, Disjunctive Syllogism, Conjunction-Addition and the Principle of Bivalence (for every P, either P is true or P is false), we can easily derive a contradiction about a normal, non-paradoxical meaningless sentence like "Green ideas sleep furiously."*

Let's symbolize "Green ideas sleep furiously" as G. Given Bivalence, we've got our first premise:

1. Tr(G) v F(G)

Symbolizing (1), above, we've got our second premise:

2. ~Tr(G)

From 1, 2 and Disjunctive Syllogism, we can conclude:

3. F(G)

Symbolizing (2), above, we get:

4. ~F(G)

And finally, of course, from 3, 4 and Conjunction-Addition, we conclude:

5. F(G) & ~F(G)

So, given Bivalence and a couple of basic logical rules, the claim that meaningless sentences aren't true or false entails contradictions. Perhaps the very notion of meaninglessness as a separate category from truth and falsity is inconsistent!

But wait. Even if we're willing to give up on the claim that any sentence anywhere is meaningless, what about questions. Surely questions exist. Can questions be true or false? How about bits of burning candle wax? Are they true? No? Are they false? Also no? Well, if G symbolized not a meaningless declarative sentence but a question or a bit of burning candle wax, we could use precisely the same five-step proof to derive an outright contradiction about the semantic status of the question or the bit of burning candle wax.

Clearly, something has gone horribly wrong in our reasoning.

Here's what it is:

When we say "meaningless statements aren't true," we might be making one of two claims:

1-Let M(P) mean "P is meaningful." For every P, if ~M(P), then ~Tr(P).

or

2. Meaningless sentences aren't the sort of thing to which truth talk meaningfully applies.

If you mean 1, you're confused. (It's significant that no dialetheist has ever used the proof above as an argument for the existence of true contradictions. And if that argument were available with them, why would they bother to swim in the murkier waters of semantic paradoxes?) When we try to symbolize a meaningless statement and perform logical operations on them, we're engaged in a nonsensical category mistake, of exactly the same sort that we'd be engaged in if we tried to symbolize and perform logical operations on a big of burning candle wax.

If you think meaningless sentences aren't true, and when you say that, you actually mean to assert of every meaningless sentence the negation of the claim that that sentence is true, you are necessarily saying something meaningless. After all, given the universal intersubstitutivity of P and Tr(P) for every P, if you say that "Green ideas sleep furiously" is not true, meaning ~Tr(P)--where P is "Green ideas sleep furiously"--then you are, in effect, asserting ~P. As the philosophers of the Vienna Circle were so fond of pointing out, the negation of nonsense is nonsense.

Unless you're willing to accept that green ideas fail to sleep furiously--and that there are true contradictions about the truth-value of every meaningless sentence--when you say that "meaningless statements aren't true", you'd better mean it in sense 2.

Now, like I said before, none of this helps the partisan of the meaninglessness view against revenge paradoxes crafted to fit the details of the view. (For example, given the discussion above, one might wonder about the following sentence, which we could call The Babbler: "This sentence is not the sort of thing to which truth talk meaningfully applies.") And that's fair enough.

Still, whether or not they are ultimately viable when we really look into the ins and outs of revenge paradoxes, intuitive difficulties and so on, meaninglessness solutions can't be batted away with the blunt instrument of pointing out that meaningless sentences aren't true.





*At least, that's one that most people take to be meaningless. (E.g. another commonly heard response to claims that Liar sentences are meaningless is "wait, you don't mean meaningless the same way that 'Green ideas sleep furiously' is meaningless, do you?") If, however, you hold semantic views on which 'green ideas sleep furiously' comes out as meaningless, please accept the following as a substitute:

Sentence S1: 'Green swimming red night fun fun fun!'**

**"But wait," I can hear some of you saying, "Sentence S1 isn't even well-formed!"

Well, I'd argue that any invocation of "well-formedness" as a consideration here misses several points at once. "Well-formed" means something fairly specific for symbolic formulas. It's not clear what it's significance is supposed to be when we start throwing it around with reference to natural language sentences. The closest natural language equivalent of the formation rules of formal systems would be the rules of grammar, and conformity to those is clearly neither necessary nor sufficient for meaningfulness. If someone accuses another person of having done something wrong, and the person being accused responds with Sentence S2:

Sentence S2: "Like hell I did!"

....everyone knows what is meant. If the accuser, trying to catch the accused person in an inconsistency, formalized Sentence S2 with a Greek letter, did the same with some of his other statements and and drew out some logical implications, no one would think the accuser was in the grips of any kind of deep conceptual confusion.

Now, someone trying to desperately hold on to some significant role for natural language "well-formedness" could try to say that the difference is that there are grammatically "well-formed" sentences that mean the same thing as Sentence S2, whereas no grammatically well-formed sentence means the same thing as Sentence S1, but, of course, by definition, no grammatically well-formed sentence *ever* means the same thing as any meaningless sentence, because meaningless sentences don't mean anything. That's what we mean when we call them "meaningless."

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 some....as 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 language....is 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.

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."

or

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

or

"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."

Back

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