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!")

## Friday, November 30, 2007

## 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.

...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.

"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.

## 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:

"...so 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.

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:

"...so 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"

#

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.

(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.

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