Chris Mortensen, in his book "Inconsistent Mathematics," says this about motivations for dialetheism (or 'strong paraconsistency,' in Mortensen's terminology):

"...the idea finds roots in an older view, that change especially motion is contradictory, which can be traced through Engels and Hegel to Zeno and Heraclitus, and has recently been revived, e.g. by Priest..."

The "e.g." strikes me as a bit odd. Who else besides Priest advocates this view, exactly?

## Thursday, August 21, 2008

## Monday, August 11, 2008

### Wikiality?

The wikipedia page on dialetheism contains* the following, from the section on "Formal Consequences," immediately after a brief run-through of the standard proof that contradictions explode in classical logic:

"Any system in which any formula is provable is trivial and uninformative; this is the motivation for solving the semantic paradoxes. Dialethesists solve this problem by rejecting the principle of explosion, and, along with it, at least one of the more basic principles that lead to it, e.g. disjunctive syllogism or transitivity of entailment, or disjunction introduction."

Now, on a nit-picky level, I object to say that "this is the motivation for solving the semantic paradoxes," as if there weren't any other motivations, but I suppose that's debatable. (It's certainly not *my* motivation for wanting to solve them, but one could maybe argue that I'm just eccentric that way and that the use of the definite article there is still basically accurate.) More importantly, though, I'm pretty sure that the last bit is flat-out wrong. Certainly, there are paraconsistent logics (e.g. the ones most used for the computer database applications) where disjunction introduction is eliminated, but (a) there seems to be no obvious reason why dialetheism being true would mean that disjunction introduction wasn't universally truth-preserving, and (b) to the best of my knowledge, there aren't any dialetheists who reject disjunction introduction (much less the transitivity of entailment), whereas all the ones I know about reject disjunctive syllogism, for obvious reasons...if a statement P can be both true and false, then P could be true, P v Q would also be true (since one of its disjuncts would be true), ~P could be true and Q could just be false. On the face of it, it seems hard to see how disjunctive syllogism *could* be valid given the assumption that there are true contradictions, or, given this, what motivation there would be for a dialetheist to reject disjunction introduction. In fact, even if some pragmatically useful formal systems disregard it, I don't know of the existence of s*any* logical monist, dialetheist or otherwise, who don't think that disjunction introduction isn't present in whatever they think the One True Logic is, or any logical pluralists who think that there aren't any logics adequate for at least some contexts that contain disjunction introduction.

Anyone have any information otherwise? Is this just a gap in my knowledge of the field? Any dialetheists out there who reject disjunction introduction? Anyone know about any that reject it?

*'Contains,' in this context, of course means 'contains on Monday, August 11th, 2008.' We are talking about Wikipedia here, so it could be edited to say something entirely different at any mnute.

"Any system in which any formula is provable is trivial and uninformative; this is the motivation for solving the semantic paradoxes. Dialethesists solve this problem by rejecting the principle of explosion, and, along with it, at least one of the more basic principles that lead to it, e.g. disjunctive syllogism or transitivity of entailment, or disjunction introduction."

Now, on a nit-picky level, I object to say that "this is the motivation for solving the semantic paradoxes," as if there weren't any other motivations, but I suppose that's debatable. (It's certainly not *my* motivation for wanting to solve them, but one could maybe argue that I'm just eccentric that way and that the use of the definite article there is still basically accurate.) More importantly, though, I'm pretty sure that the last bit is flat-out wrong. Certainly, there are paraconsistent logics (e.g. the ones most used for the computer database applications) where disjunction introduction is eliminated, but (a) there seems to be no obvious reason why dialetheism being true would mean that disjunction introduction wasn't universally truth-preserving, and (b) to the best of my knowledge, there aren't any dialetheists who reject disjunction introduction (much less the transitivity of entailment), whereas all the ones I know about reject disjunctive syllogism, for obvious reasons...if a statement P can be both true and false, then P could be true, P v Q would also be true (since one of its disjuncts would be true), ~P could be true and Q could just be false. On the face of it, it seems hard to see how disjunctive syllogism *could* be valid given the assumption that there are true contradictions, or, given this, what motivation there would be for a dialetheist to reject disjunction introduction. In fact, even if some pragmatically useful formal systems disregard it, I don't know of the existence of s*any* logical monist, dialetheist or otherwise, who don't think that disjunction introduction isn't present in whatever they think the One True Logic is, or any logical pluralists who think that there aren't any logics adequate for at least some contexts that contain disjunction introduction.

Anyone have any information otherwise? Is this just a gap in my knowledge of the field? Any dialetheists out there who reject disjunction introduction? Anyone know about any that reject it?

*'Contains,' in this context, of course means 'contains on Monday, August 11th, 2008.' We are talking about Wikipedia here, so it could be edited to say something entirely different at any mnute.

## Thursday, August 7, 2008

### The Standard for Question-Begging

In a context of trying to refute the case for dialetheism, it's common and (I'd argue) entirely reasonable to accuse proponents of various solutions to the semantic paradoxes of begging the question when they assume consistency and work from there. Something I'm less sure of is when the opposite is the case...if, in making the case for dialetheism, the dialetheist assumes at least the conceptual possibility of true contradictions, when, if ever, does this beg the question against the orthodox camp, which, after all, firmly disbelieves in this conceptual possibility? And if it doesn't, why not? Or if so, how can one make sure one isn't doing this when making the case for dialetheism?

Please discuss.

Please discuss.

## Monday, August 4, 2008

### Can Dialetheists Make Sense of Monaletheias?

A few weeks ago I was in Australia, at the Fourth World Congress of Paraconsistency, and I saw Graham Priest give a talk entitled "Inclosures, Vagueness and Self-Reference." The point of the talk was, as far as I remember, was that he now thinks the paradoxes of vagueness are in the same 'family' of paradoxes as the so-called 'paradoxes of self-reference' (i.e. the semantic and set-theoretic paradoxes), since (he now thinks) they all fit within the Inclosure Schema. I won't be talking about that in this post, but for anyone who's interested in finding out what the IS is or how it fits into Priest's views, I'd recommend that they read his book "Beyond the Limits of Thought."

Before getting to the point of this post, let's define some terms. Since true contradictions are called 'dialetheias,' and the theory that true contradictions exist is called dialetheism, my friend Ryan decided that a good word for the opposite view, that there are no true contradictions, would be 'monaletheism.' Dialetheists believe that a single statement can have as many as two truth-values, whereas a monaletheist is someone who believes that, whether or not there are gappy statements with zero truth-values, the maximum number of truth-values that any one statement can have is one. I prefer to talk about 'monaletheism,' rather than the Law of Non-Contradiction, since if the LNC is just the formula that (∀α)¬(α ∧ ¬α), or even (∀α)¬◊(α ∧ ¬α), then a dialetheist can happily accept that the LNC is true. It just means that every time they assert a contradiction (α ∧ ¬α), they also have to accept the truth of the further contradiction [(α ∧ ¬α)] ∧ ¬(α ∧ ¬α)]. Moreover, in many paraconsistent logics--including Priest's favored logic LP--(∀α)¬(α ∧ ¬α) is a basic rule, derivable from other important basic rules like the law of identity. In fact, in Australia, in a talk entitled "Making Sense of Paraconsistent Logic - Classical Logic, Paraconsistent Logic and the Nature of Logic," Koji Tanaka said that this was a feature of all the paraconsistent logics that he approved of. When I presented my paper later that afternoon, I saw that Koji was in the audience, so when I got to this point I described this as "a feature of all logics that Koji likes," and he nodded enthusiastically.

Anyway, that's monaletheism. So by analogy to true contradictions as "dialetheias," I'll be calling statements with only one truth value 'monaletheias.' A monaletheist doesn't believe in the existence of dialetheias, but a (non-trivialist) dialetheist certainly believes in the existence of monaletheias. (So does the trivialist, technically, but of course the trivialist also disbelieves in monaletheias, which the regular dialetheist does not.) In fact, dialetheists typically believe that the vast majority of statements are monaletheias. For example, 'the Axis powers won World War II' seems to be just false and not true, and 'the Allied powers won World War II' seems to be just true and not false, so those are both monaletheias. I won't be talking much about them in this post, but we could analogously call statements without truth-values (if, unlike Priest, you think that a statement can be gappy without being glutty) nonaletheias.

OK, back to the Priest talk on vagueness and inclosure. One of his examples kind of bugged me, although at the time I couldn't quite articulate why. I'm going to take this out of context, because I'm not interested in the point he was ultimately building toward in that paper, but in whether this this example is actually a serious problem for the dialetheist. Moreover, I'm doing this from memory, so I may be misrepresenting Priest's comments. Still, whether Priest said it or not, it's an interesting point.

One of his power point slides contained a Liar-type sentence that said of itself that it was a false monaletheia. "This sentence is false and not true." If I'm remembering correctly, Priest did a few quick logical derivations to reduce this to a standard Liar, and concluded that it was both true and false. Although of course he wasn't using this terminology, the implicit problem he was dealing with was that this result seems to show that the sentence in question is both a monaletheia and not a monaletheia. (To get the same effect even more starkly, one could try standard Liar paradox reasoning on "this sentence is not true and not a dialetheia." If you think that the usual reasoning from Liars to true contradictions is sound and if, like Priest, you believe that gaps entail gluts, so the only way to not be a dialetheia is to be a monaletheia, "this sentence is true and not a dailetheia" both is and is not a monaletheia.) Priest's response in the Australia talk was to say that he didn't think this was a problem for dialetheism. "After all, the point of dialetheism was never to be consistent, it was just to contain inconsistencies..."

I think this response may be too quick, and miss the real potential objection here, which is roughly this:

If you want to even be able to express the claim that most statements are monaletheias (i.e. that 'the rate of true contradictions is very low, so we are entitled to assign a very low epistemic probability to any particular contradiction,' which is the foundation stone of the 'classical recapture,' the prohibition against multiplying contradictions beyond necessity and whatever plausibility moderate dialetheism has) you need to be able to make sense of the idea of a monaletheia, i.e. that we can assert the truth of some statements in a way that rules out our also accepting their falsity, or vice versa. If, on the other hand, you believe, based on the usual reasoning, that we can derive contradictions from Liar-like sentences, then applying uniform standards means that you have to allow the possibility that something can both be a monaletheia and not be a monaletheia, as in the two cases discussed above. If that's a possibility, then there is no meaningful concept of a monaletheia.

Anyone else see a problem here? If you do, do you see any way that the dialetheist could (non-arbitrarily) get around it without sacrificing a hugely important part of the traditional case for dialetheism? It looks to me like they can't.

Before getting to the point of this post, let's define some terms. Since true contradictions are called 'dialetheias,' and the theory that true contradictions exist is called dialetheism, my friend Ryan decided that a good word for the opposite view, that there are no true contradictions, would be 'monaletheism.' Dialetheists believe that a single statement can have as many as two truth-values, whereas a monaletheist is someone who believes that, whether or not there are gappy statements with zero truth-values, the maximum number of truth-values that any one statement can have is one. I prefer to talk about 'monaletheism,' rather than the Law of Non-Contradiction, since if the LNC is just the formula that (∀α)¬(α ∧ ¬α), or even (∀α)¬◊(α ∧ ¬α), then a dialetheist can happily accept that the LNC is true. It just means that every time they assert a contradiction (α ∧ ¬α), they also have to accept the truth of the further contradiction [(α ∧ ¬α)] ∧ ¬(α ∧ ¬α)]. Moreover, in many paraconsistent logics--including Priest's favored logic LP--(∀α)¬(α ∧ ¬α) is a basic rule, derivable from other important basic rules like the law of identity. In fact, in Australia, in a talk entitled "Making Sense of Paraconsistent Logic - Classical Logic, Paraconsistent Logic and the Nature of Logic," Koji Tanaka said that this was a feature of all the paraconsistent logics that he approved of. When I presented my paper later that afternoon, I saw that Koji was in the audience, so when I got to this point I described this as "a feature of all logics that Koji likes," and he nodded enthusiastically.

Anyway, that's monaletheism. So by analogy to true contradictions as "dialetheias," I'll be calling statements with only one truth value 'monaletheias.' A monaletheist doesn't believe in the existence of dialetheias, but a (non-trivialist) dialetheist certainly believes in the existence of monaletheias. (So does the trivialist, technically, but of course the trivialist also disbelieves in monaletheias, which the regular dialetheist does not.) In fact, dialetheists typically believe that the vast majority of statements are monaletheias. For example, 'the Axis powers won World War II' seems to be just false and not true, and 'the Allied powers won World War II' seems to be just true and not false, so those are both monaletheias. I won't be talking much about them in this post, but we could analogously call statements without truth-values (if, unlike Priest, you think that a statement can be gappy without being glutty) nonaletheias.

OK, back to the Priest talk on vagueness and inclosure. One of his examples kind of bugged me, although at the time I couldn't quite articulate why. I'm going to take this out of context, because I'm not interested in the point he was ultimately building toward in that paper, but in whether this this example is actually a serious problem for the dialetheist. Moreover, I'm doing this from memory, so I may be misrepresenting Priest's comments. Still, whether Priest said it or not, it's an interesting point.

One of his power point slides contained a Liar-type sentence that said of itself that it was a false monaletheia. "This sentence is false and not true." If I'm remembering correctly, Priest did a few quick logical derivations to reduce this to a standard Liar, and concluded that it was both true and false. Although of course he wasn't using this terminology, the implicit problem he was dealing with was that this result seems to show that the sentence in question is both a monaletheia and not a monaletheia. (To get the same effect even more starkly, one could try standard Liar paradox reasoning on "this sentence is not true and not a dialetheia." If you think that the usual reasoning from Liars to true contradictions is sound and if, like Priest, you believe that gaps entail gluts, so the only way to not be a dialetheia is to be a monaletheia, "this sentence is true and not a dailetheia" both is and is not a monaletheia.) Priest's response in the Australia talk was to say that he didn't think this was a problem for dialetheism. "After all, the point of dialetheism was never to be consistent, it was just to contain inconsistencies..."

I think this response may be too quick, and miss the real potential objection here, which is roughly this:

If you want to even be able to express the claim that most statements are monaletheias (i.e. that 'the rate of true contradictions is very low, so we are entitled to assign a very low epistemic probability to any particular contradiction,' which is the foundation stone of the 'classical recapture,' the prohibition against multiplying contradictions beyond necessity and whatever plausibility moderate dialetheism has) you need to be able to make sense of the idea of a monaletheia, i.e. that we can assert the truth of some statements in a way that rules out our also accepting their falsity, or vice versa. If, on the other hand, you believe, based on the usual reasoning, that we can derive contradictions from Liar-like sentences, then applying uniform standards means that you have to allow the possibility that something can both be a monaletheia and not be a monaletheia, as in the two cases discussed above. If that's a possibility, then there is no meaningful concept of a monaletheia.

Anyone else see a problem here? If you do, do you see any way that the dialetheist could (non-arbitrarily) get around it without sacrificing a hugely important part of the traditional case for dialetheism? It looks to me like they can't.

## Saturday, August 2, 2008

### Short Story and Set Theory

I recently sold my alternate history short story T-Shirts, Tentacles and the Melting Point of Steel to Atomjack Magazine, and it's online today.

I only mention it here because I did throw in a couple of references to my day job in the story itself--where the protagonist "co-wrote three papers on the paradoxes of transfinite set theory"--and into the author bio at the end, which includes, "and just for the record, Ben...thinks that the set-theoretic paradoxes are best solved by re-interpreting set theory in mereological terms.

I only mention it here because I did throw in a couple of references to my day job in the story itself--where the protagonist "co-wrote three papers on the paradoxes of transfinite set theory"--and into the author bio at the end, which includes, "and just for the record, Ben...thinks that the set-theoretic paradoxes are best solved by re-interpreting set theory in mereological terms.

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