Wednesday, October 8, 2008

Classical Logic and Inconsistent but Non-Trivial Fiction

This is a short excerpt from my paper on "What You Don't Need Paraconsistency or Pluralism For." The basic claim of the paper, as mentioned in the last post, was that while it is certainly true that if dialetheism is right, classical logic is wrong, since rules like Disjunctive Syllogism won't be universally truth-preserving, none of the traditional (non-dialetheist) motivations for paraconsistency give the classical monist (i.e. the logical monist who thinks that classical logic is the correct logic) any reason to abandon that position in favor of either logical pluralism or paraconsistent monism, since they can happily accept the relevant claims and accommodate them in their classical monist framework. Here's the bit about fiction:

"Start with inconsistent fiction. That it is possible for the writers of fiction to create inconsistencies, whether on purpose (as in Graham Priest’s whimsical short story ‘Sylvan’s Box,’ in which Priest himself, appearing as a character, finds that the late Richard Routley owned a box that both contained and did not contain a statuette, at the same time and in the same sense) or, as is vastly more common, by accident, is clear enough. To take a classic example, in Arthur Conan Doyle’s Sherlock Holmes stories, Watson’s war wound is always referred to in the singular, strongly suggesting that he only has one of them, but sometimes it is on his left shoulder (and thus, presumably not on his right shoulder) and sometimes it is on his right shoulder (and thus not on his left shoulder). Moreover, in both the Conan Doyle case and the Priest case, not everything is true in the worlds of the stories. It is not true in the world of ‘Sylvan’s Box’ that Graham Priest agrees with everything in this paper, and it is not true in the world of Conan Doyle’s Sherlock Holmes stories that Professor Moriarty’s ultimate aim was to raise up Cthulu and Azathoth to destroy humanity.

"How can the classical monist make sense of this? Easily, and in pretty much the same way that a conscientious paraconsistentist would have to make sense of it. Regardless of what your view is on fictional characters--whether they have no ontological status at all, or are abstract objects, or are Meinongian objects that fail to have the property of existing but have all sorts of other properties, like ‘being referred to in philosophy papers’--statements like, for example, “Sherlock Holmes lives on 221B Baker Street” are, strictly speaking, false. Go to 221B Baker Street in London, and you will not find Holmes or Watson there, but a bank, one with a special employee in charge of responding to correspondence written to Holmes. To the extent that we regard such statements as true--and this is true regardless of our stance on the ontological status of fictional characters--it is because we read them as containing implicit fiction operators. Sherlock Holmes, if he exists as an abstract object, has, for example, the property of being written about by Arthur Conan Doyle, but he most certainly does not have the property of actually living on 221B Baker Street. He only has this property in fiction.

"Once we realize this, however, the apparent conflict between classical monism and our obvious ability to non-trivially reason about fictional inconsistencies disappears. If Dr. Watson had the property of both being written about by Arthur Conan Doyle and not being written about by Conan Doyle, in precisely the same sense, that would be a real contradiction symbolized as (α ∧ ¬α), that would indeed entail any and every β in classical logic. If, however, Dr. Watson has the property of having in his war wound on his left shoulder and having it on his right (i.e. not on his left) shoulder in fiction, then acknowledging that that’s true commits us not to (α ∧ ¬α), but F(α ∧ ¬α), which is a very different thing.

"When we extend classical logic, to include such operators I see no reason why we should mimic the rules about what goes on outside of them when we make the rules for what goes on within their scope. A classical monist can happily admit that there are cases where F(α ∨ β) and F(¬α) are both true, but F(β) is not. More radically still, we can allow inferences within the scope of the fiction operators to be not only ‘paraconsistent’ but non-adjunctive. That is to say, F(α) and F(¬α) need not always entail F(α ∧ ¬α). This preserves the intuitive difference between the ‘Sylvan’s Box’ case, in which it is true in the world of the story that the box is both empty and non-empty, and the Sherlock Holmes case, in which both ‘Watson’s war wound is on his left shoulder rather than his right shoulder’ and ‘Watson’s war wound is on his right shoulder rather than his left shoulder’ are both true, but ‘Watson’s war wound is on both of his shoulders’ does not seem to be true. Notice that even here, we have not revised classical logic one whit, as evidenced by the fact that even if F(α) and F(¬α) do not jointly entail F(α ∧ ¬α), they continue to jointly entail (F(α) ∧ F(¬α))."

Saturday, October 4, 2008

What You Don't Need Paraconsistency or Pluralism For

Yesterday, I gave a Miami Forum talk entitled "What You Don't Need Paraconsistency Or Pluralism For." Here's the abstract:

"If dialetheism is right, classical logic is wrong, at least for the context of reasoning about the domain of the inconsistent. As such, classical monism--the claim that there is One True Logic, and that logic is classical--must be rejected in favor of either paraconsistent monism or some sort of logical pluralism. Many paraconsistent logicians, however, reject the claim that there are real contradictions ‘out there in the world,’ but think that there are good reasons short of that to reject classical monism in favor of some sort of paraconsistent approach. I argue that standard defenses of this claim fail to hit their target, examining and rejecting claims that classical logic somewhow gets negation wrong, and showing how standard motivations for paraconsistency from inconsistent fiction, counterpossible conditionals and so on offered by Greg Restall, J.C. Beall, Newton Da Costa and others can be accommodated in a classical monist framework."

Thursday, August 21, 2008

Along The Same Lines

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?

Monday, August 11, 2008


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.

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.

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.

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.

Monday, July 28, 2008

Cats and Contradictions

In an effort to re-start discussion, what follows is a quote from the current draft of my dissertation proposal, illustrating why I reject Tarski/Azzouni-type solutions to the semantic paradoxes.

"To see why approaches that concede the ‘inconsistency’ of natural language but hope to create a consistent language, or isolate a consistent fragment of natural language, miss the mark, imagine the following scenario. Take a world in which Shröedinger’s Cat was not a thought experiment but a real experiment. Scientists discovered, or thought they had discovered, a cat that really was in simultaneous, superimposed states of being alive and being dead. Moreover, this discovery was taken to not only revolutionize physics but also logic. The cat was alive and not alive, so we had a true contradiction.
Imagine, then, that clever logicians responded by creating an artificial language in which the aliveness predicate could never be applied to cats, or carefully formulated rules prevented it and its negation from ever being applied to the same cat, or perhaps even where talk of cats was eliminated entirely. Perhaps some clever new quantifier would allow us to say most of what we want to say about cats in non-paradoxical contexts--that certain saucers of milk were depleted at certain times, or that certain creatures purred as they were petted--without actually referring to cats. These logicians then declared that although languages with sufficient expressive resources to discuss cats are inconsistent, their special artificial language was now consistent. A sensible response might be that the problem has not been solved but avoided.

"Now, there may be good solutions in this world. For example, a sensible monaletheist could point out that superposition and conjunction are very different things, that mathematically superposition is represented by linear combination, and that this sort of linear combination might not in fact make true the conjunction of the a statement describing one of these states and its negation. Surely, however, the beginning of wisdom would be the recognition that a language without the expressive resources to discuss cats would simply be a bad medium in which to discuss this thorny problem."

Blogging Schedule

In the months since I took my quals, I've let my blogging lapse, only posting occasionally, and usually not very substantively.


In the future (starting today), you can expect new material every Monday and Thursday.

Friday, May 16, 2008

Qual Follow-Up

OK, just updating to say that I did indeed pass my quals! I'll try to hammer out a proposal over the summer, and then it's on to the dissertation.

Meanwhile, in place of a real post, for now here are the questions I had on the first day...

Qualifying Exam Questions for Ben Burgis

Part I: Answer two of the questions below:

1. Compare and contrast Jon Barwise and John Etchemendy's approach to the liar paradox in terms of non-well-founded sets to the one offered by Saul Kripke in his theory of truth. Indicate the similarities and the main differences between these two approaches. Which difficulties do they face?

2. Explain how the dialetheist attempts to solve the liar paradox. What are the alleged benefits of the solution? Which problems must it overcome? Explain how the dialetheist attempts to solve the inconsistency found in naive set theory. Does this solution work? Why or why not?

3. Is truth a vague predicate? If so, does this help addressing the liar paradox? How?

....and here are the ones from the second day:

Qualfying Exam Questions for Ben Burgis
Part II: Answer two of the questions below:

1. The dialetheist suggest that we should change the underlying logic of our theories to a paraconsistent one. Can we make sense of the idea of changing a logic? In order to change a logic, don't we need a logic to assess such a change? Can this process get off the ground? If so, how?

2. Explain some of the main similarities between the liar paradox and the set-theoretic paradoxes. What are some of the main differences between them? What are the prospects of developing a unified solution to both paradoxes?

3. Set theory with an unrestricted comprehension schema is prima facie a plausible theory: it's simple, intuitive, and very powerful. All things considered, it's perhaps the best theory of sets we have. But the theory is also inconsistent. The indispensibility argument would then force us to conclude that we ought to believe in the existence of mathematical objects with inconsistent properties. Does this offer a reductio of the indispensibility argument? Why or why not? Does the fact that such a theory is inconsistent pose a problem for a realist interpretation of the theory? Why or why not? (To answer these questions, discuss in particular the approach to set theory developed by Penelope Maddy.)

Thursday, April 24, 2008


My qualifying exams are on Monday and Tuesday, from 10 in the morning to 2 in the afternoon each day.

[A prominent philosopher of language, when visiting Miami, at whose University--one of the top couple on the Leiter list--the grad student just turn in a portfolio of their best papers, responded to my description of the qual process with a tone of genuine shock. 'You have a *sit-down, closed book* qualifying exam? That's a bit passe, isn't it?' So it is. I plan, if I pass, on being insufferably smug about it in any and all future conversations with any graduate students not subjected to this little rite of passage. "In my day..."]

In any case, I expect the next few days to be taken up with an orgy of studying, reviewing and re-reviewing over notes and so on, so this will almost certainly be my last pre-qual post. Assuming I survive, as I start to organize all of this material to the form it will ultimately take in my dissertation proposal, I'll probably start posting here more.

Meanwhile, I end with a quote that I've been trying to take to heart in these last days before the quals (thanks to N. for reminding me of it):

"I must not fear. Fear is the little-death that brings total obliteration. I will face my fear. I will permit it to pass over me and through me. And when it has gone I will turn the inner eye to see its path. Where the fear has gone there will be nothing. Only I will remain."

--Bene Geserit Litany, from "Dune" by Frank Herbert

Sunday, April 13, 2008

Dialetheism in the Leiter Reports least tangentially, in that Matt Burnstein is quoted as asking if there was a true contradiction involving Graham Priest's Melbourne to CUNY.

Wednesday, April 2, 2008


This morning I found out that my paper on "Paraconsistent Tense Logic, the Metaphysics of Change and the Epistemic Consequences of Dialetheism" was accepted for presentation at the Fourth World Congress of Paraconsistency this summer at the University of Melbourne. Should anyone be interested, here's the abstract:

"Graham Priest has argued that there are some true contradictions, but that the statistical frequency of true contradictions is very low, and that as such the epistemic probability of any particular contradiction being true is very low. This claim is essential to his justification for the ‘classical re-capture.’ At the same time Priest has identified some concrete extra-semantic candidates for the status of true contradictions in analysis of the metaphysics of change. Expressed in terms of a paraconsistent logic (his own LP) outfitted with tense operators like P, which can be read as ‘it was the case that,’ Priest argues for 'Zeno’s Law,' the principle that (α & P¬α) entails the disjunction of (α & ¬α) or P(α & ¬α). Despite his repeated claims to the contrary, it will become clear that Priest is so deeply committed to the tensed theory of time that his analysis falls apart once the tenseless theory is substituted. More importantly, Priest’s argument for 'Zeno’s Law' exhibits a methodology which undermines his claim that the statistical frequency of true contradictions is very low. A closer examination of this point should demonstrate that there is no good reason why arguments at least as good in more mundane contexts couldn’t turn up enough true contradictions to overturn the claim that the statistical frequency of true contradictions is very low. As such, if dialetheism is correct, we are not justified in generally assigning low epistemic probabilities to contradictory outcomes in our arguments, and the ‘classical re-capture’ fails."

Monday, March 24, 2008

A Thought About Dialetheism and the Curry Paradox

Sorry about the lag between posts.....

Meanwhile, here's something that I've been thinking about. Here's a simple form of Curry's Paradox:

"If this sentence is true, then β."

Plugging it into the T-Schema, we get the result that that sentence is true if and only if, if it's true, then B, or formally:

Tr<α> ↔ (Tr<α>→β)

The logical principle of absorption (or contraction) says that any time we have something of the form α→(α→β), we can infer straight from there to α→β. Or, if you want to do the same thing more slowly, you can just do a conditional proof--all you'll need is Modus Ponens a couple of times, and a logic that let's you use the same premise more than once--and, one way or the other, you get the result Tr<α>→β. But from this and the right-to-left version of the biconditional above, we can infer Tr<α>. Here, of course, we now have in our possession Tr<α> and Tr<α>→β, so we can just plug in one last instance of Modus Ponens and get β, for any and every arbitrary β. Explosion without even having to get a contradiction on the way.

Just as the ordinary semantic paradoxes, like the Liar, are prima facie sound arguments for dialetheism (the position that some, but not necessarily all, contradictions are true) from intuitively plausible premises via intuitively reasonable steps, so that to show that they are unsound you have an uphill battle to explain why the premises are wrong or what's wrong with the reasoning, Curry represents a prima facie sound argument for trivialism (the position that everythign is true.) Non-trivialist dialetheists will want to avoid this at all costs--the whole project of carving out a plausible-sounding version of dialetheism is to show how some but not all contradictions can be true--and, from my point of view, the interesting thing is that their options here aren't that different from the options confronting a defender of the Law of Non-Contradiction when defusing an ordinary semantic paradox like the Liar. It seems to me that there are three ways a dialetheist (or any one else, of course) could deal with the Curry paradox:

(1) They could deny that the original sentence was a truth-bearer, e.g. on Kripke's grounds that sentences ascribing truth to other sentences are meaningful if and only if the series eventually grounds out in a sentence that's actually about external extra-semantic reality in some way, or of course on whatever other grounds.

(2) They could institute some sort of formal rules a la Tarski to ban the expression of the sentence in the first place.

(3) They could deny that the T-Schema holds universally, and make an exception for Curry.

(4) They could tinker with the logical rules that get us from Tr<α> ↔ (Tr<α>→β) to β.

Although I think Graham Priest actually goes with (4) in In Contradiction and elsewhere, I think this is probably the least plausible response. After all, on the face of it, the logical rules in question still look universally truth-preserving whether or not propositions can be simultaneously true and false, so the dialetheist has no special right (given their assumptions) to change them that anyone else does, and I don't see why anyone does. One can simply declare that "my conditional is not the conditional of classical logic, and given that, you can't make the inference from α→(α→β) to α→β with it," but (a) I'm deeply skeptical that this can be explained in any way that blunts its radically counter-intuitive edge, and (b) it looks like this is a "solution" bought via the loss of expressive power, since the → in this logic simply won't capture the notion of "if, then" in even the minimal way that → does in classical logic. Worse yet, if the whole motivation for this artifical restriction of the conditional is the avoidance of Curry problems, then it looks to me like the dialetheist who picks this option is engaging in an ad hoc manuever and begging the question against the trivialist.

The other options, however, look even less promising. Priest has been savage in his criticisms of "ad hoc exceptions to the T-Schema" for the Liar and other semantic paradoxes, and no one has been clearer in explaining why (2) doesn't solve or explain anything but merely represents a decision not to talk about it. That leaves us with (1). I think that this is the most promising option, since it represents a more than purely formal solution, and, if the independent grounding given is good enough, the one that looks least like it's assuming what needs to be proved.

The problem, of course, is that the alleged meaningfulness of ordinary paradox-producing sentences like the Liar would be an almost inevitable casualty of any explanation of why the Curry sentence wasn't meaningful, so the dialetheist who took option (1) would be sacrificing a huge part of the positive case for dialetheism.

Monday, February 18, 2008

Negation and Denial

I want to post on more recent reading soon, but meanwhile I have one last footnote to the discussion of Barwise and Etchemendy. I didn't include this in the main line of discussion, both because it might be of more general interest and because it can be intelligibly separated from the rest of their argument.

Remember that the oddest upshot of their modelling of propositions as hyper-sets is that (a) classical bivalent logic is correct, such that everything is false if it is not true and true if it is not false, (b) the Liar Sentence expresses a truth-evaluable proposition, and (c) the Liar Sentence somehow gets to be false without also being true. Various purely formal moves nominally validated this status for both the Russellian and Austinian ways of thinking about propositions, but I think the authors end the book all-too-aware of how capricious and counter-intuitive all this sounds. Thus, they end by gesturing in the direction of the distinction between negation and denial. They claim that if they had factored in denial as well as assertion and negation, it would have been clear that the logic of their notion of propositions was entirely classical, but that it would “involve us in untangling one of the most basic conflations in the logical literature, the conflation of negation and denial” and that this “would take us far from the topic of the book.” It's hard, on a snarky level, not to think of Fermat and proofs to long to include in the margins, but never mind that right now. There's still an interesting issue about what role this distinction could contribute here.

The problem is that Barwise and Etchemendy themselves tell us very little about the distinction, or what role they think it should play. They tell us that speech-act theorists are always telling logicians to take the distinction more seriously. Fair enough. But how exactly would it be helpful here?

In one of the best turns of phrase in the book, they say that just as ignoring relativistic effects doesn't cause any problems on a trip to the corner supermarket, but knowledge of those effects becomes vitally important when approaching the speed of light, “when approaching sentences like the Liar, we risk paradox if we ignore the difference between negation and denial.”

Excellent. Sadly, we never find out what exactly the import of that distinction is here.

If anyone has any suggestions to make in the comments, I'd be glad to hear them. Tentatively, though, here's my position:

Is there a distinction? Yes. Is that distinction relevant or useful for the purposes Barwise and Etchemendy are gesturing towards? Absolutely not.

Of course, the distinction between negation and denial could be very relevant to a discussion of the semantic paradoxes if, for example, we were working within a deviant logic that denied the Law of the Excluded Middle and posited extra possibilities 'between' P and ~P. (This is precisely what Barwise and Etchemendy repeatedly deny that they are suggesting.) If, however, we are assuming classical bivalent logic, then denial and negation are still distinct, but the category of propositions we are logically or epistemically warranted in denying will, it seems to be, clearly be necessarily co-extensive with the category of propositions we are logically or epistemically warranted in asserting the negations of. As such, for someone trying to fit the square peg of granting that the Liar is truth-evaluable and that every sentence that is not true is false and vice versa into the round hole of denying dialetheism, it doesn't look like this distinction can be of any use.


Meanwhile, my friend Ryan pointed me towards another webcomic that references dialetheism , although as far as I'm concerned there's nothing in the comic itself that's quite about dialetheism.

Sunday, February 3, 2008

Barwise and Etchemendy, Pt 2 (Austinian Case)

So when we left off our story, Barwise and Etchemendy were modelling Russellian propositions with Aczelian hypersets, and the Liar Sentence came out false, but this fact wasn't included in "the world," because if it was, that would violate the "coherence condition" preventing a set-theoretic proposition-object and its "dual" from both being present in a model.

They describe this consequence as "counter-intuitive." I'd describe it as absolutely incoherent.

In any case, when they turn to Austinian propositions, one of the big advantages of the shift was supposed to be that this "counter-intuitive" consequence can be avoided. Remember, Austinian propositions include the situations they are about--so, e.g. if you are at a card game and you mistake the player holding the 3 of Hearts for Jill and claim that "Jill has the 3 of Hearts," the proposition comes out false even if by coincidence Jill actually is playing cards across town and she does happen to have the 4 of Hearts.

As such, the proposition being expressed by any given use of the LS is about some specific situation, since all propositions are on the "Austinian" model. How does this help?

Well, as in the Russellian case, the LS always comes out false. We can see why by analogy to the behavior of the Truth-Teller ("This sentence is true.") In situations that don't include any semantic facts, the Truth-Teller is false, since there is nothing to make it true. In situations that include semantic facts, the Truth-Teller is sometimes true and sometimes false, depending on what those semantic facts are...i.e. in a situation that includes the semantic fact that the Truth-Teller of that situation is true, it's true, and in a situation that includes the semantic fact that the Truth-Teller of that situation is false, it's false. Where these "semantic facts" about the Truth-Teller are supposed to come from, since they obviously don't supervene on the non-semantic facts, is a bit of a mystery to me, but whatever. Let's move on.

The Liar behaves in exactly the same way, with one major difference. The relevant semantic facts are always excluded from the situation used to determine its truth value. Why? Well, if we allowed them to included, that would give us the result that each Liar was both true and false, and that would violate the coherence rule. So it's in "the world," but not the part of it used to evaluate the truth-value of the proposition.

So the Liar of a situation that doesn't have any semantic facts in it is false--just false and not true, which is a neat trick--and the semantic fact that "the Liar of Situation 1 is false" is included not in Situation 1 but in Situation 2. Sure, Situation 2 has a Liar of its own, which is made false because there is nothing in Situation 2 to make it true (after all, the semantic fact that the Liar of Situation 1 is false is about the Liar of Situation 1, not this new Liar, so it is irrelevant), and its falsehood is a fact not of Situation 2 but of Situation 3. On and on forever.

Two comments seem to be in order about this picture.

The first is that, at the beginning of the book, Barwise and Etchemendy rejected Tarski's hierarchy-of-artificial-languages solution because it was arbitrary, didn't really solve anything on the intuitive level, etc. I agree with them on that. It is interesting, though, given that, that the situation-theoretic hierarchy they posit seems to be structurally exactly parallel to Tarski's languages model.

Secondly, and more importantly, it seems to me that the one fact being excluded from the situations is the only one that cannot be plausibly excluded from them. It is, in fact, the ONLY thing that belongs in them. If the Liar Sentence really does express a proposition, then the one and only thing its about is its own truth value. That semantic fact is the one and only fact of any kind that belongs in its Austinian "situation," so a Liar that really was 'about' a situation without semantic facts, or even without this one relevant semantic fact, is simply an incoherent non-possibility.

As such, taken as a defense of the Law of Non-Contradiction--and, as an attempt to explain why the Liar does not have its prima facie property of being true if false and false if true, that's what it amounts to, even without a dialetheist who actually asserts that it is both true and false as the conscious target target--it boils down to the following argument.

"There are no counter-examples to the LNC, therefore the Liar is not a counter-example to the LNC."

That's a perfectly valid instance of universal instantiation, but those of us who want to defend the LNC in the context of an actual argument are going to have to do better than that.

Friday, February 1, 2008

Barwise and Etchemendy, Pt 1 (Russellian Case)

Last weekend, I read Jon Barwise and John Etchemendy's book "The Liar: An Essay on Truth and Circularity." The post is in two parts, divided up in a way that I hope will be intuitive.

The book is, as the name suggests, an attempt to get around the Liar Paradox. I'm sure anyone likely to glance at this book already knows plenty about it, but just for easy reference, here's the Liar Sentence again.

LS: This sentence is false.

Now, in all fairness to them, the book came out in the same year as "In Contradiction," so you can't blame them for not arguing convincingly against dialetheism, which was after all not much on the scene in 1987. On the other hand, to the extent that they are trying to "solve" the semantic paradoxes, though, they are at least trying to show that e.g. the Liar Sentence does not have its prima facie feature of being true if it is false and false if it is true. In practice, the set of arguments that would beg that question is necessarily co-extensive with the set of arguments that would beg the question against the dialetheist's position that the Liar Sentence really *is* both true and false.

Barwisee and Etchemendy argue that sentences like the LS, or, more precisely, the propositions expressed by those sentences, are false, but that, remarkably, they are not also true. And this despite the fact that they assume bivalance--every proposition is either true or false, and there are no truth value gaps.

How do they accomplish this minor miracle?

First things first, they model propositions as set-theoretic objects. This doesn't sound possible for self-referential sentences, given that standard Zermello-Frankel set theory forbids self-membership. As such, they go with Peter Aczel's "universe of hypersets," an exciting-sounding phrase for an alternate set theory (which contains the Zermello-Frankel universe of well-founded sets) in which circularity is permitted. (The sets in the orthodox hierarchy exist in Aczel's conception of the universe of sets, they just don't exhaust it. "Hypersets" are just sets outside of the hierarchy.) On the face of it, the adoption of Aczel's set theory to shed light on the semantic paradoxes sounds like it would let set-theoretic paradoxes through the back door, since these are kept out of orthodox set theory by the cumulative hierarchy of sets where sets can only have sets beneath them in the hierarchy as members. Not so, Barwise and Etchemendy claim, since Aczel has shown that we don't need the cumulative hierarchy to ban things like the Russell Set (the set of sets that are not members of themselves), we just need the set/class distinction. The Russell Set is not as set at all but a class, and there's no class of sets that are not members of themselves.

(If this sounds dangerously arbitrary, dear reader, I'm with you. In fact, here's a question I haven't been able to get a good answer to yet. If anyone wants to help me out in the comments, please jump in, since I suspect that I'm missing something important here. If we invented a new term that was neutral between sets and classes, like "grouping," where a grouping can be either a set or a class, then can't we re-instate the paradox by reference to a Russell Grouping? If not, why not?)

In any case, using Azcel hypersets, they establish models of propositions for both the "Russellian" and "Austinian" conceptions of propositions. (It's not clear how much of a historical/exegetical claim they are making about either Russell's or Austin's actual views in using these terms.) True Russellian propositions are made true by the world as a whole, and true Austinian propositions are made true by particular situations. (For example, the Austinian proposition "Jill has the 3 of Hearts," about a particular game, is false if the speaker has mistaken Lucy for Jill, but by coincidence Jill has the 3 of Hearts in another game across town.) Barwise and Etchemendy prefer the Austinian conception, although they first model Russellian propositions not just because problems in that model set up the Austinian model, but because (like the relationship of Zermello-Frankel sets and Aczel's hypersets) the Russellian sets are ultimately contained in the universe of Austinian sets.

Now, once we start the formal modelling--and this is true for both the modelling of Russellian propositions and later for the model of Austinian propositions--we get to my first and most important problem with the book.

(1) They blatantly beg the question, but making one of the rules of the model that no set-theoretic proposition-object and its "dual" will be included. Translated out of the set-theoretic context, this means that they set it up as a rule in advance that no proposition will be true and false. "We included a formal rule to ban inconsistency and, amazingly, no inconsistencies were generated by he model!"

This rule has, as you might expect, some funny consequences once the whole thing gets going. In the Russellian case, the result is that, since the world does not make the LS true, it is false, but the fact that its false is not included in "the world." Yes, you read that right. The world does not include the fact that it is false. Why not? Because, if it did, then the LS would be true as well as false, since it claims that it is false and the world which makes it true or false would include the fact that it is indeed false.


Now, they trumpet the fact that the fact that the LS is false gets to be included in "the world" as a great advantage of the Austinian alternative. On the face of it, it would be....except...well....we'll get to that next post.

Monday, January 28, 2008

Sunday, January 27, 2008

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.

Wednesday, January 23, 2008

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)


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. 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?