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.


Daniel Lindquist said...

I'm not sure that your list of ways to avoid Curry's paradox is exhaustive. (Or else I'm not quite grasping what falls under 1 and 4.) Wouldn't Hartry Field's position in "Truth and the Unprovability of Consistency" (available on his website) be a way to avoid Curry's paradox which doesn't fit neatly into any of your four categories?

Field's position seems to me to be most similar to 1, since he allows truth-value gaps in a way similar to what I understand of Kripke, but he does not seem to need to rely on any talk of "grounding" to justify this. (Or maybe he does talk about "grounding" without using the term, and I've simply not noticed it. I'm weak on Kripke; perhaps Field's willingness to grant most sentences of the form "A or not A", but deny that these hold due to their logical form, is just the sort of thing Kripke wants to get at with his ground-talk). Field thinks that avoiding various semantic paradoxes requires truth-value gaps -- motivates paracompleteness -- but he doesn't talk about "non-semantic reality" -- reasonably enough, since the topic he's considering is formalizations of mathematics; if we are only concerned with "non-semantic reality" then I am not sure we can have this in view.

But it also seems like Field's position should fall under 4, since Field thinks the usual treatment of validity is erroneous -- validity and necessary truth-preservation do not actually coincide.

Field argues that Curry's paradox shows that you can't have the standard introduction/elimination rules for the conditional along with the two parts of the T-schema (T(a)>a and a>(T)a, T-Elim and T-Intro, where > is the conditional). But what is novel about Field's approach is to maintain Modus Ponens as a valid rule of inference while not maintaining (and not denying) that it is necessarily truth-preserving -- in some cases Field thinks Modus Ponens is not truth-preserving, but it does not fail to preserve truth. So at the least, if Field's approach is a kind of 4, then it seems rather different from anything Priest considers. Indeed, Field argues that Priest's attempt to avoid Curry's paradox requires him to hold that Modus Ponens fails to preserve truth -- to affirm that in some cases Modus Ponens can lead to a false conclusion from true premises. That certainly seems unappealing, and Priest would want to avoid it. (Put another way, 4 seems unattractive because it seems unlikely to work -- Field runs through the options open to a a dialetheist, and they aren't ones Priest would cotton to.)

(I was lead to Field's paper by the discussion on this post, which concerns a modified Curry -- a "Curry for validity". JC Beall suggests that Field's approach to Curry is already able to handle this modified form, and I think that's right; I'm not sure that all ways to avoid the standard Curry work for this modified version, though. Logics with wonky conditionals usually retain standard notions of validity, and that's all this version of Curry seems to be abusing. So if your way of avoiding Curry is to modify how your conditional works, then this modified Curry seems to still show the truth of trivialism, which is absurd. So Curry and conditionals can come apart.)

Anyway, Field makes the same point you do about Priest and Curry's paradox -- Priest's approach to Curry is unattractive, and any attractive solution to Curry takes a lot of the wind out of dialetheism's sails. This seemed worth pointing out.

Ben said...

Thanks, Daniel.

I would point out that option (1) was, "(1) They could deny that the original sentence was a truth-bearer, e.g. on Kripke's ground...."

As you describe it, it sounds as if Field's solution would fit neatly into that, no matter how un-Kripke-an his favored grounds for denying that the Curry sentence is a truth-bearer.

Pseudonym said...

Why don't you allow anonymous comments? What could someone say that would offend you? Don't bother answering this question if it is too dull. The God-man is the only paradox one need worry about.

Ben said...

Hmm? I didn't actually remember turning off the anonymous comments option, but I guess I'm glad I did. It helps discourage spambots selling internet gambling and penis enlargement pills on any random web site they happen upon. Not, as they say on Seinfeld, that there's anything wrong with any of that, I'd just prefer not to wade through it on my way to the relevant comments.

Certainly it has nothing to do with fear of offensive comments, since (a) the subject (philosophical logic) is, by its nature, not one that normally provokes angry and highly personal discussion, (b) as your own choice of username cleverly suggests, it's hard to see how forcing people to register usernames would do anything to discourage offensive comments, (c) even in the very distant possible worlds where philosophical logic was a common topic of flamewars, I'm not sure how my knowing who you are would stop anyone from making nasty comments (what, I'm going to take my revenge on you using the awesome and ferocious power of a Philosophy grad student?), and (d) I think most people who know me would probably tell you that I'm not terribly easily offended. Not, mind you, that you should take (d) as a challenge.

Anyway, the God-man, huh? Do you actually think that Christianity's doctrine of the incarnation (I am getting my doctrine names right, aren't I?) entails some sort of logical impossibility or are you thinking of something else?

(As far the interface between religious doctrine and logical possibility goes, I actually take the Stone Paradox and things like that fairly seriously, although of course the solution is much more obvious than the trickier paradoxes involving our basic semantic notions--just reject theism, or even moderate your theism enough to say that God isn't literally *all*-powerful, just almost-all-powerful, which would still be pretty impressive from a human perspectivel.)

Colin said...

I agree, why are philosophers so hung up on the notion that God must be all-powerful... wouldn't it be enough if God were just the most powerful thing in existence? That's still pretty powerful.

Daniel Lindquist said...

"I agree, why are philosophers so hung up on the notion that God must be all-powerful... wouldn't it be enough if God were just the most powerful thing in existence? That's still pretty powerful."

Presumably because they care about "God" at all because of the God of Western Monotheism, who is explicitly declared to be omnipotent in all the interesting strains of the tradition. If philosophers talk of a "God" who is not omnipotent, but instead is simply that-than-which-no-greater-can-be-found, then they're talking about something other than God. Which might be interesting in a purely metaphysical sense, but I bet most folk find "God" philosophically interesting because of the various major religions which have shaped the life of the West.

Ben said...

Well, I'm not sure what you mean by "interesting" strains here, but on the question of the definition of God, if someone said "I believe in an immaterial, eternal entity that created everything other than itself in existence, that controls who goes where in the afterlife, that has issued various commandments as to how humans should live, and is immensely, unfathomably powerful, far, far more powerful than any other entity that has ever existed or will ever exist by orders of magnitude, but I don't believe that this entity is quite literally all-powerful"...

....your response would be, "OK, but you don't believe in God, right?"

Daniel Lindquist said...

"Well, I'm not sure what you mean by "interesting" strains here"
Judaism, Christianity, Islam. Maybe a few others, to lesser extents, at various times. Religions that actually have enough followers to have a visible effect on public life, which is what makes them "interesting". If it weren't for such religions as these, I don't think philosophers would express much of an interest in any questions about "God". And so the religions' dogmatic conceptions greatly influence what philosophers find it worthwhile to talk about when it comes to "God".

If someone told me they subscribed to such an odd metaphysical doctrine as the one you described, I'd probably try to figure out why only omnipotence seemed problematic to them, if all the rest of the things on that list seemed hunky-dory. I suspect it might come down to a verbal quibble -- if they think that being "all-powerful" implies being able to perform impossible actions, then in denying that God is omnipotent they aren't saying anything that, say, Aquinas would disagree with. If they really do mean that they believe in a near-God-like entity who cannot bring about some genuinely possible states of affairs: Well, that strikes me as pretty weird. I don't know what would motivate a position like that. So I'd ask them about various things to try and puzzle it out.

I would not say "But you don't believe in God, right?" because I don't see what the point of such a question would be. Given that this person has such odd beliefs, I don't know what they would make of the question. And so their response is useless to me -- if they say "yes" I don't know what they endorsed; if they say "no" I don't know what they rejected. And in any case that sort of question strikes me as likely to cause offense. You don't make friends by declaring people heretics.

It also occurs to me that having false beliefs about X doesn't entail not believing that X exists. If someone believes that there is a person such that they are currently the President of the United States, the son of a previous president, and also a lizardman from the center of the Earth, then I'm not inclined to say they do not believe that GW Bush exists. They just think he's a reptilian monster in disguise, and not a genuine human. There might be situations where it's more helpful to treat them as not believing that GW Bush exists, since they don't believe any such human as GW Bush exists, but that's not the sort of view I would be initially inclined to attribute to them. So in your example, I'd at most be inclined to object that they are wrong about God, not that they believe in some other markedly-similar entity. Our disagreement would be over the divine attributes, not whether or not the world included either or both of two Very Powerful Entities.

Ben said...

OK, see, here I was gearing up to argue about whether "God" was a definite description or a rigid designator, but it sounds like we might actually agree on that. (The reference question is of course independent of theism vs. atheism, since you can presumably rigidly designate both real and fictional entities.)

I do wonder about the viability of the possibility/impossibility move on omnipotence. On the face of it, 'God can do anything' entails logical impossibilities the same way that naive set theory's comprehension axiom 'for every every property, there is a set of objects that satisfies that property' entails logical impossibilities. Just as we don't see 'for every property except for the ones that entangle us is paradoxes when we try to assemble sets for them, there is a set of objects that satisfy that property' as a particularly satisfying alternative to the naive axiom, I'm not sure why 'God can do anything except for create stones heavy enough to generate a contradictory consequence when combined with the simple form of this rule' is a particularly satisfying alternative to standard-issue omnipotence.

J said...

.....Just as the ordinary semantic paradoxes, like the Liar, are prima facie sound arguments for dialetheism.....

You and your circle of anti-rationalists repeat this routinely, but have really not dealt with the objections to the Liar (or really, even to the interpretation of the Liar).

"This sentence is a lie" = "here is a sentence, and it contains a lie." So, is there an existence generalization?, Something like a sentence exists, and contains a lie. Put that way, does not seem paradoxical.

What does it refer to? It could be read as meaningless (ie does not refer to any real paradoxical situation), OR it's something like "this sentence exists (whatever that means), and it states a lie." A lie about what? That the sentence exists? The sentence does exist, so yes, to say it doesn't would simply be a lie. Or, "it's true this sentence exists, yet this sentence does not state truth." Not really paradoxical, or even well-formed, when you think about it. But the analytical biz depends on this sort of semantic obscurantism, so it's taken to be a big deal.

The real paradoxes arise from problems of computability, and undecidebility. If there are arguments that cannot be proven to be T v F within predicate logic via a reductio (and there are a few, ala Church or was it Turing, etc.--if not Goedel), then the system of computation appears to have a glitch (really it's not that big of deal either, when some type of restriction on self-reference is included). But undecidability a bit more daunting than ye olde Epimenides...........

Ben said...

As you know the Liar sentence isn't "this sentence is a lie," it's "this sentence is false" or (equivalently, to get around truth-value-gap dodges) "this sentence is not true."

I'm not sure why I'm supposed to be an anti-rationalist. Far from not dealing with objections to the Liar (assuming this means proposed solutions to the paradox), dealing with the ones that I don't think work, and trying to suggest lines of attack that might work, has been my primary purpose here.

I certainly don't think the argument from the Liar sentence is a actually a sound argument for dialetheism. (I'm not a dialetheist.) Hence my use of the phrase 'prima facie.' It looks like a sound argument, in that the premises are all initially plausible and the inferences from them are impeccable. There's a reason why the ordinary response of non-philosophers is not, 'why should I think that's a meaningful sentence?' It's irritated puzzlement, an intuition that there must be something wrong with it, befuddlement about what exactly it could be, and a general sense that someone else can worry about it.

Of course, given the falsity of the conclusion, one of the premises must indeed be wrong (and the premise that the Liar is a meaningful sentence is a good one to pick), but it's not obvious, and it requires argument. After all, self-referential sentences can pretty clearly sometimes be meaningful, e.g.

S1: "This sentence has seven words in it."
S2: "This sentence has eleven words in it."

S1 looks meaningful and clearly true, and S2 looks meaningful and clearly false.

Similarly, it seems to pretty routinely be the case that meaningful sentences can have no other function than to ascribe truth value to a sentence, e.g.

S3: S1 is true.
S4: S2 is true.

S3 looks meaningful and clearly true, and S4 looks meaningful and clearly false.

Now, if you want to say that a sentence that does both of these things (refer to itself, and do nothing but ascribe a truth-value to a sentence) cannot be meaningful, then, given that neither of those things on its own is normally a block to meaningfulness, the truth of the meaninglessness claim is pretty non-obvious and we have to provide some kind of substantive argument to back it up. To say that you can't meaningfully do both *because this allows us to make paradoxical statements* is both thoroughly unsatisfying and nakedly question-begging.

If 'semantic obscurationism" means that we pretend for the sake of playing the analytic game not to understand obvious things, then given that it's far from obvious, this is pretty clearly not what's going on here. If it simply means 'not agreeing with me, since I'm right,' then I guess there's nothing much to say about that.

J said...

There's no special knowledge required to understanding the Liar (or arguably to philosophy itself---and many humans--even non-philosophy majors!---have passed symbolic logic courses (even earned A's)). Arguments from Authori-tay are not valid anyway; it's sort of like an old-school chess open: X-offski doesn't win simply by flashing his ELO rating, and when some unnamed, unrated Y-levsky defeats X-offski, he, uh, beats him.

And for that matter, that X has a PhD in Wittgenstein or Derrida studies does not imply that X can solve the Liar (or Halting problem), or even balance his checkbook, for that matter. It might just mean X paid off the right peeps.

The objections to the Liar (as even stated on the Wiki) are not just to be shrugged off: Prior's point for one still seems quite relevant (and relates to descriptions). Assertions usually imply an existence claim, do they not?? "This" means something like "this sentence exists, or even, "this sentence points to some state of affairs" (in WittSpeak). So what's the relation of the non-truth to that? This sentence exists, and states a lie (about what??).

I am not denying paradox, but the Liar as given in this form does not illustrate paradox. AS stated, it's either a simple contradiction (this sentence is T and not-T) or meaningless/not-well-formed-- not paradoxical (doesn't Kripke sort of suggest that as well? Without relating to a state of affairs, the Liar's sort of just sheeits and giggles. Undecidability and the Halting problem, however, are not, whatsoever.

J said...

When well-formed (as with say Quine's variation--“Yields falsehood when preceded by its quotation,” yields falsehood when preceded by its quotation.), the semantic paradox does not really shake the foundations of Reality as a whole, either, except to some narcissists or religious nuts who mistake logic (and philosophy, really) for some heavenly abode.

One could argue, sort of constructively, that the syntax can be tweaked to create these sort of paradoxical glitches (whether in terms of natural or artificial language), but that does not means the entire system, or proof procedure is called into question. Besides, given a certain descriptivist account of language (and dare we say empirical reality), the real effects of such glitches are negligible (or perhaps the paradox fetishist should show why they are not negligible. OK we might respect that knights and knaves dude, but even the well-formed paradoxes are sort of constructs---X sets up some strange combinations on a board, but they don't arise during the game. Keynes over Krip-ke)

Ben said...


I have literally no idea what the hell you're even talking about with all this stuff about no special knowledge and arguments from authority. Who do you take yourself to be arguing against? What's the point? Has anyone here been making arguments from authority? All I remember arguing was that it was a tricky issue and the correct answer was far from obvious, so it shouldn't be brushed off as obscurationist nonsense too quickly.

As far as Prior goes, we've been over that:

At least as you describe his solution, it's simply irrelevant. "This sentence is true and this sentence is false" is a contradiction that can just be false, but "It is true that this sentence is false" is still paradoxical. To get the first, you have to argue that all sentences are implied conjunctions of the apparent sentence and the Truth-Teller ("This sentence is true"), which (a) runs pretty deeply contrary to data in terms of what sentences *seem* to be about, and (b) it's a bit uncomfortable to make the Truth-Teller part of every sentence, given that it looks damn near impossible to assign a non-arbitrary truth-value the Truth-Teller, and as even Graham Priest says in "In Contradiction," it seems to be as good a candidate for truth-value-gap status as anything.

As far as Kripke goes, what he says is that if you abandon classical logic in favor of a 3-valued Kleene logic, and impose a groundedness condition for truth, such that series of sentences that do nothing but ascribe truth-valuest to sentences ultimately have to ground out, and only assign falsehood to things whose negation ends up true, then the paradoxical sentences will always end up with ?'s next to them on the truth-table. Intuitively, there's something very appealing about what Kripke does, but giving up classical logic in favor of something much inferentially weaker is a pretty steep price to pay.

Of course, what Kripke says is utterly incompatible with what Prior says (if the Liar was just a contradiction, then we could save bivalance and just declare it false), both in intuitive foundations and formal details, so I'm not sure why you gesture at both of them as if somehow in combination they proved that no particularly difficult issues were at stake here.

Again, I have no idea why you think I'm "shrugging off" solutions to the paradox. I've provided some detailed arguments against solutions that I don't think work, and I've indicated in a general sense the sort of solution that I do think would work. Why on earth do you think anyone is shrugging anything off?

Who thinks that semantic paradoxes shake reality to its foundations? If they are unsolvable without loss of expressive power (which I certainly don't concede), then they are sound arguments for dialetheism, but even that would shake up our description of reality, not the thing itself. If you don't find it an interesting problem area, you certainly have a right not to be interested, but I don't really understand your rationale for declaring the people who do work on it (excluding Prior and Kripke....?) to be necessarily narcissists or religious nuts.

J said...

Compare Quine's paradox with the Liar: note the lack of "this". The word "This" is a demonstrative--it points at something. That presents another issue, related to description, and reference really. With the Quine formation there is a paradox, and one doesn't have to worry about the reference issue (Prior, however, DOES recognize the reference issue, and I suspect upholds descriptivism as well).

I am not really interested in the specifics of Kripke---anyone who understands how SK misread and mangled Russellian descriptions should sign a petition to have his tenure revoked---YET the point on grounding does seem somewhat "referential." A sentence with demonstratives--like "This sentence is...."--- and definite articles wants to POINT; it does semantics, even when logicians don't want it to.

At least with Quine's reformulation, that demonstrative issue has been avoided. Not sure if it's a "real" paradox, but closer than just the simple Liar.