Quantum logicians* claim that there are cases where the behavior of true statements about the properties of subatomic particles fails to conform to the distribution of conjunction over disjunction in classical logic (i.e. the rule that lets us go from P & (Q v R) to (P & Q) v (P & R)). Now, for there to be a counter-example to this law, we'd need a case where P was true, (Q v R) was true, but both Q and R had some status other than "true."**
After all (holding the truth of P constant in all of these cases), if Q was true and R wasn't, then the premise of the relevant instance of Distribution would be true, but so would the conclusion. The same would be true if R was true and Q wasn't. And, of course, it would still be true if P and Q were both true. If, on the other hand, Q and R were both false, then once again, we wouldn't have a counter-example to Distribution, because the premise would be false.
Fair enough, you might think, but that just shows you that the old bivalent conception of truth is wrong, and that's exactly the sort of thing we should expect to be shown once we've really absorbed the quantum revolution, really exposed the ancient dogmas encoded in classical logic to the searing light of empirical revision.
OK. Maybe. But postulating a third truth-value, by itself, doesn't clarify much here. What third truth-value would get the job done? A natural first thought is that what we're talking about here is a truth-value gaps--i.e. the joint absence of the two classical values--but that's not going to get it done. If Q is neither true nor false, and R is neither true nor false, then why should (Q v R) be true rather than neither true nor false itself? Now, if we think of third truth-value not as a gap but as a glut--the joint presence of the two classical values--the situation might seem to be a little bit better. Now, after all, the premise of the instances of Distribution where Q and R are both oddly-valued comes out true (whatever else it might be). The problem, of course, is that the conclusion also comes out true.
One might say that the third value is not a matter of being definitely neither or definitely both but being in some sense vague or ambiguous or indeterminate between the two. Fine. But why, then, wouldn't both the premise and the conclusion come out as vague or ambiguous or indeterminate or whatever? If it's ambiguous whether or not Q is true, and ambiguous whether or not R is true, but P is unambiguously true, shouldn't it be ambiguous whether (Q v R) is true, and also ambiguous whether (P & Q) is true, whether (P & R) is true, and whether ((P & Q) v (P & R)) is true?
One might set up the truth tables differently here, but its hard to see how one could do so, in a principled way and without opening oneself up to some "change of meaning" charges. Of course, some people routinely level those charges against all heterodox proposals about the behavior of logical connectives, but it would be much harder to answer them here. To see why, think of it likes this:
In classical logic, "either P or Q is true" and "at least one of the following things is true: P, Q" are different ways of saying the same thing. Now, if a heterodox logician comes along and says "sometimes it's ambiguous whether P is true, and it's equally ambiguous whether Q is true," and then concludes that in those cases it's ambiguous whether at least of the two is true, then the "change of meaning" charge seems unfair. It seems more natural to say that they mean the same thing by "or" as the classical logician, but that they admit possibilities that the classical logician rejects. If, by contrast, they say that it's ambiguous whether P is true, and ambiguous whether Q is true (and not epistemically ambiguous, but in terms of its objective truth-status), but that "either P or Q" is unambiguously true, it really does start to seem like they're using "or" in a new way.
...or maybe not. A more radical move yet would be to simply reject truth-functionality entirely here. Just as "for any collection of numbers, there is a sum of those numbers" is true (and might even seem so obvious as to follow from the meaning of "number" or "collection") so long as we restrict our focus to finite (and countably infinite) collections of numbers, but it breaks down when we get to uncountably infinite collections (like the collection of all real numbers), to which addition simply doesn't apply, one could argue that "the truth-value of disjunctions is a function of the truth-values of their disjuncts" holds when we restrict our attention to normal situations, but that it breaks down when we turn our attention to the outer edges of logical possibility that are physically actualized by quantum weirdness.
OK, fair enough, but if they do choose to take that line, it's surely incumbent on the quantum logician to give us a clear account of exactly what exactly the distinction is between normal and non-normal situations. If the distinction is simply a matter of shifting truth-values, then this isn't a proposal about a break-down of truth-functionality, it's simply about non-standard truth-functionality, and given the failure of the third truth-value to transfer from the disjuncts to the disjunction, the change of meaning question looms large. If the distinction between normal and non-normal situation is about something other than truth-values--e.g. we have a situation where Q and R are both false but where (Q v R) somehow manages to be true, or where P, Q and R are all true but (P & Q) v (P & R) somehow fails to be true--then they really owe us a very clear explanation of how the inclusive "or" of formal logic can retain its customary meaning at the same time as two false disjuncts somehow jointly yield a true disjunction, or two true disjuncts can fail to yield a true disjunction, and exactly what the difference is between situations where logical connectives behave in this strange way and the situations in which they don't, and exactly how to distinguish between which situations are which.
Now, from an orthodox perspective, it's tempting to conclude from the whole mess that the proposal that Distribution fails in quantum contexts is just deeply confused, and that might even be the right answer here, but I'd be far more interested in hearing attempts to resolve it and explain just how the trick can be turned--e.g. exactly how we can conceptualize a third truth-value that would plausibly behave in the right way, or how to make sense of the idea that the standard truth-values would in the relevant situations stop combining in the standard ways.
Thoughts?
*For our purposes here, the phrase "quantum logicians" refers to full-on, 1970s-Putnam-style, realist, monist quantum logicians, not the namby-pamby kind that just take quantum logic to be an interesting mathematical representation of certain experimental results and leave it at that. The latter might be far easier to plausibly argue for, but it's also far more boring.
**Note that, for the sake of simplicity, in everything that follows I'm assuming that conjunction behaves in the standard way. If anyone wants to get into that in the comments, and provide a quantum-logical motivation for questioning that, that's fine too.
Wednesday, January 27, 2010
Monday, January 25, 2010
Dialetheism and Scientific Reasoning
As paraconsistentists of all stripes are fond of pointing out, historically well-accepted scientific theories have often entailed various inconsistencies. There seems to be some sense of “accept” in which working scientists are often willing to accept implicitly inconsistent theories (or pairs of mutually inconsistent theories, like classical formulations of relativistic and quantum physics) while they search for consistent alternatives. Partisans of what one can call “paraconsistency-lite”—the view that classical logic is wrong in holding that everything follows from every contradiction, but not because this “explosion” of inferences has true counter-examples (i.e. not because any contradictions are true)—are particularly fond of these sorts of examples. After all, by their lights, the scientists’ attitude is exactly the rationally correct one: accept the inconsistency, don’t accept triviality, and don’t lose sight of the fact that the inconsistent theory you’ve provisionally accepted must ultimately be wrong, because contradictions are never true. People who, for example, accepted Bohr’s theory of the atom in the formulation that relied on Maxwell’s equations, were, implicitly and without having the logical formalism to explain why they were right, proto-paraconsistentists.
Or so goes the argument. I have a lot of issues with the whole line of thought, not least of which is the fact that, while I’m as naturalistic in my epistemic stance as the next guy (actually, in my Department, I tend to be way more naturalism-friendly than the next guy), I think that this sort of thing tends to devolve into a sort of exaggerated hyper-naturalism where the empirical sciences are taken not just as pretty good tools for getting to the truth in the long run, but as nearly flawless paragons of rationality in all of their day-to-day epistemic practices—the speed at which the scientific community actually arrives at a conclusion must be the correct speed, etc. There’s a lot to be said about this—I’ve blogged here about these sorts of “retroactive implicit paraconsistency” arguments more than once before, and I certainly will again—but right now I want to focus on another subject entirely.
From the perspective of the paraconsistency-lite crowd, the attitude of those scientists willing to temporarily swallow inconsistency while looking for the right theories (which they assume must be consistent) makes perfect sense, but what about dialetheism? (To switch from beer to cigarettes, we can think of the views we’ve been discussing as Paraconsistency Lights and dialetheist positions as Paraconsistency Reds.) I’ve read many passages in which Graham Priest basically makes the classical Paraconsistency Lights argument and leaves at that, but only one where he makes the obvious Paraconsistency Reds twist on it.
It comes on pp. 149-150 of my edition of Doubt Truth To Be A Liar:
“It is here that the impact of paraconsistent logic is revisionary—indeed, revolutionary. The Law of Non-Contradiction has been well-entrenched in Western thought—and so science—since the canonization of Aristotle, whose defence of the Law has rarely been challenged. Hence, scientists and philosophers have not been prepared to brook the thought that an inconsistent theory of any kind might be true. But subscribing to the law is not rationally mandatory, as I argued in Chapter 7, and as the development of paraconsistent logics has played a large role in showing. Once this fact is digested, scientists may—justifiably—take a different attitude to inconsistent theories of the appropriate kind. Indeed, they may even develop inconsistent theories, if these have the right empirical consequences, just as paraconsistent logicians have developed inconsistent theories of semantics to handle the paradoxes of self-references.”
…and that’s pretty much all he has to say about it. (To be fair, in a couple of other places in DBTL he gives a couple of hypothetical examples of ways this might happen, but this is pretty much what he has to say on the epistemic issue.) So, to sum up:
(1) Scientists have been in the thrall of the Aristotelian dogma of non-contradiction, which explains their bias that inconsistent theories can never be true, but
(2) Philosophical work on the self-references paradoxes, the development of inconsistency-tolerant formalisms, etc., shows us that this dogma should be rejected, so
(3) Now, one hopes, they'll start being less dogmatic.
This is certainly a far cry from the extreme deference to scientific practice you tend to get with the paraconsistency-lite foks, and it raises a lot of strange and awkward questions. Given that, as his less radical comrades are always pointing out, temporary inconsistency is not that rare a situation in the history of science, and that these are presumably situations where inconsistency is pulling in one direction (showing that the theory must be false) but happy empirical consequences are pulling in the other (giving the scientific community good reason to continue to provisionally accept the theories in question), what sort of retroactive advice should someone who agrees with Priest give scientists in this position? What scientific progress would have been lost by researchers agreeing with Priest, and what future progress would be lost if the contemporary scientific community were to be won over to his position?
To switch gears for a second, consider a traditional criticism that defenders of neo-Darwinian biological orthodoxy level against “Intelligent Design” creationists: that, if their views were generally adopted, researchers would be content to put God in the gaps in their understanding, and future explanatory progress would be stunted.
Now, here’s my question for readers more sympathetic to dialetheism than I am:
Does an analogous criticism apply to what Priest is advocating here? If not, why not? What’s the difference?
Certainly, as a sociological matter, most dialetheists are atheists (although, as I’ve mentioned before, I think that the appropriation of dialetheism by at least some philosophically aware Christian theologians is a historical inevitability), so part of their answer here might just be that there is no good reason to believe that God exists (the central philosophical arguments for God all being fairly spectacularly unconvincing), whereas the paradoxes of self-reference do give us good reason to believe in true contradictions.
Fair enough. If so, though, how about the counterfactual? If there were a good, convincing argument for the existence of God (e.g. some bright young grad student at Notre Dame came up with an absolutely unanswerable re-formulation of the Ontological Argument), would Intelligent Design theory be a rationally acceptable option for biological science, or would the stunting-future-inquiry objection still be a good one, and as such would the stance of those theologically liberal Christians who accept God but categorically reject ID be the right one? If so, is there an epistemically relevant difference between the ID creationist’s suggestion that gaps in our understanding should at least sometimes simply be accepted as evidence of divine intervention and Priest’s suggestion that inconsistent theories should at least sometimes not just be provisionally accepted while everyone assumes they can’t possibly be true and tries to find a consistent alternative, but actually accepted as true? If so, what is it?
Or so goes the argument. I have a lot of issues with the whole line of thought, not least of which is the fact that, while I’m as naturalistic in my epistemic stance as the next guy (actually, in my Department, I tend to be way more naturalism-friendly than the next guy), I think that this sort of thing tends to devolve into a sort of exaggerated hyper-naturalism where the empirical sciences are taken not just as pretty good tools for getting to the truth in the long run, but as nearly flawless paragons of rationality in all of their day-to-day epistemic practices—the speed at which the scientific community actually arrives at a conclusion must be the correct speed, etc. There’s a lot to be said about this—I’ve blogged here about these sorts of “retroactive implicit paraconsistency” arguments more than once before, and I certainly will again—but right now I want to focus on another subject entirely.
From the perspective of the paraconsistency-lite crowd, the attitude of those scientists willing to temporarily swallow inconsistency while looking for the right theories (which they assume must be consistent) makes perfect sense, but what about dialetheism? (To switch from beer to cigarettes, we can think of the views we’ve been discussing as Paraconsistency Lights and dialetheist positions as Paraconsistency Reds.) I’ve read many passages in which Graham Priest basically makes the classical Paraconsistency Lights argument and leaves at that, but only one where he makes the obvious Paraconsistency Reds twist on it.
It comes on pp. 149-150 of my edition of Doubt Truth To Be A Liar:
“It is here that the impact of paraconsistent logic is revisionary—indeed, revolutionary. The Law of Non-Contradiction has been well-entrenched in Western thought—and so science—since the canonization of Aristotle, whose defence of the Law has rarely been challenged. Hence, scientists and philosophers have not been prepared to brook the thought that an inconsistent theory of any kind might be true. But subscribing to the law is not rationally mandatory, as I argued in Chapter 7, and as the development of paraconsistent logics has played a large role in showing. Once this fact is digested, scientists may—justifiably—take a different attitude to inconsistent theories of the appropriate kind. Indeed, they may even develop inconsistent theories, if these have the right empirical consequences, just as paraconsistent logicians have developed inconsistent theories of semantics to handle the paradoxes of self-references.”
…and that’s pretty much all he has to say about it. (To be fair, in a couple of other places in DBTL he gives a couple of hypothetical examples of ways this might happen, but this is pretty much what he has to say on the epistemic issue.) So, to sum up:
(1) Scientists have been in the thrall of the Aristotelian dogma of non-contradiction, which explains their bias that inconsistent theories can never be true, but
(2) Philosophical work on the self-references paradoxes, the development of inconsistency-tolerant formalisms, etc., shows us that this dogma should be rejected, so
(3) Now, one hopes, they'll start being less dogmatic.
This is certainly a far cry from the extreme deference to scientific practice you tend to get with the paraconsistency-lite foks, and it raises a lot of strange and awkward questions. Given that, as his less radical comrades are always pointing out, temporary inconsistency is not that rare a situation in the history of science, and that these are presumably situations where inconsistency is pulling in one direction (showing that the theory must be false) but happy empirical consequences are pulling in the other (giving the scientific community good reason to continue to provisionally accept the theories in question), what sort of retroactive advice should someone who agrees with Priest give scientists in this position? What scientific progress would have been lost by researchers agreeing with Priest, and what future progress would be lost if the contemporary scientific community were to be won over to his position?
To switch gears for a second, consider a traditional criticism that defenders of neo-Darwinian biological orthodoxy level against “Intelligent Design” creationists: that, if their views were generally adopted, researchers would be content to put God in the gaps in their understanding, and future explanatory progress would be stunted.
Now, here’s my question for readers more sympathetic to dialetheism than I am:
Does an analogous criticism apply to what Priest is advocating here? If not, why not? What’s the difference?
Certainly, as a sociological matter, most dialetheists are atheists (although, as I’ve mentioned before, I think that the appropriation of dialetheism by at least some philosophically aware Christian theologians is a historical inevitability), so part of their answer here might just be that there is no good reason to believe that God exists (the central philosophical arguments for God all being fairly spectacularly unconvincing), whereas the paradoxes of self-reference do give us good reason to believe in true contradictions.
Fair enough. If so, though, how about the counterfactual? If there were a good, convincing argument for the existence of God (e.g. some bright young grad student at Notre Dame came up with an absolutely unanswerable re-formulation of the Ontological Argument), would Intelligent Design theory be a rationally acceptable option for biological science, or would the stunting-future-inquiry objection still be a good one, and as such would the stance of those theologically liberal Christians who accept God but categorically reject ID be the right one? If so, is there an epistemically relevant difference between the ID creationist’s suggestion that gaps in our understanding should at least sometimes simply be accepted as evidence of divine intervention and Priest’s suggestion that inconsistent theories should at least sometimes not just be provisionally accepted while everyone assumes they can’t possibly be true and tries to find a consistent alternative, but actually accepted as true? If so, what is it?
Thursday, January 21, 2010
This Is Only Sort Of A Post
....and mostly only here to re-assure you, gentle reader, that missing two scheduled posts in a row doesn't actually mean this blog has gone gentle into that good night night. Just normal wear and tear to the schedule from flying back to Miami on Monday, starting to teach on Tuesday, fighting with the power company and gearing up for (what I really hope will be) the final round of dissertation revisions in the next couple weeks.
That said, I should be back to the regular schedule on Monday. So, for anyone who's actually reading this, I'm open to audience requests. Anything you'd be particularly interested in hearing about for the next few weeks?
That said, I should be back to the regular schedule on Monday. So, for anyone who's actually reading this, I'm open to audience requests. Anything you'd be particularly interested in hearing about for the next few weeks?
Wednesday, January 13, 2010
Monday, January 11, 2010
A Question About Truth And Sets
The Liar Paradox is sometimes seen as a problem for "naive truth theory" roughly parallel to the problem for naive set theory posed by Russell's Paradox. In the current literature, however, there seems to be an interesting difference. When it comes to semantic paradoxes, it's not hard to find arguments for truth-value gaps, truth-value gluts or even (as in, for example, J.C. Beall's case) a combination of both solutions. When it comes to the set-theoretic paradoxes, though, lots of paraconsistentists argue for re-capturing the inferential power and intuitive appeal of naive set theory by simply embracing set-membership gluts when it comes to cases like the Russell Set. Other than Penelope Maddy's "Proper Classes," though--an article that's high on technical detail and low on philosophical motivation--it seems pretty hard to find anyone who proposes that we just rehabilitate naive set theory by embracing set-membership *gaps* when it comes to cases like the Russell Set or the set of all ordinal numbers, etc.*
Am I wrong about this? Are there legions of gappy naive set theorists out there I haven't heard of?
If not, does anyone have any thoughts about *why* there don't seem to be any (or almost any), given the level of structural symmetry between the two types of paradoxes?
*And, really, that doesn't seem to be *exactly* what Maddy is up to in that article, although she's at least in the vicinity of that position.
Am I wrong about this? Are there legions of gappy naive set theorists out there I haven't heard of?
If not, does anyone have any thoughts about *why* there don't seem to be any (or almost any), given the level of structural symmetry between the two types of paradoxes?
*And, really, that doesn't seem to be *exactly* what Maddy is up to in that article, although she's at least in the vicinity of that position.
Wednesday, January 6, 2010
Of Cockroachs and Kangaroos
Dialetheists, since they think some contradictions are true, don't think that, e.g. Disjunctive Syllogism is truth-preserving. After all, if P and ~P can both be true, then we can have a situation where P is true (and as such (P v Q) is true as well), and ~P is true, but Q is (just) false. This has the advantage for them that it gives them a principled reason to reject the inference from contradictions to triviality (which relies on DS), but it has the extremely counter-intuitive effect of forcing them to deny that an intuitively obviously-valid-seeming inferential move used a thousand times a day in ordinary reasoning is valid. To use a shop-worn example, even a dog uses Disjunctive Syllogism after a fashion when following a scent trail to a fork in the road, sniffing one fork, not finding the scent, then immediately bounding down the second without bothering to check it first.
As such, dialetheists generally attempt what's called the "classical re-capture," the idea being that they have some principled reason to continue to reason classically in at least some domains, despite not regarding some of the classical rules they're reasoning with as being technically valid.
In the past, I've mentioned the probabilistic argument for the classical re-capture, which is the main one that Priest uses in In Contradiction. A conversation over lunch at the APA helped clarify for me another formulation of classical re-capture that doesn't rely on any statistical claims, the "where they live" argument.
It goes like this:
Contradictions live in these four houses--semantic phenomena involving self-reference, mathematics (where we have the set-theoretic paradoxes and the dialetheic take on arithemtic incompleteness), instants of change, and the law (where we can both have and not have certain legal rights), etc. Or, actually, these days, for Priest at least, I guess they live in five houses, since he's recently moved to embrace vagueness paradoxes as instances of Inclosure. We haven't discovered contradictions yet in any of the other houses, so we can probably assume we're in a contradiction-free zone when we visit them.
An analogy someone gave me is this:
When driving around the back roads of rural Australia, you need to watch out for kangaroos that hop onto the road and cause messy accidents. When driving around Los Angeles, you don't need to worry about that. It's not logically impossible that a kangaroo could somehow make it to L.A., but it would be silly to alter your driving practices in any special way because of kangaroo danger there, whereas there are bits of Australia where that would be far from irrational.
Similarly, when reasoning in contexts known to contain true contradictions, you need to reason paraconsistently, since you might run into counter-examples to some of the classical laws. When reasoning in contexts not known to contain true contradictions, you don't need to worry about it.
I'm deeply skeptical about this analogy. In the case of kangaroos, after all, we can tell a causal story about why they're present in Australia but they aren't likely to make it to L.A. It's not clear what kind of story about why true contradictions are present in these domains but its not likely that they'd crop up in others we could tell on the Priestian account, given the variety and heterogeneous nature of the 4 to 5 "houses" listed above.
Moreover, they're hard to miss. To combine the analogies for a moment, if there's a kangaroo jumping around your house, you'll notice it in short order. Maybe instead, we should think about cockroaches. Cockroaches like to live in dark, damp, out of the way places, and you can have cockroaches in your house for a long time before you notice one scurrying around in the periphery of your vision. This automatically makes cockroaches a better analogy if there are such things as true contradictions, given that In Contradiction wasn't published until 1982, following a few thousand years of western philosophy pretty much systematically failing to notice the existence of true contradictions.
Now, imagine that three houses are all laid out in similar ways and inhabited by three very similar people with very similar habits and possessions. Imagine that cockroaches are discovered in the house on the left and in the house on the right, and the inhabitants of those houses start taking various defensive measures--being extra-careful about leaving food out where the cockroaches could get at it, applying bug spray to dark corners, etc. Now, if there's some sort of principled causal story about the differences between the houses that makes the house in the middle less likely to have cockroaches--i.e., as in the classic civil defense video, the house in the middle is kept much cleaner than the house on the left or the house on the right--then it might make perfect sense for the owner of the middle house not to bother taking these extra measures. If, on the other hand, we have no such story to tell--the house in the middle is no cleaner than the ones on the left and the right, the owner is no less likely to leave crumbs on the counter, there are no fewer dark and damp spaces, and so on--then the mere fact that we haven't discovered any cockroaches in the middle house yet seems utterly irrelevant to the question of whether we should take such measures.
As such, dialetheists generally attempt what's called the "classical re-capture," the idea being that they have some principled reason to continue to reason classically in at least some domains, despite not regarding some of the classical rules they're reasoning with as being technically valid.
In the past, I've mentioned the probabilistic argument for the classical re-capture, which is the main one that Priest uses in In Contradiction. A conversation over lunch at the APA helped clarify for me another formulation of classical re-capture that doesn't rely on any statistical claims, the "where they live" argument.
It goes like this:
Contradictions live in these four houses--semantic phenomena involving self-reference, mathematics (where we have the set-theoretic paradoxes and the dialetheic take on arithemtic incompleteness), instants of change, and the law (where we can both have and not have certain legal rights), etc. Or, actually, these days, for Priest at least, I guess they live in five houses, since he's recently moved to embrace vagueness paradoxes as instances of Inclosure. We haven't discovered contradictions yet in any of the other houses, so we can probably assume we're in a contradiction-free zone when we visit them.
An analogy someone gave me is this:
When driving around the back roads of rural Australia, you need to watch out for kangaroos that hop onto the road and cause messy accidents. When driving around Los Angeles, you don't need to worry about that. It's not logically impossible that a kangaroo could somehow make it to L.A., but it would be silly to alter your driving practices in any special way because of kangaroo danger there, whereas there are bits of Australia where that would be far from irrational.
Similarly, when reasoning in contexts known to contain true contradictions, you need to reason paraconsistently, since you might run into counter-examples to some of the classical laws. When reasoning in contexts not known to contain true contradictions, you don't need to worry about it.
I'm deeply skeptical about this analogy. In the case of kangaroos, after all, we can tell a causal story about why they're present in Australia but they aren't likely to make it to L.A. It's not clear what kind of story about why true contradictions are present in these domains but its not likely that they'd crop up in others we could tell on the Priestian account, given the variety and heterogeneous nature of the 4 to 5 "houses" listed above.
Moreover, they're hard to miss. To combine the analogies for a moment, if there's a kangaroo jumping around your house, you'll notice it in short order. Maybe instead, we should think about cockroaches. Cockroaches like to live in dark, damp, out of the way places, and you can have cockroaches in your house for a long time before you notice one scurrying around in the periphery of your vision. This automatically makes cockroaches a better analogy if there are such things as true contradictions, given that In Contradiction wasn't published until 1982, following a few thousand years of western philosophy pretty much systematically failing to notice the existence of true contradictions.
Now, imagine that three houses are all laid out in similar ways and inhabited by three very similar people with very similar habits and possessions. Imagine that cockroaches are discovered in the house on the left and in the house on the right, and the inhabitants of those houses start taking various defensive measures--being extra-careful about leaving food out where the cockroaches could get at it, applying bug spray to dark corners, etc. Now, if there's some sort of principled causal story about the differences between the houses that makes the house in the middle less likely to have cockroaches--i.e., as in the classic civil defense video, the house in the middle is kept much cleaner than the house on the left or the house on the right--then it might make perfect sense for the owner of the middle house not to bother taking these extra measures. If, on the other hand, we have no such story to tell--the house in the middle is no cleaner than the ones on the left and the right, the owner is no less likely to leave crumbs on the counter, there are no fewer dark and damp spaces, and so on--then the mere fact that we haven't discovered any cockroaches in the middle house yet seems utterly irrelevant to the question of whether we should take such measures.
Monday, January 4, 2010
How Not To Solve Curry
A thought inspired by a conversation at the Smoker at the APA:
Curry is, on the face of it, the same kind of problem for dialetheists that the Strengthened Liar is gap theorists. If you try to get around the Simple Liar ("this sentence is false") by saying that it's neither true nor false, you're going to have a hard time convincingly applying that solution to the Strengthened Liar ("this sentence is not true"), since, after all, if it's neither true nor false, it's not true, so it is true.
Similarly, when the dialetheist solves the Stregthened Liar by biting the bullet, embracing the contradiction and using paraconsistent logic to limit the damage, that solution is going to be utterly inapplicable to Curry sentences like
Sentence C: "If sentence C is true, the earth is flat."
Plugging this into the T-Schema, we get:
1. Tr(C) ↔ (Tr(C) →E)
Applying the definition of ↔, we get:
2. [Tr(C) → (Tr(C) → E)] & [(Tr(C) → E) → Tr(C)]
By Conjunction-Elimination, we get:
3. Tr(C) → (Tr(C) → E)
Now, from here, an easy conditional proof, using only Modus Ponens (MP), will get us to Tr(C) → E. Start by assuming the antecedent:
4. Tr(C)
Plug 3 and 4 into Modus Ponens to get:
5. Tr(C) → E
With one last application of Modus Ponens, to 4 and 5, we've got the consequent:
6. E
As such, by conditional proof, we have:
7. Tr(C) → E
Now, going back to 2, we apply Conjunction-Elimination a second time to get:
8. (Tr(C) → E) → Tr(C)
From 7, 8 and Modus Ponens, we get:
9. Tr(C)
Finally, from 7, 8 and Modus Ponens, we get:
10. E
So, regardless of whether you take C to be true, false, both, neither, tralse, frue or whatever, the mere statement of its truth conditions entails that the earth is flat. (Or, more to the point, entails everything, since the consequent could be anything.) For the dialetheist whose program is to embrace a few contradictions here and there while retaining the ability to reason non-trivially, this is every bit as much of a problem as it is for everyone else. Moreover, it's quite obviously a Liar-like semantic paradox, and the logical rules we appealed to (centrally Modus Ponens and Conditional Proof, which together seem like the most basic, intuitively obvious rules about conditionals that you could ever hope for) aren't ones whose justification relies in any way on the assumption that there are no true contradictions. (Contrast to the proof that contradictions entail everything, which relies on Disjunctive Syllogism, which in turn does rest on that assumption, since if there are true contradictions, then it's possible for P to be true and hence for (P v Q) to be true, and for ~P to be true as well, but for Q to be (just) false.*) As such, one might be forgiven for wondering if Curry shows about dialethetic approaches to the semantic paradoxes (even more specifically, the semantic paradoxes revolving around self-referential truth-talk) exactly what the Strengthened Liar shows about the (simplest versions of) gappist solutions to the same paradoxes: that they're simply a non-starter, because there are clear examples of the relevant category of paradoxes relative to which they don't help.
Of course, technically, someone could doggedly insist that the Simple Liar was neither true nor false, but diagnose the Strengthened Liar in some completely different way, but at the very least, they would owe us a really good explanation of the disconnect. Similarly here.***
Now, keeping in mind that any dialetheist's favored logic had better be paraconsistent (if they don't want every contradiction they believe in to entail everything), one paraconsistent option is relevance logic, where validity-as-mere-truth-preservation is rejected and a tighter sense of "follows from" is insisted on. Moreover, the proof machinery of some relevance logics--requirements about discharging certain premises, multiple uses counting the same as others, etc.--might invalidate the above proof, without forcing the dialetheist relevance logician to bite the considerable intuitive bullet of giving up on Conditional Proof or Modus Ponens.
That won't work.
The point, after all, of the more tightly disciplined inference rules of that relevance machinery is precisely that truth-preservation isn't enough, that inference needs to be held to a higher standard. Now, there's an interesting and respectable case to be made for that claim, but it doesn't strike me as being very relevant to why Curry is a problem.
If the dialetheist relevance logician grants that the statement of the truth conditions for Curry is true--Tr ↔ (Tr →E)--and, to be clear, they'd damn well better accept that, given that T-Schema-absolutism is central to the argument for true contradictions from the Liar, and the dialetheist can hardly start claiming when it comes to Curry that problematic sentences are meaningless or non-truth-evaluable--and they grant that all of the steps in the proof are truth-preserving, whether they take them to be valid just seems to be beside the point. The relevance logic proof machinery might stop you from having to come to every conclusion, but the proof still shows us that every conclusion is true.
As such, if you want to be entitled to reject the substantive thesis of trivialism--everything is true--any solution to Curry in terms of relevance requirements for validity simply won't cut it.
*Technically, given the logical equivalence of P→Q with (~P v Q) in classical logic**, one could argue that Modus Ponens--being logically equivalent to Disjunctive Syllogism--does rest on the assumption that there are no true contradictions. I'm not aware, however, of any dialetheists who take this route, and choose to live without the inerential power of Modus Ponens, rather than simply arguing against the equivalence of P→Q with (~P v Q).
**Sometimes critics say things about this like "in classical logic, the conditional is (~P v Q)" or "classical logic claims that P→Q means (~P v Q)," but I tend to think that this runs rough-shod over a rich problem area of philosophical debate about meaning. Intuitively, after all, two things can be true under all the same circumstances but still mean quite different things. ("Consider "the sun has a shape" and "the sun has a size.") There are philosophical theories of meaning that deny this intuition, but it is, at any rate, a contentious issue.
***Graham Priest thinks he has just such an explanation: the Incosure Schema. The Liar fits, the set-theoretic paradoxes fit, Curry doesn't, so it's OK that his solution to the first two doesn't apply to the third. I find this fairly unconvincing, for a variety of reasons, not least of which is that the subject matter of the first and third seem to have a lot more in common with each other than the first does with the second. A fuller critique of the IS is a subject for another day--and, actually, a paper I'm working on--but I'd say for now simply that if Curry doesn't fit, that looks like a pretty good counter-example to the claim that the IS captures the relevant cluster of paradoxes, given that Curry is a classic semantic paradox involving a self-referential truth-value ascription.
Curry is, on the face of it, the same kind of problem for dialetheists that the Strengthened Liar is gap theorists. If you try to get around the Simple Liar ("this sentence is false") by saying that it's neither true nor false, you're going to have a hard time convincingly applying that solution to the Strengthened Liar ("this sentence is not true"), since, after all, if it's neither true nor false, it's not true, so it is true.
Similarly, when the dialetheist solves the Stregthened Liar by biting the bullet, embracing the contradiction and using paraconsistent logic to limit the damage, that solution is going to be utterly inapplicable to Curry sentences like
Sentence C: "If sentence C is true, the earth is flat."
Plugging this into the T-Schema, we get:
1. Tr(C) ↔ (Tr(C) →E)
Applying the definition of ↔, we get:
2. [Tr(C) → (Tr(C) → E)] & [(Tr(C) → E) → Tr(C)]
By Conjunction-Elimination, we get:
3. Tr(C) → (Tr(C) → E)
Now, from here, an easy conditional proof, using only Modus Ponens (MP), will get us to Tr(C) → E. Start by assuming the antecedent:
4. Tr(C)
Plug 3 and 4 into Modus Ponens to get:
5. Tr(C) → E
With one last application of Modus Ponens, to 4 and 5, we've got the consequent:
6. E
As such, by conditional proof, we have:
7. Tr(C) → E
Now, going back to 2, we apply Conjunction-Elimination a second time to get:
8. (Tr(C) → E) → Tr(C)
From 7, 8 and Modus Ponens, we get:
9. Tr(C)
Finally, from 7, 8 and Modus Ponens, we get:
10. E
So, regardless of whether you take C to be true, false, both, neither, tralse, frue or whatever, the mere statement of its truth conditions entails that the earth is flat. (Or, more to the point, entails everything, since the consequent could be anything.) For the dialetheist whose program is to embrace a few contradictions here and there while retaining the ability to reason non-trivially, this is every bit as much of a problem as it is for everyone else. Moreover, it's quite obviously a Liar-like semantic paradox, and the logical rules we appealed to (centrally Modus Ponens and Conditional Proof, which together seem like the most basic, intuitively obvious rules about conditionals that you could ever hope for) aren't ones whose justification relies in any way on the assumption that there are no true contradictions. (Contrast to the proof that contradictions entail everything, which relies on Disjunctive Syllogism, which in turn does rest on that assumption, since if there are true contradictions, then it's possible for P to be true and hence for (P v Q) to be true, and for ~P to be true as well, but for Q to be (just) false.*) As such, one might be forgiven for wondering if Curry shows about dialethetic approaches to the semantic paradoxes (even more specifically, the semantic paradoxes revolving around self-referential truth-talk) exactly what the Strengthened Liar shows about the (simplest versions of) gappist solutions to the same paradoxes: that they're simply a non-starter, because there are clear examples of the relevant category of paradoxes relative to which they don't help.
Of course, technically, someone could doggedly insist that the Simple Liar was neither true nor false, but diagnose the Strengthened Liar in some completely different way, but at the very least, they would owe us a really good explanation of the disconnect. Similarly here.***
Now, keeping in mind that any dialetheist's favored logic had better be paraconsistent (if they don't want every contradiction they believe in to entail everything), one paraconsistent option is relevance logic, where validity-as-mere-truth-preservation is rejected and a tighter sense of "follows from" is insisted on. Moreover, the proof machinery of some relevance logics--requirements about discharging certain premises, multiple uses counting the same as others, etc.--might invalidate the above proof, without forcing the dialetheist relevance logician to bite the considerable intuitive bullet of giving up on Conditional Proof or Modus Ponens.
That won't work.
The point, after all, of the more tightly disciplined inference rules of that relevance machinery is precisely that truth-preservation isn't enough, that inference needs to be held to a higher standard. Now, there's an interesting and respectable case to be made for that claim, but it doesn't strike me as being very relevant to why Curry is a problem.
If the dialetheist relevance logician grants that the statement of the truth conditions for Curry is true--Tr
As such, if you want to be entitled to reject the substantive thesis of trivialism--everything is true--any solution to Curry in terms of relevance requirements for validity simply won't cut it.
*Technically, given the logical equivalence of P→Q with (~P v Q) in classical logic**, one could argue that Modus Ponens--being logically equivalent to Disjunctive Syllogism--does rest on the assumption that there are no true contradictions. I'm not aware, however, of any dialetheists who take this route, and choose to live without the inerential power of Modus Ponens, rather than simply arguing against the equivalence of P→Q with (~P v Q).
**Sometimes critics say things about this like "in classical logic, the conditional is (~P v Q)" or "classical logic claims that P→Q means (~P v Q)," but I tend to think that this runs rough-shod over a rich problem area of philosophical debate about meaning. Intuitively, after all, two things can be true under all the same circumstances but still mean quite different things. ("Consider "the sun has a shape" and "the sun has a size.") There are philosophical theories of meaning that deny this intuition, but it is, at any rate, a contentious issue.
***Graham Priest thinks he has just such an explanation: the Incosure Schema. The Liar fits, the set-theoretic paradoxes fit, Curry doesn't, so it's OK that his solution to the first two doesn't apply to the third. I find this fairly unconvincing, for a variety of reasons, not least of which is that the subject matter of the first and third seem to have a lot more in common with each other than the first does with the second. A fuller critique of the IS is a subject for another day--and, actually, a paper I'm working on--but I'd say for now simply that if Curry doesn't fit, that looks like a pretty good counter-example to the claim that the IS captures the relevant cluster of paradoxes, given that Curry is a classic semantic paradox involving a self-referential truth-value ascription.
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