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, March 24, 2008
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.
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.
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.
Yep.
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.
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.
Yep.
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.
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.
...so 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?
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.
...so 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?
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