In an effort to re-start discussion, what follows is a quote from the current draft of my dissertation proposal, illustrating why I reject Tarski/Azzouni-type solutions to the semantic paradoxes.
"To see why approaches that concede the ‘inconsistency’ of natural language but hope to create a consistent language, or isolate a consistent fragment of natural language, miss the mark, imagine the following scenario. Take a world in which Shröedinger’s Cat was not a thought experiment but a real experiment. Scientists discovered, or thought they had discovered, a cat that really was in simultaneous, superimposed states of being alive and being dead. Moreover, this discovery was taken to not only revolutionize physics but also logic. The cat was alive and not alive, so we had a true contradiction.
Imagine, then, that clever logicians responded by creating an artificial language in which the aliveness predicate could never be applied to cats, or carefully formulated rules prevented it and its negation from ever being applied to the same cat, or perhaps even where talk of cats was eliminated entirely. Perhaps some clever new quantifier would allow us to say most of what we want to say about cats in non-paradoxical contexts--that certain saucers of milk were depleted at certain times, or that certain creatures purred as they were petted--without actually referring to cats. These logicians then declared that although languages with sufficient expressive resources to discuss cats are inconsistent, their special artificial language was now consistent. A sensible response might be that the problem has not been solved but avoided.
"Now, there may be good solutions in this world. For example, a sensible monaletheist could point out that superposition and conjunction are very different things, that mathematically superposition is represented by linear combination, and that this sort of linear combination might not in fact make true the conjunction of the a statement describing one of these states and its negation. Surely, however, the beginning of wisdom would be the recognition that a language without the expressive resources to discuss cats would simply be a bad medium in which to discuss this thorny problem."
Monday, July 28, 2008
Blogging Schedule
In the months since I took my quals, I've let my blogging lapse, only posting occasionally, and usually not very substantively.
Sorry.
In the future (starting today), you can expect new material every Monday and Thursday.
Sorry.
In the future (starting today), you can expect new material every Monday and Thursday.
Friday, May 16, 2008
Qual Follow-Up
OK, just updating to say that I did indeed pass my quals! I'll try to hammer out a proposal over the summer, and then it's on to the dissertation.
Meanwhile, in place of a real post, for now here are the questions I had on the first day...
Qualifying Exam Questions for Ben Burgis
Part I: Answer two of the questions below:
1. Compare and contrast Jon Barwise and John Etchemendy's approach to the liar paradox in terms of non-well-founded sets to the one offered by Saul Kripke in his theory of truth. Indicate the similarities and the main differences between these two approaches. Which difficulties do they face?
2. Explain how the dialetheist attempts to solve the liar paradox. What are the alleged benefits of the solution? Which problems must it overcome? Explain how the dialetheist attempts to solve the inconsistency found in naive set theory. Does this solution work? Why or why not?
3. Is truth a vague predicate? If so, does this help addressing the liar paradox? How?
....and here are the ones from the second day:
Qualfying Exam Questions for Ben Burgis
Part II: Answer two of the questions below:
1. The dialetheist suggest that we should change the underlying logic of our theories to a paraconsistent one. Can we make sense of the idea of changing a logic? In order to change a logic, don't we need a logic to assess such a change? Can this process get off the ground? If so, how?
2. Explain some of the main similarities between the liar paradox and the set-theoretic paradoxes. What are some of the main differences between them? What are the prospects of developing a unified solution to both paradoxes?
3. Set theory with an unrestricted comprehension schema is prima facie a plausible theory: it's simple, intuitive, and very powerful. All things considered, it's perhaps the best theory of sets we have. But the theory is also inconsistent. The indispensibility argument would then force us to conclude that we ought to believe in the existence of mathematical objects with inconsistent properties. Does this offer a reductio of the indispensibility argument? Why or why not? Does the fact that such a theory is inconsistent pose a problem for a realist interpretation of the theory? Why or why not? (To answer these questions, discuss in particular the approach to set theory developed by Penelope Maddy.)
Meanwhile, in place of a real post, for now here are the questions I had on the first day...
Qualifying Exam Questions for Ben Burgis
Part I: Answer two of the questions below:
1. Compare and contrast Jon Barwise and John Etchemendy's approach to the liar paradox in terms of non-well-founded sets to the one offered by Saul Kripke in his theory of truth. Indicate the similarities and the main differences between these two approaches. Which difficulties do they face?
2. Explain how the dialetheist attempts to solve the liar paradox. What are the alleged benefits of the solution? Which problems must it overcome? Explain how the dialetheist attempts to solve the inconsistency found in naive set theory. Does this solution work? Why or why not?
3. Is truth a vague predicate? If so, does this help addressing the liar paradox? How?
....and here are the ones from the second day:
Qualfying Exam Questions for Ben Burgis
Part II: Answer two of the questions below:
1. The dialetheist suggest that we should change the underlying logic of our theories to a paraconsistent one. Can we make sense of the idea of changing a logic? In order to change a logic, don't we need a logic to assess such a change? Can this process get off the ground? If so, how?
2. Explain some of the main similarities between the liar paradox and the set-theoretic paradoxes. What are some of the main differences between them? What are the prospects of developing a unified solution to both paradoxes?
3. Set theory with an unrestricted comprehension schema is prima facie a plausible theory: it's simple, intuitive, and very powerful. All things considered, it's perhaps the best theory of sets we have. But the theory is also inconsistent. The indispensibility argument would then force us to conclude that we ought to believe in the existence of mathematical objects with inconsistent properties. Does this offer a reductio of the indispensibility argument? Why or why not? Does the fact that such a theory is inconsistent pose a problem for a realist interpretation of the theory? Why or why not? (To answer these questions, discuss in particular the approach to set theory developed by Penelope Maddy.)
Thursday, April 24, 2008
Quals
My qualifying exams are on Monday and Tuesday, from 10 in the morning to 2 in the afternoon each day.
[A prominent philosopher of language, when visiting Miami, at whose University--one of the top couple on the Leiter list--the grad student just turn in a portfolio of their best papers, responded to my description of the qual process with a tone of genuine shock. 'You have a *sit-down, closed book* qualifying exam? That's a bit passe, isn't it?' So it is. I plan, if I pass, on being insufferably smug about it in any and all future conversations with any graduate students not subjected to this little rite of passage. "In my day..."]
In any case, I expect the next few days to be taken up with an orgy of studying, reviewing and re-reviewing over notes and so on, so this will almost certainly be my last pre-qual post. Assuming I survive, as I start to organize all of this material to the form it will ultimately take in my dissertation proposal, I'll probably start posting here more.
Meanwhile, I end with a quote that I've been trying to take to heart in these last days before the quals (thanks to N. for reminding me of it):
"I must not fear. Fear is the little-death that brings total obliteration. I will face my fear. I will permit it to pass over me and through me. And when it has gone I will turn the inner eye to see its path. Where the fear has gone there will be nothing. Only I will remain."
--Bene Geserit Litany, from "Dune" by Frank Herbert
[A prominent philosopher of language, when visiting Miami, at whose University--one of the top couple on the Leiter list--the grad student just turn in a portfolio of their best papers, responded to my description of the qual process with a tone of genuine shock. 'You have a *sit-down, closed book* qualifying exam? That's a bit passe, isn't it?' So it is. I plan, if I pass, on being insufferably smug about it in any and all future conversations with any graduate students not subjected to this little rite of passage. "In my day..."]
In any case, I expect the next few days to be taken up with an orgy of studying, reviewing and re-reviewing over notes and so on, so this will almost certainly be my last pre-qual post. Assuming I survive, as I start to organize all of this material to the form it will ultimately take in my dissertation proposal, I'll probably start posting here more.
Meanwhile, I end with a quote that I've been trying to take to heart in these last days before the quals (thanks to N. for reminding me of it):
"I must not fear. Fear is the little-death that brings total obliteration. I will face my fear. I will permit it to pass over me and through me. And when it has gone I will turn the inner eye to see its path. Where the fear has gone there will be nothing. Only I will remain."
--Bene Geserit Litany, from "Dune" by Frank Herbert
Sunday, April 13, 2008
Dialetheism in the Leiter Reports
...at least tangentially, in that Matt Burnstein is quoted as asking if there was a true contradiction involving Graham Priest's Melbourne to CUNY.
Wednesday, April 2, 2008
Australia
This morning I found out that my paper on "Paraconsistent Tense Logic, the Metaphysics of Change and the Epistemic Consequences of Dialetheism" was accepted for presentation at the Fourth World Congress of Paraconsistency this summer at the University of Melbourne. Should anyone be interested, here's the abstract:
"Graham Priest has argued that there are some true contradictions, but that the statistical frequency of true contradictions is very low, and that as such the epistemic probability of any particular contradiction being true is very low. This claim is essential to his justification for the ‘classical re-capture.’ At the same time Priest has identified some concrete extra-semantic candidates for the status of true contradictions in analysis of the metaphysics of change. Expressed in terms of a paraconsistent logic (his own LP) outfitted with tense operators like P, which can be read as ‘it was the case that,’ Priest argues for 'Zeno’s Law,' the principle that (α & P¬α) entails the disjunction of (α & ¬α) or P(α & ¬α). Despite his repeated claims to the contrary, it will become clear that Priest is so deeply committed to the tensed theory of time that his analysis falls apart once the tenseless theory is substituted. More importantly, Priest’s argument for 'Zeno’s Law' exhibits a methodology which undermines his claim that the statistical frequency of true contradictions is very low. A closer examination of this point should demonstrate that there is no good reason why arguments at least as good in more mundane contexts couldn’t turn up enough true contradictions to overturn the claim that the statistical frequency of true contradictions is very low. As such, if dialetheism is correct, we are not justified in generally assigning low epistemic probabilities to contradictory outcomes in our arguments, and the ‘classical re-capture’ fails."
"Graham Priest has argued that there are some true contradictions, but that the statistical frequency of true contradictions is very low, and that as such the epistemic probability of any particular contradiction being true is very low. This claim is essential to his justification for the ‘classical re-capture.’ At the same time Priest has identified some concrete extra-semantic candidates for the status of true contradictions in analysis of the metaphysics of change. Expressed in terms of a paraconsistent logic (his own LP) outfitted with tense operators like P, which can be read as ‘it was the case that,’ Priest argues for 'Zeno’s Law,' the principle that (α & P¬α) entails the disjunction of (α & ¬α) or P(α & ¬α). Despite his repeated claims to the contrary, it will become clear that Priest is so deeply committed to the tensed theory of time that his analysis falls apart once the tenseless theory is substituted. More importantly, Priest’s argument for 'Zeno’s Law' exhibits a methodology which undermines his claim that the statistical frequency of true contradictions is very low. A closer examination of this point should demonstrate that there is no good reason why arguments at least as good in more mundane contexts couldn’t turn up enough true contradictions to overturn the claim that the statistical frequency of true contradictions is very low. As such, if dialetheism is correct, we are not justified in generally assigning low epistemic probabilities to contradictory outcomes in our arguments, and the ‘classical re-capture’ fails."
Monday, March 24, 2008
A Thought About Dialetheism and the Curry Paradox
Sorry about the lag between posts.....
Meanwhile, here's something that I've been thinking about. Here's a simple form of Curry's Paradox:
"If this sentence is true, then β."
Plugging it into the T-Schema, we get the result that that sentence is true if and only if, if it's true, then B, or formally:
Tr<α> ↔ (Tr<α>→β)
The logical principle of absorption (or contraction) says that any time we have something of the form α→(α→β), we can infer straight from there to α→β. Or, if you want to do the same thing more slowly, you can just do a conditional proof--all you'll need is Modus Ponens a couple of times, and a logic that let's you use the same premise more than once--and, one way or the other, you get the result Tr<α>→β. But from this and the right-to-left version of the biconditional above, we can infer Tr<α>. Here, of course, we now have in our possession Tr<α> and Tr<α>→β, so we can just plug in one last instance of Modus Ponens and get β, for any and every arbitrary β. Explosion without even having to get a contradiction on the way.
Just as the ordinary semantic paradoxes, like the Liar, are prima facie sound arguments for dialetheism (the position that some, but not necessarily all, contradictions are true) from intuitively plausible premises via intuitively reasonable steps, so that to show that they are unsound you have an uphill battle to explain why the premises are wrong or what's wrong with the reasoning, Curry represents a prima facie sound argument for trivialism (the position that everythign is true.) Non-trivialist dialetheists will want to avoid this at all costs--the whole project of carving out a plausible-sounding version of dialetheism is to show how some but not all contradictions can be true--and, from my point of view, the interesting thing is that their options here aren't that different from the options confronting a defender of the Law of Non-Contradiction when defusing an ordinary semantic paradox like the Liar. It seems to me that there are three ways a dialetheist (or any one else, of course) could deal with the Curry paradox:
(1) They could deny that the original sentence was a truth-bearer, e.g. on Kripke's grounds that sentences ascribing truth to other sentences are meaningful if and only if the series eventually grounds out in a sentence that's actually about external extra-semantic reality in some way, or of course on whatever other grounds.
(2) They could institute some sort of formal rules a la Tarski to ban the expression of the sentence in the first place.
(3) They could deny that the T-Schema holds universally, and make an exception for Curry.
(4) They could tinker with the logical rules that get us from Tr<α> ↔ (Tr<α>→β) to β.
Although I think Graham Priest actually goes with (4) in In Contradiction and elsewhere, I think this is probably the least plausible response. After all, on the face of it, the logical rules in question still look universally truth-preserving whether or not propositions can be simultaneously true and false, so the dialetheist has no special right (given their assumptions) to change them that anyone else does, and I don't see why anyone does. One can simply declare that "my conditional is not the conditional of classical logic, and given that, you can't make the inference from α→(α→β) to α→β with it," but (a) I'm deeply skeptical that this can be explained in any way that blunts its radically counter-intuitive edge, and (b) it looks like this is a "solution" bought via the loss of expressive power, since the → in this logic simply won't capture the notion of "if, then" in even the minimal way that → does in classical logic. Worse yet, if the whole motivation for this artifical restriction of the conditional is the avoidance of Curry problems, then it looks to me like the dialetheist who picks this option is engaging in an ad hoc manuever and begging the question against the trivialist.
The other options, however, look even less promising. Priest has been savage in his criticisms of "ad hoc exceptions to the T-Schema" for the Liar and other semantic paradoxes, and no one has been clearer in explaining why (2) doesn't solve or explain anything but merely represents a decision not to talk about it. That leaves us with (1). I think that this is the most promising option, since it represents a more than purely formal solution, and, if the independent grounding given is good enough, the one that looks least like it's assuming what needs to be proved.
The problem, of course, is that the alleged meaningfulness of ordinary paradox-producing sentences like the Liar would be an almost inevitable casualty of any explanation of why the Curry sentence wasn't meaningful, so the dialetheist who took option (1) would be sacrificing a huge part of the positive case for dialetheism.
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.
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