A new member of my department, Marion Ledwig, alerted me to this very nice list of conferences in Europe for 2006-07. Speaking of which, I am very sorry to be missing GAP.6 [Gesellschaft fur Analytische Philosophie] two weeks from now, and especially their Carnap workshop organized by my man Steve Awodey.
In other news, I now feel a bit less guilty/ fraudluent about claiming philosophical logic as one of my Areas of Specialization on my CV: I just heard back from the Journal of Philosophical Logic that they'll be publishing a paper of mine. (The paper is the same one I presented at the 2005 Eastern APA, on formal semantics for languages containing confused/ ambiguous terms; if you want to look at it and help me improve it before the final submission, it's on my webpage.)
8/22/2006
Israel, Lebanon, and the Knobe Effect
Despite the title, this post is not about politics. The Knobe Effect is roughly the following: people consider foreseen side effects to be (more) intentional (or on purpose) if those side effects are bad than if they are good. That is, if you do something that has a beneficial foreseen side-effect, you won't be seen as bringing about that side-effect on purpose, but you would if the side effect was harmful or bad. This result has been shown to be experimentally robust in several groups of subjects.
Disputes concerning the Knobe effect arise in the interpretation of this experimental finding. Knobe himself takes these results to show that our concept of intentional action is essentially tied by our moral sensibilities -- somewhat surprising, since we don't usually think of intention and morality as closely linked. Other philosophers have suggested more 'deflationary' readings of the experimental results; for example, we want to blame someone for bringing about a foreseen, bad side effect of their actions -- and as a general rule of thumb, we only legitimately blame people for things they do on purpose. So on this interpretation, the Knobe effect is seen as a sort of confabulation or rationalization for our practices of praising and blaming -- not as bearing on the very concept of intention itself. Several papers by Knobe and co-authors are available on Knobe's webpage, along with papers responding to his work. If you prefer your philosophy in blog form, there has been a great deal of discussion of this work over at Experimental Philosophy.
Recent events in Lebanon provide an example of the type of situation in which the Knobe Effect appears. Israel intends to destroy Hezbollah's military capabilities, and used various forms of military force as a means to that end. Since much of the Hezbollah forces are located in places with high civilian population density, one foreseen side effect of Israel's use of force to disarm Hezbollah is a tragically high number of civilian casualities.
On NPR, I heard a high-ranking Israeli military official justify his country's military action by saying, in effect: We Israelis are not aiming to hurt any civilians -- our goal is only to stop Hezbollah from launching strikes into Israeli territory. There's two things I wanted to say about this:
(1) If only this high-ranking Israeli offical had read the work of Knobe et al., he would have known that this excuse would not carry much water, if any at all -- we are blamed for foreseen bad side-effects, even if they are unintentional.
(2) My reaction/ intuition in this case is against Knobe's stronger interpretation of the experimental results, and with the deflationists': I think the defense official has a perfectly good grasp of the concept of purpose or intentional action, even when he says "We're not harming Lebanese civilians on purpose." This doesn't sound like "John is a married bachelor" or "This is a square circle" to me.
Disputes concerning the Knobe effect arise in the interpretation of this experimental finding. Knobe himself takes these results to show that our concept of intentional action is essentially tied by our moral sensibilities -- somewhat surprising, since we don't usually think of intention and morality as closely linked. Other philosophers have suggested more 'deflationary' readings of the experimental results; for example, we want to blame someone for bringing about a foreseen, bad side effect of their actions -- and as a general rule of thumb, we only legitimately blame people for things they do on purpose. So on this interpretation, the Knobe effect is seen as a sort of confabulation or rationalization for our practices of praising and blaming -- not as bearing on the very concept of intention itself. Several papers by Knobe and co-authors are available on Knobe's webpage, along with papers responding to his work. If you prefer your philosophy in blog form, there has been a great deal of discussion of this work over at Experimental Philosophy.
Recent events in Lebanon provide an example of the type of situation in which the Knobe Effect appears. Israel intends to destroy Hezbollah's military capabilities, and used various forms of military force as a means to that end. Since much of the Hezbollah forces are located in places with high civilian population density, one foreseen side effect of Israel's use of force to disarm Hezbollah is a tragically high number of civilian casualities.
On NPR, I heard a high-ranking Israeli military official justify his country's military action by saying, in effect: We Israelis are not aiming to hurt any civilians -- our goal is only to stop Hezbollah from launching strikes into Israeli territory. There's two things I wanted to say about this:
(1) If only this high-ranking Israeli offical had read the work of Knobe et al., he would have known that this excuse would not carry much water, if any at all -- we are blamed for foreseen bad side-effects, even if they are unintentional.
(2) My reaction/ intuition in this case is against Knobe's stronger interpretation of the experimental results, and with the deflationists': I think the defense official has a perfectly good grasp of the concept of purpose or intentional action, even when he says "We're not harming Lebanese civilians on purpose." This doesn't sound like "John is a married bachelor" or "This is a square circle" to me.
8/13/2006
Now broadcasting from the desert
I have not posted in a long time -- I've been busy with my move from Pittsburgh to Las Vegas, and with all the craziness that attends moving cross-country and starting your life over. But we are starting to settle in, so blogging may pick up again soon.
Several months ago, Doug Patterson asked me if I could contribute something to a volume he's editing for OUP called Alfred Tarski: Philosophical Background, Development, and Influence. I was honored, since many of the other contributors breathe rarified logico-philosophical air, so I cobbled an article together out of various bits of my dissertation. I now have a draft of the paper, boringly entitled "Tarski's Nominalism," and I would greatly appreciate any and all feedback from interested readers. To help you determine whether you are an 'interested reader,' I've cut-and-pasted a bit of the intro:
"This essay aims to answer three related questions about Tarski's self-described 'nominalism with a materialistic taint' through an examination of Carnap’s 1941 dictation notes. First, what is Tarski’s view? Second, what are the rationales for his view? Finally, how does Tarski attempt to reconcile his nominalist philosophical scruples with mathematics, since mathematics deals with paradigmatically abstract objects, such as numbers and sets, whose rejection is a standard sine qua non of modern nominalism?"
As a brand-new Pitt HPS alumnus, I wanted to sing the praises of a couple members of the incoming class. First, Jonah Schupbach, of Berkeley, Bacon, & Bird blog-fame, just had a paper published in Philosophy of Science on one of my favorite topics, the evidential and explanatory role of unification in science. And Jason Byron (nee Baker) has written an article, forthcoming in BJPS, that argues for a point that I became convinced of while doing work for my dissertation. Just to get a sense for what the logical empiricists were thinking about in 1940, I flipped through a few journals from the 1930s that they were reading and publishing in, especially Erkenntnis and Synthese. I was very surprised to find that there were lots of articles on philosophical details related to biology -- some of them dealing rather closely with the science. This was surprising because the usual story among current philosophers of biology is that philosophy of science completely (or almost completely) ignored biology until the late 60s. Jason has now done the detailed spade-work needed to substantiate that impression I had.
Several months ago, Doug Patterson asked me if I could contribute something to a volume he's editing for OUP called Alfred Tarski: Philosophical Background, Development, and Influence. I was honored, since many of the other contributors breathe rarified logico-philosophical air, so I cobbled an article together out of various bits of my dissertation. I now have a draft of the paper, boringly entitled "Tarski's Nominalism," and I would greatly appreciate any and all feedback from interested readers. To help you determine whether you are an 'interested reader,' I've cut-and-pasted a bit of the intro:
"This essay aims to answer three related questions about Tarski's self-described 'nominalism with a materialistic taint' through an examination of Carnap’s 1941 dictation notes. First, what is Tarski’s view? Second, what are the rationales for his view? Finally, how does Tarski attempt to reconcile his nominalist philosophical scruples with mathematics, since mathematics deals with paradigmatically abstract objects, such as numbers and sets, whose rejection is a standard sine qua non of modern nominalism?"
As a brand-new Pitt HPS alumnus, I wanted to sing the praises of a couple members of the incoming class. First, Jonah Schupbach, of Berkeley, Bacon, & Bird blog-fame, just had a paper published in Philosophy of Science on one of my favorite topics, the evidential and explanatory role of unification in science. And Jason Byron (nee Baker) has written an article, forthcoming in BJPS, that argues for a point that I became convinced of while doing work for my dissertation. Just to get a sense for what the logical empiricists were thinking about in 1940, I flipped through a few journals from the 1930s that they were reading and publishing in, especially Erkenntnis and Synthese. I was very surprised to find that there were lots of articles on philosophical details related to biology -- some of them dealing rather closely with the science. This was surprising because the usual story among current philosophers of biology is that philosophy of science completely (or almost completely) ignored biology until the late 60s. Jason has now done the detailed spade-work needed to substantiate that impression I had.
7/04/2006
Something else about Mary the neuroscientist
After writing my recent post about Sue Barry, the real-life example of Frank Jackson's Mary the neuroscientist, I received a very interesting email from... Sue Barry. Prof. Barry's email was generous and insightful, and I felt fortunate that she took the time to write me. I just wanted to mention a couple other striking things (for philosophers, at least) about her case, which came up in both her email to me and the NPR story about her that (at least here in Pittsburgh) was broadcast a day or two after my original post.
1. Sue had studied descriptions of what stereoscopic vision was like before she acquired it herself. She thought, back then, that she could imagine (at least roughly) what having stereoscopic vision was like. After getting such vision, though, she discovered that what she imagined it was like was completely different from the actual experience.
2. In describing the difference between her previous and current visual perceptions, Sue says that she can now perceive space, whereas she (now realizes that she) couldn't before. This strikes me as interesting, because we (and here I mean both philosophers working on particular problems in epistemology and philosophy of mind, as well as non-philosophers) usually think of the objects of perception as things, or attributes of things, or the like. (Consider: 'What do you see?' Compare "I see an apple" with "I see space".)
Finally, one thing that comes out clearly in the New Yorker article, the NPR story, and the email is that Prof. Barry is a generous, magnanimous person -- and that she is now getting a great deal of pleasure from an aspect of her perceptual system that most of us take for granted.
1. Sue had studied descriptions of what stereoscopic vision was like before she acquired it herself. She thought, back then, that she could imagine (at least roughly) what having stereoscopic vision was like. After getting such vision, though, she discovered that what she imagined it was like was completely different from the actual experience.
2. In describing the difference between her previous and current visual perceptions, Sue says that she can now perceive space, whereas she (now realizes that she) couldn't before. This strikes me as interesting, because we (and here I mean both philosophers working on particular problems in epistemology and philosophy of mind, as well as non-philosophers) usually think of the objects of perception as things, or attributes of things, or the like. (Consider: 'What do you see?' Compare "I see an apple" with "I see space".)
Finally, one thing that comes out clearly in the New Yorker article, the NPR story, and the email is that Prof. Barry is a generous, magnanimous person -- and that she is now getting a great deal of pleasure from an aspect of her perceptual system that most of us take for granted.
7/02/2006
Happy birthday
This blog started one year ago today. Blogging has actually been a more rewarding experience than I expected, mostly because it's put me in contact with smart and interesting folks (in both cyberspace and meatspace) that I otherwise would not have met. It's repeatedly been extremely helpful to hear other people's reactions to what's bouncing around inside my head.
I'm not sure what the future holds for Obscure and Confused Ideas. On the one hand, I start my first academic job in just under 2 months, and everyone tells me that the first year is pretty brutal -- so if anything is squeezed out by time pressures, it might be blogging. On the other hand, it'll be the first time in seven years that I won't be surrounded by 30 or so other people interested in history and philosophy of science, so I may have to bounce my ideas off the online community instead of my current offline community. We'll see.
Just so this post contains something other than insufferable self-absorption, I'm linking to a Colloquium Bingo game card, produced by the grad students at Johns Hopkins's Program in the History of Science, Medicine, and Technology. And one other thing: I was looking over the Experimental Philosophy blog again recently, and was filled with a mixture of admiration and envy -- for it looks to an outsider like me that they really have a genuine online research community there. People post new papers-in-progress, which are given careful and serious feedback my several people, and the authors engage in a substantive conversation about their work. It appears to be a model of what people wish the web would do -- link up people, in a real and almost intimate way, separated by thousands of miles. I wonder why Experimental Philosophy has succeeded here, while other blogs with the same basic idea (for example, Philosophy of Biology, which has apparently disappeared) have not done as well.
I'm not sure what the future holds for Obscure and Confused Ideas. On the one hand, I start my first academic job in just under 2 months, and everyone tells me that the first year is pretty brutal -- so if anything is squeezed out by time pressures, it might be blogging. On the other hand, it'll be the first time in seven years that I won't be surrounded by 30 or so other people interested in history and philosophy of science, so I may have to bounce my ideas off the online community instead of my current offline community. We'll see.
Just so this post contains something other than insufferable self-absorption, I'm linking to a Colloquium Bingo game card, produced by the grad students at Johns Hopkins's Program in the History of Science, Medicine, and Technology. And one other thing: I was looking over the Experimental Philosophy blog again recently, and was filled with a mixture of admiration and envy -- for it looks to an outsider like me that they really have a genuine online research community there. People post new papers-in-progress, which are given careful and serious feedback my several people, and the authors engage in a substantive conversation about their work. It appears to be a model of what people wish the web would do -- link up people, in a real and almost intimate way, separated by thousands of miles. I wonder why Experimental Philosophy has succeeded here, while other blogs with the same basic idea (for example, Philosophy of Biology, which has apparently disappeared) have not done as well.
6/27/2006
Analyticity in model-theoretic languages
Part of why I am drawn to philosophy of science and logic is that I like to operate with clean and neat formulations of apparently messy concepts -- and these two sub-disciplines of philosophy embrace such tastes more than other sub-fields. Of course, I am not claiming that ethicists and metaphysicians are muddle-headed; most think about their sub-discipline's topics with far more clarity and rigor than I can. I am merely expressing a personal preference for studying deontic logic instead of the most recent form of consequentialism.
Enough autobiography -- I mention it only to explain my motivation for this post. And my point is this: if we adopt the usual formalization of an interpreted language (viz., the model-theoretic one), then we apparently cannot capture the notion of analyticity -- at least in the way Carnap, who is widely recognized as the champion of analyticity, conceives of it.
Conceiving of a language in model-theoretic terms is one widely-used way of introducing precision into a philosophical endeavor. Most readers probably can recite the definition of a model-theoretically understood langauge by heart, but for the innocent:
A language L consists of a ordered triple, where
- L carries grammatical information: which symbols belong to the language, which strings of symbols count as sentences, which grammatical category each symbol belongs to, etc.;
- M is a model =< D, f >, where the domain of discourse D is a set of individuals, and f is an interepretation function, which assigns an individual in D to each proper name in L, sets in D to one-place predicates, sets of ordered pairs drawn from D to two-place predicates, and so on; and
- r specifies the truth-values of certain compound sentences, given the truth-values of their components -- in other words, r basically specifies the truth-tables.
So much for the model-theoretic conception of language; what about analyticity? Carnap, throughout his career, identifies the analytic truths as those sentences that are true merely in virtue of the language one speaks. That is, if we specify that I am speaking a particular language, in the course of that specification, I might present enough information that the truth-values of certain sentences within that language are fixed. (For an obvious example: if I specify what 'and' and 'not' mean in my language via the usual truth tables for those words, any sentence of the form 'p and not-p' comes out false merely in virtue of the rules governing the language I am using.)
Now, after all that rehearsal of material most readers probably know well, I can get to my point. In a model-theoretically characterized language, the truth-values of ALL sentences are determined by the specification of that language. For example, the truth-value of atomic sentences such as 'Fb' are true iff the individual named by 'b' is in the extension of the set associated with 'F' (i.e. 'Fb' is true iff f(b) is an element of the set f(F)). And Carnap certainly never wanted every sentence of a (non-contradictory) language to be analytic.
The problem then is: one of my favorite tools for 'precisification' in philosophy -- model-theoretic languages -- apparently affords no way to characterize one of the concepts I'm most interested in: analyticity. What to make of this? The first, obvious thing to say is: "Of course there couldn't be any explication of analyticity in such languages, because such languages are extensional, and Carnap and Quine (who represent opposing positions in debates over analyticity) both basically agree that analyticity is an intensional notion."
This is right, but I think there is something further to note: in a straightforward sense, every sentence in a (classical) model-theoretic language has its truth-value determined by the specification of the language. That is, by specifying the language, we fix the truth-values of all the sentences in such a language. That seems odd -- the model-theoretic way of specifying a language that has proved very useful in certain situations, but it likely cannot be a fundamental and/or universally applicable one.
One further point: Carnap, Quine, and the other primary antagonists in battles over analyticity all agree that if there is any such thing as analytic truth, then the (so-called) logical truths are paradigm instances of analytic truths, i.e., truth in virtue of meaning (if you are thinking of "Two Dogmas" and don't believe me, look at Word & Object, sec. 14, fn.3, p.65). But the model-theoretic conception of language characterizes the logical truths as a class of sentences that are true across a set of related langauges. That is, to know whether a sentence is a logical truth in one model-theoretic language, you have to check whether that sentence is true in a bunch of other model-theoretic languages that share certain features with the first one.
So, one might think that the way to cash out analyticity in the idiom of the philosophical logician is to use something like Kripkean possible world semantics (which are used, with some variations, in modal, deontic, epistemic, and temporal logics). But these are usually not given linguistic interpretations, and it's not clear to me that it's possible to give a decent one... though I'd love to be wrong. Any thoughts?
Enough autobiography -- I mention it only to explain my motivation for this post. And my point is this: if we adopt the usual formalization of an interpreted language (viz., the model-theoretic one), then we apparently cannot capture the notion of analyticity -- at least in the way Carnap, who is widely recognized as the champion of analyticity, conceives of it.
Conceiving of a language in model-theoretic terms is one widely-used way of introducing precision into a philosophical endeavor. Most readers probably can recite the definition of a model-theoretically understood langauge by heart, but for the innocent:
A language L consists of a ordered triple
- L carries grammatical information: which symbols belong to the language, which strings of symbols count as sentences, which grammatical category each symbol belongs to, etc.;
- M is a model =< D, f >, where the domain of discourse D is a set of individuals, and f is an interepretation function, which assigns an individual in D to each proper name in L, sets in D to one-place predicates, sets of ordered pairs drawn from D to two-place predicates, and so on; and
- r specifies the truth-values of certain compound sentences, given the truth-values of their components -- in other words, r basically specifies the truth-tables.
So much for the model-theoretic conception of language; what about analyticity? Carnap, throughout his career, identifies the analytic truths as those sentences that are true merely in virtue of the language one speaks. That is, if we specify that I am speaking a particular language, in the course of that specification, I might present enough information that the truth-values of certain sentences within that language are fixed. (For an obvious example: if I specify what 'and' and 'not' mean in my language via the usual truth tables for those words, any sentence of the form 'p and not-p' comes out false merely in virtue of the rules governing the language I am using.)
Now, after all that rehearsal of material most readers probably know well, I can get to my point. In a model-theoretically characterized language, the truth-values of ALL sentences are determined by the specification of that language. For example, the truth-value of atomic sentences such as 'Fb' are true iff the individual named by 'b' is in the extension of the set associated with 'F' (i.e. 'Fb' is true iff f(b) is an element of the set f(F)). And Carnap certainly never wanted every sentence of a (non-contradictory) language to be analytic.
The problem then is: one of my favorite tools for 'precisification' in philosophy -- model-theoretic languages -- apparently affords no way to characterize one of the concepts I'm most interested in: analyticity. What to make of this? The first, obvious thing to say is: "Of course there couldn't be any explication of analyticity in such languages, because such languages are extensional, and Carnap and Quine (who represent opposing positions in debates over analyticity) both basically agree that analyticity is an intensional notion."
This is right, but I think there is something further to note: in a straightforward sense, every sentence in a (classical) model-theoretic language has its truth-value determined by the specification of the language. That is, by specifying the language, we fix the truth-values of all the sentences in such a language. That seems odd -- the model-theoretic way of specifying a language that has proved very useful in certain situations, but it likely cannot be a fundamental and/or universally applicable one.
One further point: Carnap, Quine, and the other primary antagonists in battles over analyticity all agree that if there is any such thing as analytic truth, then the (so-called) logical truths are paradigm instances of analytic truths, i.e., truth in virtue of meaning (if you are thinking of "Two Dogmas" and don't believe me, look at Word & Object, sec. 14, fn.3, p.65). But the model-theoretic conception of language characterizes the logical truths as a class of sentences that are true across a set of related langauges. That is, to know whether a sentence is a logical truth in one model-theoretic language, you have to check whether that sentence is true in a bunch of other model-theoretic languages that share certain features with the first one.
So, one might think that the way to cash out analyticity in the idiom of the philosophical logician is to use something like Kripkean possible world semantics (which are used, with some variations, in modal, deontic, epistemic, and temporal logics). But these are usually not given linguistic interpretations, and it's not clear to me that it's possible to give a decent one... though I'd love to be wrong. Any thoughts?
6/14/2006
Sue Barry, a real-life Mary the neuroscientist
I'm sure this will be noted all over the philosophical regions of the blogosphere, but in the latest issue of the New Yorker, there is an example of a real person who basically fits Frank Jackson's famous example of Mary the neuroscientist -- though in this case, it is not color vision, but stereoscopic vision, that the person gains. The person is named Sue Barry, and she actually is a neurobiologist. Unfortunately, the article is not online.
(For those unfamiliar with Frank Jackson's thought-experiment, Mary is a neuroscientist of color who knows all the neuroscientific theories associated with color vision (even those theories that have not yet been discovered and formulated) -- but she is raised in a completely monochrome/ black-and-white environment. If Mary suddenly sees colors one day, does she have a fundamentally new experience? Does she learn anything? A recent book, There's Something about Mary (publisher's page, review in NDPR), is entirely devoted to issues involving this thought-experiment.)
(For those unfamiliar with Frank Jackson's thought-experiment, Mary is a neuroscientist of color who knows all the neuroscientific theories associated with color vision (even those theories that have not yet been discovered and formulated) -- but she is raised in a completely monochrome/ black-and-white environment. If Mary suddenly sees colors one day, does she have a fundamentally new experience? Does she learn anything? A recent book, There's Something about Mary (publisher's page, review in NDPR), is entirely devoted to issues involving this thought-experiment.)
5/18/2006
Quine on logical truth, again
In a previous post, I asked about the relationship between Quine's definition of logical truth and the now-standard (model-theoretic) one. Here's the second installment, which I decided to finally post after sitting on it for a while, since Kenny just posted a nice set of thoughts on the very closely related topic of logical consequence.
The standard definition is:
(SLT) Sentence S is a logical truth of language L = S is true in all models of L.
Quine's definition is:
(QLT) S is a logical truth = "we get only truths when we substitute sentences for [the] simple [=atomic] sentences" of S (Philosophy of Logic, 50).
(Quine counts open formulas, e.g. 'x burns', as sentences; specifically, he calls them 'open sentences.' I'll follow his usage here.)
So what does Quine think is the relationship between the standard characterization of logical truth and his own? He argues fot the following equivalence claim:
(EQ) If “our object language is… rich enough for elementary number theory ,” then “[a]ny schema that comes out true under all substitutions of sentences, in such a language, will also be satisfied by all models, and conversely” (53).
In a nutshell, Quine argues for the 'only if' direction via the Löwenheim-Skolem theorem (hence the requirement of elementary number theory within the object language), and for the ‘if’ direction by appeal to the completeness of first-order logic. I'll spell out his reasoning in a bit more detail below, but I can sum up my worry about it here: in order to overcome Tarski’s objection to Quine’s substitutional version of logical truth, Quine appeals to the Löwenheim-Skolem theorem. However, for that appeal to work, Quine has to require the object language to be rich enough to fall afoul of the incompleteness results, thereby depriving him of one direction of the purported equivalence between the model-based and sentence-substitution-based notions of logical truth.
The 'only if' direction
Quine presents an extension of the Löwenheim-Skolem Theorem due to Hilbert and Bernays:
“If a [GFA: first-order] schema is satisfied by a model at all, it becomes true under some substitution of sentences of elementary number theory for its simple schemata” (54).
A little logic chopping will get us to
If all substitutions of sentences from elementary number theory make A true, then A is satisfied in all models.
-- which is what we wanted to show. (I think this argument is OK .)
The 'if' direction
Quine takes as his starting premise the completeness result for first-order logic:
(CT) “If a schema is satisfied by every model, it can be proved” (54).
(Quine then argues that if a schema can be proved within a given proof calculus whose inference rules are “visibly sound,” i.e., “visibly such as to generate only schemata that come out true under all substitutions” (54), then such a schema will of course ‘come out true under all substitutions of sentences,’ Q.E.D.) This theorem is of course true for first-order logic; however, Quine has imposed the demand that our object language contain the resources of elementary number theory (explicitly including both plus and times, so that Presburger arithmetic—which is complete—is not in play). And once our object language is that rich, then Gödel’s first incompleteness theorem comes into play. Specifically, in any consistent proof-calculus rich enough for elementary number theory, there will be sentences [and their associated schema] that are true, i.e., satisfied by every model, yet cannot be proved -- providing a counterexample to (CT). So the dialectic, as I see it, is as follows: to answer Tarski's objection to the substitutional version of logical truth, Quine requires the language to be rich enough to capture number theory. But once Quine has made that move, the crucial premise (viz., CT) for the other direction of his equivalence claim no longer holds.
I think I must be missing something -- first, Quine is orders of magnitude smarter than I am, and second, while Quine is fallible, this does not seem like the kind of mistake he's likely to make. So perhaps someone in the blogosphere can set me straight.
And I have one more complaint. Quine claims that his "definition of logical truth agrees with the alternative definition in terms of models, as long as the object language is not too weak for the modest idioms of elementary number theory. In the contrary case we can as well blame any discrepancies on the weakness of the language as on the definition of logical truth" (55). This defense strikes me as implausible: why would we only have (or demand) a well-defined notion of logical truth once we reach number theory? Don't we want a definition to cover all cases, including the simple ones? If the model-theoretic definition captures all the intuitive cases, and Quine's only when the language is sufficiently rich, isn't that a good argument against Quine's characterization of logical truth?
------
(And for those wondering what Quine thinks is the advantage of his characterization of logical truth over the model-theoretic one, the answer is: Quine's uses much less set theory. "The evident philosophical advantage of resting with this substitutional definition, and not broaching model theory, is that we save on ontology. Sentences suffice,… instead of a universe of sets specifiable and unspecifiable. … [W]e have progressed a step whenever we find a way of cutting the ontological costs of some particular development" (55). Quine recognizes that his characterization is not completely free of set theory, given the proof of the LS theorem, so he says his "retreat" from the model-based notion of logical truth "renders the notions of validity and logical truth independent of all but a modest bit of set theory; independent of the higher flights" (56).)
The standard definition is:
(SLT) Sentence S is a logical truth of language L = S is true in all models of L.
Quine's definition is:
(QLT) S is a logical truth = "we get only truths when we substitute sentences for [the] simple [=atomic] sentences" of S (Philosophy of Logic, 50).
(Quine counts open formulas, e.g. 'x burns', as sentences; specifically, he calls them 'open sentences.' I'll follow his usage here.)
So what does Quine think is the relationship between the standard characterization of logical truth and his own? He argues fot the following equivalence claim:
(EQ) If “our object language is… rich enough for elementary number theory ,” then “[a]ny schema that comes out true under all substitutions of sentences, in such a language, will also be satisfied by all models, and conversely” (53).
In a nutshell, Quine argues for the 'only if' direction via the Löwenheim-Skolem theorem (hence the requirement of elementary number theory within the object language), and for the ‘if’ direction by appeal to the completeness of first-order logic. I'll spell out his reasoning in a bit more detail below, but I can sum up my worry about it here: in order to overcome Tarski’s objection to Quine’s substitutional version of logical truth, Quine appeals to the Löwenheim-Skolem theorem. However, for that appeal to work, Quine has to require the object language to be rich enough to fall afoul of the incompleteness results, thereby depriving him of one direction of the purported equivalence between the model-based and sentence-substitution-based notions of logical truth.
The 'only if' direction
Quine presents an extension of the Löwenheim-Skolem Theorem due to Hilbert and Bernays:
“If a [GFA: first-order] schema is satisfied by a model at all, it becomes true under some substitution of sentences of elementary number theory for its simple schemata” (54).
A little logic chopping will get us to
If all substitutions of sentences from elementary number theory make A true, then A is satisfied in all models.
-- which is what we wanted to show. (I think this argument is OK .)
The 'if' direction
Quine takes as his starting premise the completeness result for first-order logic:
(CT) “If a schema is satisfied by every model, it can be proved” (54).
(Quine then argues that if a schema can be proved within a given proof calculus whose inference rules are “visibly sound,” i.e., “visibly such as to generate only schemata that come out true under all substitutions” (54), then such a schema will of course ‘come out true under all substitutions of sentences,’ Q.E.D.) This theorem is of course true for first-order logic; however, Quine has imposed the demand that our object language contain the resources of elementary number theory (explicitly including both plus and times, so that Presburger arithmetic—which is complete—is not in play). And once our object language is that rich, then Gödel’s first incompleteness theorem comes into play. Specifically, in any consistent proof-calculus rich enough for elementary number theory, there will be sentences [and their associated schema] that are true, i.e., satisfied by every model, yet cannot be proved -- providing a counterexample to (CT). So the dialectic, as I see it, is as follows: to answer Tarski's objection to the substitutional version of logical truth, Quine requires the language to be rich enough to capture number theory. But once Quine has made that move, the crucial premise (viz., CT) for the other direction of his equivalence claim no longer holds.
I think I must be missing something -- first, Quine is orders of magnitude smarter than I am, and second, while Quine is fallible, this does not seem like the kind of mistake he's likely to make. So perhaps someone in the blogosphere can set me straight.
And I have one more complaint. Quine claims that his "definition of logical truth agrees with the alternative definition in terms of models, as long as the object language is not too weak for the modest idioms of elementary number theory. In the contrary case we can as well blame any discrepancies on the weakness of the language as on the definition of logical truth" (55). This defense strikes me as implausible: why would we only have (or demand) a well-defined notion of logical truth once we reach number theory? Don't we want a definition to cover all cases, including the simple ones? If the model-theoretic definition captures all the intuitive cases, and Quine's only when the language is sufficiently rich, isn't that a good argument against Quine's characterization of logical truth?
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(And for those wondering what Quine thinks is the advantage of his characterization of logical truth over the model-theoretic one, the answer is: Quine's uses much less set theory. "The evident philosophical advantage of resting with this substitutional definition, and not broaching model theory, is that we save on ontology. Sentences suffice,… instead of a universe of sets specifiable and unspecifiable. … [W]e have progressed a step whenever we find a way of cutting the ontological costs of some particular development" (55). Quine recognizes that his characterization is not completely free of set theory, given the proof of the LS theorem, so he says his "retreat" from the model-based notion of logical truth "renders the notions of validity and logical truth independent of all but a modest bit of set theory; independent of the higher flights" (56).)
5/02/2006
Knowledge via public ignorance
Yesterday Rohit Parikh gave a very interesting talk at Carnegie Mellon on a kind of modal epistemic logic he has been working on recently with several collaborators, cleverly called topologic, because it carries interesting topological properties. The first thing Parikh said was "I like formalisms, but I like examples more." In that spirit, I wanted to describe here one simple example he showed us yesterday, without digging into the technicalia, because it generates a potentially philosophically interesting situation: someone can (under suitable circumstances) gain knowledge merely via other people's declarations of ignorance.
Imagine two people play the following game: a natural number n>0 (1,2, ...) is selected. Then, one of the players has n written on his or her forehead, and the other player has n+1 written on his forehead. Each player can see what is written on the other's forehead, but cannot see what is written on their own. The game allows only two "moves": you can either say "I don't know what number is on my forehead" or state what you think the number on your forehead is.
So, for example, if I play the game, and I see that the other person has a 2 written on her forehead, I know that the number on my own forehead is either a 1 or a 3, but I do not know which. But here is the interesting part: if my fellow game-player wears a 2, and on her first move says "I don't know what my number is," then I know what my number is -- at least, if my fellow game-player is reasonably intelligent. Why? If I were wearing a 1, then my interlocutor would say, on her first move, "I know my own humber is a 2" -- because (1, 2) is the first allowable pair in the game. Thus, if she says "I don't know what my own number is" on her first move, then I know my number can't be 1, so it must be 3. This same process of reasoning can be extended: by playing enough rounds of "I don't know" moves, we can eventually successfully reach any pair of natural numbers, no matter how high. We just have to keep track of how many rounds have been played. (This may remind the mathematically-inclined in the audience of the Mr. Sum-Mr. Product dialogue.)
What is interesting to me about this is that the two players in such a game (and the other examples Prof. Parikh described) can eventually come to have knowledge about the world simply via declarations of ignorance. These cases prompt two questions for me:
(1) Is this type of justification for a belief different in kind from the others philosophers busy themselves with? Or is this just a completely normal/ standard/ etc. way of gathering knowledge, which differs only superficially from other cases? (I'm not qualified to answer this, since I'm not an epistemologist.)
(2) Are there any interesting real-world examples where we achieve knowledge via collective ignorance in (roughly) this way? (Prof. Parikh suggested that there might be a vague analogy between what happens in these sorts of games and game-theoretic treatments of evolution, but didn't have much further to say.)
Imagine two people play the following game: a natural number n>0 (1,2, ...) is selected. Then, one of the players has n written on his or her forehead, and the other player has n+1 written on his forehead. Each player can see what is written on the other's forehead, but cannot see what is written on their own. The game allows only two "moves": you can either say "I don't know what number is on my forehead" or state what you think the number on your forehead is.
So, for example, if I play the game, and I see that the other person has a 2 written on her forehead, I know that the number on my own forehead is either a 1 or a 3, but I do not know which. But here is the interesting part: if my fellow game-player wears a 2, and on her first move says "I don't know what my number is," then I know what my number is -- at least, if my fellow game-player is reasonably intelligent. Why? If I were wearing a 1, then my interlocutor would say, on her first move, "I know my own humber is a 2" -- because (1, 2) is the first allowable pair in the game. Thus, if she says "I don't know what my own number is" on her first move, then I know my number can't be 1, so it must be 3. This same process of reasoning can be extended: by playing enough rounds of "I don't know" moves, we can eventually successfully reach any pair of natural numbers, no matter how high. We just have to keep track of how many rounds have been played. (This may remind the mathematically-inclined in the audience of the Mr. Sum-Mr. Product dialogue.)
What is interesting to me about this is that the two players in such a game (and the other examples Prof. Parikh described) can eventually come to have knowledge about the world simply via declarations of ignorance. These cases prompt two questions for me:
(1) Is this type of justification for a belief different in kind from the others philosophers busy themselves with? Or is this just a completely normal/ standard/ etc. way of gathering knowledge, which differs only superficially from other cases? (I'm not qualified to answer this, since I'm not an epistemologist.)
(2) Are there any interesting real-world examples where we achieve knowledge via collective ignorance in (roughly) this way? (Prof. Parikh suggested that there might be a vague analogy between what happens in these sorts of games and game-theoretic treatments of evolution, but didn't have much further to say.)
4/29/2006
Modal logic workshop at CMU
I spent yesterday at a workshop devoted to modal logic at Carnegie Mellon University. Rather than rehearse everything that happened, I'll simply point interested parties to the workshop webpage, which has abstracts for the talks. Mostly local folks presented their work, but Johan van Bentham from Amsterdam and Stanford was here, along with Rohit Parikh, who'll be presenting on Monday as well.
I certainly learned a lot, and even though my brain was hurting afterwards, I enjoyed myself too.
I certainly learned a lot, and even though my brain was hurting afterwards, I enjoyed myself too.
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