| Your Luck Quotient: 71% |
 You have a high luck quotient. More often than not, you've felt very lucky in your life. You may be randomly lucky, but it's probably more than that. Optimistic and open minded, you take advantage of all the luck that comes your way. |
Saturday, November 18, 2006
What Does 'Randomly Lucky' Mean?
Three Pods of Pepper
On November 24, Sikhs around the world will be celebrating the martyrdom of Guru Tegh Bhadur Sahib, the Ninth Guru, also called more poetically Dharam di chadaar, the Shield for Faith. There is an interesting story associated with the event. The Guru went to Delhi, having been approached by some Hindus, asking him to help intercede with the Emperor Aurangzeb, a Muslim, who was attempting to convert them by force. In the course of his journey, he was arrested and brought before the Emperor Aurangzeb, who was Muslim. The Emperor told the Guru of a vision he had had, in which God had said to him that he should convert all the world to Islam; and he laid out his policy for doing so, whereby those who converted would receive land, money, and preferment.
"So you see," the Emperor said, "you would do well to convert. Your reputation for goodness is already known far and wide; with my backing you could gather many disciples and become a great teacher of Islam. Become Muslim, and receive your heart's desire."
The Guru reflected on this, and then asked the Emperor for a hundred pounds of black pepper. When it was brought he set it on fire, and let it burn for a whole day until it seemed to be nothing but ash. Then he had people sift the ashes. Out of all the pepper that had been burned, three little pods of pepper remained.
Then the Guru said to the Emperor, "O Emperor, you who are mighty among men, you wish to make one religion out of two religions." He spoke, of course, of Islam and Hinduism. "But God, O Emperor, wishes to make three religions out of two. For just as three pepper pods were saved out of that fire, so shall three great religions exist in India, Muslim and Hindu and Sikh."
At this, the Guru was imprisoned and given three choices: to embrace Islam, to perform a miracle to show his worthiness as a Guru, or to prepare for death. And he replied, "A miracle is not within the will of God. For me there is only one religion, that of God; and whoever belongs to it, be he Muslim or Hindu or Sikh, has his place not with the temporal and changing but with the eternal and undying. As for those who attempt to convert others by force, they are charlatans. I do not fear physical death. The Emperor may do as he pleases. "
And so the Emperor did; the Guru was tortured and beheaded. He was succeeded as Guru by his son, Gobind Singh.
"So you see," the Emperor said, "you would do well to convert. Your reputation for goodness is already known far and wide; with my backing you could gather many disciples and become a great teacher of Islam. Become Muslim, and receive your heart's desire."
The Guru reflected on this, and then asked the Emperor for a hundred pounds of black pepper. When it was brought he set it on fire, and let it burn for a whole day until it seemed to be nothing but ash. Then he had people sift the ashes. Out of all the pepper that had been burned, three little pods of pepper remained.
Then the Guru said to the Emperor, "O Emperor, you who are mighty among men, you wish to make one religion out of two religions." He spoke, of course, of Islam and Hinduism. "But God, O Emperor, wishes to make three religions out of two. For just as three pepper pods were saved out of that fire, so shall three great religions exist in India, Muslim and Hindu and Sikh."
At this, the Guru was imprisoned and given three choices: to embrace Islam, to perform a miracle to show his worthiness as a Guru, or to prepare for death. And he replied, "A miracle is not within the will of God. For me there is only one religion, that of God; and whoever belongs to it, be he Muslim or Hindu or Sikh, has his place not with the temporal and changing but with the eternal and undying. As for those who attempt to convert others by force, they are charlatans. I do not fear physical death. The Emperor may do as he pleases. "
And so the Emperor did; the Guru was tortured and beheaded. He was succeeded as Guru by his son, Gobind Singh.
Disgust as a Moral Emotion
For someone who is on record as saying that disgust is worthless as a moral compass, Ophelia Benson seems to be a bit addicted to appealing to it in morally tinged arguments. In recent posts at the Butterflies and Wheels Notes and Comments blog she has managed to call some of Swinburne's arguments 'disgusting', 'truly revolting', asking, "Why doesn't everybody for miles around just tell him 'That's disgusting' until he's so embarrassed he stops saying it?" In an earlier earlier post she talks about disgust at a mixture of abuse and sycophancy, in a yet earlier post she talks about how preaching against condom-abuse is disgusting, in an even earlier post she calls a comment by Karen Armstrong disgusting, and so it goes on back through the archives; and in every single case these comments can easily be shown to be doing moral work.
I don't bring it up in order to pick on Ophelia Benson. The moves she is making are certainly not out of the ordinary, and she would certainly be a less interesting to read if she took more trouble to be consistent. Rather, I think the quandary into which Benson has backed herself -- being officially against treating disgust as morally informative while being practically unable to stop treating it as such -- is just what the official position gets you. The official position arises from thinking that the real question with regard to disgust is "Does disgust have any role to play in moral reasoning?" when it actually is "What role does it play?" The inconsistency is not so much a flaw as the reassertion of good sense against the absurdity of the official view.
I would suggest that there are several false assumptions that tend to get in the way of clear thought about this subject.
(1) There is only one kind of disgust. This assumption, that all disgust can be treated as if there were simply one kind, seems to be very common; but it also seems to be very dubious. I grab an apple and am about to bite into it, when I suddenly notice it is crawling with worms; a nausea-laced revulsion results. I pick up a book and read a graphic scene in which a pedophile rapes a child, which also creates an intense, visceral revulsion. And note that there's nothing about the latter that's necessarily more 'abstract' or less visceral. But it's not at all clear that the two should be treated as simply one kind of disgust. In other words, how we should think about disgust appears to vary depending on what we are disgusted at. There's a sort of object-dependence to its significance. If it's a disgust due to revulsion at something put in our mouths, or almost put in our mouths, that's very different from disgust at the thought of a person being violated.
(2) Disgust is a reaction to animality. This has become popular among people who follow Paul Rozin's research on disgust; it plays a major role, for instance, in Nussbaum's attack on disgust as a moral emotion. Now, I happen to like a lot of what Rozin does in this regard, and would never say that the research itself is problematic. But Rozin seems sometimes to be rather sloppy when expressing some of his results, and saying things like, "Anything that reminds us we are animals elicits disgust" is as sloppy as you can get. Taken strictly it is obviously false; I just reminded you that we are animals by mentioning the subject, and I doubt you had a disgust reaction simply in virtue of that. And Rozin himself is quite aware that a lot of our disgust reaction is culturally conditioned. Sexual play, foreplay, and the like, for instance, reminds people of their animality over a far greater range of behavior than it elicits disgust And when we try to cash out what features of animality elicits disgust, we really don't have a much more precise view than that they are the features of animality that are gross -- hardly a profound insight. The traits related to disgust are not traits that we most conspicuously share with other animals, if we set aside disgust itself as an indicator. There is, of course, the fact that physical objects of disgust tend to be contaminating in some way. But, again, watching someone eat a beating heart can't be contaminating in quite the same sense that eating one myself would be; and watching someone eat a beating heart isn't contaminating in the sense that the disgust that comes from association with a moral monster is; and in neither case is it clear (without simply begging the question) why we would feel disgust rather than, say, shame or humor at the association. To stretch that far, contamination has to take on figurative senses; what we have is a word useful for discussing all these different kinds of disgust, but very little more. We're missing some key factor here. [As a side issue, with regard to physical disgust why don't people do more research on fun disgust? Perhaps it's just due to being part of the Nickolodeon generation; but having fun with our sense of disgust is a big part of many cultures, although the precise details of it vary from place to place. A lot of the comments about disgust made by researchers make disgust sound so solemn and serious -- which it obviously can be -- but it's like treating fear as a purely negative flight response without considering the fact that people can enjoy being scared. People can enjoy dabbling in the disgusting on occasion. It's not all revulsion. And this is itself an interesting thing.]
Now, I do think that Rozin and others like him are on the right track, and are genuinely uncovering features of disgust; I just think we must guard against premature conclusions about what they've actually shown, such as we find in Nussbaum and elsewhere. It seems to be less that disgust is a reaction against the inhuman than that it itself creates a line between humanity and inhumanity that otherwise might not exist. We can feel sick and revolted at lots of things that aren't animal or contaminating in any strict sense of the term; and many things that are animal or contaminating in a strict sense of the term don't always elicit disgust. Rozin is certainly aware of this himself, since he thinks of disgust in terms of a sort of evolution -- there's a level of disgust devoted to keeping contaminated food out of the body; there's a level of disgust devoted to avoiding animal grossness and mortality, however that's to be understood; and there's a level of disgust devoted to putting out of one's mind things that are found culturally offensive. While Rozin isn't always as careful as he should be in discussing them, the levels simply cannot be conflated. One level is purely sanitary in nature; another Rozin goes so far as to say is the sign of civilization. And that seems along the right lines. Even this stage view of disgust may be attributing more unity to disgust than actually exists; the distinction between physical disgust and socio-moral disgust seems much sharper than this suggests, even though they clearly share some important commonalities (and thus are both kinds of disgust).
(3) If some kinds of disgust are morally wrong, disgust can't be relevant to (good) moral reasoning. This also seems to be a common assumption, at least judging from how people argue. How this is to be squared with the fact that almost everything else related to moral reasoning can take morally wrong form is always left unclear. For instance, some kinds of thought are morally wrong; but it doesn't follow from this that thoughts are not relevant to moral reasoning. It would take a rather elaborate argument to eliminate disgust as a potential source for good moral reasoning; one that is never made.
In any case, how would disgust play a role in good moral reasoning? My own rough view in this regard tends Humean. That is, we start with the inchoate reactions that are developed simply from growing up human, and this inchoate state is fine for a certain level of moral maturity; however, a great part of our moral education involves cultivating these reactions so that they tend to hit the moral mark in cases farther and farther removed from the original ones. In other words, we adapt the response to general rules. The general rules, note, are not doing all the work here; for one thing, if they were, there would be nothing to adapt. Rather, what they are doing is setting up dams that guide the course of the stream; even given the dams, it's the stream itself that does the work of going where it needs to be. And so it is with disgust. Reactions of moral disgust, at least, should be treated with respect as a reason to worry, even if nothing else. It may well be -- and it has turned out before -- that the reaction was unjustifiable, due to a failure to cultivate the sense of disgust properly. But failures to react with moral disgust are in some cases equally unjustifiable, and equally due to a failure to cultivate the sense of disgust properly. It's a bit much to ask that our emotions be infallible guides, particularly since we don't demand the same of the reasoning by which we shape them. But that's precisely what people demand in their attacks on the role of disgust in moral reasoning. To put it in terms that Benson uses in the essay linked to above, it is certainly true that "Ew, ick, gross" and "That's wrong" don't mean the same thing; but that's no good reason for saying that the the sense behind "Ew, ick, gross" has no role to play in the judgment behind saying, "That's wrong."
I don't bring it up in order to pick on Ophelia Benson. The moves she is making are certainly not out of the ordinary, and she would certainly be a less interesting to read if she took more trouble to be consistent. Rather, I think the quandary into which Benson has backed herself -- being officially against treating disgust as morally informative while being practically unable to stop treating it as such -- is just what the official position gets you. The official position arises from thinking that the real question with regard to disgust is "Does disgust have any role to play in moral reasoning?" when it actually is "What role does it play?" The inconsistency is not so much a flaw as the reassertion of good sense against the absurdity of the official view.
I would suggest that there are several false assumptions that tend to get in the way of clear thought about this subject.
(1) There is only one kind of disgust. This assumption, that all disgust can be treated as if there were simply one kind, seems to be very common; but it also seems to be very dubious. I grab an apple and am about to bite into it, when I suddenly notice it is crawling with worms; a nausea-laced revulsion results. I pick up a book and read a graphic scene in which a pedophile rapes a child, which also creates an intense, visceral revulsion. And note that there's nothing about the latter that's necessarily more 'abstract' or less visceral. But it's not at all clear that the two should be treated as simply one kind of disgust. In other words, how we should think about disgust appears to vary depending on what we are disgusted at. There's a sort of object-dependence to its significance. If it's a disgust due to revulsion at something put in our mouths, or almost put in our mouths, that's very different from disgust at the thought of a person being violated.
(2) Disgust is a reaction to animality. This has become popular among people who follow Paul Rozin's research on disgust; it plays a major role, for instance, in Nussbaum's attack on disgust as a moral emotion. Now, I happen to like a lot of what Rozin does in this regard, and would never say that the research itself is problematic. But Rozin seems sometimes to be rather sloppy when expressing some of his results, and saying things like, "Anything that reminds us we are animals elicits disgust" is as sloppy as you can get. Taken strictly it is obviously false; I just reminded you that we are animals by mentioning the subject, and I doubt you had a disgust reaction simply in virtue of that. And Rozin himself is quite aware that a lot of our disgust reaction is culturally conditioned. Sexual play, foreplay, and the like, for instance, reminds people of their animality over a far greater range of behavior than it elicits disgust And when we try to cash out what features of animality elicits disgust, we really don't have a much more precise view than that they are the features of animality that are gross -- hardly a profound insight. The traits related to disgust are not traits that we most conspicuously share with other animals, if we set aside disgust itself as an indicator. There is, of course, the fact that physical objects of disgust tend to be contaminating in some way. But, again, watching someone eat a beating heart can't be contaminating in quite the same sense that eating one myself would be; and watching someone eat a beating heart isn't contaminating in the sense that the disgust that comes from association with a moral monster is; and in neither case is it clear (without simply begging the question) why we would feel disgust rather than, say, shame or humor at the association. To stretch that far, contamination has to take on figurative senses; what we have is a word useful for discussing all these different kinds of disgust, but very little more. We're missing some key factor here. [As a side issue, with regard to physical disgust why don't people do more research on fun disgust? Perhaps it's just due to being part of the Nickolodeon generation; but having fun with our sense of disgust is a big part of many cultures, although the precise details of it vary from place to place. A lot of the comments about disgust made by researchers make disgust sound so solemn and serious -- which it obviously can be -- but it's like treating fear as a purely negative flight response without considering the fact that people can enjoy being scared. People can enjoy dabbling in the disgusting on occasion. It's not all revulsion. And this is itself an interesting thing.]
Now, I do think that Rozin and others like him are on the right track, and are genuinely uncovering features of disgust; I just think we must guard against premature conclusions about what they've actually shown, such as we find in Nussbaum and elsewhere. It seems to be less that disgust is a reaction against the inhuman than that it itself creates a line between humanity and inhumanity that otherwise might not exist. We can feel sick and revolted at lots of things that aren't animal or contaminating in any strict sense of the term; and many things that are animal or contaminating in a strict sense of the term don't always elicit disgust. Rozin is certainly aware of this himself, since he thinks of disgust in terms of a sort of evolution -- there's a level of disgust devoted to keeping contaminated food out of the body; there's a level of disgust devoted to avoiding animal grossness and mortality, however that's to be understood; and there's a level of disgust devoted to putting out of one's mind things that are found culturally offensive. While Rozin isn't always as careful as he should be in discussing them, the levels simply cannot be conflated. One level is purely sanitary in nature; another Rozin goes so far as to say is the sign of civilization. And that seems along the right lines. Even this stage view of disgust may be attributing more unity to disgust than actually exists; the distinction between physical disgust and socio-moral disgust seems much sharper than this suggests, even though they clearly share some important commonalities (and thus are both kinds of disgust).
(3) If some kinds of disgust are morally wrong, disgust can't be relevant to (good) moral reasoning. This also seems to be a common assumption, at least judging from how people argue. How this is to be squared with the fact that almost everything else related to moral reasoning can take morally wrong form is always left unclear. For instance, some kinds of thought are morally wrong; but it doesn't follow from this that thoughts are not relevant to moral reasoning. It would take a rather elaborate argument to eliminate disgust as a potential source for good moral reasoning; one that is never made.
In any case, how would disgust play a role in good moral reasoning? My own rough view in this regard tends Humean. That is, we start with the inchoate reactions that are developed simply from growing up human, and this inchoate state is fine for a certain level of moral maturity; however, a great part of our moral education involves cultivating these reactions so that they tend to hit the moral mark in cases farther and farther removed from the original ones. In other words, we adapt the response to general rules. The general rules, note, are not doing all the work here; for one thing, if they were, there would be nothing to adapt. Rather, what they are doing is setting up dams that guide the course of the stream; even given the dams, it's the stream itself that does the work of going where it needs to be. And so it is with disgust. Reactions of moral disgust, at least, should be treated with respect as a reason to worry, even if nothing else. It may well be -- and it has turned out before -- that the reaction was unjustifiable, due to a failure to cultivate the sense of disgust properly. But failures to react with moral disgust are in some cases equally unjustifiable, and equally due to a failure to cultivate the sense of disgust properly. It's a bit much to ask that our emotions be infallible guides, particularly since we don't demand the same of the reasoning by which we shape them. But that's precisely what people demand in their attacks on the role of disgust in moral reasoning. To put it in terms that Benson uses in the essay linked to above, it is certainly true that "Ew, ick, gross" and "That's wrong" don't mean the same thing; but that's no good reason for saying that the the sense behind "Ew, ick, gross" has no role to play in the judgment behind saying, "That's wrong."
Milton Friedman (1912-2006)
Milton Friedman recently died at age 94, and the reminiscences and memorials are beginning to circulate. Thomas Sowell has a good one. Brad DeLong's is also worth reading. Others can be found at "PrestoPundit."
Jottings on Fitch's Paradox III
In my last set of sketchy thoughts, I proceeded in the indexing of the premises of the argument, as presented in the SEP, as far as premise (4):
(4') K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone, to be true.
Now, the way the argument moves from here requires that a reductio be made, and a contradiction be derived from (4). Here's the original set of premises:
(4) K(p ∧ ¬Kp) It is known that both the particular claim is true and that it is not known to be true.
(5) Kp ∧ K¬Kp The particular claim is known to be true and it is known that it is not known to be true.
(6) Kp ∧ ¬Kp The particular claim is known to be true and it is not known to be true.
(5) comes from (4) by the standard rule that a conjunction is known only if its conjuncts are known. So (5) has to be indexed as:
(5') K{∃x|∃c}p ∧ K{∃x|∃c}¬K{∃y|∃c}p It is known by someone or other that the particular claim is true and it is known by that same someone or other that it is not known, by someone or other, who may or may not be the same someone, to be true.
And here, of course, we are stymied a bit. For the only thing that can be derived from this is:
(6') K{∃x|∃c}p ∧ ¬K{∃y|∃c}p It is known by someone or other that the particular claim is true, and it is not known by someone or other (who may or may not be the same someone or other) that it is true.
And that's obviously not a contradiction.
However, let's assume that x=y here. That is, let's assume that the same someone or other is involved all the way through. Even this, as it turns out, is not quite enough to give us a contradiction. The astute reader will have noticed that I have been ignoring the |c side of the index, and pretending that we can assume that all the relevant circumstances or conditions of knowing are the same throughout. This, of course, is not quite legitimate, and a more accurate indexing will take this into account. I would suggest the following so far:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other (given some set of circumstnaces).
(2') (p ∧ ¬K{∃y|∃c}p) → ◊K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other (given the original set of circumstances), it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person and who may or may not be in the same circumstances).
(3') ◊K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(4') K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other in some circumstances that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(5') K{∃x|∃d}p ∧ K{∃x|∃c}¬K{∃y|∃c}p It is known by someone or other in some circumstances that the particular claim is true and it is known by that same someone or other that it is not known, by someone or other, who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(6') K{∃x|∃d}p ∧ ¬K{∃y|∃c}p It is known by someone or other that the particular claim is true, and it is not known by someone or other (who may or may not be the same someone or other and may or may not be in the same circumstances) that it is true.
It can easily be seen that, taking this into account, the assumption that x and y are the same person(s) is not enough to yield a contradiction and make the reductio viable. Suppose that x and y are both Tom, and suppose that p is the claim that the sky is blue. And suppose the set of conditions c is describable as 'being a baby who has never seen the sky and cannot understand any testimony about it' and the set of conditions d is describable as 'being a young man who has seen the sky'. The result then is:
It is known by Tom, when he was a young man who had seen the sky, that the sky is blue; and it is not known by Tom, when he was a baby who had never seen the sky and could not understand any testimony about it, that the sky is blue.
Which is not a contradiction in the slightest; it's exactly what you'd expect if Tom originally (as a baby) didn't know that the sky was blue, and came to know at some point (by the time he was a young man) that the sky was blue. In other words, it's exactly what you'd expect if we can come to know things through learning about them.
So to have a genuine contradiction here, not only must we assume that x is y; we must assume that d is c, as well. So what has been shown to be contradictory is the following:
(4'*) K{∃x|∃c}(p ∧ ¬K{∃c|∃c}p) It is known by someone or other in some circumstances that both the particular claim is true and that it is not known by the same someone or other in those same circumstances to be true.
(4'*) is necessarily false. This leads us into the rest of the argument.
(7') ¬K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) It is not known by the same person under a given set of conditions both that the particular claim is true and that it is not known by the same person under the same set of conditions to be true.
(8') ¬K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) (7) is necessary.
(9') ¬◊K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) It is not possible that this is known by the same person under a given set of conditions: both that p is true and that it is not known by that same person under that same set of conditions to be true.
(10') ¬∃p(p ∧ ¬K{∃x|∃c}p) There is no particular claim for which it is true both that the claim is true and that it is not known by someone under some set of conditions to be true.
(11') ∀p(p → Kp) For any particular claim, if it is true, it is known to be true by someone under some set of conditions.
(7') we get from the reductio of (4'*); (8') we know because (4'*) was not just false but impossible. (8') tells us that it is impossible for anyone to know both a given claim and that they don't know it (setting aside all differences of conditions, e.g., different times of life). So (9') follows from (8'). It still contradicts (3') only if we assume that x is y and that c is d. But it does do that, given that assumption.
Can we get (10') from (9'), though? We can get the unindexed (10) from (9) by using the negation of (3) to deny the consequent of (2), which by modus tollens gives us the negation of (1). We've been assuming as well that x is y and that c is d, that is, that there is only one knower involved, and only one set of conditions for knowing. But anyone who has gone this far, of course, can, instead of following the trail back to the negation of (1), simply hold that the argument up to (9') shows that it is impossible for us to make this assumption, at least legitimately, for non-omniscient knowers. Nothing precludes such a response; and it is certainly more plausible than the argument itself.
Thus KP doesn't seem to result in the paradox. What creates the paradox is the assumption that KP requires that a given claim is true only if it can possibly be known by any particular knower under any set of conditions. But, of course, this assumption is manifestly false. Not only is the alleged requirement absurd, it's not what people mean when they say that every truth is such that it is possible for it to be known.
Which raises the question of what people really do mean, or, at least, what they can consistently mean, when they use principles like KP. Another post, although I can't promise that it will necessarily be the next one.
And keep in mind that these are all sketchy thoughts, and that none of them should be taken as any sign that I know what I'm talking about.
(4') K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone, to be true.
Now, the way the argument moves from here requires that a reductio be made, and a contradiction be derived from (4). Here's the original set of premises:
(4) K(p ∧ ¬Kp) It is known that both the particular claim is true and that it is not known to be true.
(5) Kp ∧ K¬Kp The particular claim is known to be true and it is known that it is not known to be true.
(6) Kp ∧ ¬Kp The particular claim is known to be true and it is not known to be true.
(5) comes from (4) by the standard rule that a conjunction is known only if its conjuncts are known. So (5) has to be indexed as:
(5') K{∃x|∃c}p ∧ K{∃x|∃c}¬K{∃y|∃c}p It is known by someone or other that the particular claim is true and it is known by that same someone or other that it is not known, by someone or other, who may or may not be the same someone, to be true.
And here, of course, we are stymied a bit. For the only thing that can be derived from this is:
(6') K{∃x|∃c}p ∧ ¬K{∃y|∃c}p It is known by someone or other that the particular claim is true, and it is not known by someone or other (who may or may not be the same someone or other) that it is true.
And that's obviously not a contradiction.
However, let's assume that x=y here. That is, let's assume that the same someone or other is involved all the way through. Even this, as it turns out, is not quite enough to give us a contradiction. The astute reader will have noticed that I have been ignoring the |c side of the index, and pretending that we can assume that all the relevant circumstances or conditions of knowing are the same throughout. This, of course, is not quite legitimate, and a more accurate indexing will take this into account. I would suggest the following so far:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other (given some set of circumstnaces).
(2') (p ∧ ¬K{∃y|∃c}p) → ◊K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other (given the original set of circumstances), it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person and who may or may not be in the same circumstances).
(3') ◊K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(4') K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other in some circumstances that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(5') K{∃x|∃d}p ∧ K{∃x|∃c}¬K{∃y|∃c}p It is known by someone or other in some circumstances that the particular claim is true and it is known by that same someone or other that it is not known, by someone or other, who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(6') K{∃x|∃d}p ∧ ¬K{∃y|∃c}p It is known by someone or other that the particular claim is true, and it is not known by someone or other (who may or may not be the same someone or other and may or may not be in the same circumstances) that it is true.
It can easily be seen that, taking this into account, the assumption that x and y are the same person(s) is not enough to yield a contradiction and make the reductio viable. Suppose that x and y are both Tom, and suppose that p is the claim that the sky is blue. And suppose the set of conditions c is describable as 'being a baby who has never seen the sky and cannot understand any testimony about it' and the set of conditions d is describable as 'being a young man who has seen the sky'. The result then is:
It is known by Tom, when he was a young man who had seen the sky, that the sky is blue; and it is not known by Tom, when he was a baby who had never seen the sky and could not understand any testimony about it, that the sky is blue.
Which is not a contradiction in the slightest; it's exactly what you'd expect if Tom originally (as a baby) didn't know that the sky was blue, and came to know at some point (by the time he was a young man) that the sky was blue. In other words, it's exactly what you'd expect if we can come to know things through learning about them.
So to have a genuine contradiction here, not only must we assume that x is y; we must assume that d is c, as well. So what has been shown to be contradictory is the following:
(4'*) K{∃x|∃c}(p ∧ ¬K{∃c|∃c}p) It is known by someone or other in some circumstances that both the particular claim is true and that it is not known by the same someone or other in those same circumstances to be true.
(4'*) is necessarily false. This leads us into the rest of the argument.
(7') ¬K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) It is not known by the same person under a given set of conditions both that the particular claim is true and that it is not known by the same person under the same set of conditions to be true.
(8') ¬K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) (7) is necessary.
(9') ¬◊K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) It is not possible that this is known by the same person under a given set of conditions: both that p is true and that it is not known by that same person under that same set of conditions to be true.
(10') ¬∃p(p ∧ ¬K{∃x|∃c}p) There is no particular claim for which it is true both that the claim is true and that it is not known by someone under some set of conditions to be true.
(11') ∀p(p → Kp) For any particular claim, if it is true, it is known to be true by someone under some set of conditions.
(7') we get from the reductio of (4'*); (8') we know because (4'*) was not just false but impossible. (8') tells us that it is impossible for anyone to know both a given claim and that they don't know it (setting aside all differences of conditions, e.g., different times of life). So (9') follows from (8'). It still contradicts (3') only if we assume that x is y and that c is d. But it does do that, given that assumption.
Can we get (10') from (9'), though? We can get the unindexed (10) from (9) by using the negation of (3) to deny the consequent of (2), which by modus tollens gives us the negation of (1). We've been assuming as well that x is y and that c is d, that is, that there is only one knower involved, and only one set of conditions for knowing. But anyone who has gone this far, of course, can, instead of following the trail back to the negation of (1), simply hold that the argument up to (9') shows that it is impossible for us to make this assumption, at least legitimately, for non-omniscient knowers. Nothing precludes such a response; and it is certainly more plausible than the argument itself.
Thus KP doesn't seem to result in the paradox. What creates the paradox is the assumption that KP requires that a given claim is true only if it can possibly be known by any particular knower under any set of conditions. But, of course, this assumption is manifestly false. Not only is the alleged requirement absurd, it's not what people mean when they say that every truth is such that it is possible for it to be known.
Which raises the question of what people really do mean, or, at least, what they can consistently mean, when they use principles like KP. Another post, although I can't promise that it will necessarily be the next one.
And keep in mind that these are all sketchy thoughts, and that none of them should be taken as any sign that I know what I'm talking about.
Friday, November 17, 2006
'Half Monk, Half Hit Man'
I just finished watching the newest Bond film, Casino Royale. I highly recommend it, and I do mean highly. This is easily one of the best, and perhaps the best, Bond film of the past few decades. Daniel Craig plays a Bond just beginning to get his bearings, but this allows him to show both more brutality and more humanity. 007, after all, is not a superhero; he is not an action hero; he is a killer who is in Her Majesty's secret service, the loyal but ruthless attack dog sent after the worst and most elusive of Britain's enemies, who is so good at eliminating them that he can do it with style.
My primary two criticisms are: (1) The exposition during the poker game was heavy-handed and purposeless -- most people are not that ignorant of poker, and those who are wouldn't be interested enough to care. But this is a minor irritation in an otherwise great series of scenes. (2) Whoever decided the music volume should be hit over the head -- while it's OK for much of the film, it occasionally becomes intrusive and loud where it should become soft and subtle, letting the images speak for themselves.
Other than that, it was quite good. It's a bit violent and brutal in parts, but an excellent reinvention of the franchise.
My primary two criticisms are: (1) The exposition during the poker game was heavy-handed and purposeless -- most people are not that ignorant of poker, and those who are wouldn't be interested enough to care. But this is a minor irritation in an otherwise great series of scenes. (2) Whoever decided the music volume should be hit over the head -- while it's OK for much of the film, it occasionally becomes intrusive and loud where it should become soft and subtle, letting the images speak for themselves.
Other than that, it was quite good. It's a bit violent and brutal in parts, but an excellent reinvention of the franchise.
Wednesday, November 15, 2006
Philosophia Naturalis
To celebrate St. Albert's Day, I thought I'd provide a few links to recent science-oriented posts in the blogosphere that I've found interesting.
* Coturnix at "A Blog Around the Clock" discusses circadian rhythms.
* Michelle at "The Culture of Chemistry" introduces us to the meaning of the term 'eutectic'.
* Jennifer Ouellette discusses wireless energy transfer at the "Industrial Physics Forum" weblog.
* Andrew Jaffe at "Leaves on the Line" reviews Paul Davies's The Goldilocks Dilemma.
* Jacques Distler at "Musings" discusses the role of rigor in physics.
* Mark at "Chemistry World blog" points to the link between Damascus swords and carbon nanotubes.
* Derek Lowe at "In the Pipeline" discusses chemical disposal in scientific contexts.
* Carl Zimmer at "The Loom" talks about the sequencing of the bee genome.
* Coturnix at "A Blog Around the Clock" discusses circadian rhythms.
* Michelle at "The Culture of Chemistry" introduces us to the meaning of the term 'eutectic'.
* Jennifer Ouellette discusses wireless energy transfer at the "Industrial Physics Forum" weblog.
* Andrew Jaffe at "Leaves on the Line" reviews Paul Davies's The Goldilocks Dilemma.
* Jacques Distler at "Musings" discusses the role of rigor in physics.
* Mark at "Chemistry World blog" points to the link between Damascus swords and carbon nanotubes.
* Derek Lowe at "In the Pipeline" discusses chemical disposal in scientific contexts.
* Carl Zimmer at "The Loom" talks about the sequencing of the bee genome.
The Sun
You scored as XIX: The Sun. This is the happiest card in the deck. It is full of joy and optimism, everything is right with the world. We are as innocent children playing in the fields without care. The Sun brings success, well-being and happiness in all spheres - material, emotional, spiritual -wherever our desires lay.When this card appears in a Tarot spread it indicates success, joy and happiness. Obstacles will be overcome, goals achieved.When badly aspected, it can indicate a stagnation through over-indulgence, too much of a good thing.
Which Major Arcana Tarot Card Are You? created with QuizFarm.com |
I keep trying to remember whether Charles Williams ever mentions it, and I can't recall that he ever does. Which is not surprising, because the whole book is devoted to The Fool almost exclusively. By the way, that's The Fool on the horse in the card above; a common symbolism with Tarot cards is The Fool's Journey -- in which The Fool goes through all the Major Arcana as a symbol for life.
Notice, by the way, that I'm The Sun only narrowly. I almost was Death (XIII). Death, of course, is the card of change, the letting-go that allows for birth and prevents stagnation -- sorrowful hope and hopeful sorrow. Which is in some ways the exact opposite of The Sun.
Curiously, the first time I ever came across Tarot cards was reading a Nancy Drew mystery, in which the names of the three of the cards -- Death, the Fool, and The Devil -- are important clues. That, and The Greater Trumps, are about the only uses of Tarot symbolism that I've ever come across that are even remotely interesting, and those weren't particularly great. You'd think the pretty pictures would inspire something worthwhile. But, no, they don't, which goes to show that pretty pictures, like good looks, don't get you very much.
Albertus Magnus
Today is the Feast of St. Albert the Great (1206-1280), who, it is said, was so well respected in his time that he was often called 'Albert the Great' while he was still alive; and was called by one of his students "the wonder and miracle of our times" (nostri temporis stupor et miraculum). He was, famously, the teacher of St. Thomas Aquinas (whom he outlived). The tale goes that once some of Albert's other students were making fun of Thomas -- a big man from the beginning, and lost in his own world -- by calling him the 'dumb ox'; to which Albert replied, "This dumb ox will one day bellow so loudly that the sound of it will fill the world."
Also famously, Albert is the patron saint of scientists, and while he has some typical medieval limitations, his biography is filled with stories that show why. (As the saying goes, his defects were the defects of his age, his merits were his alone.) One of the most famous ones is when, having heard that ostriches eat iron, he went out and tried to feed iron to an ostrich (he wasn't able to get the ostrich to eat it). Here's another story in Albert's own words:
The formicaleon (the lish of Job 4:11) is called the ant-lion, which is also called murmicaleon. To begin with, this animal is not an ant as some say. For I have a great deal of experience of it and have shown my colleagues that this animal has very much the shape of the tick, and it hides itself in the sand, digging in it a hemispherical cup, at the bottom of which is the ant-lion's mouth; and when the ants, bent on gain, cross the pit, it seizes and devours them. This we have very often watched.
Also famously, Albert is the patron saint of scientists, and while he has some typical medieval limitations, his biography is filled with stories that show why. (As the saying goes, his defects were the defects of his age, his merits were his alone.) One of the most famous ones is when, having heard that ostriches eat iron, he went out and tried to feed iron to an ostrich (he wasn't able to get the ostrich to eat it). Here's another story in Albert's own words:
The formicaleon (the lish of Job 4:11) is called the ant-lion, which is also called murmicaleon. To begin with, this animal is not an ant as some say. For I have a great deal of experience of it and have shown my colleagues that this animal has very much the shape of the tick, and it hides itself in the sand, digging in it a hemispherical cup, at the bottom of which is the ant-lion's mouth; and when the ants, bent on gain, cross the pit, it seizes and devours them. This we have very often watched.
MacDonald on Imagination
By those who consider a balanced repose the end of culture, the imagination must necessarily be regarded as the one faculty before all others to be suppressed. "Are there not facts?" say they. "Why forsake them for fancies? Is there not that which may be known? Why forsake it for inventions? What God hath made, into that let man inquire."
We answer: To inquire into what God has made is the main function of the imagination. It is aroused by facts, is nourished by facts, seeks for higher and yet higher laws in those facts; but refuses to regard science as the sole interpreter of nature, or the laws of science as the only region of discovery.
[George MacDonald, The Imagination: Its Function and its Culture.]
The funny thing -- if you find it funny -- is that today, as in MacDonald's time, there are people who would be utterly shocked by a statement like this, seeing it as an attack on science. But as MacDonald points out later in the essay, if scientists spent all their time focus on the ascertained, if they spent all their time looking backwards at what science has established, they would never make any discoveries, because to make discoveries you have to look forward at what might (or might not) be the case. As MacDonald says, "It is the far-seeing imagination which beholds what might be a form of things, and says to the intellect: 'Try whether that may not be the form of these things.'"
We answer: To inquire into what God has made is the main function of the imagination. It is aroused by facts, is nourished by facts, seeks for higher and yet higher laws in those facts; but refuses to regard science as the sole interpreter of nature, or the laws of science as the only region of discovery.
[George MacDonald, The Imagination: Its Function and its Culture.]
The funny thing -- if you find it funny -- is that today, as in MacDonald's time, there are people who would be utterly shocked by a statement like this, seeing it as an attack on science. But as MacDonald points out later in the essay, if scientists spent all their time focus on the ascertained, if they spent all their time looking backwards at what science has established, they would never make any discoveries, because to make discoveries you have to look forward at what might (or might not) be the case. As MacDonald says, "It is the far-seeing imagination which beholds what might be a form of things, and says to the intellect: 'Try whether that may not be the form of these things.'"
Thought for the Day
People don't usually want to change the world for the better; they want the world to have been changed for the better.
Tuesday, November 14, 2006
Jottings on Fitch's Paradox II
Obviously the chief problems raised by Fitch's Paradox surround the so-called "Knowability Principle":
(KP) p → ◊Kp If a claim is true, it is possible that it is known to be true.
However, I want to set these questions aside, and focus on a clarificatory point. We know that every known is known by somebody under certain conditions. So I think we might get a better handle on this sort of argument if we go through it again, this time indexing our epistemic operators to the epistemic agents. That is, let's keep track of who knows what. The way I'll do this is by inserting an index after each epistemic operator. For instance, instead of (KP) as I've written it above, we'll have an indexed (KP'):
(KP') p → ◊K{∃x|c}p If a claim is true, it is possible that it is known to be true by someone, i.e., by at least one person (under conditions).
This would contrast with a different, and much less plausible principle:
(KP'') p → ◊K{∀x|c}p If a claim is true, it is possible that it is known to be true by everyone (under conditions).
Obviously the two are very different. And we can do things with the |c side, too. For instance:
(KP''') p → ◊K{∃x|∃c}p If a claim is true, it is possible that it is known to be true by someone (under at least some particular set of conditions).
Which contrasts with:
(KP'''') p → ◊K{∃x|∀c}p If a claim is true, it is possible that it is known to be true by at least one person (under any set of conditions).
And, needless to say, we can mix and match. Now, I don't think any of this can be said to be unreasonable. To be known, a thing must be known by somebody; and that person knows it either under some particular set of conditions or under every set of conditions. But when we start doing this, it suddenly becomes clear that our previous argument was glossing over a lot that might turn out to be important for understanding what's really going on in the argument.
So let's begin indexing. First of all, when people accept KP, what specified version(s) of it are they really accepting? Any version which involves ∀c seems unlikely. So let's assume that we are talking about ∃c. That is, we aren't demanding that if a thing is true it is possible that it is known under any set of conditions; we just mean that if a thing is true, there is at least some set of conditions under which it is possible to know it. Suppose that p = "The sky is blue during the day". What appeals to people in (KP) is not the claim that
If the sky is blue during the day, it is possible that it is known (by someone/everyone) in every circumstance that the sky is blue during the day.
That is pretty clearly false; since it is possible for the sky to be blue and yet people only to know that it is under certain circumstances (for instance, people who have lived in caves all their lives, never seeing the sky). So let's assume that we are only talking about things that are known in some circumstance or other, i.e., ∃c.
The second question is: Who are the knowers we are assuming. Do we mean:
If the sky is blue during the day, it is possible that it is known by someone in some circumstance or other that the sky is blue during the day?
Or do we mean:
If the sky is blue during the day, it is possible that it is known by everyone in some circumstance or other that the sky is blue during the day?
The latter commits us to there being some circumstance in which everyone could know that the sky is blue; and this doesn't seem what people want (KP) to say. (Likewise, I'll assume for the rest of the post that ∃c is the only version in view.) So the proper specification would seem to be:
(KP''') p → ◊K{∃x|∃c}p If a claim is true, it is possible that it is known to be true by someone or other (under at least some particular set of conditions).
So we've got that out of the way. Now let's go on to the first stage of the argument:
(1) p ∧ ¬Kp A particular claim is true, and it is not known to be true.
(2) (p ∧ ¬Kp) → ◊K(p ∧ ¬Kp) (1) implies that it is possible to know both that the particular claim is true and that it is not known to be true.
(3) ◊K(p ∧ ¬Kp) It is possible to know that both the particular claim is true and that it is not known to be true.
First with (1). (1) is intended to tell us that there is a truth, and it is not known. Do we mean that it is not known by at least some particular person, or that it is not known by anyone?
It's hard to say, actually, since people who make this claim say things that sometimes suggest one, sometimes the other. Also, we have to keep in mind that we can't assume (at this stage) that the same 'someone or other' or 'particular person(s)' are in view. So I'll use a different variable, and for now I'll assume ∃y, i.e., at least one person (who may or may not be the same person(s) in (KP''')), but let's not forget the possible alternative. So:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other.
Now to find (2'). This is supposed to follow from (1') and (KP'''). But the variables are tricky here. So I suggest:
(2') (p ∧ ¬K{∃y|∃c}p) → ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other, it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person).
And (3') has to follow from (1') and (2'). So:
(3') ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone, to be true.
Put it all together:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other.
(2') (p ∧ ¬K{∃y|∃cp) → ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other, it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person).
(3') ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone, to be true.
So far, so good. But what happens when we move on to (4)? Remember in the first post when I noted that it wasn't at all surprising that (4) led to a contradiction? Well, what about (4')?
(4') K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone, to be true.
Suppose that p = "The sky is blue," and suppose that it is true. Then (4') tells us that someone knows both that the sky is blue and that it is not known by someone (who may or may not be the same person).
And that seems less likely to yield a contradiction doesn't it? In fact, we know it doesn't, where the particular knower(s) we are describing by 'x' are not the same knower(s) we are describing by 'y'. For instance, suppose that at least one of the x-knowers is Tom, and that at least one of the y-knowers is Cindy, and that Tom is not Cindy. Then from (4') we can conclude that
It is known by Tom both that the sky is blue and that this is not known by Cindy.
Perhaps Cindy has never seen the sky and has been misinformed that the sky is mauve. So it would seem that (4') can't lead to a contradiction; there are non-contradictory instances of it.
But we know where the argument is heading, and it is toward a contradiction. Maybe it does raise a contradiction, even though (4'), as such, implies no contradiction. How so? Things are not nearly so simple as they had seemed at first; but it may well be that we don't need (4') to yield a contradiction in every instance; the paradox possibly just needs some instance of (4') to yield a contradiction. A paradox that holds only for some instances of (4') is a paradox still. Will it turn out to be so? Stay tuned for our next crazy installment. If your head is spinning a bit, good. But the roller coaster is just starting. I'll continue the indexing in a future post; and we'll also get into lots more issues, and come back to (KP) in particular, further on. And keep in mind that these are all sketchy thoughts.
(KP) p → ◊Kp If a claim is true, it is possible that it is known to be true.
However, I want to set these questions aside, and focus on a clarificatory point. We know that every known is known by somebody under certain conditions. So I think we might get a better handle on this sort of argument if we go through it again, this time indexing our epistemic operators to the epistemic agents. That is, let's keep track of who knows what. The way I'll do this is by inserting an index after each epistemic operator. For instance, instead of (KP) as I've written it above, we'll have an indexed (KP'):
(KP') p → ◊K{∃x|c}p If a claim is true, it is possible that it is known to be true by someone, i.e., by at least one person (under conditions).
This would contrast with a different, and much less plausible principle:
(KP'') p → ◊K{∀x|c}p If a claim is true, it is possible that it is known to be true by everyone (under conditions).
Obviously the two are very different. And we can do things with the |c side, too. For instance:
(KP''') p → ◊K{∃x|∃c}p If a claim is true, it is possible that it is known to be true by someone (under at least some particular set of conditions).
Which contrasts with:
(KP'''') p → ◊K{∃x|∀c}p If a claim is true, it is possible that it is known to be true by at least one person (under any set of conditions).
And, needless to say, we can mix and match. Now, I don't think any of this can be said to be unreasonable. To be known, a thing must be known by somebody; and that person knows it either under some particular set of conditions or under every set of conditions. But when we start doing this, it suddenly becomes clear that our previous argument was glossing over a lot that might turn out to be important for understanding what's really going on in the argument.
So let's begin indexing. First of all, when people accept KP, what specified version(s) of it are they really accepting? Any version which involves ∀c seems unlikely. So let's assume that we are talking about ∃c. That is, we aren't demanding that if a thing is true it is possible that it is known under any set of conditions; we just mean that if a thing is true, there is at least some set of conditions under which it is possible to know it. Suppose that p = "The sky is blue during the day". What appeals to people in (KP) is not the claim that
If the sky is blue during the day, it is possible that it is known (by someone/everyone) in every circumstance that the sky is blue during the day.
That is pretty clearly false; since it is possible for the sky to be blue and yet people only to know that it is under certain circumstances (for instance, people who have lived in caves all their lives, never seeing the sky). So let's assume that we are only talking about things that are known in some circumstance or other, i.e., ∃c.
The second question is: Who are the knowers we are assuming. Do we mean:
If the sky is blue during the day, it is possible that it is known by someone in some circumstance or other that the sky is blue during the day?
Or do we mean:
If the sky is blue during the day, it is possible that it is known by everyone in some circumstance or other that the sky is blue during the day?
The latter commits us to there being some circumstance in which everyone could know that the sky is blue; and this doesn't seem what people want (KP) to say. (Likewise, I'll assume for the rest of the post that ∃c is the only version in view.) So the proper specification would seem to be:
(KP''') p → ◊K{∃x|∃c}p If a claim is true, it is possible that it is known to be true by someone or other (under at least some particular set of conditions).
So we've got that out of the way. Now let's go on to the first stage of the argument:
(1) p ∧ ¬Kp A particular claim is true, and it is not known to be true.
(2) (p ∧ ¬Kp) → ◊K(p ∧ ¬Kp) (1) implies that it is possible to know both that the particular claim is true and that it is not known to be true.
(3) ◊K(p ∧ ¬Kp) It is possible to know that both the particular claim is true and that it is not known to be true.
First with (1). (1) is intended to tell us that there is a truth, and it is not known. Do we mean that it is not known by at least some particular person, or that it is not known by anyone?
It's hard to say, actually, since people who make this claim say things that sometimes suggest one, sometimes the other. Also, we have to keep in mind that we can't assume (at this stage) that the same 'someone or other' or 'particular person(s)' are in view. So I'll use a different variable, and for now I'll assume ∃y, i.e., at least one person (who may or may not be the same person(s) in (KP''')), but let's not forget the possible alternative. So:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other.
Now to find (2'). This is supposed to follow from (1') and (KP'''). But the variables are tricky here. So I suggest:
(2') (p ∧ ¬K{∃y|∃c}p) → ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other, it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person).
And (3') has to follow from (1') and (2'). So:
(3') ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone, to be true.
Put it all together:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other.
(2') (p ∧ ¬K{∃y|∃cp) → ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other, it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person).
(3') ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone, to be true.
So far, so good. But what happens when we move on to (4)? Remember in the first post when I noted that it wasn't at all surprising that (4) led to a contradiction? Well, what about (4')?
(4') K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone, to be true.
Suppose that p = "The sky is blue," and suppose that it is true. Then (4') tells us that someone knows both that the sky is blue and that it is not known by someone (who may or may not be the same person).
And that seems less likely to yield a contradiction doesn't it? In fact, we know it doesn't, where the particular knower(s) we are describing by 'x' are not the same knower(s) we are describing by 'y'. For instance, suppose that at least one of the x-knowers is Tom, and that at least one of the y-knowers is Cindy, and that Tom is not Cindy. Then from (4') we can conclude that
It is known by Tom both that the sky is blue and that this is not known by Cindy.
Perhaps Cindy has never seen the sky and has been misinformed that the sky is mauve. So it would seem that (4') can't lead to a contradiction; there are non-contradictory instances of it.
But we know where the argument is heading, and it is toward a contradiction. Maybe it does raise a contradiction, even though (4'), as such, implies no contradiction. How so? Things are not nearly so simple as they had seemed at first; but it may well be that we don't need (4') to yield a contradiction in every instance; the paradox possibly just needs some instance of (4') to yield a contradiction. A paradox that holds only for some instances of (4') is a paradox still. Will it turn out to be so? Stay tuned for our next crazy installment. If your head is spinning a bit, good. But the roller coaster is just starting. I'll continue the indexing in a future post; and we'll also get into lots more issues, and come back to (KP) in particular, further on. And keep in mind that these are all sketchy thoughts.
Monday, November 13, 2006
Links and a Few Notes
* Chernoff Faces: a way to illustrate statistical data as cartoon faces. (ht: Mixing Memory)
* Michael Vendsel is beginning a series on Barth's interpretation of Anselm at Cynthia Nielsen's "Per Caritatem". Part I; Part II. While I think Barth's interpretation is flawed, one of the nice things about it is that it takes the first section of the Proslogion seriously; and Barth's interpretation of the origin of id quo maius cogitari non potest is, I think, at least in the ballpark.
* An interesting article on Benedict XV's attempt to preserve peace in Europe, and the contrasts with Wilsonian internationalism.
* I think I've linked to the Nietzsche Family Circus generator before, but this one is the funniest combination I've come across, and I had to share it. One wonders what pictures in particular inspired Dolly's comment.
* Suzanne McCarthy has been doing a series of posts on the "Junia, of note among the apostles" verse of Romans 16.
* At "The Inklings" you can read George Orwell's 1945 review of C. S. Lewis's That Hideous Strength.
* The Pope discusses faith and reason again. (ht: Sacramentum Vitae)
* In order not to feel compelled to devote another post to A. C. Grayling's comments on religion, should Grayling comment on religion again, I will simply note here the general structure Grayling's comments seem to share so far, allowing for some minor variations:
a) He starts out by stating something that a little bit of research would show to be clearly false or at least dubious;
b) Using this to transition into comments about how stupid and ignorant religious people in general are;
c) During which he lectures everyone on what various words really mean, without considering how they are actually used by ordinary English speakers;
d) And ends with a plea for mutual respect and tolerance.
And that seems to be about it; put that way, there doesn't seem to be much more that needs to be said.
* Theodore Dalrymple discusses what makes Samuel Johnson great. (ht: LTA)
* I recently had someone come to my weblog with the search question, "Is the earth a perfect sphere?" The answer, of course, is that it is not. It is an oblate spheroid -- and only an approximate one of those.
* Don't forget the Carnival of the Citizens.
* Also, don't forget the Cliopatria Award nominations.
* Michael Vendsel is beginning a series on Barth's interpretation of Anselm at Cynthia Nielsen's "Per Caritatem". Part I; Part II. While I think Barth's interpretation is flawed, one of the nice things about it is that it takes the first section of the Proslogion seriously; and Barth's interpretation of the origin of id quo maius cogitari non potest is, I think, at least in the ballpark.
* An interesting article on Benedict XV's attempt to preserve peace in Europe, and the contrasts with Wilsonian internationalism.
* I think I've linked to the Nietzsche Family Circus generator before, but this one is the funniest combination I've come across, and I had to share it. One wonders what pictures in particular inspired Dolly's comment.
* Suzanne McCarthy has been doing a series of posts on the "Junia, of note among the apostles" verse of Romans 16.
* At "The Inklings" you can read George Orwell's 1945 review of C. S. Lewis's That Hideous Strength.
* The Pope discusses faith and reason again. (ht: Sacramentum Vitae)
* In order not to feel compelled to devote another post to A. C. Grayling's comments on religion, should Grayling comment on religion again, I will simply note here the general structure Grayling's comments seem to share so far, allowing for some minor variations:
a) He starts out by stating something that a little bit of research would show to be clearly false or at least dubious;
b) Using this to transition into comments about how stupid and ignorant religious people in general are;
c) During which he lectures everyone on what various words really mean, without considering how they are actually used by ordinary English speakers;
d) And ends with a plea for mutual respect and tolerance.
And that seems to be about it; put that way, there doesn't seem to be much more that needs to be said.
* Theodore Dalrymple discusses what makes Samuel Johnson great. (ht: LTA)
* I recently had someone come to my weblog with the search question, "Is the earth a perfect sphere?" The answer, of course, is that it is not. It is an oblate spheroid -- and only an approximate one of those.
* Don't forget the Carnival of the Citizens.
* Also, don't forget the Cliopatria Award nominations.
Jottings on Fitch's Paradox I
None of this is to be taken as more than a few sketch thoughts.
Fitch's Paradox is a paradox that has occupied quite a few people in epistemology. Discovered by Fitch in 1963, the result is usually characterized along the lines of "If all truths are knowable, all truths are known." What makes it a paradox is that the knowability principle (KP) is usually considered to be very plausible:
(KP) p → ◊Kp If a claim is true, it is possible that it is known to be true.
But if KP is true, and Fitch's argument is right, then it follows that all truths are known. And it's difficult to determine what could be wrong with Fitch's argument. So here's the argument as portrayed in the SEP article linked to above, with a rough English translation to the right.
(1) p ∧ ¬Kp A particular claim is true, and it is not known to be true.
(2) (p ∧ ¬Kp) → ◊K(p ∧ ¬Kp) (1) implies that it is possible to know both that the particular claim is true and that it is not known to be true.
(3) ◊K(p ∧ ¬Kp) It is possible to know that both the particular claim is true and that it is not known to be true.
(1) is just a particular instance of the claim that there is at least one truth that is not known. (2) follows by KP from (1). (3) follows from (1) and (2). Now suppose that it is not only possible to know (p ∧ ¬Kp) but that this is actually known:
(4) K(p ∧ ¬Kp) It is known that both the particular claim is true and that it is not known to be true.
(5) Kp ∧ K¬Kp The particular claim is known to be true and it is known that it is not known to be true.
(6) Kp ∧ ¬Kp The particular claim is known to be true and it is not known to be true.
(4) is our supposition; (5) follws from (4) by the standard epistemic principle that a conjuction is known only if its conjucts are known; and (6) follows from the standard epistemic principle that if p is known, p is true. We have a contradiction here, of course. So (4)) is false.
(7) ¬K(p ∧ ¬Kp) It is not known both that the particular claim is true and that it is not known to be true.
(8) ¬K(p ∧ ¬Kp) (7) is necessary.
(9) ¬◊K(p ∧ ¬Kp) It is not possible that this is known: both that p is true and that it is not known to be true.
(10) ¬∃p(p ∧ ¬Kp) There is no particular claim for which it is true both that the claim is true and that it is not known to be true.
(11) ∀p(p → Kp) For any particular claim, if it is true, it is known to be true.
We know that (7) is true because we showed that (4) led to a contradiction. Because (4) led to a contradiction, however, (7) must be necessarily true, which gives us (8). (8) tells us that it is necessary that something is not true; and whenever it is necessary that something is not true, it is not possible for it to be true. So, given (8), we have (9). (9) contradicts (3). From this it follows that there is not some particular claim for which (p ∧ ¬Kp) is true. That's what (10) says; and from (10) we can get (11) directly. So if something is true, it is known to be true.
The general consensus is that this is very odd. I am, as I said, going to have a few sketchy thoughts on this; but I'll save those for another post. Right now, I want to make two brief points.
The first is that it is not at all surprising that (4) is necessarily false. Suppose that p = "The sky is blue". (4) then claims that this is true:
Both of these are known: The sky is blue and it is not known that the sky is blue.
It is not in the least surprising that that this leads to a contradiction. But this suggests that the whole argument from (4) to (11) is right, because if (4) is necessarily false, (7) is necessarily true; and the rest of the argument seems to follow in good order.
The second is a matter of translation. I already said that the result is usually put in terms of knowability: KP is usually read, "If p is true, it is knowable." I translated differently, as you can see. This is because I think "knowable" is a very bad translation of the double operator, ◊K. To see this, think about what we really mean when we say the following two things:
This is knowable: The sky is blue.
It is possible that this is known: The sky is blue.
The two are not equivalent, and for good reason. The English word 'knowable' in all but a very small handful of uses hides a third operator, a temporal operator -- an incipit, to be exact:
This can come to be known (can begin to be known): The sky is blue.
So when I say that some claim is knowable, I usually don't mean that it is possible that it is known; I mean that it is possible that it could come to be known. So I think there's reason to stay away from the added complications that are introduced by the word 'knowable'. Using 'knowable' makes it sound even more paradoxical; but (1) it doesn't need to be made to sound more paradoxical; and (2) it is misleading.
Both of these points will be seen again. More on Fitch's Paradox in a future post.
Fitch's Paradox is a paradox that has occupied quite a few people in epistemology. Discovered by Fitch in 1963, the result is usually characterized along the lines of "If all truths are knowable, all truths are known." What makes it a paradox is that the knowability principle (KP) is usually considered to be very plausible:
(KP) p → ◊Kp If a claim is true, it is possible that it is known to be true.
But if KP is true, and Fitch's argument is right, then it follows that all truths are known. And it's difficult to determine what could be wrong with Fitch's argument. So here's the argument as portrayed in the SEP article linked to above, with a rough English translation to the right.
(1) p ∧ ¬Kp A particular claim is true, and it is not known to be true.
(2) (p ∧ ¬Kp) → ◊K(p ∧ ¬Kp) (1) implies that it is possible to know both that the particular claim is true and that it is not known to be true.
(3) ◊K(p ∧ ¬Kp) It is possible to know that both the particular claim is true and that it is not known to be true.
(1) is just a particular instance of the claim that there is at least one truth that is not known. (2) follows by KP from (1). (3) follows from (1) and (2). Now suppose that it is not only possible to know (p ∧ ¬Kp) but that this is actually known:
(4) K(p ∧ ¬Kp) It is known that both the particular claim is true and that it is not known to be true.
(5) Kp ∧ K¬Kp The particular claim is known to be true and it is known that it is not known to be true.
(6) Kp ∧ ¬Kp The particular claim is known to be true and it is not known to be true.
(4) is our supposition; (5) follws from (4) by the standard epistemic principle that a conjuction is known only if its conjucts are known; and (6) follows from the standard epistemic principle that if p is known, p is true. We have a contradiction here, of course. So (4)) is false.
(7) ¬K(p ∧ ¬Kp) It is not known both that the particular claim is true and that it is not known to be true.
(8) ¬K(p ∧ ¬Kp) (7) is necessary.
(9) ¬◊K(p ∧ ¬Kp) It is not possible that this is known: both that p is true and that it is not known to be true.
(10) ¬∃p(p ∧ ¬Kp) There is no particular claim for which it is true both that the claim is true and that it is not known to be true.
(11) ∀p(p → Kp) For any particular claim, if it is true, it is known to be true.
We know that (7) is true because we showed that (4) led to a contradiction. Because (4) led to a contradiction, however, (7) must be necessarily true, which gives us (8). (8) tells us that it is necessary that something is not true; and whenever it is necessary that something is not true, it is not possible for it to be true. So, given (8), we have (9). (9) contradicts (3). From this it follows that there is not some particular claim for which (p ∧ ¬Kp) is true. That's what (10) says; and from (10) we can get (11) directly. So if something is true, it is known to be true.
The general consensus is that this is very odd. I am, as I said, going to have a few sketchy thoughts on this; but I'll save those for another post. Right now, I want to make two brief points.
The first is that it is not at all surprising that (4) is necessarily false. Suppose that p = "The sky is blue". (4) then claims that this is true:
Both of these are known: The sky is blue and it is not known that the sky is blue.
It is not in the least surprising that that this leads to a contradiction. But this suggests that the whole argument from (4) to (11) is right, because if (4) is necessarily false, (7) is necessarily true; and the rest of the argument seems to follow in good order.
The second is a matter of translation. I already said that the result is usually put in terms of knowability: KP is usually read, "If p is true, it is knowable." I translated differently, as you can see. This is because I think "knowable" is a very bad translation of the double operator, ◊K. To see this, think about what we really mean when we say the following two things:
This is knowable: The sky is blue.
It is possible that this is known: The sky is blue.
The two are not equivalent, and for good reason. The English word 'knowable' in all but a very small handful of uses hides a third operator, a temporal operator -- an incipit, to be exact:
This can come to be known (can begin to be known): The sky is blue.
So when I say that some claim is knowable, I usually don't mean that it is possible that it is known; I mean that it is possible that it could come to be known. So I think there's reason to stay away from the added complications that are introduced by the word 'knowable'. Using 'knowable' makes it sound even more paradoxical; but (1) it doesn't need to be made to sound more paradoxical; and (2) it is misleading.
Both of these points will be seen again. More on Fitch's Paradox in a future post.
Sunday, November 12, 2006
Neither Fish nor Fowl nor Good Red Herring
| What American accent do you have? Your Result: The Midland "You have a Midland accent" is just another way of saying "you don't have an accent." You probably are from the Midland (Pennsylvania, southern Ohio, southern Indiana, southern Illinois, and Missouri) but then for all we know you could be from Florida or Charleston or one of those big southern cities like Atlanta or Dallas. You have a good voice for TV and radio. | |
| The West | |
| The South | |
| Boston | |
| North Central | |
| The Inland North | |
| Philadelphia | |
| The Northeast | |
| What American accent do you have? Take More Quizzes | |
Epitaph, Intended for Himself
The following is by James Beattie (1735-1803). You can read more of Beattie's poetry here.
Escaped the gloom of mortal life, a soul
Here leaves its mouldering tenement of clay,
Safe where no cares their whelming billows roll,
No doubts bewilder, and no hopes betray.
Like thee, I once have stemm'd the sea of life;
Like thee, have languish'd after empty joys;
Like thee, have labour'd in the stormy strife;
Been grieved for trifles, and amused with toys.
Yet, for a while, 'gainst Passion's threatful blast
Let steady Reason urge the struggling oar;
Shot through the dreary gloom, the morn at last
Gives to thy longing eye the blissful shore.
Forget my frailties, thou art also frail;
Forgive my lapses, for thyself mayst fall;
Nor read, unmoved, my artless tender tale,
I was a friend, O man! to thee, to all.
A Poem Draft
Karta Purakh
all-enlightening boundless being
True is His Name
never captured
never dying
never by the concept tamed
forming all
all things seeing
every seeing eye He makes
to see Him in a given guise
the wisest knowing Him as light
the sweetest knowing Him as grace
the bravest knowing Him as might
in the everlasting Name
His Name is True
never failing
never bounded by guise or word
He knows no bounding
all expressing
makes each spirit in its kind
builds the souls of living nations
makes them all to sing and live
the bounded masks unbound creation
every living mind He makes
to know Him in a vital guise
the purest heart as vital goodness
the softest heart as touching beauty
the warmest heart as fire truest
every guise is but a glimpsing
none can capture all His Name
all this world is fiercely burning
all this land is mired in flame
save it Lord with showered blessing
through every door that may deliver
save it Lord in every way
unbounded God within His people
moves about in living form
the unformed speaks in bounded creature
takes a shape in living works
the noumenal by all uncaptured
captures all
takes all by storm
through the lovers of the True
all-enlightening boundless being
True is His Name
never captured
never dying
never by the concept tamed
forming all
all things seeing
every seeing eye He makes
to see Him in a given guise
the wisest knowing Him as light
the sweetest knowing Him as grace
the bravest knowing Him as might
in the everlasting Name
His Name is True
never failing
never bounded by guise or word
He knows no bounding
all expressing
makes each spirit in its kind
builds the souls of living nations
makes them all to sing and live
the bounded masks unbound creation
every living mind He makes
to know Him in a vital guise
the purest heart as vital goodness
the softest heart as touching beauty
the warmest heart as fire truest
every guise is but a glimpsing
none can capture all His Name
all this world is fiercely burning
all this land is mired in flame
save it Lord with showered blessing
through every door that may deliver
save it Lord in every way
unbounded God within His people
moves about in living form
the unformed speaks in bounded creature
takes a shape in living works
the noumenal by all uncaptured
captures all
takes all by storm
through the lovers of the True
Carnival of the Citizens Call for Submissions
Richard at "Philosophy, et cetera" has begun the call for submissions:
Do think seriously about contributing something.
The inaugural Carnival of Citizens will be held here in two weeks' time. Submissions -- due by November 23 -- should be written in a generous and deliberative spirit, to engage rather than insult those who might disagree with the author. (See the guidelines for more details, and let me know if they are at all unclear.)
Submissions are invited on any topic relevant to public debate. For example, suppose you have the attention of a friendly and reasonable "opponent": what would you want to discuss with them? To advance your own view, what crucial insight or argument would you bring to their attention? To better understand their view, what would you like further explained?
Alternatively: you might write a post explaining why you deviate from the "party line" on some particular issue....Or you may be able to think up other possible approaches that align with the carnival's civic values -- if so, go for it!
While I have high hopes for the new carnival, it can only succeed by securing the good will and participation of diverse bloggers, spanning the entire spectrum of political perspectives. So please help "spread the word"!
Do think seriously about contributing something.
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