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"This Reply Is Nonsense!": Reply to Margaret Cuonzo, "Subjective Probability and Paradox" ANTON ALTERMAN Independent Scholar altermana@hra.nyc.gov One of the paradoxical things about paradoxes is that the only way to talk about them intelligently is to say why they don't exist. Although no one knows whether intelligent life exists on other planets, respectable scientists can talk about it as if it exists, whether it does or not. But with paradox, the only thing to say is that it doesn't exist, and why; and anyone who has a substantive theory that explains why paradoxes really do exist is to be avoided at cocktail parties and philosophy conferences. Theories of paradox are therefore couched in terms that leave plenty of room for the possibility that the whole thing is based on a misunderstanding. Though from time to time some brave soul like Zeno or Kripke may insist that "this really is a paradox", the more usual attitude is that though the premises seem to be true and the argument valid, surely one of these is an illusion. As for the conclusion, it is usually held to be certainly either false or contradictory. All other things being equal, it is false that Achilles cannot catch the tortoise, or that a man with 10,000 hairs on his head is bald; and it is a priori false that some proposition can be true and false at once, or that no set can both be and not be a member of itself. It is the very definition of logical validity that such propositions should not follow from true premises. So either we must admit that our logical truths do not always hold - an alternative most people find unappetizing - or find something wrong with the argument as a whole. This situation opens up the possibility of evaluating the degree to which we believe the components of a logical paradox, a possibility that is addressed in Margaret Cuonzo's original and sometimes amusing paper. Her idea is to apply Bayesian criteria to paradoxical arguments, thereby providing a means of judging the degree of difficulty a paradox presents. She assigns arguments a "paradoxicality rating" according to the stock we put in the premises, the validity of the argument, and the conclusion. Her formula is to multiply the "combined subjective probability" of the premises (to use her phrase) by that of the logical argument and then by a factor inversely related to the probability of the conclusion. Thus the lower the Bayesian value of the conclusion, the higher the paradoxicality rating. While I can see nothing wrong in principle with this procedure, I believe it presents difficulties that at least suggest the need for refinements. Let me first remark on some properties of the formula, before turning to more general considerations. First, when Cuonzo refers to the "combined subjective probability of the premises" she must mean the product of their probabilities, for she says that "if the subjective probability of a premise were 0... the argument has a paradoxicality rating of 0". The average Bayesian value of the premises is surely not very meaningful, and adding their values could produce a paradoxicality rating greater than one, which would surely be paradoxical; so the product is perhaps the only way to go. Nevertheless it is not without problems. For example, a valid argument with a false conclusion and 25 premises each having a fairly high Bayesian value of .9 would have the very low paradoxicality rating of .07; with 100 virtually certain premises each rated .99 such an argument would have a paradoxicality rating of only .37. The more epistemically certain premises we add, the less paradoxical the false conclusion becomes. Whether this is a good result or not I'm not sure, but it seems odd. The strength of an argument is not normally thought to be inversely related to the number of premises if each of them is an extremely well grounded belief, but perhaps this represents an unjustified faith in our own infallibility. Another issue arises if we follow Cuonzo's interpretation of the formula. She says that "sound arguments' paradoxicality rating will always be 0, because of the conclusion", and indeed "all arguments with true conclusions get paradoxicality ratings of 0". But this seems to go against the Bayesian nature of the rating system; for the truth of the conclusion is far from guaranteeing it a high subjective probability. Indeed, some arguments whose conclusions are true on at least some readings are normally treated as paradoxes. The counterintuitive conclusions of Newcomb's Problem, Kripke's rule-following paradox, Hume's problem, the Grue paradox and the Raven paradox are surely not categorically false. All of these yield conclusions that violate our intuitions about the inferences we can make from previous instances, yet unlike the Sorites paradox, which also violates such intuitions, these paradoxes depend not on the falsity of the conclusion but on its truth. Perhaps this issue can be handled without a change in Cuonzo's formula. Rather, we must stick to the Bayesian reading and consider the subjective certainty we have regarding the conclusion, not its truth. Then a true conclusion that inspires doubt will receive a value of less than 1, and the argument will receive its due as a paradox. We must also consider the possibility that that we might have a low degree of belief in a valid argument: for instance, Peirce's Law ((p > q) > p) > p directly yields a theorem equivalent to the law of the excluded middle; yet I assume most non-logicians would not have a high degree of confidence in Peirce's Law and might even find it paradoxical, as it seems to suggest that any proposition that implies another proposition thereby implies its own truth. So validity does not count any more than truth, and the second term of the formula must once again be concerned with our degree of belief in the reasoning, not its correctness. Interpreted in this way, I think the formula is a reasonable measure of our subjective willingness to accept the conclusion of an argument based on its premises and validity. But let me now turn to some more general concerns with this sort of approach. First, the use of such a formula presupposes that we can measure and compare Bayesian values of different arguments with one another. Is this necessarily true? It is important to remember that most paradoxes require quite a bit of context-building to set up the paradox. In other words, even if you can reduce many such arguments to the syllogistic form that Cuonzo provides, the belief we have in the the premises or the conclusion can be increased or decreased by telling a story. For example, no one believes offhand the conclusion that finding a red apple increases the likelihood that all ravens are black. But suppose we add some supporting claims: it is not irrational to assume that there are a numerable set of objects in the world at any one time; suppose we know that there are ravens but don't know what color they might be; then each item we find that is not a non-black raven very slightly decreases the chances that we will find a non-black raven, and since we know ravens exist, it increases the chance that they will be black. This may be fallacious logic, but nevertheless it may increase our tendency to believe the conclusion. Or take Sainsbury's example of the Barber Paradox: "In a certain remote Sicilian village, approached by ascent up a precipitous mountain road, the barber shaves all and only those villagers who do not shave themselves." Of course it is readily apparent that there can be no such barber so long as he himself is one of the villagers, but as Sainsbury points out, "the story may have sounded acceptable: it turned our minds, agreeably enough, to the mountains of inland Sicily". (Paradoxes, 2nd ed., p.2) So to get commensurable paradoxicality ratings one might have to evaluate our degrees of belief in the components of contextual or ancillary arguments. This could obviously lead to the possibility that truly commensurable arguments involve rating our entire belief systems. I think Cuonzo could constrain this overflow; the challenge would be to do it without imposing overly artificial boundaries. Next, some paradoxes do not depend on arguments at all; unlike the Sorites or Lottery paradoxes, which can perhaps be reduced to arguments, the Liar, Moore's Paradox and many others have to be augmented into arguments. The paradox, we might say, emerges directly from the presentation or utterance. "It's raining and I don't believe it" is not the conclusion of an argument, though one might construct an argument with the paradoxical conclusion: "Moore believes it is raining and he does not believe it is raining". This seems a bit forced to me, and I wonder if it might be better to say that such paradoxes are so obscene that the only rating we can give them is X. Finally, I worry that the rating system tends to indirectly validate the notion that there exist fundamental flaws in the structure of rational thought, language or even reality. By assigning positive values to the arguments based on degrees of belief, it points away from the idea that paradoxes are just linguistic or cognitive confusions. Perhaps if anything should be assigned it is negative values based on degrees of perplexity. To explain this point further, I have to say a word about my own views. I think of some paradoxes as arguments with true conclusions that are counterintuitive, but can be explained with a little effort; and these I take to be problematic only to the extent that it takes some work to overcome our cognitive prejudices. These paradoxes might be rated on a Bayesian scale, but only because they are real arguments. The rest I take to be arguments or utterances that are fundamentally absurd but twist language or logic in ways that are hard to pin down. Here, almost any degree of belief we have in the argument as a whole is tantamount to a degree of cognitive dysfunction. Take for example the Sorites argument. It fails immediately upon providing a numeric limit for heaps or baldness, whereas in a continuum with no defined limits, we can proceed in discrete increments forever without being able to say which increment crossed a border. The argument therefore depends on our failure to apply a readily accessible fact about spatial organization. Or take the Liar Paradox, which Sainsbury assigns a paradoxicality rating of 10 out of 10. Since no paradox arises for the Truthteller, all the Liar reveals is that no utterer or proposition can sensibly predicate falseness of itself. For anything to predicate falsity, senselessness, irrationality or improbability of itself is like uttering, "I believe p and not-p". It simply classifies the content or the utterer as unintelligible. To rate it highly paradoxical seems to be a roundabout way of pointing out that we tend to be very confused by the mode of presentation. We think we have found a conundrum internal to rationality itself; but there is no such conundrum, just confusion. Again, as Cuonzo points out, it seems quite plausible that the phrase "class of all classes that are not members of themselves" refers to some logical possibility. But the paradox shows that there can be no such class. It is just a lot more difficult to understand why than to see why there are no Sicilian barbers who do and do not shave themselves. Should we rate this as highly paradoxical, or our grasp of class concepts as relatively primitive? Perhaps both, but my feeling is that the latter is really the issue. Wittgenstein says: "This was our paradox: no course of action could be determined by a rule, because every course of action can be made out to accord with the rule" (PI 201). Though Kripke takes this and runs with it, Wittgenstein's own attitude is different: "It can be seen that there is a misunderstanding here from the mere fact that in the course of our argument we give one interpretation after another..." Wittgenstein thinks of the paradox about rule-following as a confusion about how a rule guides our actions; we think of the rule as a "machine" in which "all the movements have already been determined", and then see it as paradoxical when the student wants to continue a series in a way that seems not to have been determined. But the paradox was only introduced by thinking of the rule in this mechanical way, which forces us to give "one interpretation after another" to explain the machine's movements. And I think Wittgenstein would say that the cognitive value of the paradox is the same as that of any other confusion between the physical and conceptual. We might compare paradoxes to magic tricks; these too inspire varying degrees of belief, which might be rated according to the ingenuity of the trick in deceiving us; or from the other side, according to our gullibility in being deceived. Either way can be defended, but I wonder again whether Cuonzo's Bayesian rating system does not tend to obscure the epistemological problem. |