LV :2 VI p THESiS 3—4”- ' I‘M“ r- ' ‘fi 4 it, ,. , .J. 6 .. This is to certify that the dissertation entitled Moore's Problem and the Prediction Paradox: New Limits for Epistemology presented by Roy A. Sorensen has been accepted towards fulfillment of the requirements for Ph . D. degree in Eggs—QM!— 5. Major professor Date 18 May 1982 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 WRE'S PROBLEM AND THE PREDICTION PARADOX: NEW LIMITS FOR EPISTEMOImY By Roy A. Sorensen A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DCXYI'OR OF PHILOSOPHY Department of Philosophy 1982 ABSTRACT HERE'S PRDBLEN AND THE PREDICPICN PARAIIDX: NEW LIMITS FOR EPISTEMOLCEY By Roy A. Sorensen Ludwig Wittgenstein once exclaimed that the nost inportant philosophical discovery made by G. E. Moore was of the oddity of sentences like 'It is raining but I do not believe it'. This dissertation can be viewed as a partial vindication of Wittgenstein's enthusiasm. However, my direct target is the prediction paradox. In the first chapter, the history of the prediction paradox is covered in detail. With the help of some new variations of the prediction paradox, I then argue in Chapte II that the paradox has not yet been solved. Chapter III contains my solution to Moore's problem. My concept of an epistemic blindspot emerges from this chapter and is used to establish new kinds of limits on knowledge in Chapter IV. In the following chapter I argue that the prediction paradox is a symptom of our unfamiliarity with these limits. Thus the prediction paradox is part of a general epistemological problem rather than an isolated logical problem. I try to make this claim more plausible in Chapter VI by applying the lessons learned about these new limits for epistemology to the more traditional philosophical problems associated by A. Sorensen with predictive determinism. Along the way I show that disagreement amongst ideal thinkers is possible. I use this possibility to argue against emulation theories of moral problem solving, like ideal observer theories, conventionalism, and the Rawlsian appeal to the original position. I conclude the chapter by using epistemic blindspots as counterexamples to predictive determinism and retrodictive determinism Having shown how pre-decisional blindspots have been illicitly employed to support the thesis that decisions are uncaused in Chapter VI, I argue in Chapter VII that post-decisional blindspots are involved in Newcomb's problem. .A solution to this problem is then proposed. In my concluding remarks I provide a general characterization of my approach to the philosophical problems that have concerned me in this dissertation and a brief summation of its results. For Julia Lynn Driver ii ACKMVLEIXMENTS I thank the many people who have helped me become better at philosophy through their conversations with me. Of all these people, Herbert E. Hendry has helped me the most, and so it is to him that I am the most grateful. iii INTROIIJCTICN CHAPTERI CHAPTER II CHAPTER III CHAPTER IV CHAPTER V CHAPTER VI CHAPTER VII TABLEOFmNTENTS HISTORY OF IIT-IE PREDICTION PARAIIDX . . mtes O O O I O O O O O O I O O O I O CRITICISMS (F PAST PROHBALS . . . . mtes O C O O O O O C C O O I O O O O PURE NDOREAN PROPOSITIONS: A SOLUTION 'IO DOORE'S PROBLEM . . . . mtes O I O O O O O O O O O O O O O O NEWLIMITSFOREPISTEDDILBY..... mtESoooooooooooooooo THE BLINIBPOI' FALIACY: A SOLUTION '10 THE PREDICTIQV PARADOX PRE-DEKIISIONAL BLINDSPOI‘S AND PREDICTIVE DETERMINISM . . . . . . . mtes O O C O O O O O O O O O O O O O POST-DECISIQQAL BLINIEPOTS: A SOLUTION 'IO NEMCOMB'S PROBLEM . . . mtes O O O O O O O O O O O O O O O O CMHJDIm W O O O O O O O O O O O O O O O O BIBLImRAPHY iv 64 86 87 95 96 110 131 132 139 140 141 INTROIIJCTION The most popular variation of the prediction paradox involves a teacher who tells his students that there will be a suprise test next week. A.clever student objects that the test is impossible. He first notes that the test cannot be given Friday since the students would then know on Thursday evening that that test must be on Friday. The test cannot be given on Thursday since the student would then know on wednesday evening that the test is either on Thursday or Friday, and they have already eliminated Friday. In a like manner, the remaining days of the week are eliminated thereby "proving” that the test cannot be given. The first two commentators on the prediction paradox agreed with the clever student and considered the paradox to be veridical. Following commentators were more sophisticated. Most have either thought that the clever student's argument contains an equivocation or have thought that the teacher's announcement is, contrary to appearances, self-referential. A recent few have thought that the prediction paradox shows that we must reject the principle that one knows only if one knows that one knows. Still others have tried to place the paradox in the same family as Moore's problem. Moore's problem is the problemiof explaining the oddity of sentences like 'It is raining but I do not believe it‘. Since I agree with those who think that the prediction paradox is related to Moore's problem, I try to solve the latter in the hope of solving the former. My analysis of Nbore's problem yields a definition of an 1 2 epistemic blindspot. Roughly, an epistemic blindspot is a consistent propostion which cannot be know by a certain people at certain times. we all have epistemic blindspots though they have been almost entirely unnoticed even by philosphers. Epistemic blindspots are counterexamples to the principle that I can know whatever you can know and to the principle that if I can know something at a certain time, then I can know it at another time. Thus these blindspots show that there are unfamiliar limits to knowledge. I argue that the prediction paradox is a symptom of our unfamiliarity with these new limits for epistemology. Tb>further support my claim that our unfamiliarity with these limits are responsible for some philosophical problems, I try to show that much of the work done on the topic of predictive determinism is flawed by this unfamiliarity. One exanple of an epistemic blindspot is that a person cannot know what his decision is immediately before he makes the decision. This blindspot has been used to support the thesis that decisions are uncaused. Roughly, the argument is that if decisions are caused, then they are in principle predictable in which case it would be possible to know what one's decision will be immediately before making it. Since it would then be the case that one can know something that one cannot know, we must reject the supposition that decisions might be caused. Although I try to refute this argument, I do use blindspots as counterexanples to predictive determinism and for that matter, retrodictive determinism. In addition, I show how the sentences Moore was interested in suggest a way for ideal thinkers to disagree. The possibility of this kind of 3 disagreement undermines enulation theories of moral problem solving. emulation theory of moral problem solving is a theory which inplies that there is an agent or group of agents such that for any moral question, one can correctly answer the question by agreeing with the answer of that agent or group of agents. Ideal observer theories, conventionalism, and Rawls' original position device are exanples. In addition to the pre-decisional blindspot mentioned above, there is an interesting post-decisional blindspot involved in Newcomb's problem. This problem involves a chooser and a predictor. The chooser is shown in two boxes. One box is transparent and contains one thousand dollars. The other box is opaque and contains one million dollars if and only if the predictor has predicted that the chooser will decide to take only the opaque box. Newcomb's problem is the problem of determining whether one should take only the opaque box or both boxes. I argue that once the role of this post-decisional blindspot is understood, Newcomb's problem is solved. The general theme of this work is that several recent philosophical problems are due to our unfamiliarity with certain peculiar epistemological limits. As we gain familiarity with these limits, these problem are solved and we are given reason to hope that contributions to other philosophical problems can be made by further study of these limits. CHAPTERI HIS‘I'OIW OF THE PREDICTICN PARAIIDX Although Quine reports that the prediction paradox had some currency from 1943 onward, it first appeared in philosophical literature in 1948 in D.J. O'Connor's "Pragmatic Paradoxes". The military comnander of a certain camp announces on a Saturday evening that during the following week there will be a ”Class A blackout". The date and time of the exercise are prescribed because a ”Class A blackout" is defined in the announcement as an exercises which the participants cannot know is going to take place prior to 6:00 pm on the evening in which it occurs. It is easy to see that it follows from the announcement of this definition that the exercise cannot take place at all. It cannot take place on Saturday because if it has not occurred on one of the first six days of the week it must occur on the last. And the fact that the participants can know this violates the condition which defines it. Similarly, because it cannot take place on Friday last available day and is, therefore, invalidated for the same reason as Saturday. And by similar arguments, Thursday, WEdnesday, etc., back to Sunday are eliminated in turn, so that the exercise cannot take place at all.1 O'Connor considers the argument cogent. He points out that the definition of a "Class A black-out" is consistent but goes on to claim that it is pragmatically self-refuting. He conpares the definition to the following sentences: (1) I remember nothing at all. (2) I am.not speaking now. (3) I believe there are tigers in Mexico but there aren't any there at all. Although (1)-(3) are consistent, they "could not conceivably be true in any circumstances“.2 Further, (1)-(3) are all statements in the first person which refer to the contenporary behaviour or state of mind of the speaker. In other words, they are all statements involving what Russell calls "egocentric particulars" and Reidhenbach calls "token reflexive" words. That their peculiarities are ciosely connected with this can be seen from the fact that the peculiarties disappear if we substitute "you" or ”he” for "I" or allow the statement to refer to past or future conditions of the speaker. But not all pragmatic paradoxes are of this kind,...3 In "Mr. O'Connor's 'Pragmatic Paradoxes'," L. Jenathan Cohen argues that pragmatic paradoxes are consistent propositions which are falsified by their own utterance. Public announcement of (4) A.”Class Aiblackout" will take place during the fellowing week, makes it false. In a footnote Cbhen adds: If the camp commander intended to stage a suprise exercise on one day during the week and yet wanted to warn his troops of his intention, he would have to make an announcement somewhat like one or other of the following: Either "One day next week there will be a surprise exercise. A surprise exercise is an exercise about which, unless it takes place on the last day of the period for which you are warned, you will be in doubt as to when it is to happen until 6:00 pn on the evening in which it occurs" Or "One day next week there will be an exercise. Unless it take place on Saturday you will be in doubt as to when it is to happen until 6:00 pm on the evening in which it occurs.” In the former case he utters a prediction and a definition, in the latter two 6 predictions. Owing to the irreversibility of the time series, if it is known that an event will take place on either t1 or t2 0r...tn-1.4 In "Pragrnatic Paradoxes", Peter Alexander objects to Cohen's treatment of (1) and (4). (4) is not paradoxical at all since any announcement of an intention is implicitly recognised to be conditional on the possiblity of carrying out that intention. Even if I make a simple statement like ”I will go to the cinema tomorrow” I mean, although I do not state, that I shall do so if I am not in any way prevented. Thus Professor O'Connor's statement, which can be abbreviated to read "A 'Class A Blackout' will be carried out next week" M for completeness, to read ”If the conditions of a 'Class A Blackout' can be realized, a 'Class A Blackout' will be carried out next week.” Now this seems to raise no other difficulties than are raised by any conditional statement whose condition is unrealisable, like, for instance, "If I can live without air I will not breathe all day tomorrow but, similarlyy men might cease next week to be able to realize that if the blackout had not occurred by Friday it must occur on Saturday, and then the condition would realizable. Any problens raised by these statements do not appear to be similar to those raised by the other statements with which I have dealt (1)-(3) nor to be properly called “paradoxical”.5 The first publication devoted exclusively to the prediction paradox was Michael Scriven's "Paradoxical Announcements" in 1951. Whereas O'Connor regarded the paradox as rather frivolous and Alexander considered it interesting but of no great concern, Scriven is deeply impressed by the prediction paradox. Scriven puts (4) in the same class as (5).and (6). (5) You are going to have a surprise at lunch-time tomorrow. You are are going to have steak and eggs. 7 (6) I'll wage you can't find the roots of the equation x2+5x-24=0 within thirty seconds. The roots are 3 and -8. Although the person who says (5) or says (6) does not contradict himself in the usual sense, his saying (5) or (6) is pointless since he has undermined part of what he says. Scriven goes on to insist that the unexpectedness of the exercise be given a logical rather than a psychological interpretation. The drill is unexpected by the participants in the sense that they cannot produce a proof that it will occur on a given day. Scriven argues that a solution to the paradox requires that one distinguish between publicly uttered statements and ordainments. Ordainments are guarantees, as when the dates of performances and meetings are announced. As a private prediction (7) There will be a Class A Blackout next Saturday, is proper, but it cannot be used as an ordainment for the drill participants. Oonstrued as an ordainment, (7) guarantees a blackout which will on the one hand have an unspecified date, and on the other hand, have a specified date. This incompatibility forces one to conclude that either the blackout will occur on Saturday and not be Class A, or it will not occur Saturday and will be Class A. Neither conclusion is proper; we would only be led to these conclusions if we inferred from the self-refuting character of the announcement that there was a mistake. Since no proper conclusion can be drawn from (7) 8 as an ordainment, a Saturday blackout will be a Class A blackout, making (7) correct. Scriven next considers (8) There will be a Class A blackout next week. He claim that this announcement is also self-refuting since if the blackout does not occur before Saturday, it will be equivalent to (7) on Saturday morning. Saturday is therefore not a real possibility or else [(8)] is self-refuting. In general, a Class-A blackout cannot occur on the last day of any sequence of nights during which it is ordained or else the governing announcement will be self-refuting. The first five nights of the week rm form such a sequence: at the next stage, the next four. An thus the nights of the reversed week fall one by one: falling with the last is the point of the ordainment. Now if the governing announcement is [(8)] which is self-refuting, and a blackout occurs on any night of the week, the statement [(8)] will be verified. And if publicly stated, it would still be correct. Conclusion. At first we thought that the reductive proof showed a Class-A blackout to be inpossible while in fact any blackout that took place was a Class-A blackout. Now we have come to see that the suicide of the announcement as an ordainment is accmpanied by its salvation as a statement.6 Scriven's proposal deviates sharply from the proposals of his predessesors. Whereas O'Connor and Cohen held that the paradox was veridical, Scriven classifies it as falsidical. In the next issue of M, O'Connor reported that he converted to Alexander's view . Since Alexander believed that the alleged paradox is dissolved by his paraphrase, O'Connor's conversion deepens his disagreement with Scriven. Scriven believes that there is a paradox. 9 Apparently in the hope of undermining Alexander's proposed dissolution, Paul Weiss reformulated O'Connor's paradox. .A headmaster says, "it is an unbreakable rule in this school that there be an examination on an unexpected day." The students argue that the examination cannot be given on the last day of the school year, for if it had not been given until then, it could be given only on that day and would then no longer be unexpected. Nor, say they, can it be given on the next to the last day, for with the last day eliminated, the next to the last day will be the last, so that the previous argument holds, and so on and so on. Either the headmaster gives the examination on an expected day or he does not give it at all. In either case he will break an unbreakable rule; in either case he must fail to give an examination on an unexpected day.7 weiss explains that O'Connor's formulation makes it possible for the announcement to be rescinded, so that the nonoccurrence of the blackout can be predicted. weiss' stipulation that the rule is unbreakable corrects this flaw. In addition, weiss believes that it is more appropriate to call the paradox "the prediction paradox". Since this name has a plurality of users, I have adopted it as well. Weiss attenpts to solve the paradox by assimilating it to the problem of logical fatalism. By the law of excluded riddle, all proposition about the future, it is now either true or false.. But then there is nothing one can do to change the truth values of these propositions. Thus the law of excluded middle seens to imply that we are not free. For example, tomorrow I will either eat cereal or not. But given that either 'I will eat cereal tomorrow' is true now or false now, there is nothing I can do to avoid eating cereal if it is 10 true now that I will and there is nothing I can do which will bring it about that I eat cereal tomorrow it is now false. According to‘Weiss and many others, Aristotle tried to avoid logical fatalism by denying that (9) implies (10). (9) It is true that p or not-p. (10) Either it is true that p or it is true that notfip. According to this view, contingent propositions about the future lack truth values. weiss then claims that the prediction paradox arises from confusing the collective and the distributive senses of 'or'. (9) is an example of the collective 'or' while (10) is an example of the distributive 'or'. When we predict we refer to a range of possibilities which are as yet undistinguished one from the other. They are connected by means of a collective "or", prohibiting the separation of any one of them from the others, without the introduction of some power or factor not included in the concept of the range. Since predictions always refer to a range and never to the specific determinations of it produced in fact, the predictions must be supplemented by history or the imagination if we are to select and eliminate first one and then another alternative. What is selected and eliminated in history or in the imagination will be something distinct, focused on, actualized, connected with others by means of a distributive "or”. If we avoid confusing these two meanings of "or", our paradox, I think, will disappear.8 This distinction is obscure but the basic outline of weiss' solution can be discerned. When we are asked to consider whether the examination could be given on the last day, we imagine ourselves in the future and thus shift from the realm of the possible to the realm of the actual. The disjunction of examination dates is distributive 11 'meanings of "or", our paradox, I think, will disappear.8 This distinction is obscure but the basic outline of weiss' solution can be discerned. When we are asked to consider whether the examination could be given on the last day, we imagine ourselves in the future and thus shift from the realm of the possible to the realm of the actual. The disjunction of examination dates is distributive in the realm of the actual but is collective in the realm of the possible. The shuttling back and forth in time invites confusion between realms and thus confusion between kinds of disjunctions. Another popular version of the prediction paradox is the Hangman. A.man is sentenced to hang on one of the fellowing seven noons but must be kept in ignorance until the morning before the execution. The man argues that he cannot be hung on the last day since he would know after the penultimate noon. Having eliminated the last day, the rest are eliminated in the familiar way. In his "On a so-called Paradox", W. V. Quine blocks the elimination by showing that the announcement corresponding to the one-day case is not self-contradictory. Given that the judge says (11) Ybu will be hanged tomorrow noon and will not know the date in advance, Quine claims that the man should reason as follows: "we must distinguish four cases: first, that I shall be hanged tomorrow noon and I know it now (but I do not); second, that I shall be unhanged tomorrow noon and know it now (but I do not): third, that I shall be unhanged tomorrow noon and do not know it now; and fourth, that I shall be hanged tomorrow noon and do not know it now. The latter two alternatives are the open possibilities, and the last of all would fulfill the decree. 12 Rather than charging the judge with self-contradiction, therefore, let me suspend judgement and hope for the best. 9 Since the base step of the induction is fallacious, Quine concludes that there is no paradox. The first attempt to assimilate the prediction paradox to the self-referential paradoxes appeared in R. Shaw's "The Paradox of the Unexpected Examination". Shaw insists that " 'knowing' that the examination will take place on the morrow" must be 'lmouing' in the sense of ”being able to predict, provided the rules of the school are not broken".10 Shaw complains that "If instead one adopted a vague common-sense notion of 'knowing' , then one could perhaps agree with Professor Quine that an unexpected examination could take place even in a one-day term; but to my mind, this would be evading the paradox rather than resolving it."11 Given that 'unexpected' means 'not deducible from certain specified rules of the school', Shaw believes he can formulate two rules for the school described in Weiss' prediction paradox. Rule 1: An examination will take place on one day of next term. Rule 2: The examination will be unexpected, in the sense that it will take place on such a day that on the previous evening it will not be possible for the pupils to deduce from Rule 1 that the examination will take place on the 12 morrow Although a last day examination can be eliminated since it would violate Rule 2, an examination on any other day would satisfy Rules 1 and 2. By adding a third rule, the possibility of an examination on the last two days can be eliminated. 13 Rule 3: The examination will take place on such a day that on the previous evening it will not be possible for the pupils to deduce from Rules 1 and 2 that the examination 13 will take place on the morrow. If only two days remain in the term, the pupils can deduce by Rule 1 that the examination is on one of the two remaining days. By Rule 2, they can eliminate the last day, leaving the next to the last day as the only possibility. Since this deduction would violate Rule 3, the last two days are not possible examination days. However, an examination on any other day of the term would satisfy Rules 1, 2, and 3. In general, the last n days of the term.are eliminated by appealing to Rule 1 and n additional rules of the form Rule n + 1: The examination will take place on such a day that on the previous evening it. will not be possible for the pupils to deduce from the conjunction of rules 1, 2, . . ., n, that the examination will take place on the morrow. The n + 1 rules are incompatible with an n + 1 day term. Shaw concludes that original paradox arose by taking in addition to Rule 1, Rule 2*: The examination will take place on such a day that on the previous evening the pupils will not be able to deduce from Rules 1 and 2* that the examination will 14 take place on the morrow. By applying rules 1 and 2*, one can eliminate every day of the term. Once we realize that 2* is self-referential, the paradox is resolved. Ardon Lyon complains that Shaw's choice of the rules for the school is an evasion rather than solution of the paradox. Lyon points out that mere self-referentiality is not sufficient for paradox. For 14 example, 'This sentence is written in black ink' is perfectly all right. Dyon reject's Quine's analysis on the grounds that Quine's criterion for knowing implies that we cannot know anything about the future. According to Lyon, the paradox rests on an equivocation. Shaw's Rule 2* can mean either S1 or 82, but not both. S1 The examination will be unexpected in the sense that . . . it will not be possible for the pupils to deduce from Rules 1 and S1 that the examination will take place on the morrow, unless it takes place on the last day. S2 The examination will be unexpected in the sense that . . . it will not be possible for the pupils to deduce from Rules 1 and 82 that the examination will take place on the last day.15 Lyon argues that if one reads Rule 2* as S1, like a sensible person should, then the clever student's argument is fallacious. And even if one reads it as $2 . . . it can have no possible application, must always remain false, for nothing, including setting the examination earlier, would make it true that the boys would be unable to deduce on the eve of the last day that it would occur on the morrow, _i_f_ the master were to wait that long. For R1 and 82 applied together on the eve of the last day give us: (1) The examination must take place tomorrow. (2) (The examination will be unexpected in the sense that) it is not possible to deduce from (1) and (2) that it will take place on the morrow. clearly contradict each other, as opposed to Quine's solution. 16 Shaw concludes that the paradox arises from taking Rule 2* to mean S1 and $2 at the same time. 15 In 1960, David Kaplan and Richard Montague published "A Paradox Regained" in the Notre Dame Journal of Formal Logic, the first publication on the paradox to appear outside of M. They begin their rigorous development of Shaw's self-referential approach by letting M, T, and W respectively stand for 'K is hanged on Monday', 'K is hanged on Tuesday‘, and 'K is hanged on Wednesday'. 'Ks(x)' stands for 'K knows on Sunday afternoon that sentence x is true'. 'Ign', 'Kt', and T“ are treated analogously. The variable 'x' takes names of sentences as substituends . So Kaplan and Montague introduce a system of names of expressions. If E is any expression, then E is the standard name of E, constructed according to one of various alternative conventions. They suggest that one might either construe E as the result of enclosing E in quotes, or identifying E with the numeral corresponding to the Godel number of E, or regarding E as a structural-descriptive name of E. Kaplan and Montague are now in a position to express the judge's decree, D1: nsdrs-ws-Ksn'nv -M&T&-W&-Km(T)v -M&-T&W&-Kt(W) They use the sentence 'I(S1, 82) ' to indicate that S1 logically implies 32' Kaplan and Montague then express the principles K appeals to for the impossibility of D1, as: (A1) (-M&-T) “*Kt (-M&-T‘) (A2) [I(-M & -T, w?) & Kt(-M & -rr)] ‘* Kw?) 16 They use the sentence 'I(§1, S2) ' to indicate that 81 logically implies 82. Kaplan and Montague then express the principles K appeals to for the impossibility of D1, as: (A1) (-M a 4r) + Kt (4473) (A2) mm, W) & xy-fiT-‘IBI + Kai) (A1) and (A2) are special cases of the principles of knowledge by memory and the deductive closure of knowledge, respectively. Although dubious in full generality, ”we can hardly deny K the cases embodied in (A1) and (A2), especially after he has gone through the reasoning above."17 We can also assume that K knows (A1) and (A2). (A3) [fin(m—2) Since K assumes that (A1) and (A2) logically imply -W, he tries to argue that he cannot be hanged on Wednesday noon. (A4) mm, a?) s. wig-313)] “gm-v7) To exclude Tuesday, K uses the following analogues of (A1) and (A2): (A5) -M fig“ (J4), (A6) [Ian—Tin. 'F) s the?) a Ign(-W)]+ mud?) Additional analogues to (A1) and (A2) are used to eliminate Monday, and thus to show that D1 cannot be fulfilled. Kaplan and Montague note that K has committed the fallacy Quine pointed out when applying (A2) . (A2) only implies W when conjoined with D1, so (A2) must be replaced by (A2.) m—M & -T & D1,?) & Ktt-M" & -T) & Hawaii) 17 Thus we need to also assume KtFD1). But this seems unreasonable, especially in light of K's attempt to prove that the decree will not be fulfilled. However, Kaplan and Montague argue that Quine's formulation fails to capture the self-referential aspect of the decree. For the sake of brevity, Kaplan and Montague use the two day version of the hangman paradox to show how the self-referential aspect can be expressed. (1)133 sum 5. -'r s. -Ks(fi3“—>’E)] v [(-M s. T) s. -Ign('b'§—TTI")]] K excludes Tuesday and then anday be appealing to the following analogues of (A1)-(A4): (31) —M +Km(-‘M’) (32) "[[I(‘ir D3 T) 5' IS“(—’i)]+1$n(03 T) (B3) KS(B1 & 82) (B4) [1031 s 32. 03+ M) & Ks(‘éT&"é§)1~> Kym» According to Shaw, the decree is genuinely paradoxical, not merely incapable of fulfillment. However, Kaplan and Montague argue that the decree is merely incapable of fulfillment since the supposition that D3 can be fulfilled leads to absurdity. Suppose as before that K is hanged on Tuesday noon and only then. In this possible state of affairs, -M and T are ture. The hangman must now establish 'Ififlfil- Tb apply his : earlier line of reasoning, he must show that D3+T, considered on mnday afternoon, is a non-analytic sentence about the future. But D3+T is in fact analytic, for as K has shown, -D3 follows logically from general epistemological principles, and hence so does D3->T.18 18 A.paradoxical decree would result if the judge tried to make the decree capable of fulfillment by adding a stipulation: Unless K knows on Sunday afternoon that the present decree is felee, one of the following conditions will be fulfilled: (1) K is hanged on anday noon' is true, or (2) K is hanged on Tuesday noon but not on Monday noon, and on Monday afternoon K does not know on the basis of the present decree that 'K is hanged on Tuesday noon' is true.19 Kaplan and Montague are able to show that this version is a complicated variation of the Liar paradox leading to the conclusion that the decree can and cannot be fulfilled. They go on to consider a one-day version of this variation: Unless K knows on Sunday afternoon that the present decree is false, the following condition will be fulfilled: K will be hanged on Monday noon, but on Sunday afternoon he will not know on the basis of the present decree that he will be hanged on Monday afternoon. 20 Finally, they consider a version in which "the number of possible dates of execution can be reduced to zero”. Here the judge asserts: K knows on Sunday afternoon that the present decree is false.21 Another branch of the self-referential approach was introduced by G. C. Nerlich in his ”Unexpected Examinations and Unprcvable Statements". After expressing his view that the prediction paradox is neither trivial nor easy to solve, Nerlich suggests that . . . it is a quite unique kind of ordinary language problem, having some connection with the situation posed by Goedel's famous sentence, to the effect that the sentence itself cannot be proved. 19 It will be clear, when I have dealt with the paradox, why I think it is of sore importance to logic—of more importance than the cotparatively simple Grelling paradox, for example.22 Nerlich reviews Shaw's treatment of the paradox. Shaw provided a non-self—referential formulation of the school rules and a self-referential formulation. He then argued that the first formulation is not paradoxical since no unexpected examination can be given during the term and that the second formulation is paradoxical. Nerlich insists that both formulations are paradoxical. After all, if an examination is given Wednesday, it would not be expected. Thus Shaw's first formulation shows that self-reference is not an essential feature of the prediction paradox. Nerlich next considers Lyon's claim that the paradox rests on an equivocation. Lyon argued that the sensible interpretation of the announcement is (a) rather than (b): (a) it will not be possible to deduce from the statement when the examintion will occur at any time prior to its occurrence, m it occurs on the last day. (b) it will not . . . , whether or not it occurs on the last dey. Nerlich objects that the announcement cannot mean (a) since there is a perfectly proper and strict sense of 'unexpected' in which (a) is equivalent to the 'the examination will occur unexpectedly, unless it occurs expectedly on the last day. ' Since the announcer can plainly mean strictly what he said, that the examination will be unexpected, the equivalence of (a) and the above ensures that the announcer does not mean (a). 20 On the other hand, denying that the announcement means (a) is not tantamount to asserting that it means (b). One is only denying that the examination will occur on any day sudh that on a previous day, the examination date could be deduced. Nerlich further argues that (a) is not equivalent to the announcement because . . . there are tests which actually reqeire the rejection of the "unless" clause and such tests occur daily. The trial emergency stop in every driving test is a case in point. The trial is improper if the order does not take the candidate unawares, so it cannot be allowed to occur expectedly even at the end of the test. Yet proposing such a trial is not proposing anything contradictory.23 Nerlich admits that his own solution is "rather bizarre". He first points out that at each stage of the student's argument a negation of a statement of the form 'Examination on -day' is derived. But after deriving a negation for eadm of the alternatives, the students have no basis for thinking one day rather than another is the examination date. So if the examination is given on one of the days, it will be unexpected. Tb falsify the announcement, the students must derive a statement which excludes . . . "a day such that it is not possible to deduce from the head's statement, at any time prior to the day, that the examination he§_been arranged for that day.24 Since only negations are derived, the announcement is not falsified. The possibility of deriving an examination date from.a contradiction should be ignored since it would be of no use to the pupils. 21 So due to the fact that it entails not, e.g., Examination on Wednesday, but something else (a contradiction), the statement is self-consistent. This is a hard saying. However, let us look again at the curious logical features of this everyday remark. The statement is partly about an examination and partly about its own logical consequences, gig. that the examination date is not among them . . . . The only way in which this metalogical statement can be falsified is by proving that the examination h_a_s_ been arranged for a certain day. It is this that the students attempt to do but fail to do, producing only days on which it seems _n_o_t_ possible to hold it. And that is because in the attempt, they are forced to use the very premise (or set of premises) which they hope to falsify.25 Nerlich admits that this alone is insufficient to account for the odd state of affairs since reductio ad absurdum arguments also use the premises the arguer hopes to falsify. He claims that the oddity is due to the fact that the key premise states that it cannot be used that way, for it says that only false statements can be deduced. Nerlich goes on to further claim that in so far as it is about provability, the prediction paradox resembles Goedel 's incompleteness proof. Central to the proof is a sentence, G, which is true only if G is rot provable. If the logical system is consistent, then G must be undecidable. For if G is proved, then G is also unproved, and if the negation of G is proved, then G is proved. So here consistency is incompatible with completeness. Nerlich claims that the same holds true for the announcement in the prediction paradox. By implying that 22 there is a true but unprovable alternative, the announcement is, as it were, describing itself as incotplete. But just that remark about incotpleteness seets to make the system now complete, and therefor contradictory. Yet, as we have seen, it is really neither complete ror inconsistent.26 Nerlich concludes that when one's sole source of information seems to impeach himself, one does not know what to make of it. This is just what the teacher wants. He manages to say nothing by contradicting himself. In The British Journal for the Philoscphy of Science, Martin Gardner compared the prediction paradox to Langford ' 3 Visiting Card Paradox. Langford's paradox consists of a visiting card on the front of which is written 'The assertion on the other side of this card is true' while on the back is written 'The assertion written on the other side of this card is false'. Tb show the analogy between the prediction paradox and the Langford paradox, Gardner constructs a "New Prediction Paradox”. Here, one puts a card in an envelope and instructs the receiver to send it to a mutual friend only after writing on its (as yet blank) back 'Yes' or 'No' according to whether the receiver feels justified in predicting that the mutual friend will find that 'No' has been written o1 its back. In "A Comment on the New Prediction Paradox" , Karl Popper agrees that Gardner has established a close analogy between the two paradoxes. As a friendly amendment, however, Popper argues that Gardner's paradox can be formulated in such a way that it is free of the idea of negatim (common to the Liar 23 and Langford paradoxes). Here, one instructs the receiver to write 'Yes' in a blank rectangle to the left of one's signature if, and only if, the receiver feels justified in predicting that when it is sent back, the rectangle will still be blank. In the first issue of the American Philoscphical Quarterly, Brian Medlin first expresses disappointment with all of the previous contributions to the problem except Shaw's. Nerlich is first criticized for offering a solution which merely reformulates the paradox. Medlin then moves on to formalize the paradox. Although he never mentions bbntague and Kaplan, his approach and major results duplicate their work. However, Medlin does defend the stronger thesis that the prediction paradox rather than an offshoot of it is a paradox of self-reference. I will return to Medlin shortly when I describe Jonathan Bennett's criticisms of the self-referential approach. In the next issue of American Philosophical Quarterly, Frederic Fitch's "A Goedelized Formulation of the Prediction Paradox" appeared. Fitch first argues that the announcement is merely self- contradictory. He then modifies the prediction paradox by weakening the notion of surprise so that an expected last day examination counts as a surprise examination. Fitdi shows that this prediction is consistent and considers it a resolutim of the paradox. Third, Fitch develops Nerlich's suggestion by modifying the prediction in the prediction paradox so that it is an undecidable proposition equivalent to Goedel's. 24 The first general criticism of the self-referential approach appeared in Jonathan Bennett's review of the articles written by Shaw, Dyon, Nerlich, Medlin, and Fitch. Bennett's first criticism of the attempt to solve the prediction paradox by showing that it has an element of self-referentiality is that Nerlich's objections have not been satisfactorily answered. Nerlich first argued that Shaw illegitimately assumed that all self-reference is improper. Medlin conceded that some cases of self-reference may be proper, citing R. M. Smullyan's "Languages in which Self-Reference is Possible" (T§e_ Jburnal ofeSymbolic Logic, 1957), but denies that self-reference is proper in the case in question. Medlin formulates the announcement as: (M) The information concerning dx [the day on which the examination occurs] is not sufficient to allow determination of x at any stage before the examination is actually given.27 The impropriety of (M) is then argued fbr on the grounds that The proposition (M) says something about the proposticns in a non-empty set S; namely, that the conjunction of all these propositions does not constitute a premiss of sufficient power to permit the determination of x at any stage before the examination is given. . . . But if (M) is in S, then what (M) says is (roughly) that (M) does not permit us to determine x. This kind of self-reference is circular. It invites us the question, ngee_does not permit us to determine x?" we do not understand (M) until we know what (M) is about, which set S happens to be. If (M) is itself in S, then we shall never know this and never understand an.” 25 Bennett objects to this argument since if it is valid, one could prove that 'No universal proposition entails that all men are mortal' is unintelligible. Nerlich's second objection was that self-reference is not essential to the paradox. Medlin formulates Nerlich's objection with the help of the following (using _ standing above symbol for propositional negation, and letting pi be the propostion 'The examination occurs on the ith day'). (I) (p1 V 132 m3) & pi & pj (1753': 19.2133) (M1) From (I) it is not possible to determine x, even given as additional information one of E1 , W‘ (M2) From (I) & (M1) it is not possible to determine x, even given as additional information 5, (M3) From (I) & (M1) & (M2) it is not possible to determine x. (C) (M1) & (M2) & (M3)- Nerlich argues that self-reference is not essential to the prediction paradox because (I) & (C) imply a contradiction by steps parallel to the self-referential cases. In reply, Medlin argues that Nerlich's own proposed solution keeps the paradox alive with the help of self-reference. Medlin explains that Nerlich argues in favor of the compatibility between (I) & (C) and p2 on the grounds that there is no sound deduction from the former to the latter. But if this is to be taken as providing a model for (I) & (C), then we must interpret (C) as saying of itself that it does not, with (I), constitute sufficient information for the determination of x. The statement for which p2 does provide a model is 26 (M4) The conjunction (I) &(C) does not constitute sufficient information for the determination of x. Unlike (C), the statement (M4) is true. It is true because (C) is false. Nerlich confuses (M4) with (C). He is then led to say that (C) is true because it is false. we should note in passing that the case p1 provides a model for (M4). So does the case p3: that is why Nerlich finds that even an examination on d3 is unexpected.29 Despite Medlin's report that Nerlich agrees with all of Medlin's comments about Nerlich's analysis, Bennett dismisses Medlin's attempt to meet Nerlich's objection as ad hominem. Even if the above criticism of Nerlich's constructive analysis succeeds, it does not show that Nerlich's destructive analysis fails. After noting the common diagnosis that the Lyon ambiguity in the announcement is the source of our puzzlement, Bennett concludes: Perhaps there is that ambiguity and perhaps it might puzzle someone; but it has nothing to do with the fact which makes the announcement teasing to everyone, namely the fact -- noted by Fitch on page 161 - that "in practice the event may nevertheless occur on some one of the specified set of days, and when it does occur it does constitute a sort of surprise." But 5§e§_puzzle cannot be handled by someone who thinks that "the Prediction Paradox can be formulated in a . . . way that makes no use of epistemological or pragmatic concepts” (p. 161).” Bennett's review is followed by James Cargile's review of Kaplan and Montague, Gardner, and Pepper. Cargile dismisses Gardner's "new prediction paradox" as not being a genuine paradox. Although Langford presents some similar paradoxes, the visiting card paradox is due to 27 Jourdain. Cargile concludes that Langford has already shown that the alleged paradoxes of Gardner and Popper have already been dealt with in Lewis and Langford's chapter o1 logical paradoxes in their SMIic Iggic. Cargile summarizes "A Paradox Regained" as variations on the there: A: "K knows that A is false." He points oit that this is an old theme appearing in Buridan's fliemata. In Cargile's opinion, . . . These "knower"-type paradoxes are just Liar-family paradoxes in which knowing is involved only in that it entails truth. "K knows that p is false” is logically equivalent to "p is false and K knows it.” So A is fundamentally the same as B: "B is false and K knows it." B is just a case of the Conjunct-Liar, ”This conjunction is false and q," which makes possible a semblance of proving the falsity of any q you please. Similarly with C: "K does rot know that C is true," which appears to be true but unknowable by K. It is fundamentally the same as D: ”Either D is false, or D is true but K does not know it," which is a case of the Disjunct-Liar.31 So unlike Bennett, Cargile is quite sympathetic to the self- referential approach . According to R. A. Sharpe, the prediction paradox arises if both parties know and apply the rules set by the teacher's announcement. For then, ell the days are eliminated. If the rules excluded all but one day, no paradox would arise. 28 Since the rule here excludes all days in the week as possible days for the examination, to choose a day at all will be a surprise in the sense of displaying ignorance of or a deliberate breaking of the rule. An element of self-reference arises from the fact that on the terms by which the paradox can occur, the master must take into account the boy's own prediction before choosing a day. Since he cannot choose days which they have predicted, they negatively affect the choice and if they have played a part in making the choice it is difficult to see how it can surprise them.32 Sharpe points out that an announcement which only excluded one day would still be self-referential but no paradox would arise. He therefore concludes that self-reference is not a sufficient condition for the paradox. It is interesting to note that the "element of self-reference" to which Sharpe alludes, is not the kind of self-reference Bennett and Cargile considered. Sharpe's conception of self-reference seems to be game-theoretic. J. M. Chapman and R. J. Butler in their "On Quine's 'So-called Paradox'”, propose a "perverse solution" taking Quine's rejection of the base step of the induction as their inspiration. Like others, they argue that if the examination has not been given by Thursday, the students can deduce that the examination has not been given by Thursday, the students can deduce that the examination is on Friday and deduce that it is not on Friday. The conclusion that the examination must be held on the last day is just as warranted as the conclusion that it cannot be held. Therefore the boys cannot predict, by a valid process of logical argument and without laying themselves open to contradiction, that 29 the examination will be held on the last day. Therefore the examination will be held on the last day. Therefore the examination, even if it is held on the last day, will be unexpected in the required sense.33 Another proposal put forth by commentators professing sympathy with Quine is "The Prediction Paradox Again". Here, James Kiefer and James Ellison first insist that the problem can only be made interesting and precise if surprise is defined in terms of deducibility. Let us use "deduce1" to mean "deduce, using as premises the nonoccurrence of the examination up to the moment of deduction, plus the truth of this announcement". Let us use "deducez" to mean "deduce, using as the premise the nonoccurrence of the examination up to the moment of deduction". Let us define ”surprise1" and in terms of ”deduce1" and "deducez" respectively.34 Kiefer and Elison interpret the prediction paradox as showing that the announcement is contradictory if given the 'surprise1' reading. However, if the announcement will true if the examination is given on any day of the week. Once this ambiguity is noted, the authors claim the paradox is resolved. They claim that if they have correctly understood Quine, he has largely anticipated their solution. Quine's suspicions about the base step of the induction are shared in Judith Schoenberg's "A the on the Logical Fallacy in the Paradox of the Unexpected Examination". The elimination argument begins with 30 the last day: if the examination has not been given by the penultimate day, then . . . . Schoenberg claims that the antecedent of this conditional illegitimately assumes that conditions laid down by the teacher have already been violated. The rest of the student's argument is ”merely a verbal play”. 80 although Schoenberg agrees with the student that the examination cannot be given on the last day, she believes that the student is arguing fallaciously when he begins his argument with the conditional 'If the examination has not been given by the penultimate day, then it must be given on the last'. . . . the premise entertains a condition under which the event cannot occur as defined, and thus cannot serve as the point of departure for a line of reasoning about the event's possibility. All it can lead to deductively is a clarification of the condi- tions under which the event cannot occur by definition.35 In "The Surprise Exam: Prediction on the Last Day Uncertain", J. A. wright launches another attack on the base step of the induction. He suggests that the usual interpretation of the announcement is to the effect that: (1) A test will be held, and any one day of a given finite set of days is possible for it. (2) It will not be possible to predict the test, with logical necessity, on the morning of that day. wright then suggests that the paradox can be avoided by reading the announcement as saying the last day: if the examination has not been given by the penultimate day, then . . . . Schoenberg claims that the antecedent of this conditional illegitimately assumes that conditions laid down by 31 the teacher have already been violated. The rest of the student's argument is “merely a verbal play". So although Schoenberg agrees with the student that the examination cannot be given on the last day, she believes that the student is arguing fallaciously when he begins his argument with the conditional 'If the examination has not been given by the penultimate day, then it must be given on the last'. . . . the premise entertains a condition under which the event cannot occur as defined, and thus cannot serve as the point of departure for a line of reasoning about the event's possibility. All it can lead to deductively is a clarification of the condi- tions under which the event cannot occur by definition.35 In ”The Surprise Exam: Prediction on the Last Day Uncertain", J. A. wright launches another attadk on the base step of the induction. He suggests that the usual interpretation of the announcement is to the effect that: (1) A test will be held, and any one day of a given finite set of days is possible for it. (2) It will not be possible to predict the test, with logical necessity, on the morning of that day. wright then suggests that the paradox can be avoided by reading the announcement as saying (A) Any one of a finite set of days is a possible day for the test to be planned. (B) It will be cancelled if it is acutally predicted on the morning of that day.36 The d'lange from "the test will be held on" to "the test is planned for” allows (A) to be cancelled rather than contradicted by (B). The change from possibility of prediction to actual prediction undermines 32 the base step of the inductim according to Wright. The teacher will only refuse to consider a Friday examination if he is certain that a student will come to him with a prediction. Although it is highly probably that a student will do this, it is rot certain. Thus the students cannot eliminate a Friday examination with certainty. Later in 1967, M. J. O'Carroll published "Improper Self-Reference in Classical Logic and the Prediction Paradox” in IQLique et Analyse. O'Carroll claims that although the prediction paradox has received much attention, it has not been correctly formulated. He argues that the teacher is really claiming that the students cannot deduce the day of the examination without there also being a counterdeduction that is not that day. O'Carroll also claims that the conclusion to be drawn is . . . either it is rot true that there is an exam on one and only one afternoon "next week" 93 the the teacher's statement . . . falls oitside the field of valid application of tmdvalued, non-levelled logic.37 Two years after his sympathetic review of the self-referential approach to the prediction paradox, James Cargile rejected this approach in favor of a game-theoretic approach . He conceives the problem as involving rational agents, one of which is trying to make a choice that cannot be predicted by others even though all the rational agents have the same relevant information. Cargile stipulates that the teacher has no means of randomizing his choice and that this is common knowledge. Besides knowing that the teacher prefers to give a surprise test, the students know that it is common knowledge that both 33 teacher and students are ideally rational agents. Since Cargile is interested in the two day version of the prediction paradox, it is also common knowledge that the test must take place either Thursday or Friday. Cargile believes this situation leads to a puzzle because . . . the following would appear to be an essential truth about ideal rationality: If two ideally rational agents are asking independently whether a give proposition is true and if both have exactly the same relevant data and exactly the same knowledge about what is relevant, then they will both reach the same conclusion. The conclusion may be "Yes" or "No” or "insufficient data to determine" or "the question is unclear," etc., but it must be the same for both. For suppose that the two agents arrive at different answers, X and Y. The X cannot be a better answer than Y on the information given, since that would contradict the assumption that both agents are ideally rational -- that is, think as well as is possible in every case. But then the answer "X" is no better than answer than "Y" is determinable on the information given and is clearly a better answer than X.of Y, whidh contradicts the assumption that both agents will give the best possible answer on the information available to them.38 The teacher will think that the students might be surprised by a Thursday test just in case the students will. If the teadher thinks that there is no chance that a Thursday test will be surprising, then the students will know this as well, because they will have arrived at the same conclusion. If the teacher concludes that he cannot know whether there is a chance, then the students will know this. Cargile's point is that someone can surprise someone else only if the surpriser and the surprisee disagree about something at some time. SInce the teacher and the students are ideally rational agents with 34 the same relevant information, such a disagreement is impossible (given the principle mentioned above). Cargile tries to solve the problem by introducing a third ideally rational agent; a judge to adjudicate the students' claim. Cargile argues that the students know that the test will be given on Thursday only if the judge would agree that they know. The students cannot know that the test will be given on Thursday because the teacher will only give the test on Thursday if he knows that the students do not know that the test will be given Thursday. If the judge ruled in favor of the students, he would be ruling against the judgment made by an ideally rational agent, the teacher. Since the students cannot satisfy this criterion of knowledge, they do not know. Cargile concedes, however, that the students can have justified confidence in the test being held Thursday. Indeed, since the standards for certainty fluctuate from context to context, cargile is willing to allow that the students are certain that the test will be on Thursday in other, less stringent contexts. Six months after publishing Cargile's article, the Journal of Phileeepgy published Robert Binkley's "The Surprise Examination in Modal Logic". Binkley's main achievement was to provide a rationale for Quine's claim that the prisoner cannot eliminate the last day because he does not know that the announcement is true. Binkley points out that the announcement corresponding to the single day version, 35 (11) Ybu will be hanged tomorrow noon and will not know the date in advance, resembles the sentences G. E. Moore was so puzzled by: (12) It is raining but I believe it is not raining, (13) It is raining but it is not the case that I believe it is raining. In Knowlegge and Belief, Jaakko Hintikka argued that these sentences cannot be believed by perfect logicians even though (12) and (13) are consistent. Since the prediction paradox is a paradox for perfect logicians, Binkley points out that Hintikka's explanation of the incredibility of (12) and (13) can be extended to the question of why the prisoner cannot know (11). The prisoner cannot believe, and therefore, cannot know (11) because (11) is logically incredible to the prisoner. By appealing to the principle that if a perfect logician believes p, then he believes that he will believe p thereafter, Binkley is able to demonstrate that the announcement corresponding to the n + 1 day case are so incredible to the prisoner. So Binkley concludes that the prediction paradox is in the same family as more's paradox. In “Believing and Disbelieving", Kathleen Johnson ‘wu arrives at much the same conclusion as Binkley, differing only in that she sees no need to restrict the analysis to perfect logicians. About ten years after Binkley's article appeared, Igal Kvart published "The Paradox of Surprise Examination”. Kwart never mentions Binkley or wu but aside from greater care in formalization, does little else but duplicates Binkley's results. 36 When most people learn about the prediction paradox, they are inclined to accept the base step of the induction more readily than the induction step. The first commentator to plausibly follow this inclination was Craig Harrison in "The Unanticipated Examination in 'View of Kripke's Semantics for Modal Logic". Harrison considered the paradox as it arises in the following form. Student e is told by his instructor that there will be a test on either the second or the fourth of the month. The test will be unforeseen in the sense that if the test is given on the fourth of the month, then e_will not know so on the third, and if the test is given on the second, Q will not know so on the first. Although Harrison provides a formalization of the prediction paradox as it arises in this fbrm, I prefer another formalization for reasons which will become apparent in the next chapter. Let 'Kaipk' read 'a knows at day i that the test occurs on day k'. Where i