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draft version 2.3.1, uploaded March 31, 2014 [previously uploaded versions: 2.1 (Feb. 26, 2014), 2.2 and 2.3 (March 21, 2014)] Two-Context Probabilism and the Dissolution of the 'Lottery' Problem 1. Which and What 'Lottery Problem'? It is a fairly undisputed fact of life that gambling often enough leads to various sorts of problems, and while philosophers may not be facing those more classical problems, they sure enough have their own issues with gambling since Kyburg (1961) introduced the lottery problem to philosophy. In those more than 50 years a great deal of oftentimes very sophisticated literature on that topic has been produced. Yet, as diverse as that literature and as accomplished as the respective authors may be, I cannot quite get rid of the impression that a very large part of the existing literature on the lottery problem – including that of Kyburg 1961 – would seem to share the common flaw of misrepresenting certain in my opinion rather intuitively obvious aspects of the lottery case, Cf. e.g. premises 1 and 1* in Nelkin 2000, 373, 375; for the rejection and partial correction of that misrepresentation cf. Nelkin 2000, 380, 400, 401-402 and the respective passages in this work. thereby failing to solve an otherwise relatively easy to solve puzzle. These shortcomings unfortunately also apply to some of the most recent articles such as McKinnon 2011, Kroedel 2012, Littlejohn 2012 and Kroedel 2013 where, perhaps also due to their shortness, relatively little has been done to continue on the paths to the solution laid out by articles such as DeRose 1996, Nelkin 2000 and Douven 2002, not to forget the fallibilist, probabilist and contextualist approach of Cohen 1988 that will also feature prominently in this work. That, at least, is my rather provocative claim that I will try to make a case for in this paper. However, before proceeding to the first step towards the dissolution of 'the' lottery problem, we need to remind ourselves that there is more than just one general version or rendering of it, meaning that it needs to be decided which version(s) of the lottery problem to take for our analysis (I am using the term of "lottery problem" as the hyperonym to both of the following "paradoxes"). One version stays relatively true to Kyburg's work by staying centered on his concept of rational acceptability: The Lottery Paradox is generally thought to point at a conflict between two intuitive principles, to wit, that high probability is sufficient for rational acceptability, and that rational acceptability is closed under logical derivability. (Douven 2012, 55) Nelkin (2000, 374-75) consequently and quite fittingly regards these as the "rationality version[s]" of the lottery problem. In the second general rendering of the lottery problem by Hawthorne (2004) and others, however, we find that rational acceptability has been replaced by the concept of knowledge: The paradox arises from the combination of two plausible claims: first, no one can know that one's lottery ticket will lose prior to learning that it in fact has lost, and, second, the justification one has for the belief that one's ticket will lose is just as good as the justification one has for paradigmatic instances of knowledge. (Reed 2008, 217) This, in general, is what Nelkin (2000, 373-74) refers to as the "knowledge version[s]" of the paradox, sometimes also called the "Harman Paradox" (Douven 2007) or the "Harman-Vogel Paradox" (Nagel 2011). In comparing those two contexts in which the lottery situation has been placed, the latter represents a higher standards context and one that encompasses the former, because of those propositions which we regard as rationally acceptable, only some will qualify for knowledge, but of those propositions which we regard as knowledge, (virtually) all will also be deemed as rationally acceptable. The reason for this is that rational acceptability is closely overlapping with if not identical to justification In personal communication Igor Douven confirmed my suspected link between justification and rational acceptability by taking the latter to mean "justified credibility." However, to avoid further possible confusion, I will largely avoid the terms of "rational acceptability" or "justified credibility" and simply talk of "justification." and that justification is a necessary component for knowledge or 'warranted assertibility' Cf. Dewey 1941, 169: "[…] my analysis of 'warranted assertibility' is offered as a definition of the nature of knowledge […]." Russell, on the other hand, interpreted "warranted assertibility" to mean truth (cf. Dewey 1941, 169), whereas Zagzebski (1994, 65) equals "warrant" with justification. To make the confusion complete, DeRose (1996, 568ff) just establishes a vague connection between "assertability" and "knowledge" – reason enough, I believe, to do without the terms of "warrant" and "assertability.": We "accept" beliefs when they appear to be well-justified, which is a result of those beliefs being, among other things, "rational(ly formed)." For further confirmation that rational acceptability is linked to justification and not to, for example, truth, also see the following passage: Truth cannot simply be rational acceptability for one fundamental reason; truth is supposed to be a property of a statement that cannot be lost, whereas justification can be lost. The statement 'The earth is flat' was, very likely, rationally acceptable 3000 years ago; but it is not rationally acceptable today. Yet it would be wrong to say that 'the earth is flat' was true 3000 years ago; for that would mean that the earth has changed its shape. (Putnam 1981, 55) As such, rational acceptability, via its once again confirmed link to justification, would also be a part of or contained in the higher standards or more complex concept of knowledge, meaning that the lower standards rational acceptability/justification version of the lottery problem is encompassed in the higher standards knowledge version. To the question of which version(s) to take for our analysis, I will give a fairly straightforward and pragmatic answer: We do not only want our beliefs to be justified but false, we rather want them to be justified and true, i.e. we would rather want to know the outcome of lotteries and many other things. Given both this greater appeal of knowledge as well as the fact that justification is contained in knowledge and taken into account when knowledge is accounted for, I am going with the higher standards knowledge related version of the problem (for a simultaneous treatment of both versions see Nelkin 2000). So much for the question of which lottery problem. The second and more provocative question of what 'lottery problem' is owed to two other highly noteworthy circumstances. First and no matter if one adheres to the justification or to the knowledge take on it, the 'lottery' problem (or paradox or puzzle) is somewhat of a misnomer due to the respective problem pertaining not just to lotteries or, for that matter, to gambling, but to the much bigger field of knowledge of future events which, like all such knowledge, rests on a possibilistic or probabilistic basis. Cf. Nagel 2011, 2: "It is sometimes suggested that Harman’s pattern of intuitions should generate worries only about our knowledge of future events (Dretske 2005; Harman 1986), or of outcomes dependent on propositions whose support is purely statistical or probabilistic in character (Cohen 1988; Nelkin 2000)." As such, the problem we are dealing with here should be renamed to "future knowledge problem" (henceforth FKP) and traced back much further than Kyburg, namely to Hume: As to past Experience, it can be allowed to give direct and certain information of those precise objects only, and that precise period of time, which fell under its cognizance: But why this experience should be extended to future times, and to other objects, which for aught we know, may be only in appearance similar; this is the main question on which I would insist. (Hume 1777 [1748], sect. IV, part II, 34-35) However, even though the Kyburgian 'lottery problem' needs to be expanded to the Humean FKP, it would be a mistake to inflate it further than that, for instance in the way Hawthorne did: Following Vogel (1990), John Hawthorne (2004) argues that we should consider a wide class of propositions to be lottery propositions. Lottery propositions share two characteristics: they are (arbitrarily) highly probable on one’s evidence and there is an intuitive reluctance to say that one can know or warrantedly assert them. (McKinnon 2013, 7) For other generalizations also see e.g. Nelkin 2000, 388: "If the fact that Jim's belief that he will lose is based on a P-inference is the ultimate—or intermediate—explanation of why one cannot have knowledge in the lottery case, then it must generalize to non-lottery situations in which we base beliefs on probabilities" or Greco 2007, 301: "Cases a through d satisfy the weak safety condition, but so does the Lottery Case. The Lottery Case fails to satisfy strong safety, but so do cases a through d." Hawthorne is admittedly correct in claiming that lottery propositions actually belong to "a wide[r] class of propositions" because lottery propositions do belong to the wider class or category of future knowledge propositions, and he is also correct in respect to the two defining criteria of high probability and reluctance to assert knowledge. What is unfortunately omitted though is the perhaps most important criterium of future-relatedness which also contains the explanation for why these propositions are only "highly probable" (in a sense of close to but less than 1) and not of maximum certainty (probability of 1). As such, Vogel and Hawthorne, but also their respective followers Douven (2007, 328, 342) and McKinnon, who believe that it is permissible to extend Harman's Paradox (Harman 1986) or, respectively, the lottery problem to propositions about the past or present such as "The Magna Carta was signed in 1215" or "I am holding my coffee cup" (McKinnon 2013, 10), are simply wrong in that assumption (cf. McKinnon 2013, 10, n. 20 for a brief consideration of that possibility). The obvious reason for this is that in respect to both past and present events, factual truth exists and with that also the possibility of forming factual knowledge, i.e. "know[ledge] that something is the case" (Unger 1968, 157). In respect to future events such as future lottery draws, however, factual truth is neither determined nor existing (not the case yet), meaning that there can only be knowledge of possibilities and the associated intermediate probabilities (i.e. probabilities of occurrence between 1 and 0) but no knowledge of facts or the associated maximum probabilities (1 for existence and 0 for negation of existence). For a possible association of knowledge of facts with the probability of 1 cf. McKinnon 2013, 5, n. 9, on Williamson 2000: "[…] Williamson’s view of evidence and evidential probability on which knowledge that p is probability 1 on one’s evidence." It is these findings that prohibit or in any event should prohibit the conflation of propositions about the future with propositions about the past or the present, for these are not one and the same but two different classes or categories of propositions with different associated probabilities. For that reason and to avoid further confusion by way of Vogel's or Hawthorne's inappropriate conflation of these two categorically different propositions (and given my preference of the knowledge over the rationality/justification version), it is more fitting to talk of the "future knowledge problem" than of the "lottery problem" and to replace the latter complex term with "lottery case" (cf. DeRose 1996, 577) or "lottery situation" (cf. Nelkin 2000, 408) in order to signify that one only talks about an exemplification of the FKP and not of a smaller – or larger – problem. Second, the question of what 'lottery problem' is also owed to the fact that the 'problem' actually is not that much of a problem: So far, the FKP the way we encounter it in the lottery situation has persisted because, on the one hand, the future typically features more than just one possibility of how things could go; as such, each possible event is only "relatively (un)certain," i.e. each possible event can only be assigned an intermediate probability between 1 and 0. Knowledge, on the other hand, is typically regarded as relating to present or past facts to which one typically assigns the maximum probability of 1. As such, if one conjoins the two terms in the complex term of "future knowledge (of facts)," the result, among other things, is a problematic 'tension' between the intermediate probability associated with "future" and the maximum probability associated with "knowledge (of facts)" – something which has led Hume and many others to believe that future knowledge is impossible and that the corresponding term is essentially a contradiction in itself. Cf. Douven 2007, 328: "[…] the Humean skeptic who denies that any knowledge of future events is possible" and moreso Reed 2010, 234: "Hawthorne [2004, 25] connects this principle with the Humean view that knowledge and probability are of ‘contrary and disagreeing natures.’ [...] The conflict between fallibilism, which takes knowledge to be probabilistic, and this Humean view should be obvious." It is this problem that lies at the heart of the FKP. The dissolution of the problem, however, is astonishingly simple: One must not only allow for (by default fallible and probabilistic For the evident connection between fallibilism and probabilism cf. e.g. Reed 2010, 233: "[…] a fallibilist must recognize that knowledge comes in degrees ranging downward from absolute certainty. In this sense, fallibilistic knowledge is probabilistic," or 235: "[…] the way we typically make knowledge attributions is obviously compatible with the fact that knowledge can come in degrees.") knowledge of (past or present) facts, but also for knowledge of (future) possibilities, because then the FKP can be overcome Cf. Kitcher 2011, 252, in quotation of Dewey 1998, 19: "As Dewey remarks of philosophical questions, ‘We do not solve them: we get over them.’" with relative ease – and just how that can be done is what will be shown in the following chapters. 2. Lottery Propositions: Induction to Fact vs. Calculation of Probability A fairly recent lottery situation rendering that offers us a good opportunity to analyze the logical issues with the FKP is the following: The lottery problem trades on two stable intuitions. [...] But how could both intuitions be right? How can it be both (1) that we can have knowledge on inductive grounds, but (2) that we don't know we will lose the lottery, even though our inductive grounds for believing this are excellent? (Greco 2007, 299-300, in reference to Cohen 1988, 92-93) What is of interest here is the claim that we arrive at lottery propositions by way of induction or in any event not by way of deduction or entailment (cf. Cohen 1988, 92-93: "[…] conclusion highly probable without entailing the conclusion"). To evaluate that claim, let us construct a paradigmatical lottery situation or argument by way of which we can analyze whether this is true or not. Assuming that there is only one winning ticket in a fair lottery with a total of 100 tickets and assuming that someone (S) has bought one ticket, one will often find inferences to the following kind of lottery proposition: Argument No 1 (induction to fact) Premise 1: One ticket out of a total of 100 will win the lottery. Premise 2: S has one of those 100 tickets Conclusion ("factual lottery proposition A"): S will not win the lottery. Even though not explicitly stated, many such lottery propositions (cf. e.g. Nelkin 2000, 373, 375) give the impression of being a proposition about a 'future matter of fact' – something that is or in any event would be rather unfitting or contradictory given that the event specified in the lottery proposition is only a possibility with a (very high but nevertheless) intermediate probability of occurrence and not (yet) a fact with a by default maximum probability of 1. So then why did so many philosophers reach a factual conclusion and not, as would be more accurate and correct, a possibilististic one? One 'party' that is chiefly responsible for this grave omission or confusion would appear to be (a too limited understanding of) the correspondence theory of truth (italics by myself): "Narrowly speaking, the correspondence theory of truth is the view that truth is correspondence to a fact—a view that was advocated by Russell and Moore early in the 20th century. But the label is usually applied much more broadly to any view explicitly embracing the idea that truth consists in a relation to reality […]" (David 2013, introduction). Yet is that "label" also "applied broadly" enough? David (2013, introduction) lists "various concepts for the relevant portion of reality (facts, states of affairs, conditions, situations, events, objects, sequences of objects, sets, properties, tropes)." What is strangely amiss in all of these fact-like or fact-related things, however, is the decidedly non-factual "portion of reality" of possibility. Given the huge influence of the correspondence theory of truth and its factual interpretation of reality, it is therefore little wonder that the possibilistic portion of reality was and still is severely neglected. This rather unwitting and overly factual interpretation of reality and truth would appear to be the reason for why many authors end up with a factual lottery proposition, and this is what was left out in the previous argument. As such, the line of thought by way of which a factual lottery proposition is reached needs to be expanded in the following manner, namely by the inclusion of an inductive second inference: Argument No 1* (induction to fact) Premise 1: One ticket out of a total of 100 will win the lottery. Premise 2: S has one of those 100 tickets Conclusion 1/Premise 3: S will not win the lottery with a probability of 0.99. Premise 4: A possibility-to-fact transformation and rounding up clause. Conclusion 2 ("factual lottery proposition A"): S will not win the lottery. In other words: The for Hume "incomprehensible medium" Hume 1777 [1748], sect. IV, part II, 35: "But if you insist, that the inference is made by a chain of reasoning, I desire you to produce that reasoning. The connexion between these propositions is not intuitive. There is required a medium, which may enable the mind to draw such an inference, if indeed it be drawn by reasoning and argument. What that medium is, I must confess, passes my comprehension; and it is incumbent on those to produce it, who assert, that it really exists, and is the origin of all our conclusions concerning matter of fact." by way of which inferences from mere future possibilities to future facts were made would seem to lie in the by and large omitted inductive second inference from conclusion 1/premise 3 and premise 4 to conclusion 2, and especially in premise 4. The obvious reason why the second inference is an induction is that inductive inferences are defined as containing an inflation from "some" to "all" (or at least to "more") and that the increase of the probability of 0.99 (conclusion 1) to 1 (conclusion 2) features such an inflation. As such, Greco's claim that lottery propositions are based on "inductive grounds" would be correct – at least if we were to regard it as fitting to infer from possibilities and the associated intermediate probabilities to facts and the associated maximum probability, because that 'feat' can only be accomplished by way of inductive inflation of the associated probabilities. Yet why should we wish to make the inductive second inference from possibility to fact? People may have the tendency to round up (to fact and probability of 1) or down for the sake of convenience, but in my opinion it is much more fitting to stay with a simpler and (more) accurate and correct probabilistic interpretation of such lottery situations (cf. e.g. Harman 1986, 71; Cohen 1988, 92-93, 106; DeRose 1996, 568; Nelkin 2000, 400; Douven 2002, 391; Douven 2012, 55) by not going beyond the following first argument: Argument No 2 (calculation of probability, "P-inference") Premise 1: One ticket out of a total of 100 will win the lottery. Premise 2: S has one of those 100 tickets. Conclusion ("possibilistic lottery proposition B"): S will not win the lottery with a probability of 0.99. Here we no longer see any inductive inflation of probability as claimed or implied by Cohen, Greco or others. Instead, the above argument is quite similar to deductive Aristotelean syllogisms like Darii or Baroco, i.e. similar to those types of valid syllogisms where an affirmative all-proposition (A) in one premise (cf. the probability of 1 that is associated with premise 1) and a some-proposition (either I or O) in the other premise (cf. the probability of 0.99 that is associated with premise 2) lead to an according some-proposition in the conclusion (cf. the probability of 0.99 that is associated with the conclusion) and not back to an all-statement as featured in the conclusion of the arguments 1 and 1.1 which, from a syllogistic-deductive point of view, feature invalid types of inferences. Cf. Williamson 2000, 248: "[…] probabilistic evidence warrants only an assertion that something is probable." In other words: The non-inductive argument 2 gives a much more accurate rendering of the thought processes of an individual that ponders her chances of winning or losing the lottery, whereas the inductive "rounding up from possibility (associated probability of <1) to fact (associated probability of 1)" argument is a clear misrepresentation of the lottery situation, with the obvious reason for that being that factual truth does not yet exist in the lottery situation: "[…] it isn’t determinately true that your ticket is a loser. So you can’t know your ticket is a loser […]. By contrast, there is a fact of the matter as to who won the Bulls game yesterday." (DeRose 1996, 570) "[…] if Jim were to irrationally postulate a causal connection between his belief that he will lose the lottery and the fact that he will lose, he is thereby guilty of irrationality." (Nelkin 2000, 399) "It is clear that if one asks ordinary folk why such reasoning is unacceptable, they will respond by pointing out that the first premise was not known to be true." (Hawthorne 2004, 29-30) It is this so to speak common sense argument that, (P1) with the lottery situation being about future events, (P2) with factual truth in respect to the outcome of those future events being neither determined nor existing, and (P3) with truth being a necessary component of knowledge, we have to reach the conclusion that propositions about the future outcomes of lotteries cannot possibly contain knowledge of facts. A special situation in which the outcome of a lottery could be called "determined but not yet existing" is one where the person or machine that makes the draw is fractions of a millimeter away from grasping the winning ticket: No other ticket will be drawn, but the draw has not been made yet. Instead, given that lottery-like situations are undetermined probabilistic ones where things could turn out different than expected (despite being well-justified or rational in expecting to lose due to the overwhelming probability of occurrence of that event, one could nevertheless still get lucky and win the lottery), we can only have fallible (or infallible) knowledge of possibilities or of the associated intermediate probabilities here. The evident disadvantage of such (fallible) knowledge of possibilities compared to (fallible) knowledge of facts is that possibilities are less certain than facts. The advantage of knowledge of possibilities, however, is that we do not necessarily need to reject anything in our assumptions if things turn out different than probabilistically expected: […] in seeing that my reasons failed to guide me to the truth, I must reject something in my total "package" of reasons. Notice, in contrast, that this is not the case for P-inferences. If I believe that p (say, my ticket will lose) on the basis of a high statistical probability for p, and I find out that not-p (I won!), then there is nothing at all in my reasons to reject. I still believe the same odds were in effect, and I still believe that they made my losing extremely probable. I have no reason to think that my evidence failed to bear a connection to my conclusion that I previously thought it did. (Nelkin 2000, 401) Also see Nelkin 2000, 389, n. 18: "The fact that Jim makes a P-inference to his belief that he will lose makes his winning a "salient" alternative that he cannot rule out." With the non-inductive calculation of subjective intermediate probabilities furthermore being more exact and above all without categorical mistake (i.e. without the impermissible and contradictory equalization of future possibilities with so far non-existent facts), it seems to me that we should clearly favor the non-inductive and probabilistically inferred/calculated argument No 2 and the according possibilistic (as well as probabilistic) lottery proposition B over the inductive argument No 1 or rather 1* and the according factual lottery proposition A in terms of giving a fitting description of the lottery situation. 3. Two-Context Probabilism So then what about contextualism? Besides a more exact 'higher standard' non-inductive possibilistic/probabilistic interpretation of lottery aka future knowledge propositions, can we not also allow for 'lower standard' ordinary (thought and) language contexts where, by social allowance of inexactness, one also gains allowance to inductively inflate a possibility's associated probability of <1 to a fact's associated probability of 1? In that respect my answer is a clear "no" for the reason that it is this inexactness which has led to the FKP in the first place: Due to being the very smart and principle of non-contradiction minding philosophers that we are, we intuitively grasp that possibility/intermediate probability equals possibility/intermediate probability and not fact/maximum probability. As such, when someone comes along and starts portraying possibilities with an associated 0.99 probability of occurrence as inductively inflated facts with an associated probability of 1, our internal 'nonsense-detectors' start or in any event should start sounding an alarm independent of whether this has been said in a more or less exact context: Even though rounding up 0.99 to 1 or rounding down 0.01 to 0 may be prohibited in a more exact context such as this one, it is most certainly allowed and happening all the time in less exact ordinary language contexts. Confusing a possibility (no factual truth/knowledge) with a fact, however, is an impermissible categorical mistake even in lower standards contexts. As such, it is always wrong to treat a possibility as a fact independent of how many tickets there are, That increasing the total number of tickets does not solve the problem was already pointed out by Unger (1968, 161-62) in his Analysis of Factual Knowledge in a basically identical color guessing example: "This same tension is also in evidence when we consider the application of our concept of factual knowledge. For in the simple case presented, it is neither clear that the man does know nor clear that he does not. […] The magnitude of the numbers involved may help to further our willingness to say that the man knows, to apply our concept of knowledge. But sheer consideration of number will not remove the tension entirely. Thus, were there a billion white cards, and only one of another color, we are more ready to say that the man who bets that the top card is white knows full well that he will win (assuming of course that he will win). Still, we may also find ourselves saying that he cannot really know that he has won until the color of the card is actually revealed." Option two is the winner here, since there can be no factual future knowledge, only possibilistic future knowledge. meaning that attempts to regard future knowledge of possibilities (and intermediate probabilities) as knowledge of facts (and maximum probabilities) are doomed to fail as already indicated or at least intuited by a host of other authors: […] S cannot know on the basis of the statistical information concerning the number of tickets, even though the probability that he loses may be greater on the basis of the latter evidence than on the basis of the former. (Cohen 1988, 106) ‘It’s probably a loser’, ‘It’s all but certain that it’s a loser’, or even, ‘It’s quite certain that it’s a loser’ seem quite alright to say, but, it seems, you are in no position to declare simply, ‘It’s a loser’. (DeRose 1996, 568; see p. 579 for two similar passages) While Jim might say “I knew I would lose,” after hearing the winner announced, I think many of us would view his words as not strictly speaking true. Perhaps what Jim really meant (or should have meant) was that he knew he would almost certainly lose. But had he known that he would lose, then he would not have bought the ticket. (Nelkin 2000, 376-77; also see p. 380 for the rejection of both 1 and 1*) It is not clear to me that lottery entrants know what they will be able to afford in the future or that people generally know where they will be during the coming summer. [...] My favorite examples involve discussions about upcoming sporting events. People assert things, and everyone knows that those making the assertions lack knowledge. (Feldman 2007, 212, 217) […] no one can know that one's lottery ticket will lose prior to learning that it in fact has lost. (Cohen 2008, 217) If Jane were to find out that Ben’s evidence for his assertion was merely that it is highly probably true, then even if it happens to be true that Jane’s ticket will lose, she is entitled to regard Ben as having spoken improperly. (McKinnon 2013, 4) What we consequently need to do is to introduce a second kind of context where one distinguishes between knowledge of facts and knowledge of possibilities, because these two cannot be lumped together. This also explains why previous contextualist advances towards solving the FKP (cf. Cohen 1988, 1998 or Baumann 2004) proved to be insufficient, because they only featured one kind of 'subjective' and 'epistemological' context (henceforth C1) in terms of infallible vs. (varying degrees of) fallible knowledge. Beyond that, however, one also needs a second kind of ´'objective' and 'ontological' context (henceforth C2) where the relation of knowledge (henceforth K) is given room for variation in terms of "of facts" vs. "of possibilities" (instead of no more than a by default "of facts" relation). Out of this we get not the usual two but four options: infallible K of facts, fallible K of facts, infallible K of possibilities, fallible K of possibilities, Due to infallible K hardly ever to never applying, we are effectively left with not four but two general options (fallible K of facts and fallible K of possibilities). with both "infallible K" and "of facts" associated with the maximum probability of 1 and both "fallible K" and "of possibilities" associated with intermediate probabilities of between 0 and 1. Cf. de Finetti 1969, 9, as quoted by Galavotti 1996, 256: “...all the assumptions of an inference ought to be interpreted as an overall assignment of initial probabilities.” This leaves us with the following general distinctions within the two contexts of knowledge: Contexts of Knowledge: C1 (subjective, epistemological) C2 (objective, ontological) Highest Standard (p = 1) infallible K ...of (past or present) facts Lower Standard(s) (0 < p < 1) fallible K In order to count as fallible K, the subjective degree of certainty of a belief or proposition must of course not only be between 0 and 1, but also relatively close to 1. ...of (future) possibilities As such, contextualism can no longer be regarded as pertaining only to a subjective C1; instead, we need a contextualism that, besides being founded on probabilism, also accounts for a (more) objective C2 where the ontological relation of K is accounted for. Hence two-context probabilism. 4. Dissolving the Future Knowledge Problem The FKP owed its existence mainly to three circumstances: (1) The ignorance of fallibilism, (2) the ignorance of (underlying) probabilism and (3) the ignorance of C2 and especially its possibilistic aspect. Doing away with those shortcomings, and especially making the switch from knowledge of facts to knowledge of possibilities in the lottery and similar possibilistic situations, will consequently dissolve the FKP. Since "[t]he acceptance of fallibilism in epistemology [has been] virtually universal" (Cohen 1988, 91) for quite some time now and rightly so – the alternative of infallible K would transform talk of knowledge into an absurd comedy where, whenever one makes a knowledge proposal like a Moorean "I know that 'Here there are hands,'" one would be forced to give the Wittgensteinian reply of "you don't know anything!" (1969, §406) –, I feel that there is not much to say about fallibilism right now since it is such an obvious choice and since others like Cohen or Reed have already largely said what needs to be said about fallibilism (my only addition would be the minor proposal to distinguish between "agent-fallibilism" and the consequently ensuing "knowledge-fallibilism"). Regarding the C2-based "of possibilities" dissolution of the FKP, especially DeRose (1991, 1996), Nelkin (2000) and Reed (2010) would seem to have come relatively close to discovering this, because at some point or other, all three of them at least hinted at the rejection of there only being knowledge of facts and the admission of knowledge of possibilities and the associated intermediate probabilities: […] these ‘epistemic possibilities,’ […] have something to do with knowledge […]. (DeRose 1991, 581) My probabilistic thoughts and statistical reasons don’t seem to rob me of assertability, or of knowledge. And why should they? […] Assertability and knowledge can survive an abundance of merely probabilistic thought. (DeRose 1996, 577-78) […] Jim's belief is based on mere probabilities. (Nelkin 2000, 391) (1**) Jim is rational in believing he will very probably lose. (ibid., 400) The same fact that makes (1*) false also makes (1) false. (ibid., 402) It is important to note that to claim fallibilistic knowledge is probabilistic is not to claim that it is knowledge of probabilities. One could, obviously, have probabilistic knowledge of non-probabilistic propositions and infallible knowledge of probabilistic propositions. (Reed 2010, 240, n. 30) However, despite correctly underlining the importance of probability in both the knowledge and the rationality versions of the lottery paradox, despite Nelkin mentioning that "facts" make for "false" lottery propositions, and even despite Reed already talking of "knowledge of probabilistic propositions," there has been (as far as I could determine) no explicit mention that, beyond C1, there also exists C2 and that intermediate probability (in terms of knowledge of possibilities) also needs to be taken into account in C2 and not just in C1. So despite these and other probabilistic advances generally going into the right direction, the above passages would by and large still only seem to do justice to C1 and to knowledge of facts For an exemplification of that mistake see e.g. Nelkin 2000, 389: "[…] there are non-lottery cases in which one's belief that p is based on a P-inference and in which we would deny that one has knowledge." Here "knowledge" should rather read "knowledge of facts," and there are also several other passages in pp. 388-90 where "knowledge" is automatically and unwittingly reduced to "knowledge of facts," resulting in the general denial of knowledge in the lottery situation even though that denial only applies to knowledge of facts and not to knowledge of possibilities. even though the lottery situation clearly demands that, in respect to C2, knowledge must be regarded as knowledge of possibilities in order for the concept of "knowledge of lottery propositions/the future" to make any sense. We can also see that mistake in the following account of propositions about future events: Suppose a husband and wife are discussing that night’s dinner. He asks, ‘‘Will you be home for dinner?’’ She responds, ‘‘I don’t know, but you can safely assume that I will.’’ He sensibly reasons, ‘‘She’ll be home. So I should get enough fish for two people.’’ You can insist that he does know or that his premise is ‘‘really’’ some probability claim. (Feldman 2007, 224) Feldman is correct in that the above is a claim about the probability of occurrence of a certain future event and not a claim about a fact – but why should that claim fail to be knowledge? Like so many other Humean skeptics about future knowledge, Feldman's phrasing of the last sentence once again suggests that an either fallible or infallible proposition about a possibility with an intermediate probability of occurrence cannot possibly be knowledge. But that either-or separation between knowledge and possibility/intermediate probability simply does not apply, neither for the first context (where such a denial would amount to the denial of universally agreed-on fallibilism) nor for the second context (where the denial of knowledge of possibilities/intermediate probabilities has caused the 'lottery' problem aka FKP). As such and as advocated many times by Igor Douven (2002, 2007, 2012) and especially by Maria Carla Galavotti (e.g. 1996, 2003, 2011), it would have been the wiser choice of action if philosophers had paid even more attention to earlier proposals of probabilism – such as the one made by Hume himself: "If we […] put trust in past experience, and make it the standard of our future judgment, these arguments must be probable only […]" (1777, sect. IV, part II, 36). Even more encompassingly, H. Jeffreys was convinced that "probability is ‘the most fundamental and general guiding principle of the whole of science’" (Galavotti 2003, 44, in quotation of Jeffreys 1931, 7). Or to take it from Dewey (1941, 172): "The position which I take, namely, that all knowledge […] involves a sceptical element, or what Peirce called 'fallibilism.' But it also provides for probability, and for determination of degrees of probability […]." Reichenbach (1951, 47) was of a similar opinion: "Ever since the early days of logical empiricism, Reichenbach held that ‘the concept of probability was fundamental to any theory of knowledge’" (Galavotti 2011, 96). And just in case that should not have been clear enough yet, please take a look at the first word in the title of Kyburg 1961 (yes: it is "Probability"). These and other philosophers and mathematicians would by and large appear to have been correct in heavily stressing the importance of probability in the conception of knowledge – subjective probability in C1 and (subjectively rendered) objective probability in C2 –, and if epistemologists had paid more attention to their works and proposals of probabilism, we might have been able to find out much earlier that probabilistic (and furthmore gradualistic) theories of knowledge are working a lot better than their alternatives or that probabilism has been at the foundation of both fallibilism or contextualism. Add in C2 and the "of possibilities" relation of (fallible) knowledge and replace the contradictory concept of "future knowledge of facts (and associated maximum probabilities)" with "future knowledge of possibilities (and associated intermediate probabilities)," and the 'lottery' problem aka FKP is history. 5. Conclusion Where philosophers went wrong in respect to the FKP and Humean skepticism about knowledge of future events should be clear by now: The in the end fairly simple fact of the matter would seem to be that almost the entire discussion of knowledge, contextualism and probabilism has by and large and effectively only occurred in respect to C1 since it was assumed that knowledge can only ever be knowledge of facts and the associated maximum probabilities of 1 and 0 (in case of negation) but never knowledge of possibilities with underlying intermediate probabilities between 1 and 0. A possible explanation for this omission might be that philosophers somehow intuited that it was not necessary to lower the probability value of C2 to <1 since one can also achieve any overall probability from 1 to 0 by leaving the value of C2 at 1 ("of facts") and by raising or lowering only the value of C1. For the lottery and other future events situations, where respective propositions cannot possibly contain knowledge of facts due to facts and respective factual truths being neither determined nor existing, this rendered knowledge claims impossible, thereby giving rise to Humean skepticism and the 'lottery' problem – a doubly inappropriate term given that the problem is neither restricted to lotteries nor, above all, any sort of problem as soon as one allows the concept of "(fallible) knowledge of possibilities and intermediate probabilities" to enter the picture and as soon as one stops making categorically mistaken and probabilistically inaccurate-to-false inductive inferences from possibilities and intermediate probabilities to facts and the maximum probability. This dissolution of the knowledge version of the lottery problem also includes the dissolution of the rationality version due to the rationality version being about justification and justification being a component of knowledge. As such, what the discussion of the lottery situation has ultimately shown is not that we cannot have any sort of knowledge of future events. The lesson to take home rather is that, based on the assumptions that knowledge requires truth and that those facts and factual truths are neither determined nor existing in the present, we cannot have any sort of factual knowledge of future events. This, however, still leaves us with possibilistic and probabilistic knowledge of future events and the realization that, beyond C1 (infallible vs. fallible knowledge), we also need to introduce C2 (of facts vs. of possibilities). It is that realization that ultimately amounts to the dissolution of the 'lottery' problem aka FKP as well as to the limitation of Humean skepticism to factual knowledge of future events, because possibilistic and probabilistic knowledge of future events is not and never was effected by this (we after all can fallibly know the probability of occurrence of possibility future events). Secondly and as shown in ch. 2, the inferences to lottery and similar future events propositions are not, as implied by Hume and as commonly assumed, inductive, because an inductive inflation only occurs if one makes the mistake(s) of inferring from a possibility/probability of <1 to a fact and thereby typically also to a probability of 1. Instead, the inference that should be used here is a non-inductive and accurate "P-inference" or simply a calculation of probabilities by way of multiplication: If, for instance, the two assumptions of a valid argument have the assigned or associated subjective probabilities of 1 and 0.99 (c.f. the probabilistic lottery situation example in ch. 2), then the conclusion has a subjective probability of (1 * 0.99 =) 0.99. If, however, the underlying associated subjective probabilities had been 0.7 and 0.99, then the conclusion has an associated probability of (0.7 * 0.99 =) 0.693. The major difficulty here of course lies in the correct or at least moderately accurate or plausible "assignment of initial probabilities" (de Finetti 1969, 9). By "assigned probabilities" I largely mean the same as "associated probabilities," except that the former are the so to speak "initially associated" ones. Another lesson to take home is that probabilities should not be based on facts; instead, and just like possibilities, facts should rather be put on a probabilistic foundation: [...] we don’t just happen to think probabilistic thoughts in the lottery case: it seems they are forced upon us. Perhaps all we think in the newspaper case is the non-statistical and non-probabilistic: The newspaper says it; so it is so. But why wouldn’t similar, non-probabilistic reasoning come off in the lottery case […]? (DeRose 1996, 577) "Probabilistic thoughts" are indeed "forced upon us" here and the obvious reason for this is that a successful interpretation of the lottery situation and respective knowledge requires a probabilistic and not a factual framework. This even applies to purported knowledge of or inferences to facts such as "The newspaper says it; so it is so" because it would once again be more correct to rephrase this as "The newspaper says it; so there is a possibility/probability that it is so." Facts, in other words, are based on underlying probabilities. As such, knowledge in its entirety can be discussed within a probabilistic framework (we can, after all, also associate infallible knowledge of facts with an overall probability of 1), meaning that we are left with the realization(s) of probabilists that "[…] human knowledge rests on probability judgments, not on certainties" or that "[…] judgments are the result of a context-dependent human activity, which qualifies as intrinsically probabilistic" (Galavotti 1996, 253, 254). Finally, and even though something like (by default fallibilistic) "two-context probabilism" is a relatively fitting term for the position presented in this paper, I would nevertheless like to draw the reader's attention to an even wider position that I will refer to as "gradualism" and which is, in essence, the both ontological (and metaphysical) and also epistemological position that things come in degrees or 'shades of gray' and not in terms of 'either all black (e.g. probability of 1) or all white (e.g. probability of 0),' and that, as a consequence of this, one must develop an according "gradualistic ontology" (henceforth GO) as well as an according "gradualistic epistemology" (henceforth GE) where those finely varying degrees no longer come under the wheels of prejudiced, crude and overall fallacious bivalent either-or conceptualization. For now, though, "two-context probabilism" will do. References Baumann, Peter. 2004. "Lotteries and Contexts." Erkenntnis 61(2-3): 415-28. 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