Do You Know Everything That You Know?

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  • Canadian Journal of Philosophy

    Do You Know Everything That You Know?Author(s): Steven R. LevySource: Canadian Journal of Philosophy, Vol. 9, No. 2 (Jun., 1979), pp. 315-322Published by: Canadian Journal of PhilosophyStable URL: http://www.jstor.org/stable/40231097 .Accessed: 09/06/2014 21:38

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  • CANADIAN JOURNAL OF PHILOSOPHY Volume IX, Number 2, June 1979

    Do You Know

    Everything That You Know?

    Steven R. Levy, , University of California, Riverside

    In the ongoing attempt to provide a satisfactory analysis of

    knowledge numerous conditions have been proposed as necessary and sufficient - the most noteworthy being justification, truth, and belief. In addition, various epistemic principles are frequently employed. In this paper I intend to show how the seemingly innocuous justification condition, along with two relatively uncontroversial epistemic principles, can give rise to a paradoxical situation.

    Let us begin by examining an interesting principle, which emerges from time to time and which has been most recently endorsed by D.M. Armstrong.1 The principle, which Armstrong calls the principle of the conjunctivity of knowledge, states that if a person, S, knows that p and knows that q, etc., then it is rational for S to believe the

    conjunction of p and q, etc.

    [1] (KSp & Ksq ...)- RBs(p & q ...)2

    1 In Belief, Truth and Knowledge (Cambridge, 1973), p. 186.

    2 Of course it may be rational for S to believe something that he does not, in fact, believe (perhaps he has never thought about it). So the satisfaction of

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  • Steven R. Levy

    This principle should not be confused with

    [2] (RBSp&RBsq ...) / RBs(p&q...)

    which gives rise to the now famous lottery paradox' and which is thus not widely accepted.3 The principle of the conjunctivity of knowledge [1] has a stronger antecedent condition than does [2]. While "mere" rational belief in each of a series of propositions does not necessarily carry over into a rational belief of the conjunction of those propositions, it is claimed that if one has knowledge of each then it is rational for him to believe their conjunction. It may be rational for me to believe, for each particular lottery ticket, that it will lose - but it is not rational for me to believe that all of the lottery tickets will lose (or even that all of a large subset will lose). But although it is rational for me to believe that any given lottery ticket will lose, I cannot properly be said to know that any one of them will lose. There is an initial plausibility in supposing that given a large subset of the lottery tickets issued, if I knew that each would be a loser, then it is perfectly rational for me to believe that they are all losers. Aprima facie case has thus been made for [1].

    Armstrong feels that the principle of the conjunctivity of knowledge is so secure that he rests a rather important argument on it (the conclusion of which is that we must demand absolute empirical reliability for our non-inferential knowledge). But far from being secure, it seems that [1 ] can be shown to be false by a relatively simple counterexample. The counterexample takes as a presupposition that absolute certainty on S's part is not a requirement for S's knowledge

    [3] (KSP A Csp)

    This presupposition is accepted by most who are currently interested in providing requirements for knowledge and I take a defense of it

    the antecedent of [1 ] does not entail that S believes that (p & q ...) - only that such a belief would be rational. Throughout this discussion we shall also make the usual additional assumptions required by epistemic principles such as [1]: (1) that S understands the logical entailment between the components of a conjunction and the conjunction itself, (2) that S correctly deduces the conjunction as a result of his understanding of the entailment, and (3) that S believes the conjunction as a result of (1) and (2). At certain points in the discussion we shall stop short of making the third assumption for reasons that will become obvious.

    3 See, e.g., H. E. Kyburg, Probability and the Logic of Rational Belief (Middletown,1961).

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  • Do You Know Everything That You Know?

    here to be unnecessary. Armstrong himself defends such a view.4 Nevertheless, in light of our final results, [3] will be one principle for which a reexamination may be necessary.

    Consider the following case : S is given a written examination. The examination consists of a very large number of short-answer

    questions (1,000; 100,000; ...). Suppose further that S knows the answers to all of the questions on the test. The questions may be like the following:

    1. In what year were you born?

    2. In what city were you born?

    3. How many planets are currently thought to comprise our solar system?

    4. Do you in fact know everything that you think you know?

    S answers the questions, '1946', 'Los Angeles', '9', 'No', ....The answer to question 4 will, of course, be 'no' because S is not

    absolutely certain of everything that he knows and he knows that somewhere in his vast collection of beliefs there is something that he thinks that he knows which is really false. Perhaps because of

    gerrymandered boundary lines he wasn't really born in Los Angeles or perhaps another planet was recently discovered and he hasn't yet heard the news. The examination given to S just turned out to be such that he did know the answers to all of the questions. If one of the

    questions had been about the color of S's car he would not have fared so well - he would have answered that it was red when, in fact, while he was taking the test vandals had painted it green (thus ensuring that S's answer to question 4 was correct).

    Is it rational for S to believe that he scored 100 percent on the examination? Given his answer to question 4 and given that the number of questions is sufficiently large, it seems quite obvious that it need not be rational for S to believe the conjunction of all of his answers (i.e., that he scored 100 percent). The rational belief would

    4 Op. cit., p. 139.

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  • Steven R. Levy

    be that he scored very well but that on at least one occasion, he had given an incorrect answer. Here we have a case in which S knows that p and S knows that q, etc., but in which it would not be rational for S to believe the conjunction of p and q etc.5

    Note that this example would not work if question 4 were:

    4'. Do you in fact know all of the answers to this test?

    For if S answered 'yes' and this answer were correct, then it is a much more difficult matter to determine whether or not it is rational for him to believe that he scored 100 percent. We shall consider this question shortly. Any other outcome with regard to question 4' (an incorrect 'yes' or a 'no', correct or incorrect) would fail to satisfy the antecedent of[1].

    Of course it is not necessary to the counterexample that question 4 actually be included in the examination. All that is necessary is that S recognize the fact that he is fallible. S's failure to recognize this would be wholly arrogant and dogmatic.6 But the hypothesis is stated more strongly than it need be stated. It is not necessary to the counterexample that S know that he is fallible (although I think that this is something that he can know) - all that is necessary is that it is rational for S to believe that he is fallible:

    [4] RBs(Fallibles)7

    Indeed, if asked S may even be able to give a good inductive argument for his own fallibility - time and time again he has found himself in error about matters in which he had complete confidence. S's evidence for his own fallibility may even be greater than his evidence for his own birth date. So if he knows his birth date, certainly he knows that he is fallible. If it is rational for him to believe that a certain date is his birth date, then it is rational for him to believe that he is fallible.

    5 A similar point was made by Alan H. Goldman, "A note on the conjunctivity of Knowledge/' Analysis (1975).

    6 Keith Lehrer makes a related observation in "When Rational Disagreement is Impossible/' Nous (1976).

    7 With the supposition stated in this way, the counterexample works when modified so as to omit question 4 from the examination but to allow that S rationally believes that he does not know everything that he thinks he knows.

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  • Do You Know Everything That You Know?

    We can now embrace the denial of [1] as being possible (and probably true for all of us)

    [5] [(Ksp & KSq ...) -RBs(p & q ...)]

    But this leads to a very surprising result. Almost all analyses of what it is for S to know that p include a requirement to the effect that S be justified in believing that p (at least for "non-basic" propositions). And there is no analysis of justification under which a person may be justified in believing a proposition that it is not rational for him to believe. Most use the phrases 'S is justified in believing that p' and 'it is rational for S to believe that p' interchangeably Thus, for any non basic proposition p, if S knows that p then it is rational for him to believe that p:

    [6] Ksp-^RBSP

    Now suppose that [5] is true of S. This means that S knows that p and S knows that q etc., but that it is not rational for S to believe the conjunction of p and q etc. However by [6] if it is not rational for S to believe the conjunction of p and q, etc., then S cannot properly be said to know the conjunction of p and q, etc. We are left with the paradoxical result that although S knows that p and S knows that q, etc., S does not know the conjunction of p and q, etc.

    [7] Ksp - Ksq...&~Ks(p&q...)

    S does not know everything that he knows ! Of course S does know everything that he knows when we consider each proposition individually. But when taken together it can no longer be correctly said that S has knowledge.

    So from the suppositions that knowledge does not entail certainty, that it can be rational for us to believe that we are fallible, and that knowledge entails rational belief, we arrive at the puzzling conclusion that we do not know everything that we know.

    There has been an attempt to modify the principle of the conjunctivity of knowledge [1], the denial of which gives rise to the above paradox. Goldman8 argues that only when S knows that he knows each of a set of propositions is it entailed that it is rational for

    8 Op.,cit.

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  • Steven R. Levy

    him to believe their conjunction. He schematizes his claim in this way:

    [8] Ks(Ksp & Ksq ...) - *-RBs(p & q ...)

    This is analogous to the case in which S correctly answers 'yes' to question 4' on the test. The denial of [8] conflicts with two other widely accepted epistemic principles: that it is rational for S to believe a logical consequence of a proposition that he knows,

    [9] Ksp & (p - q) - RBsq

    and that truth is a necessary condition of knowledge,

    [10] Ksp- p

    However we need not be too hurried in our acceptance of [9]. Suppose that S knows that p, that p entails q, and that S correctly infers q from p. We may, perhaps, be tempted to approve any belief formed by S that q, as a result of this process is a rational belief without inquiring further into the matter. But to do so would be a mistake. Suppose that S also has a firm belief that r (it makes no difference whether or not this belief is rational). Suppose further, that r entails not-q and that S correctly infers not-q from r. In such a situation it would hardly be rational of S to believe that q. Rather, S should be required to escape the dilemma either by further inquiry or by suspending any belief that q or that not-q. Thus, even though the antecedent of [9] is satisfied, given S's other beliefs it would not be rational for him to believe q. So [8] need not be accepted because of its logical relationship with [9] and [10]. Neither, however, should it be rejected due to the failure of [9].

    A consideration of [8] gives rise to some intriguing questions. Excluding a situation such as described above the conditional as a whole seems plausible. But how does the consequent emerge from the antecedent? My example of the examination demonstrates that the knowledge operator cannot be factored out of a conjunction. Can it be distributed into a conjunction ? Nothing that has been said so far precludes this possibility. Furthermore, it seems reasonable to suppose that it can. Suppose that on the examination described above the answer to question 1 is p and the answer to question 2 is q and so on. If we can accurately say of S that he knows that p and q, etc., then it is hard to imagine how we can avoid saying that he knows that p and he knows that q, etc., i.e. that he knows the answer to each question. It is not hard to imagine how S can know that he scored 100

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  • Do You Know Everything That You Know?

    percent on the test without knowing the answer to each question. Supposing that the test were a multiple choice one and that S had procured a copy of the grading key before taking the examination, it is easy to see that S can know that he scored 100 percent on the test without knowing (in some sense) the answer to each question. But this equivocates on what it means to know the answer to a question. Certainly if S knows that he scored 100 percent then he knew the answer to each question in the sense that he knew which choice would be counted as correct. However S may still know that he scored 100 percent but may have forgotten which answer went with which question. In this case we may say that S knows that he scored 100 percent but that he doesn't know (or no longer knows) the conjunction of p and q etc. This just shows that knowing that one scored 100 percent, is not equivalent to knowledge of the conjunction of the answers to the questions on the test. But it seems that if it is true that S does know the conjunction then S knows each of the conjuncts. Returning to principle [8] and distributing the knowledge operator of the antecedent yields

    [11] KsKsp&KsKsq ...

    It has already been shown that S's knowing p and S's knowing q, etc., doesn't entail that it is rational for him to believe their conjunction. Does the fact that he knows that he knows each of these entail that it is rational for him to believe their conjunction? Not so long as [3] and [4] are accepted. If we allow that knowledge does not entail certainty, then although S knows that he knows that p - he need not be certain that he knows that p. If the conjunction is long enough and S rationally believes that he is fallible, then it still might not be rational for S to believe that there is not at least one conjunct which, although he thinks that he knows that he knows it, he is mistaken . So for exactly the same reasons that it is not rational for S to believe everything that he knows, it is not rational for S to believe everything that he knows that he knows.

    In fact, I think that iterating the knowledge operator doesn't even entail a greater degree of certainty on S's part for the known proposition . What an iterated knowledge operator does is to indicate an etiologic difference in S's beliefs. That is, a person may come to know a fact in one way, perhaps by being told - but he comes to know that he knows in quite another, perhaps by psychological analysis, positive reinforcement, or divine revelation! At any rate, nesting knowledge operators as in [8] does not appear to be a promising way of strengthening principle [1], because this would require that before we can have a rational belief of everything that we

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  • Steven R. Levy

    know, we must first acquire (by whatever way one can determine whether or not he has knowledge) knowledge of our knowledge. This is an exercise independent of our original knowledge and is much too strong of a condition. Without such independent corroboration of our knowledge, principle [8] still allows that we do not know everything that we know.

    It is difficult to accept the conclusion that we do not know everything that we know because it seems that the statement that S knows that p and S knows that q etc., means nothing if not that S knows that p and q, etc. But the alternative to accepting this conclusion is to deny one of the three suppositions of the paradox. Most are unwilling to deny the supposition that knowledge does not necessarily entail certainty [3]. The denial of this supposition leads lickety-split to skepticism. And although most want to avoid skepticism, few are so brash as to deny that sometimes we are in error. So the second supposition, that it is rational to believe that we are fallible [4], seems to be on safe ground. That leaves the third - that we cannot have knowledge of a (non-basic) propopsition unless it is rational for us to believe it [6]. This principle - the justification principle - seems to me to be the most suspect. If justification (rationality of belief) were not made a necessary condition of knowledge, then the paradox would no longer exist.

    Causal theories seem to provide the most promising means of analyzing knowledge in such a way as perhaps to avoid a justification condition.9 For if we can properly specify which causal processes can lead to knowledge, the function currently served by the justification condition will have been met. But much more needs to be done.

    August 1977

    9 See, for example, Alvin Goldman, "Discrimination and Perceptual Knowledge," Journal of Philosophy (1976). Goldman's analysis there is on the right track although it suffers because it contains a condition similar to a defeasibility requirement. I have examined such requirements in depth in "Defeasibility Theories of Knowledge," Canadian Journal of Philosophy (1977), and in "Misleading Defeaters," The Journal of Philosophy (1978).

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    Article Contentsp. 315p. 316p. 317p. 318p. 319p. 320p. 321p. 322

    Issue Table of ContentsCanadian Journal of Philosophy, Vol. 9, No. 2 (Jun., 1979), pp. 195-372Front MatterHypocrisy [pp. 195-210]Four Conclusions about Violence of the Left [pp. 211-246]Threats and Coercion [pp. 247-259]Grammar and Understanding [pp. 261-281]On the Relationship between "A Priori" and Necessary Statements [pp. 283-287]Must a Cause Be Contingently Related to Its Effects? [pp. 289-298]Of Philosophers, Kings and Technocrats [pp. 299-314]Do You Know Everything That You Know? [pp. 315-322]Locations [pp. 323-333]Critical NoticeReview: untitled [pp. 335-350]Review: untitled [pp. 351-356]Review: untitled [pp. 357-372]