Induction: A Consistent Gamble

Induction: A Consistent GambleAuthor(s): Keith LehrerSource: Noûs, Vol. 3, No. 3 (Sep., 1969), pp. 285-297Published by: WileyStable URL: http://www.jstor.org/stable/2214552 .

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Induction: A Consistent Gamble

KEITH LEHRER

UNIVERSITY OF ROCHESTER

The purpose of this paper is to discuss an important recent work on inductive inference, Gambling with Truth*, by Isaac Levi. We owe Professor Levi a debt of gratitude for producing a book of such excellence. His own approach to inductive inference is not only original and profound, it also clarifies and transforms the work of his predecessors. In short, the book deserves to become a classic. The book formulates new problems for research, and I hope to make some contribution to the solution of these problems. The problem that will be my primary concern is the formulation of acceptance rules for inductive logic. I shall argue that the rule Professor Levi proposes requires modification and propose an alternative analysis based on his methods.

Professor Levi is concerned to provide criteria that are conditions of rational belief. He correctly maintains that we should not equate believing that something is true with acting as though it were true. A man might believe that something is true but be unwilling to act as though it were true, and he might act as though something were true but not believe it is true. However, the theory that we use for determining when behavior is rational can be modified for use in determining when belief is rational. Bayesian Decision Theory tells us that a specific action is rational when that action maximizes expected utility. Whether the action maximizes expected utility depends on the goals and objectives of the agent, on what utilities or values he assigns to various possible outcomes of his actions. Levi argues that cognitive inquiry has its own goals and objectives, distinct from those of practical action. He concludes

* Published by A. C. Knopf, New York, 1962, 241 pages.

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that it is rational to accept that hypothesis which maximizes expected utility when the utilities in question are those appropriate to cognitive inquiry.

What it is rational to believe depends on the evidence a man has. So questions of rational acceptance are always relative to total evidence, which Levi divides into new evidence e and back- ground information b. However, Levi argues that such questions are also relative to a set of answers to some problem formulated by the investigator. This set of answers Levi calls an ultimate partition. An ultimate partition is a set of hypotheses that are individually com- patible with the evidence and such that it follows from the evidence that exactly one of the hypotheses is true. That questions about what hypothesis it is rational to believe should be relativized to an ultimate partition is a consequence both of Levi's commitment to pragmatism and his attendant conviction that justification is a local affair arising in the context of some specific problem an investigator wishes to solve.

Though I do not share Levi's penchant for leaving things to pragmatism, I agree with him that the question of what it is rational to believe may be formulated relative to some ultimate partition. It will be useful to have before us an ultimate partition, Ue, to consider. A game of chance provides a good illustration. Suppose that e tells us that there are tickets in an urn, that each ticket is colored red, white, or blue, and that exactly one of the tickets will he drawn. R says that the ticket drawn will be red, W that it will be white, and B that it will be blue. The ultimate partition Ue contains these three hypotheses. From Ue we may generate the following set of hypotheses which Levi designates Me:

hi R h2 W h3 B h4 RorW h5 W or B h6 B or R h7 RorWorB h8 R and W and B

Which of these hypotheses is it reasonable to believe relative to the evidence and ultimate partition? Any reasonable person would believe h7 and reject h8 relative to e and Ue. For, h7 follows from e, and the denial of h8 follows from e. These are minimum condi-



INDUCrION: A CONSISTENT GAMBLE 287

tions of rationality, but there are others as well. For example, suppose that we believe hl, then we should also believe h4 and h6, as well as h7, because these hypotheses follow from hi. Moreover, if one believes both h4 and also believes h6, then one should believe hi, because hl follows from h4 and h6 in conjunction with the evidence e. In short, relative to a partition and total evidence, the set of hypotheses one believes should be consistent with the evidence and should include the deductive consequences of those beliefs taken in conjunction with the evidence. With one additional qualification to be formulated below, Levi proposes this as a principle of deductive cogency. It is a restriction upon rational acceptance.

According to Levi rational acceptance is also relative to the goals and objectives of cognitive inquiry. What are those goals and objectives? To understand Levi's answer, it is useful to consider two opposing schools of thought that have influenced Levi. According to Popper, the hypothesis to be accepted should be that one which has the most content or is the strongest of all those hypotheses consistent with the evidence. Such a hypothesis is most likely to be falsified, and Popper extolls this as a virtue. What is attractive about this proposal is that we hope to acquire beliefs that are informative, that give us as much information about the world as possible. The more content a statement has, the more information it gives us-if it is true! And there is the rub. For this policy, which brashly tells us to take all the content we can consistently get, is a policy that is almost sure to lead us to error instead of truth.

It is reasonable to hope to acquire true beliefs. This suggests the policy opposite to that recommended by Popper, to wit, that one believe what is most probably true. The difficulty with this program is that it will lead to a kind of inductive scepticism, because, as Levi proves, if we seek the truth and nothing but the truth, we shall believe nothing beyond the evidence and the deductive consequences thereof. To venture inductively beyond the evidence is to risk error. Hence we arrive at a dilemma. If we seek a maximum of content in what we believe and only that, then we shall believe what is almost certainly false, and if we seek a maximum of truth in what we believe and only that, we shall believe nothing beyond the evidence. Assuming that we seek a maximum one or the other, we wind up either fools or sceptics. Not a happy set of alternatives!

Obviously, we should seek in some way to attain both objectives, to obtain true and informative beliefs. Before I read Professor



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Levi's book, I did not think there was any way to achieve the desired result. The reason for my now vanquished doubt was the assumption that the more probable a hypothesis was the less content it had, and the more content it had the less probable it was. One might even define the probability of h on e as equal to the content of the denial of h on e. So, even if one decides to multiply the content times the probability and believe those hypotheses that have the highest product, one will believe those hypotheses that have a probability of 1/2 and no others. Hardly a happy compromise!

Professor Levi's proposal for obtaining a maximum of truth and content rests on the assumption that we have independent means of measuring content and probability. Probability may be determined in one of the standard ways, for example as a measure of subjective probability. How do we measure content? According to Levi, each of the members of the partition are to be assigned equal content. The choice of an ultimate partition is determined by a cognitive problem and, therefore, the partition constitutes a set of equally good solutions to that problem. Hence, relative to the problem, the members of the ultimate partition are equally informative. This is Levi's justification for assigning the members of the ultimate partition equal values, and, when there are n members, the content of each member is 1 - 1/n. Since there is no reason to assume that the probability of members of an ultimate partition on the evidence must equal 1/n, it becomes feasible to seek a rule of acceptance that aims at maximizing truth and content. The hypothesis accepted will then be the strongest hypothesis accepted via induction from the evidence.

With the aim of formulating such a rule, Levi first defines the utility of correctly accepting a hypothesis on the evidence, U(H,e), and the utility of erroneously accepting a hypothesis on the evidence, u(H,e), in terms of the content of the hypothesis, cont (H,e), and index of caution, q, as follows:

U(H,e) = (q + s) -qcont(-H,e), where (q + s) = 1 u(H,e) = -qcont(-H,e)

He then defines the expected utility of accepting a hypothesis on evidence, E(H,e), as follows:

E(H,e) = p(H,e)U(H,e) + p(-H,e)u(H,e).

This leads to the proof of the following equality:



INDUCTION: A CONSISTENT GAMBLE 289

E(H,e) = p(H,e) -qcont(-H,e).

He subsequently defines two notions of maximality as follows:

E(H,e) is maximal in Me if and only if, for every G in Me, E(G,e) <? E(H,e).

E(H,e) is strongly maximal in Me if and only if E(H,e) is maximal, and for every G other than H, such that E(G,e) is maximal in Me, cont(H,e) < cont(G,e).

Finally, Levi proves that the policy of accepting the strongly maximal element of Me as strongest via induction yields the following rule of acceptance:

Rule (A): (a) Accept b&e and all its deductive consequences.

(b) Reject all elements ai of Ue, such that p(a1,e) < qcont(-a1,e), i.e., accept the disjunction of all unrejected elements of Ue as the strongest element in Me accepted via induction from b&e.

(c) Conjoin the sentence accepted as strongest via induction according to (b) with the total evidence b&e and accept all deductive consequences.

(d) Do not accept (relative to b, e, Ue, the probability distribution, and q) any sentences other than these in your language.

Thus, we determine which hypothesis is to, be accepted as strongest via induction by rejecting those members of the ultimate partition whose probability on the evidence is less than q times the content of the denial of that member on the evidence. We then accept the disjunction of the remaining members. The value of q must be such that 0 < q < 1. By requiring that q be greater than 0 we avoid scepticism, and by requiring that q be no greater than 1 we escape inconsistency. Thus, rational belief and the principle of deductive cogency are relativized to an index of caution as well as the evidence and an ultimate partition.

In order to investigate the implications of rule (A), let us choose the least cautious value for q, namely, q = 1, and consider the result of applying the rule relative to the ultimate partition cited above. We let R. W., and B constitute the ultimate



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partition as before. Since the evidence told us that one ticket would be picked and did not tell us whether it would be red, white, or blue, p(R,e) = p(W,e) = p(B,e) = 1/3. Moreover, with q = 1, qcont(-R,e) = qcont(-W,e) = qcont(-B,e) = 1/3. Hence, according to rule (A), none of the members of the ultimate partition may be rejected, because they are equally probable on the evidence. Thus no conclusion may be accepted that is not deducible from the evidence. I find this result of applying rule (A) quite reasonable. I shall refer to this example as case one.

Now let us consider an alteration of the evidence that will permit genuine inductive inference by application of rule (A). Sup- pose that our evidence tells us that there are four red tickets, four white tickets, and six blue tickets and that exactly one ticket will be picked. Letting R, W, and B constitute the ultimate partition as before, we note that p(R,e) = 2/7, p(W,e) = 2/7, and p(B,e) = 3/7. Since cont(-R,e) = 1/3 = cont(-W,e) = cont(-B,e), and q = 1, rule (A) tells us to reject R and W and to accept B. Relative to a minimal degree of caution, these results seem reasonable enough. This is case two.

Though the foregoing results of applying rule (A) are reasonable, the rule also yields unsatisfactory results. Suppose we alter the evidence to make acceptance of hypothesis B more reasonable. We do this as follows. First we remove three red tickets, and then we add another blue ticket, so our evidence now tells us there is one red ticket, four white tickets, and seven blue tickets. With the change, p(R,e) = 1/12, p(W,e) = 1/3, and p(B,e) = 7/12. Notice that we have strengthened the argument for accepting B over case two. In the present case p(B,e) is greater than p(-B,e), while in case two p(-B,e) is greater than p(B,e). Moreover, p(B,e) - p(R,e) is greater in this case as is p(B,e) - p(W,e). Thus, not only is the probability of B greater in this case but the probabilistic superiority of B over R, W, and -B is also greater in this case than in case two. However, according to rule (A) we may not accept B in this case even though we retain the same ultimate partition and degree of caution. The reason is that, though we may reject R, we may not reject W because p(W,e) = qcont(-W,e) = 1/3. This result is ob- tained when q has the least cautious value. Thus, even when q is assigned the least cautious value, rule (A) will not permit the rejection of W. Therefore, rule (A) has the very unsatisfactory consequence of permitting the acceptance of B in case two and disallow- ing the acceptance of B in the present case, case three.




These results show that rule (A) is not an adequate means for attaining the objectives of truth and content. For, in case three B is more likely to be true than in case two and it is also more likely to be true in comparision with R. W, and its own denial. Moreover, B is a hypothesis of greater content than the hypothesis (B v W) which is accepted as strongest via induction in case three. Here probability of truth is ignored and greater content sacrificed as a result. The reason for this failure is the insensitivity of rule (A) to differences in probability between the members of the ultimate partition. Rule (A) tells us to reject a member of the ultimate partition if and only if the probability of the member of the evidence is less than some fixed number determined by the ultimate partition and index of caution. But a member whose probability does not fall beneath the fixed level of rejection may be so much less probable than another member of the partition that it should be rejected for that reason. In short, a member of an ultimate partition may be rejected simply because of its probablistic inferiority to some other member. In case three, we should reject W as well as R because of the probablistic superiority of B.

It is a strength of Levi's system that we may modify it to obtain the results we desire by simply defining different utilities. Levi has defined the utility of correctly accepting H on e in such a way that no member of the ultimate partition will have an expected utility that is strongly maximal-no matter how high the expected utility of that member-if there is any other member of the ultimate partition whose expected utility is positive-no matter how small the expected utility of that member might be. The reason is that the expected utility of disjunctions of members of the ultimate partition is equal to the sum of the expected utilities of the members disjoined. This shows that, for all Levi says to the contrary, he does not value high content very greatly. To see this more clearly, notice that if we assign q the value 1, the least cautious value, we obtain the following equality:

U(H,e) = cont(H,e).

Thus, when one values high content as greatly as Levi's system allows, then the value of correctly accepting a hypothesis is equated with the content of the hypothesis. But this does not indicate any great preference for accepting high content rather than low content hypotheses. If one greatly preferred correctly accepting hypothesis of high content, then one's utility function should reflect this



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preference. My preference for high content leads me to value correctly

accepting a disjunction of two members of an ultimate partition only half as much as correctly accepting a single member, a three membered disjunction only a third as much as a single member, and, more generally, an n-membered disjunction only an nth as much as a single member. Consequently, I propose the following utility function:

U*(H,e) = cont(H,e)/mH u*(H,e) = -cont(-He)/mH'

where mu equals the number of members of Ue that are disjuncts of H, and equals 1 when H is a member of Ue. (When H is inconsistent with e, then mH will be 0, thus leaving the utility of such a hypothesis undefined, and the hypothesis disregarded as it deserves to be.) From this utility function, we obtain the following equalities:

E(H,e) p(H,e) - cont(-H,e) mH

E(H,e) p(H,e) - cont(-H,e),

when H is a member of Ue, and k

E(H,e) = _ E(ai,e)

when H is a disjunction of members a,, . . . ak of Ue. These equalities lead to the result that if one member of Ue is

more probable than all the rest, then the expected utility of that member is maximal and strongly maximal. If two or more members of Ue have a probability that is the highest among such members, then the expected utility of all such members, as well as disjunctions of them, is maximal, and the disjunction of them all is strongly maximal. Therefore, the policy of accepting the strongly maximal hypothesis as strongest via induction from the evidence will, when we employ U* and u* as our utilities, yield the following modification of rule (A): conditions (a), (c), and (d) are the same as in rule (A) but condition (b) now tells us to reject any member, al, of Ue if there is another member, aj, of Ue such that p(ai,e) is less than p(aj,e), and to accept the disjunction of all unrejected members of Ue as strongest via induction from the evidence.

1 This utility function was suggested to me by Mr. Charles Cardwell.




The preceding rule gives us stronger conclusions and in- creased sensitivity to differences of probability among members of the ultimate partition. By employing this rule we obtain the desired result of accepting B in case three. However, the rule, as it stands, might be considered to be inflexibly adventurous, because it does not provide any way of adjusting one's level of caution. For the benefit of the epistemically wary, it is possible to alter the preceding rule by requiring that the strongly maximal hypothesis have a greater expected utility than any other hypothesis of the same content by at least some number c which serves as a "margin of caution". If we let c = r/mH, where mH is as above and r is assigned varying values, then the policy of accepting that hypothesis as strongest via induction which is strongly maximal by the required margin of caution yields the rule I wish to advocate.

The rule I propose is as follows:

Rule (R): Conditions (a), (c), and (d) are as in rule (A). For condition (b) substitute (bR). (bR) Accept as strongest via induction from e the member or, if more than one, the disjunction of all those members, aj, of Ue such that for any other member of Ue, aj, if p(ai,e) =, p(aj,e), then p(ai,e) - p(aj,e) is at least r.

The variable r restricted to nonnegative values is to serve the function of an index of caution for rule (R) as q does for rule (A). How- ever, r gets more cautious as it gets larger, in contrast to q where the reverse was true. When r is assigned the value 1, then rule (R) has the skeptical consequence of only permitting the acceptance of what may be deduced from the evidence. However, unlike q, there is no assignment of value to r that would permit the acceptance of inconsistent conclusions. Moreover, while the most cautious assignment of value to q would permit the acceptance of the disjunction of all the members of Ue as strongest via induction by condition (b) in rule (A), an equally cautious assignment of value to r may not permit the acceptance of such a disjunction as strongest via induction by condition (bR) in rule (R). But this difference is unimportant, because condition (a) of both rules insures the acceptance of such a disjunction as a deductive consequence of the evidence.

The more fundamental differences between rule (A) and rule (R) are the following. Whatever hypothesis we accept by employing rule (A) and assigning some value to q, we may accept that hypo-



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thesis by employing rule (R) and assigning some value to r. But there are hypotheses we may accept by employing rule (R) and assigning a value to r that we cannot accept by employing rule (A) no matter what value we assign to q. The hypothesis B in case three is an example of such a hypothesis. Rule (R) permits the acceptance of a hypothesis, a member of an ultimate partition for example, because of its probabilistic superiority to other hypotheses of the same content, when rule (A) prohibits such acceptance. Thus, rule (R) allows us to accept stronger conclusions via induction than does rule (A) when there is some probabilistic justification for such acceptance. So, we may obtain conclusions of greater content with rule (R). Moreover, this result cannot possibly constitute any objection to rule (R), for example the objection that rule (R) shows an unreasonable preference for content as opposed to truth. For, by increasing the value of r we can increase our preference for truth as greatly as we please. Thus, rule (R) is a more reasonable principle of inductive inference than rule (A) for obtaining the twin objectives of truth and content.

The foregoing objection and proposal is entirely within the spirit of Levi's own system. However, I now wish to turn to a much more fundamental disagreement. Professor Levi has argued that acceptance is relative to a choice of ultimate partition and index of caution as well as to the evidence. Once a partition and degree of caution are chosen, then, having suitably restricted the values of the index of caution, it follows from both rule (A) and rule (R) that the conclusions accepted will be logically consistent with the evidence. Although the conclusions accepted on the basis of one ultimate partition and degree of evidence are consistent, the conclusions accepted on the basis of the same evidence but different ultimate partitions may be inconsistent with each other and the evidence. Now suppose an investigator chooses a number of ultimate partitions at one time and on the basis of the same evidence accepts a set of conclusions that are logically inconsistent with each other and the evidence. Should such a choice of ultimate partitions be prohibited?

Professor Levi is somewhat ambiguous on this matter, but I am inclined to think that his answer is that such results need not be prohibited. For example, suppose we have an urn with one million balls all but one of which are black. If we make a billion trials of drawing a ball from the urn, replacing the drawn ball after each draw, then, using one set of ultimate partitions, we could accept the




conclusion that each of the draws is a black ball, but, choosing another ultimate partition, we accept the conclusion that at least one of the draws will not be black. Our short run conclusions about the outcome of each draw contradict our long run conclusion about the outcome of a billion draws. In this case, Levi remarks,

... in the case of predictions concerning sampling from the urn, an investigator might very well reach conclusions with regard to the short and the long run, that could not, by being taken together, form a deductively consistent and closed body of sentences.

Levi says such cases preclude specification of a unique ultimate partition; but that is not the issue. He also says that such inconsistent conclusions are reached from different points of view, but that is not the issue either. The issue is whether a system of rational acceptance that takes truth as one of its objectives should permit the acceptance of a set of conclusions that, as a mere matter of logic, cannot possibly be true. And to this issue Levi says, "Rule (A) also admits the possibility that at certain times a person may rationally accept as true a set of sentences that is not consistent and closed," and later concludes "there seems to be no hope of elimi- nating the relativization of deductive cogency requirements to the choice of ultimate partitions." In short, according to Levi, it is not irrational to detach or choose a set of partitions with the result that the set of sentences we accept is logically inconsistent.

The idea that an inconsistent set of beliefs might be perfectly rational is truly staggering. It is difficult to argue against a philos- opher who maintains such a position, because one cannot hope to refute his arguments by proving that they lead to an inconsistent set of conclusions. For, it is open to him to reply that, though his conclusions are inconsistent, they are nevertheless quite reasonable. And what does one say then? Nevertheless, I shall hazard an argument against Levi's position which I consider to be decisive.

I assume that if it is reasonable for a man to accept a hypothesis then it is reasonable for him to accept any other hypothesis he knows to be logically equivalent to the first. Now if this is true of individual hypotheses, then it must also be true of sets of hypotheses. Thus, if it is reasonable for a man to accept any set of hypotheses, then it is also reasonable for him to accept any other set of hypotheses he knows to be logically equivalent to the first. Two hypotheses are logically equivalent if and only if they have



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exactly the same deductive consequences. Similarly, two sets of hypotheses are logically equivalent if and only if they have exactly the same deductive consequences. I also assume that there is at least one set of hypotheses that is not reasonable to accept, for example, the set containing as its sole members "The urn contains a black ball" and its contradictory "It is not the case that the urn contains a black ball". However, all inconsistent sets of hypotheses are logically equivalent. Therefore, any man who knows his logic and knows that the set of hypotheses he accepts is inconsistent also knows that the set he accepts is logically equivalent to that two membered set of contradictories. Hence, it is reasonable for him to accept an inconsistent set of hypotheses if and only if it is reasonable for him to accept that two membered inconsistent set. Obvi- ously, it would never be reasonable to accept the latter. Therefore, it would never be reasonable for a man to accept a set of hypotheses he knows to be inconsistent and logically equivalent to any other logically inconsistent set.

I conclude that any adequate system of rational acceptance will prohibit the acceptance of inconsistent sets. Moreover, though my disagreement with Levi on this point is fundamental, his position on this issue is quite inessential to his system. It is not at all difficult to impose a restriction on the application of Levi's system that will avoid the acceptance of inconsistent sets. The restriction is placed on the choice of ultimate partitions and may be formulated as follows:

CC. The set of ultimate partitions chosen for the acceptance of hypotheses relative to b&e must be so chosen that the resultant set of hypotheses accepted relative to b&e and the set of ultimate partitions is logically consistent.

This condition, once imposed, not only avoids the acceptance of inconsistent sets, it also limits the choice of ultimate partitions and thereby offers some guidance concerning that choice. Though I am convinced of the necessity and justification for relativizing acceptance to ultimate partitions, Levi offers convincing arguments in favor of this, there is need for greater clarification of the conditions restricting the choice of such partitions. Since the system has the attainment of truth as one objective, weare certainly justified in restricting the choice of ultimate partitions so as to avoid the cer- tainty of error. Condition CC is the weakest restriction that will achieve this purpose.




In the foregoing discussion, I have restricted my remarks to the problem of formulating rules of acceptance. However, there is a great deal of interest in the book besides these basic matters. Some of the moist interesting chapters in the book are those that examine the implications of such rules. The discussions of probability, generalization, and various forms of inference are brilliant and enlightening. Indeed, the problems and methods elaborated by Professor Levi in his book serve as a new foundation for the study of inductive inference.



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Induction: A Consistent Gamble