Special Issue on Epistemology || Decision Theoretic Epistemology

Decision Theoretic EpistemologyAuthor(s): Thomas D. Paxson, Jr.Source: Noûs, Vol. 14, No. 4, Special Issue on Epistemology (Nov., 1980), pp. 605-617Published by: WileyStable URL: http://www.jstor.org/stable/2215004 .

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Decision Theoretic Epistemology THOMAS D. PAXSON, JR.

SOUTHERN ILLINOIS UNIVERSI AT EDWARDSVILLE

In an effort to throw new light on the nature of epistemic justification, Professor Keith Lehrer has worked out, in recent years, a theory of justification based on one's subjective probability assignments ([2], [3], [4] and [5]). He eschews both foundations theories and appeals to explanatory coherence, relying instead on the corrected coherence of one's beliefs and subjective probability assignments when one's only goal is to believe as many truths as one can while believing the fewest falsehoods. This highly original approach deserves careful evaluation. By "epistemic justification" is meant, roughly, the justification required, but not sufficient, for knowledge. Economic, political, and moral considerations are not ger- mane. Lehrer has tried to specify conditions necessary and sufficient for a person to be epistemically justified in believing that something is the case. His work raises the question, can an adequate analysis of epistemic justification be worked out in terms of subjective probability and Bayesian Decision Theory? The attempt to answer this question requires not only that Professor Lehrer's own work be critically appraised (see [7] and [11]) but also that major alternative methods sharing the basic strategy of his program be explored. In this paper one of these alternatives is proposed.

SUBJECTIVE PROBABILITY AND EPISTEMIC JUSTIFICATION

There are many interpretations of subjective probability, but for the purposes of this paper, a person's subjective probabil- ity assignment to the state of affairs, h, should be construed as the person's belief as to the chances that h, rather than to her or his degree of belief. Nothing in the following depends critically on this interpretation, however. In the approach

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taken here, the link postulated between subjective probability and epistemic justification is supplied by Bayesian Decision Theory. Bayes employed probability and desirability matrices to show that the expected utility of an act is the sum of the products of the subjective probability assignments for each outcome of the act and the desirability of that outcome. This can be represented in the following formula, where 'E(A)' stands for "expected utility of A":

(1) E(A) = P(01, A) Des(01, A) ... + P(On, A) Des(On, A)

The rule is to compare the expected utilities of all one's options and then to choose the act which maximizes one's utility. To apply this to the epistemic context, we need to treat the utility of believing something in terms of one's beliefs regarding its chances of being true and the epistemic desira- bility of believing it when it is true and of believing it when it is false. A variety of analyses can be constructed on the basis of this schema which differ in relation to the method of determining desirabilities, probabilities, and relevant out- comes. Provisionally, let it be assumed that the only relevant outcomes with respect to our epistemic interests are two: that the belief be true and that the belief be false. Our epistemic rule will be that one should compare the expected utilities of all hypotheses and believe those that maximize one's epistemic utility. To bejustified, i.e. epistemicallyjustified, in believingh is, roughly, to believe h when so doing has greater expected epistemic utility than believing anything else with which h competes.

PROBABILITIES AFFECTING EXPECTED EPISTEMIC UTILITY

According to decision theory our actions should be guided by our beliefs as to the chances that the several outcomes of the various actions open to us will occur should we so act. Of course the closer our beliefs regarding chances are to the actual frequency with which acts of the sort are followed by these outcomes, the better will be our choices; but the believed chances, not the actual frequency distributions, are ingredi- ents in a decision. For this reason our epistemic decision theories employ subjective probability. It is not plausible to suppose that our subjective probability assignments are inde- pendent of one another, that with respect to each state of

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DECISION THEORETIC EPISTEMOLOGY 607

affairs entertained we pick out of the air or directly intuit a probability assignment. On the contrary, one comes to believe that the chances that Bill has left his office are 3/4, for example, in the basis of past experience, beliefs regarding Bill's habits, his current obligations and interests, etc. Fur- thermore most of one's probability assignments are not the consequence of any conscious inference. One simply finds that one has them. In such cases it is not at all obvious what beliefs, desires, anxieties, or experiences have led to a particular assessment of the chances that Bill has left his office. One way of interpreting the probability assignments affecting one's choices, then, is to think of them as absolute probabilities causally dependent in some way upon one's total evidence. The result is a holistic approach to the probability component of epistemic utility. Professor Lehrer has taken this approach.

Each of us, Lehrer argues can be thought of as having a system of beliefs, not just a set, because each of us has many beliefs regarding others of our beliefs. More controversially, he argues that this system of beliefs constitutes one's total evidence. One's own desires, anxieties, and the like are not as such evidence, and experience always underdetermines what one believes as a result of it. It is only when something is believed, he therefore contends, that it becomes evidence for a person ([4]: 187-8). In addition he holds that no perceptual beliefs are incorrigible; any belief is such that one may come to doubt it depending on one's other beliefs ([4]: 78-100). Justifi- cation is always a matter of the relations among one's beliefs, where 'justification" here refers not to the process of justify- ing to another person a belief one holds, but rather refers to the epistemic state of beingjustified in believing something. It is for this reason that Lehrer characterizes his view as a coher- ence theory. The system of our beliefs is, of course, not so tightly intraconnected that a change in any one probability assignment will cause all the others to change. Only some of my beliefs are germane to the question regarding Bill's loca- tion.

Professor Mark Pastin has suggested that starting with non-conditionalized subjective probability, as does Lehrer, one could employ a more empiricist probability function by isolating an initial privileged set of beliefs, e.g. perceptual beliefs and beliefs regarding past experiences ([8]: 143). Each member of this set would be given initial subjective probability assignments independently. "Final probabilities" for each

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member of the set would be assigned on the basis of this initial probability and its probability relative to the other proposi- tions in the set. Pastin has not, to my knowledge, published any systematic discussion of such a theory. For our purposes, Pastin's approach is important in suggesting how in broad outline the decision theoretic approach might be employed in the explication of a foundationalist theory of justification responsive to empiricist intuitions. The foundation set of jus- tified beliefs would, be a subset selected from one's set of perceptual beliefs and recollections on the basis of their initial probability and their probability relative to the larger set of perceptual beliefs. Other beliefs would be justified, presum- ably, on the basis of the probability assignments they have relative either to the foundation set, or to the larger set of perceptual beliefs. Initial probability assignments would be employed only with respect to the privileged set.

These two ways of determining the epistemically relevant subjective probability functions by no means exhaust the pos- sibilities. Nonetheless, they may suffice for our purposes since they complement one another so well. Lehrer developed a modified coherence theory of justification and Pastin's sug- gestions lead one to believe that a modified foundations theory could be constructed analogously. Appeal to subjective probability and Bayesian decision theory is open to both foundationalist and coherentist. Both must lay down restric- tions that eliminate epistemically irrelevant influences upon subjective probability assignments if these are to be used in an account of epistemic justification. Following Lehrer, let us stipulate that in the remainder the epistemically relevant sub- jective probability assignments of a person at a time are taken to be those that person would make then were he or she interested only in the acquisition of truth and the avoidance of error.1 In addition, the assignments must be probabilistically consistent ([4]: 202-205).

The remainder of the paper comprises two sections. In the first a modified foundationalist schema is developed, and in the second, its strengths and weaknesses are compared with those of its competitors.

A WEAK FOUNDATIONALIST SCHEMA

Epistemologists have frequently distinguished among our be- liefs two classes, the members of one class being dependent upon members of the other for their epistemic warrant. Even

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many coherence theorists aver that within an individual's sys- tem of beliefs some beliefs play a more central role than others. Foundationalists traditionally hold that those beliefs playing such a central role are (or ought to be) of a certain specifiable character. Distinguishing such epistemically priveleged beliefs and assigning them a different utility func- tion than governs one's other beliefs has interesting ramifica- tions.

Every cognitivist is persuaded that our interaction with what exists affects and informs our believings. This is not an epistemic principle, but it does guide our evaluations and choice of epistemic principles. The foundationalist typically identifies one or more types Qf belief which are thought to be such that there are especially intimate connections between

.their causal antecedents and their truth conditions. For example, it is often held that my being in pain necessarily causes my believing that I am in pain (provided that I have the necessary concepts); my being in pain on this view is "self- presenting." Or it is held that my believing that I am in pain can only be caused by my being in pain; this belief would then be incorrigible. A somewhat different but comparable stance is to hold that there are especially intimate connections between the causal antecedents of certain beliefs and their warrant conditions. It might be maintained, for example, that whatever causes my believing that there is a pencil in my hand that I see to be yellow and feel to be round, hard, and smooth, warrants my so believing. There is considerable territory between such traditional foundationalist principles, on the one hand. and classical coherentism according to which no beliefs are privileged, on the other. Guided by the conviction that our beliefs are in large measure affected by our interac- tion with what exists such that many accurately reflect the circumstances in which we live, yet persuaded that no signifi- cant number of our beliefs are incorrigible, let us consider the thesis that some sorts of beliefs have a special epistemic status which is nonetheless weaker than foundationalists tradi- tionally have thought.2

Criterial beliefs, as I shall call them, will be of two sorts. First, with respect to so-called occurrent, conscious, mental processes or states, all one's beliefs, to the effect that one has them now are criterial. This could include such beliefs as that I seem to remember giving a rose to Flora, that I am appeared to redly, that I fear the bandersnatch, etc. The second class contains all one's beliefs, with respect to simple relations

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among one's concepts, to the effect that they obtain. If for Sam Sharp to accept that there are two pairs of dice on the table is to accept that there are four dice on the table then the belief, that there being two pairs of dice on the table entails there being four dice on the table, is criterial for Sharp.3 Or gain, my concept of redness and my concept of color are such that I believe that if something is red than it is colored, and this is a criterial belief for me.

Criterial beliefs are to be given greater weight in the competition for epistemic utility than non-critical beliefs. This is ensured by assigning them their own utility function. I propose that Professor Lehrer's epistemic utility function be adapted for this purpose. Briefly, Lehrer suggests that the desirability of believing h, when h is false, may be taken as equivalent to the chance for truth given up in believing h instead of believing its strongest competitor, h* (see [4]: 206).4

(2) Des h, when h is F 1 - p(h*)

His explanation is that "In believing h to be true, we pass up the opportunity to believe any competitor of h, and hence our loss is equal to the greatest chance for truth we passed up in believing what we did" ([4]: 206). This desirability function punishes one for believing h while assigning a subjective probability to h* at least as great as one assigns to h, which is a state of affairs that will obtain if one holds inconsistent beliefs. The desirability of believingh whenh is true turns out to be the maximum desirability, 1, minus the probability of h's strongest competitor, Lehrer argues, because in believing h when h is true we lose the opportunity of acquiring the value of believing h when h is false ([4]: 206). He then constructs his epistemic utility function (4) from (3) by substitution ([4]: 206).

(3) e(h) = p(h) Des h when h is true + p(sh) Des h when h is false

(4) e(h) = {p(h)[1-p(h*)]} + {[1-p(h)][-p(h*)]} = p(h) - p (h*)

Let the expected epistemic utility of a possible criterial belief, h, for a person S, be the subjective probability S assigns h on C (or would assign h on C were S's desires only for the acquistion

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of truth and the avoidance of error)5 less the subjective probability (under the same conditions) S assigns h* on C, where C is the set of S's criterial beliefs excluding h and h*.

(5) e, (h) = p(h,C) - p(h*, C)

The proposal is to adapt Lehrer's (4) by employing it only as a measure of the epistemic utility of criterial beliefs and by using conditional probabilities rather than absolute probabilities. The latter change is a step toward Pastin's suggestion that the epistemic utility of foundational beliefs might be measured in terms of a weighted average of the proposition's absolute and conditional probabilities ([1]: 416). The weighting would be specified within a fully developed epistemological theory. On this approach, (5) would become (5'):

(5') ec(h) = (p(h) + Kp(h,d) - (p(h*) + Kp(h*, d)

2 2

In this formula K is a weighting function or constant and d the set of all one's other beliefs. I think that (5) captures much of what Pastin desired. Since the criterial beliefs are relatively independent, the probability of a criterial belief on the set of one's other criterial beliefs should not differ much from the absolute probability of that criterial belief. Perhaps more to the point, an adequate epistemological theory would have to specify the restrictions upon the subjective probabilities, p(h) and p(h, C), that would render them epistemologically signifi- cant. This crucial problem, needless to say, is not addressed in this paper. The utility function (5) and those to be discussed subsequently are schematic in the sense that they require epistemological theories that would specify the probability functions employed.

The choice ofp(h, C) rather than p(h, d) was guided in part by the greater expected independence between h and C as compared with h and d, as well as by a desire to capture more of the intuitions behind traditional foundationalism. The criterial beliefs within C derive their warrant, according to the schema proposed, from their absolute probabilities and their (probabilistic) relations to other criterial beliefs rather than to d, all one's other beliefs. On the other hand, the competitor, h*, need not itself be a criterial belief. This stipulation is to permit the possibility that a criterial belief be unwarranted because of other beliefs one holds. R. I. Sikora gives an

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example of a person who has a vivid afterimage of a figure that happens to be a dodecahedron and takes himself or herself to have an eleven-sided afterimage ([ 10]: 229). If such a person were provided with good evidence that the figure of which he or she has an afterimage is in fact a dodecahedron, it would be reasonable for that person to reject the criterial belief that he or she is experienceing an eleven-sided afterim- age.6 In other language, non-criterial beliefs may defeat the warrant possessed by criterial ones. Allowing h* to be non- criterial, then, will give the epistemic system a holistic char- acter even though the utility of criterial beliefs is determined by their probabilities conditional upon the person's criterial beliefs, i.e. by p(h, C) and p(h*, C).

Non-criterial beliefs are plausibily thought to derive some of their warrant from criterial beliefs. The approach taken here is, however, holistic and no presupposition is made to the effect that there must be linear chains of inference linking specific non-criterial beliefs with specific criterial beliefs in order for the non-criterial beliefs to be grounded. The proposal is that an epistemic utility function for non-criterial beliefs be used that requires them to be "grounded" on criterial beliefs. Such a utility function will penalize high absolute probabilities in favor of rewarding beliefs that have low absolute probabilities but high conditional probabilities. This calls for a different approach to measuring epistemic desirability. A leading candidate is the appeal to a notion of the content of a proposition. This, in turn, is related to notions of the evidential power one's beliefs have with respect to a particular non-criterial belief. One measure of such evidential power is found in the Kemeny-Oppenheim function for the measure of the evidential support that a body of evidence d provides for an hypothesis h. Professor Hintikka in an interesting 1967 article ([4]) interpreted the Kemeny- Oppenheim function as presupposing that the empirical content of a proposition is one minus its probability. If, then, we take (6) as giving the expected epistemic utility of believing h given belief that d, desirability now being understood in terms of content, then (6) will be equivalent to the simple function (7).

(6) p(h, d) cont (h) + p(-h, d) (-cont (-h))

(7) p(h, d) - p(h)

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DECISION THEORETIC EPISTEMOLOGY 613

Professor Hintikka proceeded to argue that the Kemeny and Oppenheirn formula (8) can be interpreted as a product of (7) and some normalizing function ([1]: 328-9).

(8) p(d, h) - p(d, -h)

p(d, h) + p(d, -h) This function is particularly well behaved, he observed, since it is independent of h, always changes in the same direction as (7), and varies between + 1 and -1. Adopting Hintikka's suggestion to our own concerns we might interpret epistemic utility as given not by (9)

(9) e(h, d) = p(h, d) cont (h) + p(-h, d) (-cont (h)) = p(h, d) -p(h)

but by the Kemeny-Oppenheim version of (9), namely (10),

(10) e(h, d) p(d, h) + p(d, -h)

where d is the conjunction of our background beliefs. How- ever, I propose to adapt (10) for non-criterial beliefs only and let the epistemic utility be measured by the product of (10) and the probability of h, given C, one's criterial beliefs. This yields the following epistemic utility function:

p(d, h) - p(d, -~h) (11) en1(h)

P (h, C) p(d, h) + p(d, -h)

It is obvious that h must be non-contradictory, for otherwise the expected utility of a non-criterial belief would be unde- fined. There is no epistemic utility in believing an inconsistent proposition, for in so doing one guarantees that one believes to be true something that is false. This function is epistemi- cally conservative in that it assigns low utilities to non-criterial beliefs that are probabilistically independent of one's other beliefs.

COMPARISON WITH ALTERNATIVES

The character and strengths of the proposed schema can be seen by comparing it with selected alternatives. One of these

(Clj e5hQ = p(d, h) - p(d, -h) (Cl) e(h - p(d h) + p(d, h)

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In effect, this is (10) used as the epistemic utility function for all beliefs. (Cl) is a coherentist schema that enjoins us to maximize the positive relevance among our beliefs; we are to accept those beliefs such that either their conditional prob- abilities exceed their absolute probabilities by more than the conditional probability of each competitor exceeds its absolute probability, or their conditional probabilities are 1. Thus if h1 has an absolute probability of 3/10 and a conditional probabil- ity of 8/10, while h2 has an absolute probability of 8/10 and a conditional probability of 9/10, we are enjoined to believe h1, since e(h ) = .81 but e(h2) = .38. This is counter-intuitive. (C1) does militate against rash dogmatism, but at too great a cost. What reason is there to think that hypothesis actually as- signed. high absolute subjective probabilities must be, or even for the most part are, false?

Although (C 1) must be rejected because it penalizes high absolute probability assignments less than one, this feature becomes an asset, when criterial beliefs are separated from the others and assigned a different utility function. For one thing, those beliefs that have been identified as criterial beliefs are just those that have traditionally been thought to warrant high initial absolute probabilities. The epistemic function govern- ing criterial beliefs (5) does not penalize high absolute or non-conditional probability assignments; indeed, it favors them. Thus the criticism against (C1), of which the proposed epistemic utility function for non-criterial beliefs (11) is a variant, is met by distinguishing a set of beliefs to whose members one may assign without penalty high absolute prob- abilities by means of (5). Furthermore, these criterial beliefs affect the epistemic utility of accepting non-criterial beliefs. (11) is conservative; it conserves the warrant of criterial be- liefs, extending it only to positively relevant propositions. Thus the two epistemic utility functions supplement one an- other in a useful way. It should be noted, moreover, that (1 1) does not penalize high absolute probabilities as much as does (10). For example. where p(d) = 6/10 then (10) assigns as epistemic utility of .308 where p(h) = 8/10 and p(h, d) = 9/10, while assigning a utility of .806 where p(h) = 3/10 and p(h, d) = 8/10. In comparison (12),

(12) en(h) = p(h, d) P(d, h) p(d h) p(d, h) + p(d, '-h)

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DECISION THEORETIC EPISTEMOLOGY 615

an approximation of (11), assigns a utility of .692 to the first case, p(h) + 8/10 and p(h, d) = 9/10, and a utility of .645 to the second case, p(h). = 3/10 and p(h, d) = 8. High absolute probabilities are punished by (12) but not nearly to the extent that they are by (10). Where h1, has an absolute probability of 3/10 and a conditional probability of 8/10, while h3 has an absolute probability of 5/10 and a conditional probability of 8/10 then (12) assigns belief in h1, an epistemic utility of .81 but assigns h3 an epistemic utility of only .6.

It would seem that there are ever so many possible schema that might serve. Several of these will be presented with the reasons for rejecting them. In the first place, there is the Popperian alternative to (8), namely (13) ([1]: 329).

(13) p(h, d) - p(h) p(h, d) + p(h)

This so severely penalizes high absolute probabilities that it assigns a higher epistemic utility to believing h4, where p(h4) =

5/10, p(h4, d) = 6/10, and p(d) = 6/10 than toh2, when p(h2, d) = 9/lOand p(d) = 6/10. This is unreasonable. Another possi- bility is to use (8) as a measure of desirability or content so that, when plugged into (6) we get either (14) or (15).

(1 4) p (h)p(d, h)-(p(d, -h) + {[1-p(h)] *[1- p (dh)-p(d,]-h)1 p(d, h)+ p(d, h) p(d, h)]+p(d, h)

(1 5) p(h) p(d, h) - p(d, + Ih) p) p (d, -h)- ~,h p(d, h) + p(d, -h) + -p(h)] p(d, -h) + p(d, h)

Unfortunately, neither of these will do since, given that p(d) = wherever d is negatively relevant to h, p(h) < 5/10 and p(h, d) < 1, then the utility of believingh assigned by (14) and (15) will turnout to be positive, and in the case of (14) greater than that of believing sh, while in the case of (15), where d is posi- tivelyrelevant to hand p(h, d) < 5, the utility of believingh will be negative. Both of these results are certainly odd. These difficulties are not removed by substituting conditional for absolute probabilities in (14) and (15) or by substituting a weighted average a la Pastin for the absolute probabilities in (14) and (15). For example, changing (15) to (16) or (17) does not help.

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(6) p(h, d) p(d, h) - p(d, --h) + [ p(d, -h) - p(d, h) p(d, h) + p(d, -h) p(d, -h) + p(d, h)

(17) p(h) + p(h, d) p(d, h) - p(d, Th) 2 p(d,h) + p(d, -h)

p(h) + p(h, d)], p(d, h) - p(d, h) 2 p(d, h) + p(d, h)

The functions (13) through (17) have anomalies, then, whether they are construed as schemas for coherence theories or as schemas within foundationalist schemas, and are thus unsuitable for our purposes here.

CONCLUDING REMARKS

The question remains, can an adequate analysis of epistemic justification be worked out in terms of subjective probability and Bayesian Decision Theory? The principal task remaining is the development of an epistemological theory that warrants the choice of one of these functions or one set of them as capturing the relations which obtain among one's beliefs when and only when one is fully warranted in holding them. Sub- sidiary tasks include discovering an unproblematic way of specifying subjective probabilities that are epistemically sig- nificant and discovering a way to ensure that the developed theory entails for each person at a single time that there is a unique set of beliefs a person is warranted in holding. This paper has been concerned to explore alternatives within the general framework presented by Keith Lehrer in which jus- tification is explicated in terms of the holistic coherence of one's subjective probability assignments. The two-tiered schema consisting of (5) and (1 1) resembles a Foundationist approach to justification in that members of a certain class of beliefs, criterial beliefs, are accorded priveleged status. It is not in accord with traditional foundations theories in that no criterial belief is, as such, incorrigible, indubitable, self- presenting, or in any other important way beyond suspicion. This two-tiered schema has the advantage of specifying the warrant relations between criterial and non-criterial beliefs. It is hoped that these reflections and schemata will stimulate

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further interest and work on this general approach to epis- temic justification.7

REFERENCES

[1] Jaakko Hintikka, "The Varieties of Information and Scientific Explanation" in Van Rootselaar and Staal, ed., Logic, Methodology and Philosophy of Science (Amsterdam: North Holland Publising, 1968): 311-30.

[2] Keith Lehrer, "Eviuence, Meaning and Conceptual Change: A Subjective Approach" in Pearce and Maynard, ed., Conceptual Change (Dordrecht- Holland: Reidel, 1973): 94-122.

[3] , "Induction and Conceptual Change," Synthese, 23(1971): 206-25. [4] , Knowledge (Oxford: Oxford University Press, 1974). [5] , "The Knowledge Cycle," Nous, 11(1977): 17-35. [6] Mark Pastin, "C. I. Lewis's Radical Foundationalism," Nous, 9(1975): 414-6. [7] , "Keith Lehrer's Knowledge," Nous 11 (1977): 431-6. [8] "Modest Foundationalism and Self-Warrant" in Nicholas Rescher, ed.,

Studies in Epistemology, APQ Monograph 9 (Oxford: Basil Blackwell, 1975): 141-9.

[9] R. I. Sikora, "Foundations Without Certainty," Canadian Journal of Philosophy, 8(1978): 227-45.

[10] Ernest Sosa, "Keith Lehrer: Knowledge," The Journal of Philosophy 73(1976): 812-21.

NOTES

'One technical problem with this stipulation is that beliefs regarding one's interests in acceptance, companionship, comfort, etc. would be affected were one to become interested only in the acquisition of truth and avoidance of error. This may turn out to be an important problem for the approach tojustification explored in this paper.

2As the referee for Nous noted, it is sufficient to recognize that some sorts of beliefs are more likely to be true, apart from any inferential support or counter- evidence. One need not insist on causal connections between that believed and one's believing it.

3This notion is derived from Chisholm. 4Lehrer suggests that a belief competes with all other beliefs negatively relevant

to it, where a proposition is negatively relevant to all those propositions whose probability is lower given its truth than otherwise. In. [4] he tried to define a limited range of beliefs "germane" to a given hypothesis to which competitors would be restricted. Unfortunately, there are technical problems with his proposal. Formulat- ing an adequate notion of competition is one of the major tasks facing this general approach to epistemic justification.

5The parenthetical condition, together with a consistency requirement, is used by Lehrer to purify the subjective probability assignments to render them epistemically relevant. These seem to be necessary but not sufficient conditions for the purpose.

6This example is not presented to demonstrate that criterial beliefs are corrigible, but rather to suggest that it is reasonable to adopt an epistemic system that allows the warrant for a criterial belief to be defeated. Taking oneself to be seeing an eleven-sided afterimage may not even be criterial (if one has to count the sides, for example.)

7I wish to thank the editors of Nous and the anonymous referee for helpful suggestions.

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