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Theory of Knowledge

The Significance and Priority of Evidential Basis
in Epistemic Justification

Zekiye Kutlusoy
Gazi University

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ABSTRACT:There are various approaches to epistemology as well as to the philosophy of science. The attempt to naturalize them is the newest approach. In the naturalistic framework, epistemology turns out to be identical with the philosophy of science. The main characteristic of both naturalized epistemology and naturalized philosophy of science is their methodological monism. Therefore, both of these meta-level areas of philosophy pursue only one scientific discipline to be a meta-method for themselves. There are objections to naturalism on the basis that (from a methodological point of view) naturalized philosophy is monistic.

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I. The Concept of (Epistemic) Justification

Today's Epistemology. Epistemology (theory/philosophy of knowledge) is defined in general as the branch of philosophy, concerning the nature, possibility, source, scope and limits, criteria/standards (of truth) of knowledge. Today's epistemology, however, has been interested basically in the notion of epistemic justification, since the publication of Edmund L. Gettier's article "Is Justified True Belief Knowledge?" in 1963. In this article Gettier introduces two counter-examples to the traditional tripartite account of knowledge, i.e. the standard analysis of knowledge: knowledge as justified true belief, and shows that these three conditions, the truth condition, the belief condition, and the justification condition, are not sufficient even if they are necessary, and thereby they could not give a proper definition of knowing. After Gettier's article, epistemologists have seen the justification condition as the most problematic one among these three conditions. So, within the last thirty years the justification of —giving an account (reasons/grounds) for — the belief/believing has attracted considerable attention. Now, the justification component of the classical definition of knowledge takes place at the center of recent epistemology. Consequently, various epistemic justification theories as contemporary theories of factual knowledge have been produced with respect to some different classifications of these theories — e.g. internalist theories (foundationalist and coherentist theories), and externalist ones (probabilist and reliabilist theories): While in the former ones justification of belief is internal, in the latter ones, external to the believer's mind/mental life. That is, internal justification of the belief is possible by reflection upon the believer's own conscious state of mind, i.e. upon her internal, doxastic state. In short, the justifiability of her belief is determined by what beliefs she has, and her direct cognitive access to the justification for her belief is required. On the other hand, in externalist justification some important feature of the belief justification is outside the mind of the believer. The externally justifiability of a belief may be a function of the reliability of a belief-forming process, which causes that belief to be a true one, or it may be a function of the probability of the belief. Such factors as reliability, and probability, which play justificatory role, are external, i.e. non-doxastic factors to the believer. She does not have direct access to them, while her beliefs, as being her internal, doxastic states, are directly accessible for her.

Now, context of justification has a complicated structure, and in my opinion one effective way of inquiring about the concept of justification could be to investigate it in a definite, problematic case of justification; for instance, in trying to solve a paradox of justification one could understand the notion of justification better. Therefore, as a contemporary paradox of justification the lottery paradox, which is discussed in various contexts, such as induction, defeasible reasoning, a Bayesian theory of rational decision-making, confirmation, rationality of inconsistent sets of beliefs, and default reasoning/artificial intelligence, would well help us for this purpose. What is more, by drawing our attention to the significance of the notion of epistemically evidential basis in justification, more clearly, to that the case under consideration should be an epistemic one, it would happen to have a great influence upon the mass of our epistemological intuitions.

The Lottery Paradox. The lottery paradox is originated by H. E. Kyburg in [1961, p. 197]. The background information b about the lottery under consideration here is the following: It is a fair lottery with thousand tickets, in which only one ticket will be the winner of the lottery prize after a random drawing. In this lottery case, subject S justifiably believes that ticket i will lose the lottery, for each i=1, ..., 1000. Then, symbolizing "Ticket i will lose the lottery." as Pi, and "S is justified in believing that Pi." As JBsPi, thousand individual justifications, by means of purely probabilistic justification rule PPJR that is a general justification rule, on the basis of purely probabilistic piece of justification evidence e1 that is probabilistically derived from b, are: JBsP1, JBsP2, ..., JBsP1000. These e1 and PPJR are as follows. e1 : "i(Prob(Pi, b)=.999), i=1, ..., 1000, and PPJR : Prob(P, I) >.9 (r) JBsP. For any empirical proposition P, which states some empirical fact belonging to some empirical case, and for any background information I about that case, PPJR states the justification threshold as 0.9. When this rule is applied to the lottery case on the basis of piece of evidence e1, individual justifications in the lottery case come to occurrence via modus ponens rule of logic (MP) at the end of the following steps — also the reason for each step is within parentheses on the right hand side:

1.Prob(P1, b)=.999 & Prob(P2, b)=.999 & ... & Prob(P1000, b)=.999
2. Prob(P1, b) >.9 & Prob(P2, b) >.9 & ... & Prob (P1000, b) >.9
3.[(Prob(P1, b) >.9) (r) JBsP1] & ... & [(Prob(P1000, b) >.9) (r) JBsP1000]

4. JBsP1 & JBsP2 & ... & JBsP1000

(2, 3, MP)


After this step, S is justified in believing this time that "iPi, i=1, ..., 1000, on the basis of these individual justifications, i.e. "iJBsPi (®) JBs"iPi, by means of the following conjunction principle CP as a principle of justification. For any empirical proposition Q like P; CP : (JBsP & JBsQ) (®) JBs(P & Q). Then application of this principle to the lottery case in the continuation of the above steps is as follows:

5. (JBsP1 & JBsP2 & ... & JBsP1000) (®) JBs(P1 & P2 & ... & P1000)
6. JBs(P1 & P2 & ... & P1000)

(4, 5, MP)


Furthermore, S is also justified in believing that either ticket 1 will win the lottery or ticket 2 will win the lottery or ... or ticket 1000 will win the lottery, on the grounds of the following piece of evidence e2 derived from b. e2 : $!i~Pi, i=1, ..., 1000 (º ~P1¬ô®~P2¬ô® ... ~P1000). Here, $!i~Pi: "There exists exactly one ticket that will win the lottery." (Since b states that exactly/only one ticket, not some tickets or at least one ticket, will absolutely win the lottery, this piece of evidence would support exclusive disjunctive proposition either ~P1 or ~P2 or ... or ~P1000.) Hence, in going on to analyze the paradoxical situation step by step:

7. $!i~Pi, i=1, ..., 1000
8. JBs$!i~Pi i=1, ..., 1000
9. $!i~Pi (®) $!i~Pi i=1, ...,1000
(logic rule: the exclusive disjunction implies the inclusive one)
10. JBs($!i~Pi (®) $!i~Pi) i=1, ..., 1000
(9, epistemic rule-ER: S is justified in believing a logical rule)


Now, another required justification principle, the principle of closure in justification, which is widely accepted by epistemologists, is the following one. PC : [JBsP & JBs(P (®) Q)] (®) JBsQ. By applying it to the related case:

11.[JBs$!i~Pi & JBs($!i~Pi (® ) $!i~Pi)] (®) JBs($!i~Pi) i=1, ..., 1000
12. JBs($!i~Pi) i=1, ..., 1000
(8, 10, 11, MP)
13. JBs(~"iPi) i=1, ..., 1000
(12, by logic: $i~Pi º~"iPi)
14.[JBs(~"iPi) & JBs(~"iPi)] (®) JBs(~"iPi & ~"iPi) i=1, ..., 1000
15. JBs(~"iPi & ~"iPi ) i=1, ..., 1000
(6, 13, 14, MP)


This is the paradoxical end of the analytical formulation of the lottery paradox.

II. Analytical Epistemology.

Elements of (Epistemic) Justification. As the above analysis of the lottery paradox in fifteen steps illuminates, there are different justification processes of this paradoxical case. And also, analyzing these processes it is seen that there are different justifying elements in them. Although these elements undertake significantly functional roles in the above justification procedures, two different sorts of them are distinguished: general and specific elements of justification. While general elements are not peculiar to any distinct case, specific ones, as belonging or relating only to any specific case under consideration, are relative to that case only, and thereby, state the information about and the clues to the related situation. Now, the general elements of justification are rules, principles, presumptions, and the like that are applicable to any case of justification. The specific ones, however, are pieces of evidence that only belong to a concrete case under consideration so that they can show whether that case is really an epistemic one. Furthermore, it is clear that a justification is realized by means of a justification rule/principle, and, on the basis of some justifying evidence. Therefore, the above distinction between elements appears now as the distinction between justification by means of and justification on the basis of respectively. In the lottery paradox, general justifying elements, by means of which justifications are made, are PPJR, CP, ER, and PC that are applied to the lottery case. MP also works through the whole paradoxical process, as if it were a justification rule; yet exhibits it, as a rule of logic, only some logical relationship between propositions. On the other hand, specific justification elements in the lottery paradox, on the basis of which justifications are made, are pieces of evidence that belong only to this distinct lottery case, i.e. e1and e2. Such pieces of evidence, as being derived from b, reflect certain characteristics of this particular case of the lottery, thereby inform us about the case under consideration here whether it is an epistemic one.

Analysis of Justifying Elements of the Lottery Paradox. Since the lottery paradox is a paradox of justification, in order to understand and to explain it, it is clear that one should investigate all of the elements, which enter the procedures/processes of justification in it, namely PPJR, CP, ER, PC, and pieces of evidence, e1 and e2. It is seen that epistemologists do not distinguish the exclusive disjunction from the inclusive one; for this reason, they automatically cancel the part between lines 7-11 in the above analysis, and thereby they do not concern ER and PC. Now, ER is a typical sample of epistemological rule, expressing a justification of believing a rule of logic. On the other hand, PC states (under ER) that any direct, logical consequence of a justified belief is also justified. J. Dancy formalizes this convincing principle as the principle of closure in justification: It is a closure principle under justifiably believed entailment, because, it states that something implied in justification (Q) by some other thing justifiably believed (P) remains still in the closed area of justification. It is quite plain here that when something in subject S's experience is evidence for justification of S's believing that P, it becomes evidence for justification of S's believing any consequence Q of P as well. (1) Now, we see in the literature that PC is widely accepted by epistemologists. On the other hand, CP, which is the strongest form of PC, is rejected by some of them.

Generally in their responses epistemologists have rejected CP or PPJR to avoid the lottery paradox. (2) According to them, PPJR states the notion of justification only in terms of high probability, and together with CP, leads to an open (direct) contradiction. Thus, the following lottery-paradox-set, LPS, of subject S's justified beliefs in the lottery paradox contains both a proposition and its negation:

LPS : {P1, P2, ..., P1000, P1 & P2 & ... & P1000, ~(P1 & P2 & ... & P1000)}.

However, I want to show that it would well be possible to solve the lottery paradox by rejecting none of CP and PPJR. Now, by rejection of CP the 1001st member of LPS is eliminated, and the open contradiction vanishes. Unfortunately the new set is still logically inconsistent. As one knows, from an inconsistent set any proposition, whatever it is, can be inferred so that all possible beliefs, any belief and its negation, are justified. The proponents of the idea of solving the paradox by rejecting CP avoid this difficulty by not allowing inferences from two or more members of the set, and by allowing inferences only from single members. In this way, contradictory beliefs could not be inferred from the set since each member of the set is consistent when taken alone. (As a matter of fact, they accept PC but do not accept CP. In short, they weaken PC.) However, such a drastic restriction of logical inference to only single members of the set is counterintuitive and thus intolerable. Indeed, it conflicts with our usual practices.

As to the alternative solution, i.e. rejection of PPJR, (justified) individual beliefs and the conjunctive belief of them would be excluded from set LPS, thus only its last member would be kept in the new set. Thereby, any of the (in/direct) contradiction, namely all kinds of inconsistency in LPS, would have been eliminated. Furthermore, the unrestricted right of inference would also be preserved here. So, from a formal point of view, this solution would be a full solution to the lottery paradox. Now, it is certain that any epistemic case should allow a coherent set of individually justified beliefs in the subject's belief system. That is, an individual justification of any belief must be coherent with individual justifications of the other beliefs of the subject's system of beliefs. However, in the lottery case each individual justification, say of Pi, is completely overridden by jointly individual justifications of the remaining 999 belief-propositions, P1, ..., Pi-1, Pi+1, ..., P1000. Therefore, it is quite clear that they are not acceptable. Yet, at this point, the question that should be asked is the following: In order to be able to cancel these individual justifications, why does one need to reject PPJR? Such a probabilistic model of justification with a very high degree of probability/justification, in a correct context naturally, could well be valid. Then, in such a justification model (the truth of) S's belief would be capable of being supported on the basis of some relevant probabilistic evidence.

At this stage, there might be an objection to accepting both PPJR and CP. It could be seen that in an epistemic context they did not compromise with each other, because the individual justifications on the grounds of the high probabilities of belief-propositions/events were overridden by the low probability of the conjunction of them. It is true indeed that the probability value of their conjunction is always less than their individual probabilities; it might be even less than the required justification degree. But one should consider here that, while the individual justifications are made by means of PPJR, the justification of the conjunction is made by means of CP. Both of this rule and this principle are some ways of justification. And, it is not necessary for the conjunction to be justified by means of PPJR as well. Therefore, subject S's beliefs in her system of beliefs could well be justified by various ways of justification, probabilistic or non-probabilistic. And, one justified belief in her belief system does not need to be justified once more by another rule/principle which justifies another belief in the system.

This could be seen best in responding to the paradox of the preface which was viewed as a relative of the lottery paradox. (3) There defending the non-paradoxical preface case, in contrast to the lottery case, as a case of epistemic justification would essentially mean to defend CP that was a very intuitive and sound principle of epistemic justification: It is possible for all the individual justifications of the preface case to be epistemic; that is, it is possible for all the S's (belief-)statements in her book to be justified epistemically. And, S is conscious of all of these justifications, i.e. that all justifying pieces of evidence and rules/principles are epistemic, because S is an author, probably an academician, writing an academic book. Now, during these individual justifications, each related piece of evidence and rule/principle, realizing the related justification, more clearly, each one of these justifications, as a way of justification, in relation to the related statement, are added to S's intellectual corpus one by one. The collection of these pieces of evidence and rules/principles makes up the collection of these justification ways. Therefore, it seems very plausible that, on the basis of and by means of this collection, S is justified in also believing (the conjunction of) the statements in her book jointly all together.

Then, one would propose a real solution to the lottery paradox by refuting individual justifications, moreover without rejecting PPJR. As stated above, subject S's individual beliefs are justified both by means of PPJR and on the basis of e1. Hence, if only e1, more clearly b deriving e1, is ruled out from the epistemological framework, namely, application of PPJR to the lottery case is given up, individual justifications would have been eliminated without rejecting PPJR, That is, rejection only of the lottery case itself, which is a non-epistemic one, would be sufficient. In fact, the paradoxical situation here arises from the lottery case itself, because this case, which is pictured by b, contains some incoherency in itself. In what follows this is clarified by means of a new piece of evidence derived from b.

Solution to the Lottery Paradox: Refutation of the Individual Justifications by Refuting the Lottery Case as an Epistemic One. In order to see where the problem lies in this paradox, one should approach to the chain of justifications through the paradox more closely. As it is seen, the paradoxical result in the above fifteenth line expresses the justification of S's two conjunct-contradictory beliefs (jointly). In connection with this, the whole process of the lottery paradox above could be divided into three main parts: The first part between lines 1-6 provides the justification process of one of these contradictory beliefs; the second part between lines 7-13 yields the justification process of the other contradictory belief; and the third/last one between lines 14-15 concludes the justification of their conjunction. Now, while justifications in the first part are started with the application of PPJR to the lottery case on the basis of e1, chain of the justifications in the second part starts with the justification of S's "either... or..." belief on the basis of e2. Therefore, starting points of both of these two parts are pieces of evidence in lines 1 and 7 upon which justifications of the contradictory beliefs ultimately depend. For this reason, in trying to explain the paradox, to investigate especially these pieces of evidence, more correctly, the background information b, which derives them, would be inevitable: We know that e1, which is probabilistically derived from b, as a justifying piece of evidence, assigns a very high degree of probability to events, P1, P2, ..., and P1000. Again it is derived, as another piece of evidence, i.e. e2, from b that either ~P1 or ~P2 or ... or ~P1000. Now, there is no trouble here with e2, because it is a piece of direct information about the lottery case (at the outset the lottery has been defined so that only one ticket among thousand ones would be drawn randomly as the winner of the lottery). And so subject S is immediately justified in believing it. This is an instance for the situation of direct justification. Consequently, this means that the problem does not lie within the above-mentioned second part. Therefore, the first part remains in order to show what creates the problem in this paradoxical case.

Pieces of evidence, whether probabilistic or non-probabilistic, give us here some clues to the considered situation in justification, i.e. this concrete case of the lottery which is the object (factual) side of the justification. As well as further pieces of evidence, the following piece also could be derived from b, which provides the evidential basis for justifications in this lottery case, in order to be capable of expressing probabilistically some (ontological) impossibility:

e3 : Prob(Pi, b & P1 & P2 & ... & Pi-1 & Pi+1 & ... & P1000) = 0.

Now, this piece of evidence clearly shows a significant characteristic of this case that it does not allow a coherent system/set of individual events, because the occurrence of 999 events of them prevents the occurrence of the remaining one. (In fact, that is why each individual justification is completely overridden by the others jointly.) This property, which is peculiar to this lottery case, makes the case a destructive closed system, in which P1, P2, ..., P1000 are dependent events (in the sense that: event P1 and event P2 & ... & P1000; P2 and P1 & P3 & ... & P1000; P3 and P1 & P2 & P4 & ... & P1000; ...; P1000 and P1 & P2 & ... & P999). On the other hand, all of the Pi's (i=1, ..., 1000), as being independent events, have the same position, the same power of occurrence, in the lottery case, as e1 declares (that the lottery is a fair one). Thereby, that b hides in itself incompatible pieces of evidence appears clearly. In short, (the background information of) the case here does not constitute an (epistemically) evidential basis opening the way to knowledge. Accordingly, on this basis, each one of the individual justifications in the beginning of the lottery paradox cannot be accepted. (4) And, Pi's also cannot be viewed as subject S's knowledge-claims. (S may believe them, but this basis cannot make them turned out to be knowledge.) Therefore, since the lottery case here is a so-defined one, and thereby it itself, more particularly, b is the cause of this unwanted situation of incoherency, this case is not allowable in an epistemic framework. Thus, it is concluded that such a starting point in the first line of the above analysis of the lottery paradox, at which b appears as the first time in the analysis, could never be acceptable as the departure point of an epistemologically justification process. Hence, the lottery paradox completely collapses from the very beginning.

III. What the Lottery Paradox Teaches Us.

Structure of Paradox. (The term "paradox" has a Greek origin -para: contrary to, doxos: opinion.) By considering this specific example, here, a paradoxical structure in general sense could be analyzed: A paradox, as being a systematic process with a definite beginning, a body of reasoning, and an unbelievable result, has a tripartite structure. That is, there is a specific departure-case of the paradox which tells the story lying behind the paradox. This is the first part of the paradoxical structure. In the second part, the chain of reasoning in the body of the paradox goes, by taking the starting point of the process from its departure-case. At the end, the process gets a paradoxical result consisting of generally two contradictory/contrary propositions. Therefore, a paradox includes three different structural parts some (or all, possibly) of which contain some problems giving rise to the paradox. In R. M. Sainsbury's words: "Either the conclusion is not really unacceptable, or else the starting point, or the reasoning, has some nonobvious flaw." (5) We have seen that in the lottery paradox that problematic part of these three parts was the first one, that is, the starting point.

Priority of Evidential Basis. Paradoxes are very important from a philosophical point of view. Both the occurrence of them and also proposing solutions to them are possible by means of mental activity of man. Namely, paradoxes do not exist in the nature; man mentally creates them. In the face of them man's curiosity and desire, even passion, of explaining/solving them compel the limits of the mind, thereby, man's thought progresses and the ability of thinking is developed. Man is in a really intellectual, philosophical act(ion) in trying to understand the true reason behind them, as is in introducing them: Concepts, notions, facts, qualities and relationships to each other, etc., are reconsidered in detail and in-depth, and are illuminated correctly. Therefore, paradoxes influence the intellectual development of man and add new things to the intellectual mass of man. Also the lottery paradox has had a great importance especially from an epistemological point of view, because in the course of analyzing the explanations and solutions to it, the key clues concerning some crucial subject-matters of today's epistemology have been discovered. In the end of the above analysis of the lottery paradox, however, it has come to light that since the lottery case included that kind of incoherency/incompatibility in itself, b was, in fact, just a prima facie epistemically evidential basis. Hence, this case cannot take place in the comprehension of epistemic justification. Therefore, it should be rejected, as well as the barber case of the barber paradox. (6)

Now, we could see this only by an analytical study about the paradox. Epistemologists have made some contributions to the literature of the lottery paradox and justification, but they could not see the point which should have the priority. By this study, whether the lottery case is relevant to be an epistemic one, and in determining this, the vital importance and priority of its evidential basis, have come to front. Thus, our epistemological intuitions regarding epistemic evidence also have been enlarged and strengthened by the notion of (epistemically) evidential basis. Therefore, PPJR was an acceptable rule of justification, yet in a correct epistemic framework, e.g. in a correct/relevant stochastic process. For this reason, before any procedure of justification, namely any application of justifying rules/principles, i.e. general elements of justification, to any considered definite case, one should absolutely investigate the background information about the case which provides the evidential basis of the case. So, for (epistemic) justification the requirement of epistemically evidential basis, deriving compatible pieces of evidence, and even its priority over justification in order to be considered, happen to have been emphasized. Furthermore, it has been conceived that any theory of justification should clarify the basic nature (fundamental characteristics) of epistemic evidence.

Justification Picture. Epistemic picture has both external and internal factors. On the one side, subject S's belief, as being a statement of fact, is concerned with an external fact (just empirical facts are taken into account here). On the other side, S's believing/belief is an S's internal, doxastic state. In a justification theory, whole of the complicated picture with both externalistic and internalistic dimensions, that is, justification of S's (internally) believing an external fact should be carefully described. Now, pieces of evidence derived from the evidential basis give us some clues to the considered concrete case in justification which is the object (factual) side of the justification. The knowledgeable person S, on the other hand, makes up the subject side of the justification. On this side she lives/experiences according to cognitive processes which have mental/psychological activities. Believing itself can be considered as an example of a mental/psychological action of the subject. Thus, the notion of justification of believing/belief includes justification of a proposition believed which is a product of the justification process of believing. This also, in the context of knowledge, shows that knowledge should be regarded as much of a process as a structure. However, some writers take it as if it were only a structure. In conclusion, justification means to build a bridge between the subject side (believing/belief internally) and the object side (external facts related with a specific case), since S (the believer) is connected with the factual realm in believing and her beliefs are about some facts related with the case. Albeit a theory of justification must inquire about justifying rules/principles, but in the same manner, it must inquire about justifying evidence/evidential basis. Only in doing this, the object-side-foot of the justification bridge could be built. And, epistemic justification could be differentiated from non-epistemic, like moral and prudential, ones which are usual justifications of the rightness of an action.

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(1) Jonathan Dancy, An Introduction to Contemporary Epistemology, Oxford: Basil Blackwell Ltd., 1985, p. 11.

(2) For instance, Keith Lehrer [1990, p. 235], John L. Pollock [1990, pp. 80-107, 227-228], and Paul K. Moser [1989, pp. 136-151], reject PPJR. On the other hand, Henry E. Kyburg, Jr. [1983, pp. 36-37, 232-235] and Richard Foley [1987, pp. 99-100, 236-237] repudiate CP. Peter D. Klein [1981, pp. 190-201], however, abandons both of PPJR. and CP.

(3) D. C. Makinson in "The Paradox of the Preface," Analysis 25(1965), pp. 205-207, talks about an author who is writing a book. In the course of writing, she expresses her assertions in statements of the book, and believes that each of them is true. However, at the same time, she believes rationally that not everything in the book is true, because she has learnt this from her prior experience. She has previously written other books, in which she expounded the truth of these statements, but later discovered/was told of errors in them. For this reason, believing that there are undetected errors in the present book as well, she writes in the preface to the book that she is responsible for these inevitable mistakes. This paradox is discussed mainly in the context of whether a person can be justified epistemically in believing a set of propositions that are justified to her, if she knows that this set is a logically inconsistent one.

(4) Here, by refutation of individual justifications, 999 potential Gettier counter-examples are also collapsed. Otherwise, justified Pi's in the lottery paradox could be potentially accidental truths. Let us suppose that S is justified in believing that Pi, for each i=1, ..., 1000. But she cannot follow the random drawing at the end because of some private reasons of hers, and she has no information about the result of the lottery. So, this case happens to be a case, paving the way for new 999 Gettier-type examples, because S's 999 justified beliefs of thousand ones turn out to be coincidentally true, i.e. truths by chance (through luck). Now, although S's these 999 beliefs are justified true beliefs, it cannot be said after all that S knows them. (Essentially, a famous example in an earlier work, Bertrand Russell's "stopped clock" example has again the notion of accidental truth in Human Knowledge: Its Scope and Limits, NY: Allen and Unwin, 1948, p. 154.)

(5) R. M. Sainsbury, Paradoxes, Cambridge: Cambridge Univ. Press, 1988, p. 1.

(6) The background information of the barber paradox is: There is a barber in a village, shaving all and only villagers who do not shave themselves. So, by this barber-definition/supposition, the following contradiction comes to occurrence: If the barber shaves himself, he does not shave himself; if he does not shave himself, he shaves himself. Now, the general response to this paradox is to reject the story/supposition that there is a so-defined barber, because here the so-defined case itself contains the contradiction in itself. That is, in this paradox a so-and-so barber/case, as being just a product of reason, is defined first; but in the end that contradiction appears. Therefore, there cannot exist such a barber. So, such a case is given up, and thereby the related paradox is solved. But, the difference in rejections of these lottery and barber cases should be pointed out here: The lottery case is objected from an epistemological point of view, whereas the objection to the barber case is from an ontological point of view. That is, while there cannot be such a barber, there is such a lottery. And, the objection here is not to (the possibility) of this lottery's being, the objection is to its being viewed as a justification case; and thereby, the proposed solution here is only to cease the application of rule PPJR to it.

Selected Bibliography

Foley, R., 1987. The Theory of Epistemic Rationality, Harvard Univ. Press, Cambridge.

Gettier, E. L., 1963. "Is Justified True Belief Knowledge?" Analysis, 23, 121-123.

Klein, P. D., 1981. Certainty: A Refutation of Skepticism, Univ. of Minnesota Press, Minneapolis.

Kyburg, H. E. Jr., 1961. Probability and the Logic of Rational Belief, Wesleyan Univ. Press, Middletown.

———, 1983. Epistemology and Inference, Univ. of Minnesota Press, Minneapolis.

Lehrer, K., 1990. Metamind, Clarendon Press, Oxford.

Moser, P. K., 1989. Knowledge and Evidence, Cambridge Univ. Press, Cambridge.

Pollock, J. L., 1990. Nomic Probability and the Foundations of Induction, Oxford Univ. Press, Oxford.

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