quarta-feira, 9 de maio de 2012


Obs: Paper first published in the journal Ratio, vol. 23, 2, 2010. What this paper shows, I think, is that the essential truth about the matter has been already discerned by Plato more than two thousend years ago. But if this is the case most contemporaries views are seriously mistaken... (This paper will be published in a revised form by Cambridge Scholar Publishers as part of the book Lines of Tought).

                                 A PERSPECTIVAL DEFINITION OF KNOWLEDGE
                                                                Claudio F. Costa

                   Knowledge is not simply justified true belief, but it is justified
                   true belief justifiably arrived at.
                   Robert Fogelin

In this paper an improved formulation of the classical tripartite view of knowledge is proposed and defended. This formulation makes explicit what is concealed by the symbolic version of the tripartite definition, namely, the perspectival context in which concrete knowledge claims are evaluated. Once we have done this, Gettier’s problem disappears smoothly, without the gratuitous generation of new difficulties.

Analytic philosophers have dissected the classical tripartite view of propositional knowledge as justified true belief with the following definition:

                                    (i)     (ii)       (iii)
           (Df.k1)  aKp  ≡  p & aBp & aEBp          (where p = proposition,
                                                                        a = person, B = belief,
                                                                        E = reasonable justifying

     It is well-known that this and similar formulations of the old view have given rise to a challenge to the rationality of our knowledge which is known as Gettier’s problem: there appear to be counterexamples in which all three conditions of knowledge are satisfied, even though the knowledge claimer a has in fact no real knowledge of the proposition p.  It is also well-known that counterexamples of the Gettier-type have lead to a multiplicity of answers which have typically generated new difficulties, even suggesting that the conceptual analysis of knowledge is a kind of degenerative research program without any good prospect.
     Our overall diagnosis of the situation is much more optimistic: the classical view of propositional knowledge, as presented in the formulation above, though not incorrect, is an oversimplification of conceptual structures that have always belonged to the praxis of our natural language; this formulation conceals a perspectival and potentially dialogical dimension of our knowledge evaluations which can lead to misunderstandings of the Gettierian kind. This diagnosis calls for a therapy which consists in improving the tripartite definition of knowledge in such a way that it becomes reflexive of the possibly dialogical dimension of our knowledge evaluations. Once we have achieved this, we will not only have alleviated the symptoms, as most solutions to Gettier’s problem do, but actually cured the disease by treating its real cause. Gettier’s problem will then vanish without a trace, while the analysis of propositional knowledge will achieve its full pragmatic dimension. In order to arrive at these results, we need to begin by reviving an old discussion.

The Internal Link Between the Conditions of Evidence and Truth:  Almeder’s Attempt
A natural way to solve the problem without abandoning or substantially changing the tripartite definition of knowledge was defended in the seventies by Robert Almeder.  His solution emerged from the observation that in Gettier’s counterexamples the justifying evidence given by a does not have anything to do with what makes proposition p true, while our epistemic evidence always makes p true.
     To clarify this point, consider the following Gettier-type counterexample. A knowledge claimer a believes he saw b stealing a book in the library this afternoon. This is the justification for a’s belief that the statement p is true, namely, ‘b stole a book from the library today’. However, what a really saw was c, b’s identical twin, stealing a book. Yet statement p is nevertheless true, for b actually was in the library earlier today and also stole a book, even though a did not see him do this. According to the tripartite definition of knowledge, a knows p because all three conditions are satisfied: p is true, a believes that p is true, and a has a reasonable justification for this. However, it is clear that a does not know p. But why is this so? The most intuitive answer seems to be that a’s lack of knowledge is due to the fact that the evidence given by a does not make p true, which is necessary for a to know p. Therefore, we need to introduce the requirement that the evidence given by a for p must be sufficient for the truth of p, which for Almeder can be expressed in logical terms by saying that adequate epistemic evidence must entail p. As he notes, this requirement appears to be consistent with our linguistic intuitions: it seems odd to deny this with assertions such as: ‘Your evidence (justification) is sufficient for your knowledge of p, but it does not make p true’.  Using the sign ‘=>’ for entailment, we can restate the tripartite definition in an extended form which includes Almeder’s requirement:

                                                            (i)     (ii)               (iii)
                                  (Df.k2)  aKp  ≡  p & aBp & (aEBp & (E => p)).

     The addition of the condition that E must in some sense imply the truth of p should eliminate Gettier’s counterexamples, because in these cases it is only a coincidence that the conditions of truth and of justification are conjunctively satisfied: in none of these cases does E entail the truth of p.
     Unfortunately, Almeder’s proposal has always seemed too strong, making inductive justification impossible, for in these cases the truth of proposition p does not necessarily follow from the truth of the proposition E asserting the justifying evidence, as should occur in the case of entailment.
     There is a rejoinder to this kind of solution that has been proposed by W.E. Hoffmann, which poses the problem in a striking way.  Compare the following two cases: (i) Having nothing better to do, Jones sits in the lobby of a hotel for some hours watching some very large people crossing a certain spot carrying heavy suitcases and then, concluding inductively that the floor at this spot can obviously also support his weight, confidently walks across it. (ii) In the lobby of another hotel, Smith conducts an experiment identical to that of Jones in every detail, except that the floor caves in when he tries to walk across it. Comparing the two cases shows us that since the proposition p – ‘The floor will support me’ – and the justifying evidence E are similar in both cases, and since E does not make p true in the case of Smith, then E does not entail the truth of p. However, if this reasoning is correct, Almeder’s acceptance of a requirement such as ‘E => p’ leads to the conclusion that Jones does not know p too, which is surely false.

The Epistemic Link: Fogelin’s Solution
A more successful attempt to solve Gettier’s problem by relating the conditions of justification and truth was proposed by Robert Fogelin in the nineties. His approach, unlike Almeder’s, already places the problem in a dialogical context. According to his view, the justifying evidence given by anyone claiming knowledge must be both a personal justification and an epistemic justification. A personal justification is one that satisfies the condition of epistemic responsibility, being consonant with the right epistemic standards and the information available to the person; this is what I called reasonable evidence when introducing the tripartite definition. The evidence stated in Gettier’s cases satisfies this requirement. On the other hand, an epistemic justification must also be evidence (a ground, a reason) that establishes the truth of the proposition p for us – which no Gettierian counterexample is able to do. In all of them, Fogelin says, we have ‘wider information’ than person a, and because of this we can see that the justification given by a, though personally justified, cannot make proposition p true, as it fails to satisfy the standard of epistemic justification. As he states:
We are given wider information than a possesses, and in virtue of this wider information see that a’s grounds, though responsibly invoked, do not justify p. I think this double informational setting – this informational mismatch between the evidence a is given and the evidence we are given – lies in the heart of Gettier’s problem. 
     So, returning to the counterexample, we see that the evidence given by a, that b stole a book from the library today because a saw this, is epistemically inadequate. And the reason for this inadequacy is given by the wider information that we have, for we know that c, b’s identical twin, also stole a book, and that this is what a really saw this afternoon.
     According to this view, the condition of justification in the tripartite definition of propositional knowledge should be split into a condition of personal justification (iii-p) and a condition of epistemic justification (iii-e), and based on this we obtain the following definition of propositional knowledge:

       a knows that p   ≡    (i) p is true,
                                     (ii) a believes that p is true,
                                     (iii-p) a justifiably came to believe p,
                                     (iii-e) the justifying evidence given by a
                                               establishes the truth of p.

     This formulation immunizes the tripartite definition against counterexamples of Gettier’s type because, although they satisfy (iii-p), they do not satisfy (iii-e); hence they are correctly identified as cases that fall short of knowledge.
     Although this version of the tripartite definition is intuitively acceptable, as it already reflects the perspectival and often dialogical dimension of our knowledge evaluations, it still leaves unsolved the logical problem addressed by Almeder, namely, the question of what kind of logical or internal link exists between conditions (iii-e) and (i), between justifying evidence and truth. If the word ‘establishes’ in (iii-e) means the same thing as ‘entails’, then we are reverting back to the same difficulties.
      Next I will develop a more perspicuous symbolic formulation of our epistemic intuition, one that is able to reflect the dialogical context in which most concrete knowledge claims are evaluated, as in the case of Fogelin’s solution, but that also solves the logical problem defectively addressed by Almeder, enabling us to bypass objections like Hoffmann’s.

Introducing the Dialogic Equivalence
Before reformulating Df.k1 it is useful to be more explicit about our dialogical assumptions. In order to do this, I shall call the concrete person who is evaluating the knowledge claims of a the knowledge evaluator s. Usually we speak about s allusively, using personal pronouns such as ‘we’ or ‘us’, as in ‘We are aware that a knows p’ or ‘a’s knowledge of p is known to us’, and the plural form indicates that the evaluation is accepted, or can be expected to be accepted by any reasonable person provided with the relevant information. This, of course, does not preclude that s = a, where a intends to evaluate his or her own knowledge claims in an internalized (non-proper) dialogue. Moreover, we will call ‘tj’ the time of the judgment, which here is the time at which s evaluates a’s knowledge claims. Equipped with these concepts, we can add to Df.k1 the following dialogical equivalence:

                  (DE)     sKtj(aKp)  ≡   sKtj(p & aBp & aEBp)
                                                     or (which is the same thing)
                                                     sKtj(p) & sKtj(aBp) & sKtj(aEBp),

     What DE says is intuitively clear. Let us suppose for now that the knowledge evaluator is the teacher s, who asks the schoolgirl a where the city of Angkor is located, and that a answers (correctly) p: ‘Angkor is in Cambodia’. To judge that a knows p, s must know that a knows p, and in order to know this, according to the tripartite definition, s must also know that p is true (that Angkor really is in Cambodia), that a believes p to be true (perhaps based on a’s belief-affirmative behaviour), and that a has reasonable evidence for her belief that p is true (a has presumably found this information in the schoolbook).  Bearing in mind this dialogical assumption, my procedure will be to carefully reexamine what exactly is involved in the conditions of truth and justification, searching for the right link between them.

What Might be Dialogically Involved in the Condition of Truth
The condition of truth is usually formulated in the tripartite definition as p, or ‘p is true.’ This formulation completely leaves aside what makes p true and for whom. However, there is no way of attributing truth value to p independently of judging subjects and the ways in which they arrive at this attribution. As the one who decides that p is true is the person evaluating whether or not a knows p, the condition of truth assumes that p must be true for the knowledge evaluator s.
     To understand the relevance of what should be an obvious point, let us suppose that p is the proposition ‘The earth circles around the sun’. This proposition would be considered true by s1, the astronomer Aristarchus, who dared to propose the heliocentric view in the Third Century B.C. However, a well-informed knowledge evaluator s2, living in Antiquity or in the Middle Ages (and representing his community of epistemic subjects), would undoubtedly consider p to be false, while s3, a well informed knowledge evaluator living in Europe in the Eighteenth Century (and representing another epistemic community), would again hold p to be true. The evaluation that s2 would make of a knowledge claim of p by person a would unavoidably be negative, for there can be no knowledge of false propositions, and therefore this evaluation would differ from the evaluations made by s1 and s3, which would depend on different conditions. This is so because the s’s are distinct knowledge evaluators, ascribing a different truth value to p at different times. A definition of knowledge as humble as Df.k1, suggesting that the truth value of p might be considered independently of the evidence accessible to s, leaves any spatio-temporal variation in truth evaluation, and consequently in knowledge evaluation, totally unaccounted for. This definition does not take into account that what is held to be true or false (and for this reason to be knowledge or lack of knowledge) depends on the changeable standards of a given community of epistemic evaluators.
     Nonetheless, one could still ask if what is meant by the condition of truth isn’t the ultimate truth value of p, even if it is impossible to ascribe truth value to p independently of a knowledge evaluator and the ways in which he or she comes to know it. The answer is that here this demand would lead us to epistemic scepticism, since our empirical truth attributions are almost always dependent on fallible evidential support. Only God, the infallible evaluator, by knowing the ultimate truth value of any empirical proposition, would be able to apply the tripartite definition of knowledge in order to decide with absolute certainty whether or not p is true and, consequently, whether or not a really knows p. However, this is not what we mean when we say that knowledge is ‘justified true belief’. When we evaluate knowledge claims, we are not appealing to God’s judgment of the truth value of p, but rather to our own present evaluation of this truth value, which is contextually contingent and based on our finite human cognitive powers. For this reason, what is at stake is the truth value ascribed by s to p and based on the evidential support accessible to s at the time. This truth value isn’t usually seen as the ultimate one, but rather as a mere candidate for this role, arguably the one with the highest probability. Therefore the interpretation of ‘p’ as ‘p is true for s’ is the only really sound alternative. 
     After making explicit for whom p must be true, we still need to make explicit what makes p true for s. In order to do this, we must again consider what is involved in the condition of truth as it appears in the DE. This condition appears as sKtj(p) or sKtj(that p is true). As s is a human epistemic subject, s must come to know that p is true by drawing on evidence (which might be seen as truth-makers, as facts, etc.). Thus, one could express sKtj(p) more explicitly as sKtj(that there is sufficient evidence to make p true) or as sKtj(that there is at least one piece of evidence E, such that E is sufficient to make p true).
     However, this is not yet a fully explicit presentation of what is involved in the condition of truth as viewed by the knowledge evaluator, since there is more to consider about the role of evidences. To arrive at a more complete account, we need to introduce the concept of a corpus of evidence E*, understanding this as a set of pieces of evidence that individually count decisively for or against the truth of a proposition p for some s at a certain time. Here is the definition:

     (Df.E*)  E* = a set of pieces of evidence, each considered sufficient for
                            the assignment of a truth value to a proposition p.

     This definition means that if a piece of evidence E is an element of the set E*, then E must be sufficient to render p true or to render p false.
     It is important to see that a piece of evidence E which belongs to E* can be composed of other pieces of evidence that in themselves are not sufficient for the assignment of truth value to p. The most common form of composition is by conjunction. So, for example, if I am sure that I am sitting in the same chair I sat in yesterday, because it has the same appearance and because it is located in the same place, the conjunction of these two pieces of evidence may be what I find to be sufficient evidence for the truth of the proposition.
     In order to deal more precisely with the notion of being sufficient, I will introduce the symbol ‘~>’ to represent what might be called ‘sufficiency’, defining it in the following way:
P‘~>Q means that if the antecedent P is true, the consequent Q must either be necessarily true (with a probability of 1 and logical certainty) or be probably true to a very high degree (with a probability near to 1 and practical certainty).
In this way the symbol ‘~>’ respectively captures the force of formal evidence (appropriate for knowledge claims belonging to the formal sciences) and also the force of empirical evidences (appropriate for inductive knowledge claims, such as those belonging to the empirical sciences, where the inference has strong inductive force and the consequent should be seen as practically certain).
     Given that E is the case and that E ~> p, then either p must be true or p is very probably true; and, given that E is the case and that E ~> ~p, then either p must be false or p is very probably false. Considering this, with the concept of E* we could render sKtj(there is at least one piece of evidence E, such that E is sufficient for p), as sKtj(there is an E* and E* ~> p), since E* is a set displaying individually sufficient pieces of evidence as its elements. We will come to this conclusion shortly, but before this we need to make two explanatory points about E*.
     The first point concerns the intuitive basis of the concept of a corpus of evidence for the ascription of truth value, as the following examples show.
     Firstly, let us suppose that at time t the subject s holds as true the proposition p1 ‘The temperature was below zero last night’, because of the evidence E1 ‘The snow didn’t melt’, and also because of E2, ‘p1 was stated in the weather forecast’. If s considers each of these pieces of evidence sufficient to make p1 true, and these are the only two pieces of evidence that s has, then s has a corpus of evidence E* for p1 constituted by the set {E1, E2}. In this case, each of these pieces of evidence will also be considered by s (at this time, on the assumption that his stock of beliefs is true) as making the truth of p1 highly probable, which means that s knows at time t that ‘E* & (E* ~> p)’, which means that under these circumstances s knows (or believes he knows) inductively, with practical certainty, that p1 is true.
     Now let us assume that at time t a subject s believes in the falsity of p2, a proposition stating that the earth is flat, based on at least one of the following pieces of evidence: E1 = ‘Photos taken from space show that the earth is round’, E2 = ‘There are many historical accounts of the circumnavigation of the globe’, and E3 = ‘Ships seem to sink below the horizon when they sail out of view’. In this case, for s ‘E* = {E1, E2, E3… En}’, and each of these pieces of evidence is considered by s at t – assuming the truth of his stock of beliefs at time t – to be sufficient to falsify the proposition p2. This means that s knows (or believes he knows) at time t that ‘E* & (E* ~> ~p2)’, namely, that the probability of ~p2 being true is very high or that p2 is certainly false.
     A third case is that in which s doesn’t know the truth value of the proposition. For example: Suppose that s doesn’t know whether the proposition p3 ‘Aston Rowant is bigger than Kingston Blount’ is true.  In this case, s’s corpus of evidence for p3 is empty: E* = Ø.
     Looking back to Df.E*, we come to the conclusion that a subject s can access E* in three different ways:

     (1) s does not attribute any truth value to p; in this case, E* is seen by s
as an empty set;
     (2) s has cognitive access to a non-empty set E* of justifications,
while (2) divides itself into two possibilities:
     (2a) Each element of the set, each piece of evidence, is considered by s as sufficient to make p true; in this case it is clear that s knows that ‘E* & (E* ~> p)’, which means (deductively or inductively) that s knows that p is true.
     (2b) Each element of the set, each piece of evidence, is considered by s as sufficient to make p false; in this case s knows that ‘E* & (E* ~> ~p)’, which means (deductively or inductively) that s knows that p is false.
     One could ask if there isn’t a further possibility, namely:
(2ab) E* contains at least one piece of evidence sufficient to make p true and at least one piece of evidence sufficient to make p false.
     However, this would not be a real epistemic alternative. As E* is defined as a set of individually sufficient conditions, in this case E* turns out to be an inconsistent set, making simultaneously p and ~p true for s at a certain time. Although we are often irrational in our judgments, holding inconsistent beliefs, insofar as we regard ourselves as epistemic subjects, we are supposed to be consistently rational, and this means that we should only consider one evidence as sufficient for the truth of a proposition p when we do not possess any other evidence that we consider sufficient to make p false and vice versa. Assuming the rationality of s’s judgment, the elements of E*, when achievable, must all be either evidence for the truth of p or for its falsity, but not for both.

     The second point about Df.E* concerns a better understanding of the concept of sufficiency expressed by the symbol ‘~>’. As was shown above, this symbol must either be understood as representing something like material implication in the case of formal evidence, or a strong inductive relation in the case of the usual empirical evidence. It is important to see not only that ‘E* ~> p’ must be true for s at a certain time, so that when joined with E* it allows s to conclude that p is true, but also that this conditional only works under the assumption of the truth of the background beliefs and other relevant beliefs belonging to the stock of beliefs held by s at this time, being insofar context-dependent.  To make this point clear, let us consider the following cases of conditionals where the truth of the antecedent is viewed by a subject s at a given time as sufficient for the truth of the consequent:

(i) If my car’s gas tank is full, then there is sufficient gasoline for my trip.
(ii) If the defendant’s fingerprints are found at the scene of the crime, then we will have enough proof that he is guilty.
(iii) If this is the result of the biopsy, then the surgeon can be sure that the tumour is malignant.

     In all of these cases it is possible that the antecedent is true but the consequent is nevertheless false. Let us suppose that the car’s gas tank has a leak, that the fingerprints were planted, that the tumour is a benign one of an unknown kind, indistinguishable in its histology from a malignant tumour. In all these cases, the consequent will be false, although the antecedent remains true, which would make the three conditionals false. Is this a threat to our understanding of ‘being sufficient’? Certainly not, because in all three cases, if s becomes aware of the facts that the gas tank has a leak, that the fingerprints were planted… then some of the beliefs belonging to s’s stock of beliefs will change, which would on this basis lead him to disclaim his admission of the conditionals. On the other hand, under the assumption that all the other relevant beliefs held by s (such as the belief that his car’s gas tank does not have a leak, that the fingerprints were not planted, that this is not a new kind of tumour…) are consistent with the conditionals, it follows that if the antecedent true, the consequent should very probably also be true. Hence, it seems that the sign ‘~>’ gives s enough of a sense of ‘being sufficient’ or ‘being enough’ or ‘making true’, insofar as he interprets it as making its consequent true with a very high probability, assuming the truth of the relevant beliefs belonging to the stock of beliefs held by s at the time of his evaluation.
     Now that we have explained our concept of E* it is time to return to our task of restating sKtj(p) in a precise and fully explicit way. We have seen that sKtj(p) can be rendered as sKtj(there is at least one piece of evidence E, such that E is sufficient for the truth of p), where the evidence and its role were only mentioned. Now, using E* we can restate what is contextually assumed in the condition of truth as it appears in the DE as:

                                                   (i’)  sKtj(E* & (E* ~> p)).

     Indeed, when s is aware of an E* at tj, and when for him E* has some element viewed as sufficient for the truth of p, then s concludes either deductively (using the modus ponens) or inductively (using the rule of induction) that p must be true, which amounts to the same thing as that it satisfies the conditions of truth! In this way, ‘E* & (E* ~> p)’ only makes explicit what we (as placeholders for s) implicitly mean by ‘p’ or ‘p is true’ in the condition of truth. As we will see, this analysis will suffice as a restatement of what is presupposed by the conditions of truth which does not play down the role of the knowledge evaluator.
What Might be Dialogically Involved in the Condition of Justification
What about the condition of justification? Our proposal is to use a similar strategy for its reconstruction, keeping in mind the intuitive idea that the justifying evidence given by a must in some way make p true (which was misleadingly captured by ‘aEBp & (E => p)’ in Df.k2) and our perspectival analysis of the condition of truth. When could E be sufficient for the truth of p, considering that what is involved in the condition of truth might be rendered as ‘sKtj(E* & (E*  ~> p))’? The answer springs to mind: When the evidence E given by a can be seen by s as belonging to E*! For if E belongs to E*, and p is viewed by s as true, and if s is reasonable, this means that all the elements of E* (E inclusive) are viewed as making p true. To do justice to our intuition that E must establish the truth of p in the context of the DE, we are led to introduce what might be called a requirement of epistemic adequacy for E, which can be stated as follows:
REA: The evidence E given by a for the truth of p is considered epistemically adequate (and not just reasonable) iff the evidence E comes to be regarded by s as belonging to his E* when s evaluates a’s knowledge claim for p, insofar as E* is held by s to make p true.

      In other words, REA demands that, given the circumstances in which for s ‘(E ~> p) & (E belongs to E*)’, it should also be the case that for s E makes p true under the assumption of the truth of his stock of beliefs at time tj, an assumption that must be taken into consideration, since the content of the E* accepted by s, as we have already seen, might easily change.
     It is easy to assimilate REA into the analyzed form of DE. As we have already adopted E* ~> p as part of the condition of truth, the only thing we need to do is to add to aEBp the condition that E must belong to the evidence set E*, which is accepted by s when s evaluates a’s knowledge claim. In other words: instead of Almeder’s condition E => p of Df.k2, which wrongly suggests that E makes p true necessarily and in all contexts, what must really be required is that for s at tj E belongs to E* and E* ~> p. Therefore, what is assumed by the condition of justification in a dialogical context of knowledge evaluations can be made sufficiently explicit as follows:

                                         (iii’)  sKtj(aEBp & (E belongs to E*)).

     Accepting the reformulated conditions (iii’) and (i’), REA turns out to be part of the DE, for which REA can be abbreviated as ‘(E belongs to E*) & (E* ~> p)’, which already belongs to the conjunction of the conditions (iii’) and (i’).
 DE’ and the Perspectival Definition of Knowledge
 The next step is to improve DE substituting (i’) for (i) and (iii’) for (iii), as follows:

      DE’:                           (i’)                      (ii)               (iii’) 
      sKtj(aKp)  ≡  sKtj((E* & (E* ~> p)) & aBp & (aEBp & (E belongs to E*)))

     DE’ clearly displays the logical or internal link between the conditions of justification and truth, as shown by the arrows.  We are now able to give expression to the perspectival and often dialogically reflexive formulation of the tripartite definition of knowledge, insofar as we conceal sKtj on both sides of the equivalence as redundant, even if presupposed. Here is the definition:

                                                           (i)                   (ii)               (iii)
                      (Df.k3)   aKp  ≡  (E* & (E* ~> p)) & aBp & (aEBp & (E belongs to E*)).

     This is, I believe, the view of knowledge as justified true belief with its ‘missing link’ relating conditions (iii) and (i). This definition can be somewhat simplified. Since condition (ii) is already included in aEBp, we can eschew it as redundant, formulating Df.k3 as:
                      (Df.k4)   aKp ≡  aEBp & (E belongs to E*) & E* & (E* ~> p)

     By means of these two formulas knowledge is explicitly defined in a way that reflects the epistemic perspective of the knowledge evaluators. The decisive point of this definition is that it shows the correct internal link between the justifying evidence E and the truth of p. The evidence E must be sufficient for the truth of p, but in a contextually dependent way, by means of its acceptance by s, as belonging to the set of given evidence able to make p true under the assumption of the truth of other beliefs belonging to his stock of beliefs at the time of his evaluation. But would this definition solve our problems? To answer this question we first need to see how it works.
Ordinary Cases of Application
We begin with the most common and unproblematic case. Suppose, for our first example, that the knowledge claimer a states p, namely that the temperature was below zero last night. I ask a how she knows that. Her answer is E1: she has seen that the snow did not melt during the night. In this standard case I am the knowledge evaluator s, and I have the evidence set E*, which consists of, for example, E1: ‘I saw that the snow did not melt’ and E2: ‘I heard it on the weather report’. Thus, my E* is made up of {E1, E2}, and because a believes in the justification E1 which I accept as belonging to my E*, and which at the time of my evaluation makes p true for me, I conclude that a knows p.
     Not all cases are so straightforward. There are others in which s accepts a’s knowledge claim by expanding his E* in order to include the evidence E in E* because of its coherence with s’s previous beliefs. Imagine that I am s and that person a tells me that she knows p: ‘Oxford is bigger than Witney’, because of E1, namely, because she has heard this from a tourist guide. Let us suppose that I know this for reason E2: ‘I have spent some time in both towns’. Although her justification does not belong to my E* for the truth of p, it is natural for me to expand my E* in order to accept her justification (for if I held E2 to be true, assuming my stock of beliefs, which includes my belief in the trustworthiness of a and of tourist guides, it follows that E1 also turns out to be true for me). Thus, for me E* turns out to be {E1, E2}, both justifications making p true at the time of my evaluation, and from this I conclude that a also knows p.
      There are also examples showing how dependent epistemic evaluation is on the time at which s makes his evaluation. Let us suppose that in a court of law jury a concludes that b has committed a murder, basing his conclusion solely on the sufficient evidence given by E, which is the result of a DNA test... At time t1 judge s concludes that a knows that b is the murderer, confidently including E in his E*. A few days later, at time t2, it becomes known that the DNA test was performed incorrectly, which invalidates the results. Nevertheless, new information, including b’s confession, shows beyond any doubt that b was in fact the murderer. Aware of this, the judge could not persist in his belief that jury a knew p at t1, even by accepting the truth of p, for at t2 E has ceased to belong to s’s E*. Indeed, judge s makes a new evaluation with the expectation that jury a will accept p based on evidence belonging to his present E*. Because the time at which knowledge claims are evaluated is essential to the construction of E* by s, the perspectival definition of knowledge explains spatio-temporal variations in knowledge evaluations left unexplained by the usual formulations of the tripartite definition.
     Another important point is that by making knowledge claims relative to the changeable belief stocks of knowledge evaluators, we are not compromising ourselves with any relativist view concerning truth or knowledge. This is shown by the fact that the two opposite evaluations of the same knowledge claim by judge s are asymmetrical: the old evaluation would be rejected by any other rational evaluators possessing the new information. Most of our perspectival and changeable epistemic evaluations are not incommensurable. However, it is not the task of a theory of truth evaluation to explain the structure of such variants, but rather that of a theory of verisimilitude or truth approximation employing some normative concept of ultimate truth.
     Finally, it is noteworthy to mention that sometimes the Df.k3 (or Df.k4) seems to collapse into Df.k1. This is what occurs when we ask ourselves whether we know something at the present moment. For example: I wish to evaluate my present knowledge claim of p: ‘The air-conditioning is on’. According to the traditional definition, I know p because it is true that the air-conditioning is on and because I have reasonable evidence for my belief, namely, the continuous humming sounds that I can hear. Following the perspectival definition, I know this because I have evidence E, namely, I can hear the sound of the air conditioner and because this evidence belongs to my presently accepted evidence set E*, which makes p true. In this case, however, not only are s = a and E = E*, but even the time of evaluation is the same (namely, the present moment). It seems that here Df.k3 presupposes the identity DE’ in a trivial way. But this does not necessarily render the application of our definition superfluous, for it seems to require at least a meta-cognitive awareness of s as a, and of E as E*.
     Some Unusual Cases: Gettier-Type Counterexamples
     Arriving finally at Gettier’s problem, it is not hard to understand why the proposed analysis of the intuitive view dissolves this problem once and for all. By assimilating REA into the formulation of the conditions of truth and justification, the perspectival definition of knowledge brings to light the real internal link between these two conditions. This internal link has sufficient strength to neutralize all Gettier counterexamples, since they all arise from its absence, and at the same time it is sufficiently flexible to circumvent objections such as Hoffman’s. In order to show this, we must first adjust our eyes to the new bright light outside the cave of Gettierian shadows, reconsidering some of these counterexamples.
     Consider the following well-known Gettierian counterexample : person b is a worker in a’s office, and a not only sees b coming to work in a BMW, but b has also told a that the car belongs to him and even has shown him his ownership documents. Based on this evidence E, a makes the knowledge claim p: ‘Someone in my office owns a BMW.’ However, b has lied to a: the BMW actually belongs to his sister, the ownership documents are forgeries, etc. However, p is still true, for without a’s knowledge, another worker in his office, c, really does own a BMW. Under such circumstances, it is obvious that a does not know p, since there is no relationship between the truth of p and the evidence given by a. However, according to the tripartite formulation of the classical definition, a must know p, because (i) it is true that someone in a’s office has a BMW; (ii) a believes that p is true; and (iii) a is able to offer the very reasonable evidence E for the truth of p.
     However, this is no counterexample to our dialogically conditioned definition of propositional knowledge, for it is not able to satisfy it. To make this clear we need to follow the strategy of always take into account the supposed counterexamples contextually, considering the whole of the concrete dialogical situation in which a knowledge claim is evaluated by s, with all the relevant independent information accessible to him. These counterexamples need to be considered not just in their abbreviated form, like examples used in books on epistemology, but as fully concrete cases, similar to ones that could be found in real life. In the case of the counterexample above, the question is: how do we know all these facts about the worker in the office? Well… let us suppose that, when considering who belongs to this elusive ‘we’, we are led to a judging subject s, an older worker in the office who knows all his colleagues very well and has told us this Gettierian story. He knows that c has an old BMW, and he also knows that b is a compulsive liar who drives his sister’s BMW, pretending that it belongs to him. If a had justified p by saying that c had told him this, giving him in this way the evidence E1, s would judge that a knows p, because s accepts this justifying evidence as belonging to his E*, and because he thinks that on the basis of all he knows, this E* ~> p. But taking into account that a justifies his claim to know p by giving the evidence E that b has told him, s refuses to accept this justification as an element of his E*. The knowledge claim of a is rejected by s because E belongs to E*, failing to satisfy REA.
     Now let us suppose that a justifies p by giving the true evidence E2, ‘A worker in my office told me that he owns a BMW.’ Although this justifying evidence is true, it does not satisfy Df.k3 for s for in this case, anticipating that b and not c might be the worker who told this, s does not immediately accept E2 as belonging to E*, asking a who told him p, which brings him to the same result as before. A possible objection here would be that a knowledge evaluator s might not be so well informed, judging falsely that a knows p… But in this case we would need to consider another well-informed knowledge evaluator in order to explain our awareness of the fact that b is lying and that only c really owns a BMW. Otherwise, how could this Gettierian story be grounded? The rejection of the knowledge claim turns out to be unavoidable every time we replace the usually abbreviated reports of Gettierian cases with a sufficiently detailed explanation of how the evaluation of the knowledge claim was generated.
     A second counterexample of the Gettierian type is based on perceptual evidence.  Imagine that a is a traveller who, looking out of her car window thinks she knows the truth of p: ‘There is a red barn in the field’. However, it is only by chance that what she sees is a real red barn – for with exception of this one, all the red barns in the vicinity are really only the external façades of red barns, convincing enough to fool even the most observant traveller. Although a satisfies the conditions of justified true belief as stated in the standard definition, she does not satisfy these conditions as stated in its dialogically conditioned form. For in this form we need to consider the reasons for belief in the truth of p, which always arise from the point of view of a knowledge evaluator. To see whether the example satisfies our definition in a real case, we need to consider where one got all this information! Here is a plausible story: we are only reporting what the knowledge evaluator s has told us. This person lives in the region and is well aware that all the red barns, with the exception of this one, are only barn façades… Traveller a has given s a lift. As a drives away from the river, she points out to s the beautiful red barn she has noticed in the field. Since s knows that this is the only authentic red barn and that a is a foreigner, s would not think that a really knows that she is seeing a real barn, as here the evidence E – which can be formulated as ‘I see something like a red barn’ – does not belong to s’s E*. The only evidence s would accept as belonging to E* would be a close inspection of the barn by a, or a’s having told him that she already knows about the façades and this one real exception, which is located after the bridge, as such justifications would be accepted as belonging to or being implied by his E*, and therefore as being sufficient to support the truth of p. As the justification given by a was only that she had seen a red barn, s concluded that a had said the truth only by chance, and that her merely Gettierian justification does not belong to E*. 
     We can also consider a Gettierian counterexample that involves self-evaluation, the interiorized form of dialogical evaluation. Let us suppose that a looks at her watch and sees that it is 11:15.  At this moment, a believes that she knows p, namely, that it is 11:15 a.m. After this, a looks at the clock on the church tower in the square and sees that it really is 11:15 a.m. But then a remembers that yesterday her watch was running slow. Therefore, a examines her watch carefully and concludes that its hands are not really moving and that the watch had probably stopped the night before. At this point in time a realizes that she did not really know that it was 11:15 a.m. the first time she looked at her watch: it was all just an amazing coincidence. Here again, for the standard version of the classical definition, the three conditions are satisfied: p is true, a believes that p is true, and a has a reasonable justification for her belief. But for our improved statement of the classical view, a’s knowledge claim is being evaluated in an internalized dialogical context by a knowledge evaluator a’ (s = a’) who is the same a after she realizes that her watch is not working. At this moment, the E* consists in the evidence E1, given by the Church clock, while the evidence E, that her watch shows that it is 11:15 a.m., which was given by a, cannot be accepted as belonging to the E* of the knowledge evaluator a’. Once again no knowledge is acquired, as at the time the knowledge claim is evaluated, E belongs to E*, leaving REA unsatisfied.
     A nearly Gettierian counterexample is the following : a reads in a newspaper p: ‘The famous civil rights leader M.L. has been assassinated’, which is based on E: an eyewitness report. Those close to a have the additional information of later reports to the contrary. However, p is in fact true, since the late reports are false and were fabricated by the eyewitnesses. Df.k3 allows an improved answer to this: if s is someone near to a, the additional information will make him reject E. But if s also knows about the conspiracy of the eyewitnesses, s will agree with a, for a’s E belongs to s’s E*, and to miss false information is no epistemic sin.
     Indeed, it seems impossible to build a Gettierian counterexample that cannot be met by a sufficiently careful application of the perspectival definition of knowledge, since all of them suffer from a detachment between reasonable evidence and truth.

Inductive Evidence and Defeasibility
Unlike the old solutions, the perspectival understanding of the link between epistemic evidence and truth allows us to deal satisfactorily with the problem of inductive evidence. The given empirical evidence E must be seen by s (and certainly also by a), as sufficient for the truth of p, that is, as making it true with practical certainty. Moreover, this inductive evidence belongs to a more complex inductive framework, for the empirical evidence is viewed as being sufficient for the truth of p only on the assumption of background information and other relevant beliefs belonging to the whole stock of beliefs held by s at tj, and for this reason it is always susceptible to revision. That is: a change in a stock of beliefs brought about by new experience and information can always undermine evidential support, which fully conforms to what we expect from inductive inferences.
     We can illustrate the last comments by answering Hoffman’s objection that when we hold Df.k2, which requires that E => p, we must accept that Jones does not have knowledge that the floor of the hotel will support him, since his evidence E, being fallible, does not entail the truth of p. However, as we saw in the perspectival reformulation of the tripartite definition, the link between the conditions of evidence and of truth should be expressed by a context-dependent inductive relation in which E ~> p or not, depending only on whether E is accepted by the knowledge evaluator at the time of his evaluation of a’s knowledge claim as belonging to his E*, assuming that E* ~> p. In the case of Jones, the knowledge evaluator s, being certain that the floor could bear Jones’ weight, accepts his evidence E as being sufficient for the truth of p, accepting that E belongs to E*, and that E* makes p true, which is confirmed by his safely walking across the spot. Before Smith’s accident the knowledge evaluator can conclude that both Jones and Smith know p, as this conclusion is in conformity with his E* at the time. But after the accident, because s already knows that the floor could not support Smith’s weight, s drops his belief that the evidence E given by him is sufficient for the truth of p, for at this point in time he is aware that E belongs to E* such that E* ~> p, having changed his assumptions about what evidence is sufficient for p. However, s finds no reason to drop his view that Jones knows p. The perspectival definition is flexible enough to overcome the possible objection that the establishment of the truth of p by the evidence E is too stringent a requirement, for it makes the inductive relation relative to the context of evaluation of particular knowledge claims.
     Finally, the perspectival definition of knowledge also allows us to find a better place for the condition that epistemically adequate evidence must remain ultimately undefeated. When appended to a humble form of the tripartite definition like Df.k1, the condition that the justifying evidence must remain ultimately undefeated imposes a much too heavy burden on knowledge claimers, namely, that they must know the totality of truths, for this is the only way to warrant that the given evidence has no possible defeater. The problem with this conclusion is that it leads to scepticism, since no human epistemic subject can have access to the totality of truths. However, this condition makes sense again when transformed into a requirement that must necessarily be satisfied by each piece of evidence E given by a in order to be accepted as belonging to E*; this requirement (which is already implicit in Df.k3) states that E must remain ultimately undefeated by the totality of beliefs held by s at the time of his evaluation, which is reasonable enough.

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