The Probabilistic argument.

On the other hand, the implication that mere passing of time in some cases may even create some luck sounds doubtful. One may object that consideration of time intervals and S’s actions involves even more implicit assumptions. Especially, the question is still not clarified in the light of obvious intuition that someone could be “gettiered” by Russell’s clock, in principle. It seems we have to do better.

If a statement makes sense then this can be translated from one language to another. In order to answer on a possible objection I will try to draw an analogy between the forms of Gettier’s and lottery’s cases and the mathematical language of the Theory of probability. If this attempt is success we’ll get the corresponding consistent statements.

Note that a probability P is a measure of the chance that an event will occur. The P is *a priori *prognosis, before the fact. If an event happened then we may ask what was the probability of the event in past? Same characteristic has conditional probability. The probability of an event A given that the event B occurred is the forward probability. We can determine the *a posteriori *reversed probability P (B given A) as if the event A occurred then P is the probability that one of possible condition B had place. Thus, this issue is also about past events and therefore this can be expressed in terms of forward probabilities using Bayes’s Rule.

Next, I will consider possible interpretations of three probabilistic cases: P (A), P (A and B), P (A given that B).

The simple case P (A) shows the chance that an event A will occur. It is the form of the lottery’s case, for example. Can we ask about degree of cognoscibility of possible future results of a throw of the dice and tossing the coin? Can I know more authentically that tomorrow I will see a car accident than a fall of a meteorite? Since we are discussing that “S knows that p” and “p is truth” these events are equally unknowable despite the fact that the probabilities are not equal. The prognosis is not the fact exclude only one possibility when P (A) = 1. The probability of a future single event does not indicate that it is knowable if P (A) < 1. But iff P (A) = 1 we can assert that the event will definitely happen and therefore we know it now. Note that the case P (A) = 1 does not depend from a degree of luck rather from the presence some sort of luck.

Because of that the version of the safety principle proposed by Duncan Pritchard, (SP**), seems to work properly in lottery’s style cases:

S’s belief is safe iff in most nearby possible worlds in which S continues to form her belief about the target proposition in the same way as in the actual world, and in all very close nearby possible worlds in which S continues to form her belief about the target proposition in the same way as in the actual, the belief continues to be true (see Pritchard (2007: 290-2)).

Indeed, if we take the set of all very close nearby possible worlds as a sample space comprised of the collection of all possible outcomes then we get the certain event, P (A) = 1. Here the event A is “that p is true”. Moreover, there is no vagueness about “very close nearby possible worlds”. They differ from the actual world only in the results of the experiment “that p”.

Can we draw some information about events then P (A) <1? We saw that they are likely epistemic unknowable but we can estimate the chance of S to be involved in the process of cognition of the event A in future. It will play important role in next cases.

There is some kind of coincidence in the Gettier’s cases or using Zagzebski’s account there is the coincidence of “good luck” and “bad luck” (see Zagzebski 1994, 66 p). We may describe that in two ways. Either there is conjunction of two events A and B or there is the event A given the condition event B. Implicitly we tend to look on those situations as indistinguishable. But if we take an event B as a condition then the other event A occurs *either simultaneously or later*. As we saw conditioning can work in an unexpected way. We have by definition of conditioning probability that P (A given that B) = P (A and B) / P (B), 0 < P (B) < or = 1. Therefore, always P (A given that B) = or > P (A and B). Note, that the probability here rather describes the chance for S to be “gettiered”. Certainly, it is more likely that S is “gettiered” by already stopped clock than the coincidence that S consults clock in a very same moment of stop. The participant of the Gettier’s cases operates in time and consequently the estimation of probability helps us to assess the likely degree of friendliness of epistemic environment.

We can assert that in some Gettier’s cases the mere passing of time makes the environment epistemic risky or unsafe. Moreover, we can strictly say that if there is a Gettier’s case as the coincidence a good luck event and a bad luck event then it is more likely that the one event occurred earlier than the other. This statement corresponds with the Bayes’s Rule for reversed probability as well.

Thus, we show from the probabilistic point of view that in some cases the mere passing of time keeps the degree of epistemic luck; in the others it even arises this degree.

There may still be an objection. Why we don’t care about epistemic environment in a lottery’s style case? The answer can be given as follows: we have limited class of cases by the restriction that P (A) =1. It eliminates the influence of the circumstances because there are certain events.

The other class of cases includes more possibilities. In the case of coincidence, events could not happen but they happened by describing of case. This class of cases include also events that happened certainly but coincidence with another event doesn’t occur necessarily. For example, I know that tomorrow will be sunrise. I don’t know that I will see it for sure. On the other hand, today I saw sunrise and I knew that there was sunrise and I saw it. In addition there is another case that today was sunrise even if I didn’t see it because I was sleeping.

The scheme for constructing definition.

As we’ve seen, the parameter “a moment of time t” has deep and counterintuitive connection with luck in some important cases and it can be taken as the boundary condition for the definition of knowledge. Thus, all situations that “S knows that p” are divided into two types. Either S knows in the moment t that p (t) or S knows in the moment t that it will be that p (t + dt), dt > 0.

In order to establish the working model which can be diagnosed we draw two principles. One is modification of Prtchard’s safety principle: S’s belief is safe iff in *all* nearby possible worlds in which S continues to form her belief about target proposition in the same way as in the actual world the belief continues to be true. As the consequence of this principle there is safety condition (SC): S knows that p if S’s belief is safe.

The second useful principle lies at the base of the argument of the creditable approach to the account of knowledge and follow variation of ability conditions (AC): S knows that p if S believes the truth because of the exercise of S’s relevant cognitive abilities.

Thus, the scheme for definition of knowledge is formed in a following way:

If S knows that p (t) then apply ability condition (AC)

If S knows that will be p (t + dt), dt > 0 then apply safety condition (SC).

In addition, note that there are several types of epistemic luck: content-, capacity-, evidential- and doxastic epistemic luck. Despite the fact that some of them contain coincidences they are all compatible with knowledge (see Pritchard (2005), p. 140). But veritic epistemic luck is only one bad sort of luck for knowledge. It may be worth to consider two types of veritic luck. The probabilistic veritic luck overtime and afflicts future events. The coincidental veritic luck afflicts the current events. As we’ve seen, it is not at all easy to make proper intuitive evaluation of the sort and degree of those types of luck.

The diagnosis.

The consideration of proposing frame for definition depends on our attitude towards to the understanding of the word “know”. If we tend that one cannot know about wining in the fair lottery even though the chances to win could be either 1% or 99% then safety condition has advance in the lottery case and in the case of superstitiously unreliably formed belief. In recent literature is reviewed in detail.

We have to pay attention to the cases that S in the moment t knows that p (t). By way of illustration, consider the case of Henry in Barn Façade Country but using pervious methodological intuitions about describing of case. Let’s suppose that Henry is at the border of Barn Façade Country which consist 99 fake barns and one real barn. The real barn placed randomly somewhere along the Henry’s road and the probability for every position is 1/100. If Henry will take a look randomly only one time on one of barn façades what are the chance for him to be “gettiered”? Note that if Henry will normally look at a barn façade and form a false belief that he is looking at the barn this is nothing to worry about. We want to evaluate the chance that he will look at the one real barn. Suppose that probability that Henry will look at the façade number N is 1/100 for every N then we have situation with two random value and the chance for Henry to be “gettiered” is 1/100 * 1/100 or 0,0001. Let’s change the circumstances that this country consists 50 fake barns and 50 real barns distributed randomly and equiprobably and probability for any position has a real barn or fake barn is ½. In this case the chance for Henry to be “gettiered” is 1/100 * ½ or 0,002. So, despite of friendliness of environment where fake barns are less common the chance to be involved in Gettier’s case is even higher. Those chances will change dramatically if we change slightly any probability. So, different describing or understanding of case can implicitly change the chances. This illustration demonstrates that the safety condition has a strong claim that p has to be true in *all* nearby possible worlds. This claim put us immediately in position that we cannot know about any uncertain event even if it happened already or happens right now before us.

So, we need abandon the safety condition to other alternatives. It can be ability condition or may be someone can propose more precise developing of definition in case of S knows that p (t). Indeed, if Henry deserved the credit by getting out from his car and taking precise consideration of his environment then we can account him as a knower. It seems as very strong demand to form researching strategy for a knower but really we know only a little using usual cognitive patterns in supposing of existence of unusual circumstances.

Of course, still many objections could be imagined. There are cases as meeting of a protagonist of a case with “epistemic demon” who changes any circumstances. The demon set a clock, change a data and equipment and so forth. I cannot provide entire consideration of “demon’s cases” exclude one notation that methodologic intuition appeals to be careful in the possible degree of idealization in counterexamples because it easy put us before a row of infinite sceptic demands. Anyway, further I will discuss one counterexample proposed by Pritchard which can be associated with the safety condition or ability condition as well.

Archie is a professional archer. He goes to the shooting range, picks a target, and takes a shot. Suppose that unbeknownst to Archie, he is shooting at the only target at the shooting range that is not equipped with a hidden forcefield that would repel any arrow fired at it. There are two events in this case. The first event is “Archie in the moment t chooses the right target”. The second event is “Archie is shooting the target”. At the first event Archie didn’t use properly his cognitive ability because he didn’t check the shooting range. If we suppose that it is a sports qualification with special service at the field and the service makes possible the sabotaged targets then we are in a situation where “an epistemic demon” has able to eliminate any our knowledge. If Archie just came with friends into forest to take some fun by shooting targets then it is acceptable for him to prepare carefully the shooting range. The second event is considering easily. Archie knows that he will hit the target iff from that location (relevant initial conditions) he will do it in all nearby possible worlds. Actually, even it is counterintuitive Archie’s arching ability doesn’t need to be explore deeply in this case.

Conclusion.

Eventually, we can assert that common intuition about sort and degree of luck is misleading in some important cases. There are crucial distinctions in degree of luck in cases of coincidence of events and conditioned events. It is not always obvious how they can be detected in descriptions of counterexamples. One of the methods for detecting degree of luck is to consider the passing of time and thus to be able to form the scheme for definition of knowledge by using the time axis. Therefore it is possible to make noncontradictory definition consisting of different epistemic intuitions.