Fallibilism And The Principle Of Closure Under Known Implications

The lottery puzzle stems from the same structure as the lottery paradox, which is a modern paradox credited to Professor Henry E. Kyburg Jr. The lottery puzzle, much like the lottery paradox is episodic, dealing with belief or knowledge. To understand the lottery puzzle, I will analyze the concepts of fallibilism and the principle of closure under known implications. Then I will analyze the plausibility and the strength of the possible solution to the lottery puzzle: the denial of knowledge of ordinary propositions, acceptance of the conclusion, and denial of closure under known implications. The fallibilism principle states that “if it is both highly probable and true that p, then S knows that p”(cite). In order for p to have a high probability
of truth, its truth does not have to be absolute, but instead circumstantial knowledge is enough to suffice. From previous lectures, we know that knowledge implies truth so the only condition for this premise the subject, S, must not know that p is false, even if there is a possibility of it being false. The second principle for the lottery puzzle, the widely accepted principle under known implication states that “if S knows that p and if S knows that p implies q, then S knows that q” (cite). In this principle, p is intuitive to the knowledge of q, so we are able to expand our knowledge base to related items. One condition for this premise is that the subject, S, must have prior knowledge of the association between p and q, but the principle does not restrict how we know this association. This way we cannot imply things we have no prior knowledge of. Since we know that knowledge implies truth, the truth of p extends to the truth of q. The truth of these two premises p and q, extends to the conclusion of the argument. These two principles set the stage for the lottery puzzle, in terms of content and structure. The lottery puzzle is very similar to the lottery paradox, except it deals with knowledge.
The lottery puzzle is constructed of two propositions, the ordinary and lottery propositions and a conclusion, which is identical to the structure principle of closure under known implications. The principle of closure under known implications states that “if S knows that p and if S knows that p implies q, then S knows that q” (cite). The first proposition, the ordinary proposition would be considered p and states what we already know, such as “my car is parked at the Target parking lot”. The next proposition, the lottery proposition, which would be considered q and states something that is highly likely, but we are not willing to say we know them like “if I know that my car is parked in the Target parking lot, then my car has not been stolen in the last few minutes”. Following the principle of fallabilism, the lottery proposition within the puzzle has a strong chance of being true, but it does not have to be absolutely true, there just needs to be circumstantial evidence of truth. The principle of closure under known implications states this high likelihood of the lottery proposition being true extends from the known truth of the ordinary proposition, but because of its lack of certainty we know that there is still a chance of the lottery proposition being false. This type of argument leads to a conclusion that we are not in a position to know, like for example, the conclusion to the
two prior propositions that states “therefore, I am in a position to know that my car has not been stolen in the last few minutes”. This is why it is called a lottery puzzle, because even though there is a high likelihood of the lottery proposition being true, depending on the circumstance there is a chance of it being false. For example, if the parking lot has a high rate of car thefts then these thefts happen at random, just like the lottery, and it could be your car that is chosen. So because of this, we can now whether the q proposition is true or not so therefore, we cannot the conclusion. Lecture outlines three possible solutions to the lottery puzzle: the denial of knowledge of ordinary propositions or skepticism, the acceptance of the conclusion, and the denial of the principle of closure under known implications, In this next section I will analyze the strengths and weaknesses of each solution and from there establish the most plausible solution. The first attempt at resolving this lottery puzzle is deny that we know ordinary propositions. In order to solve the puzzle, we must find a way to make the conclusion true- this means either rejecting the lottery proposition or in the case the ordinary proposition. In the previous example, this solution would require us to reject the ordinary proposition that “ I know my car is parked in the Target parking lot”. In rejecting that we know the ordinary proposition, we deny the categorical truth of this proposition. This means that by denying the truth of the categorical proposition, we deny any extension of this truth to apply to the lottery proposition- which dissolves the original paradox. This solution raises several major problems though. For example, it is not plausible to take this kind of approach to the paradox because it violates both the knowledge as the norm of assertion principle and the principle about practical reasoning. The knowledge as the norm of assertion principle states that “assert that p only if you know that p”, so if we do not know ordinary propositions then we would not be able to assert anything. The principle of practical reasoning states that “in deliberating about what to do, you should only use known propositions as premises”. If we deny knowledge of propositions, then according to this principle it would flaw our practical reasoning. The next solution suggested is to accept the conclusion, in doing so we claim that we do know the lottery proposition. This would eliminate the puzzle, because then both propositions would be true which means we can accept the truth of the conclusion. Given the previous example, if we accept that “if I know that my car is parked in the Target parking lot, then my car has not been stolen in the last few minutes” then we would accept the conclusion that “if I know that my car is parked in the Target parking lot, then my car has not been stolen in the last few minutes” without question because it logically follows the argument. This solution raises several problems as well, in the form of inconsistent beliefs and practical inconsistency. The whole idea behind lotteries, is that there must be a winner in order for the lottery to be fair. So if we were to know the lottery propositions, like the example that our car has not been stolen then we can easily say this for the other cars in the parking lot. In order for it to be a fair lottery though, someone has to win or some car needs to get stolen. So, if we know the lottery propositions, then we cannot except that it is a fair lottery because in that case there would be no winner. Also, this solution creates practical inconsistency because if we know the lottery proposition, then we would know that our car would not be stolen and therefore we would lose the lottery. In this case, we would want to be the losers but if the lottery poses some reward for the participants then you would want to “win” the lottery- and if you already have knowledge that you will not then what is the point in participating in it. The last solution would require us to deny or restrict the principle of closure under known implication, which states that “if S knows that p and if S knows that p implies q, then S knows q” (cite).
Acceptance of this solutions would lead to inconsistent assertions; we realize the association between the two propositions by accepting how one relates to the other but we deny knowledge of the second proposition. In order to accept this solution, we would have to reject either the addition closure principle, the equivalence principle or the distribution principle. The addition closure states that if we know p and deduce q from p while retaining knowledge of p then we must know q. The equivalence principle states that you know p is equivalent to q then we are in a position to know q. Lastly, the distribution principle claims that if you both p and q then you know p and you know q. None of these are possible if we accept the principle on which the structure of this lottery paradox rests on. By restricting the principle of closure, we run into even more problems like creating an ad hoc solution, a vague definition of lottery propositions and weak logical principles that do not hold for all propositions. Restriction also brings into question relevance of alternative propositions, which according the objective reading has to do with the possibility of it being realized as relevant and according to the subjective reading relevance is dependent on whether or not the subject, or
person reciting it, believes it as being probable. Out of these three possible solutions I believe that the best solution to the lottery puzzle involves accepting the conclusion as being true by accepting knowledge of the lottery propositions. The only problems that come from this solution have to do with the idea of a fair lottery ensuring that there is a winner. If we reject the idea of a fair lottery, then we can accept the conclusion of the lottery paradox because we now have knowledge over the lottery proposition. We can also accept a solution where we have knowledge of certain lottery propositions but not others because some lottery propositions are more probable than others almost to a point where we can say it’s a sure thing. Lastly, if we think of this in terms of a lottery, much like our own lottery sometimes there is not a winner- sure there could be a winning ticket that was not realized or bought but there does not always have to be a “winner”.