In general when people think of probability they most likely think of gambling. Although it is true that gamblers can make good use of probability philosophers can benefit as much or more than them. Three areas that the mathematical study of probability can shed some light on are: ethics, epistemology, and the philosophy of science.
It might seem like the probability calculus would be useless in ethics, but when you consider that much of ethics is centered around rational action it is perhaps not such an outlandish idea. The obvious application is with an ethical theory that promotes self-interest as a course of action. Under such a theory the ethical person should choose the action with the greatest expected value. The expected value of an action is calculated as:
where o is a possible outcome for that action, Pr(o) is the probability of that outcome, and V(o) is the value of that outcome to the individual. Expected value can be applied to other ethical theories as well though. For example the act-utilitarian would also recommend the action with the greatest expected value, except that they would calculate the value of an outcome as the total happiness resulting from it, not as the benefits to a single individual. Expected value could even be applied to the Kantian categorical imperative. As usual the calculation is performed in the same manner, except that the value is now dependant on how many duties one could fulfill though that outcome, and would be reduced in by number of duties would fail to be fulfilled. Unfortunately the application of the probability calculus to ethics is not without problems. One difficulty is that there is no simple way to calculate the value of an outcome. We might think, for example, that in the case of the self-interested individuals the value could be calculated in terms of money, but it turns out that people value greater quantities of money less (the principle of diminishing returns) and so money is only an approximation, and the value is equally difficult to calculate for any application. A second problem is that realistically individuals cannot perform such calculations before making every choice (the same objection that rule-utilitarianism raises to act-utilitarianism). We could devise a set of rules before hand that encourage us to act in ways that maximize the expected value, but is possible following that such rules could lead to undesirable outcomes (a normally useful rule may occasionally recommend something obviously foolish). It seems then that we are stuck either advocating breaking the rules on occasion, which leads us back to the original dilemma, namely that deciding when to break the rules is too time consuming, or that we should conclude that maximizing our expected utility isn’t always the best course of action. Finally there is the problem that all people are somewhat risk adverse, which implies that people are naturally irrational, or that we haven’t quite captured the essence of rationality with the definition of expected value.
The application of the probability calculus to epistemology is much more straightforward than its application to ethics. Bayes’ theorem for example neatly formalizes how likely a hypothesis is given evidence, or in other words to what degree we should believe it, an issue central to epistemology. Bayes’ theorem is:
which gives the probability of the hypothesis, H, given a set of evidence, E. If we accept this formula as accurate then it is easy to determine what kind of evidence we should be searching for in order to prove a hypothesis, specifically evidence which we judge the probability to be low, but which, if the hypothesis is true, the probability will be much high (Pr(E) is low and Pr(E|H) is high). This is equivalent to advocating that we look for the predictions that are most likely to be false, which is exactly what falsificationists have been recommending for some time. Another interesting application is to information theory, which allows us to determine how much information is conveyed about a set of events S by a set of possible messages M as follows:
but I won’t get into that here.
Finally, the probability calculus can be useful to an investigation of the philosophy of science. Obviously epistemology overlaps somewhat with the philosophy of science, but there are uses of probability, such as when collecting data over a large number of trials, that aren’t covered by the belief type probabilities discussed above. For example will performing a large number of trials give you information about the actual probability of an event? Bernoulli’s theorem says that it will, although the number of trials required may be quite large. It may seem though that finding out about the probability in this way is useless, because if you have enough information about a situation, and you know the relevant physical laws, then you should be able to calculate the probability of an event directly based on what values the initial variables are allowed to range over. For example if you knew all there was to know about how coins moved through the air, as well as the ranges for possible air currents and initial force applied to the coin, then you should be able to deduce that the probability of heads is 1/2 without doing any trials. However there are two types of situations where figuring out the probability a priori, so to speak, is impossible. One is any event at the quantum scale, which as far as we can tell truly are random, and thus there is no way of determining how likely it is without doing the appropriate experiments. The other is chaotic situations, where the outcome is so sensitive to initial conditions that prediction becomes impossible, and hence finding out the probability without performing trials is equally undoable.
So, in conclusion, probability is more than about betting and making Dutch books against unsuspecting rubes. It is far more interesting, and more rewarding, to study probability with an eye to how it should influence how we act, what we believe, and how we gather information.