A critical examination of the three main interpretations of probability

UnrestrictedThe three main interpretations of probability (i.e. of probability-statements) are the classical/logical, the subjectivist and the relative-frequency. My dissertation is a critical examination of each of these interpretations.; The core tenet of the classical interpretation is that whenever a set of outcomes of an event is such that a) no information is available about any of the outcomes or b) symmetrical information is available about each outcome, then each outcome must have the same probability. This is usually known as the Principle of Indifference. The logical interpretation –which descends directly from the classical interpretation– differs from the classical interpretation in that: i) outcomes of the same event may be assigned unequal probabilities and ii) probability is a logical relation between two complex propositions e and k (k is the information relevant to the occurrence of e). I discuss the classical and the logical interpretations in chapter 1.; Nowadays, the classical interpretation is not held in high consideration. The accepted view is that the principle of indifference, the core of this interpretation, faces, in certain infinite domains, a devastating paradox. In my dissertation, I contend that this is not the case. By distinguishing between a weak principle of indifference and a strong principle of indifference, I show that it’s not at all clear that there is such a paradox. Thus, the classical interpretation is unproblematic in this respect.; But the classical interpretation is not unproblematic in general. Moreover the classical interpretation is affected by the very same two problems that affect the logical interpretation. First, both the classical and the logical interpretations presuppose Kolmogorov’s axioms of probability (or an equivalent axiomatization). Without Kolmogorov’s axioms, an outcome cannot be assigned a unique probability in principle and it is impossible to establish a probabilistic hierarchy between sets of outcomes (i.e. either two sets of outcomes have the same probabilities or their probabilities are not commensurable). The problem is that it is not at all clear how to justify such a presupposition. Surprisingly, only a few philosophers have appreciated the seriousness of this issue. The second problem is that neither the classical nor the logical interpretation can provide an adequate explication of probability. In particular my analysis shows that any possible explication (within these two interpretations) is either unacceptable because it is confused or contradictory, or is unfit for the job (because it doesn’t express or relate to any of the ideas that are usually associated with probability, such as indetermination, prediction, indecision). This is obviously a major problem, since the raison d’être of the classical and the logical interpretations is to interpret, i.e. to explicate, probability.; The subjectivist interpretation (subjectivism) identifies probabilities with the numerically formulated degrees of beliefs of individual agents. Subjectivism is the object of chapter II. A large part of the appeal of this interpretation is due to the alleged fact that numerical degrees of beliefs must satisfy the axioms of Probability Theory (Kolmogorov’s axioms) on pain of irrationality. This (alleged) fact is said to be the consequence of either of two arguments, the Dutch Book Argument and an argument from representation theorems, the Representation Theorems Argument. I show (in the first part of the chapter) that both the Dutch Book Argument and the Representation Theorems Argument are flawed. This entails that subjectively interpreted probabilities cannot be proved to satisfy the axioms of probability and with that a large part of the appeal of subjectivism disappears. The second part of the chapter is devoted to Bayesian Confirmation Theory (BCT), a subjectivist doctrine according to which one’s (numerical) degrees of belief should be updated strictly by way of a probabilistic equation called Bayes’ Theorem. It is fair to say that BCT is the most important doctrine that originates from the subjectivist interpretation. The main claim (and the whole point) of BCT is that if certain (allegedly mild) conditions are respected then probabilistic induction is possible and perfectly justified. I argue that this claim is in fact false. Firstly, I reconstruct Popper’s famous criticism of BCT qua justification of probabilistic induction. Popper pointed out that the conditions under which BCT allows probabilistic induction are anything but mild, and in fact one of these conditions presuppose induction to begin with. I address several objections that have been made against his argument. Secondly, I present a new, improved version of Popper’s original argument.; Thirdly, I present a novel argument to the effect that BCT cannot provide a genuine justification of probabilistic induction.; The relative-frequency interpretation (frequentism) asserts that the probability of an outcome is its frequency of occurrence (relative frequency). Historically, frequentism has come in two variants: finite frequentism –a relative frequency is computed on the basis of a finite sequence of instances (occurrences and non-occurrences) of an outcome– and infinite frequentism –the sequence of instances of an outcome must be infinite. However, the vast majority of the proponents of frequentism (frequentists) have endorsed infinite frequentism. I discuss frequentism in Chapter 3. After clarifying why infinite frequentism is to be preferred to finite frequentism and after taking care of a few difficulties related to the former, I discuss what I take to be the central problem of infinite frequentism: the fact that probability-values can neither be verified nor falsified (rejected). I show then that this problem may be mitigated by means of an “unorthodox” infinite acceptance test of Classical Statistics that may be used as a “quasi-falsification” procedure for infinite frequentism.; Since within infinite frequentism probability-values are not logically falsifiable and since there are reasons why dealing with infinitely many instances of an outcome is undesirable, I present a new (as far as I know) form of frequentism. This new version of frequentism, I argue, has all the advantages of infinite frequentism, but lacks those two drawbacks.; Lastly, let me briefly anticipate the conclusion I reach in my dissertation. Of the three main interpretations of Probability Theory, frequentism (either in its classical form or in the novel variant I present) seems to constitute a viable option; this doesn’t mean of course that frequentism is unproblematic; it means, rather, that frequentism is the least problematic among the available interpretations