Probability, frequency and evidence - Benenson, F; Frederick C. Benenson (1976)
﻿ Chapter I of this thesis considers the frequency theory of probability and in particular its treatmant of individual events. Attempts by frequentists to give an account of ordinary statements of probability which are generally about individual events are criticized at length: Reichenbach's attempts to deal with the problem of individual events is found to be unsatisfactory in virtue of his introduction of such dubious entities a 'fictitious probabilities'; Salmon's related suggestion to treat tha problem of the single case in terms of 'weights' determined by an 'application' of his theory of probability is also found to be unsatisfactory, for the concept of probability defined by his theory can not, on his own admission, be applied to the single case. Proposals by Keynes and Popper to treat the single case along frequentist lines, while apparently more promising, turn out in the end to be too rudimentary and sketchy to fulfill even the minimal condition for an adequate semantic definition of probability - the specification of a probability function which provides an interpretation of the axioms of probability. The basic difficulty for a frequentist in assigning probabilities to individual events is (not surprisingly) found in Chapter I to be that the probability of an individual event will vary depending on what reference class is chosen to determine its probability. In Chapter II however, the dependence of the probability of individual events on the reference class chosen is seen not to present a genuine difficulty in formulating an adequate semantic definition of probability. The variability of the probabilities assigned to individual events indicates that they are relational probabilities crucially involving a reference class in one term of the relation; once recognized the relational character of the probability of individual events provides the basis of an adequate semantic interpretation of the standard axioms of probability. It is shown that the resulting theory of probability can meet the objections raised by Salmon, Von Mises and Reichenbach against the assignment of probabilities to individual events and, indeed, this theory of probability resolves contradictions found to be inherent in Reichenbach and Salmon's treatment of the single case. Chapter III begins with a discussion of a methodological principle by which the relational theory given in Chapter II can be applied to actual situations to determine tbe unique probability values often needed for the purposes of action. This is the well-known principle of choosing the narrowest reference class for which statistics are available. This principle, usually referred to in the thesis as that of choosing the narrowest available reference class, is given a preliminary (and traditional) formulation at the start of Chapter III. Von Mises' requirement of randomness is then analyzed in light of this principle and it is found that if we employ this principle to determine unique probabilities, only random classes of events can be used to assign unique probabilities to every individual member of the class. This is found to account for the view - expressed in Van Mises' requirement of randomness - that only random classes are truly probabilistic. In Chapter IV various problems concerning the principle of choosing the narrowest available reference class are considered. Some of these are familiar, e.g. Ayer's objection to frequentists' use of this principle, the difficulties caused by unlawlike predicates, and the problems raised by the quite common absence of complete statistical knowledge. Less familiar problems are also considered. In particular Chapter IV is concerned with the question of precisely what kinds of reference classes can be considered 'available' for determining the probability of individual events. A solution to this problem is devised, based, in the first instance, on the concept of an effectively calculable function introduced by Church into discussions of randomness. A reference class is said to be available for determining the probability of an individual event if and only if there exists a determinate procedure which, if carried out, would yield the result that the event belongs to that reference class. In the case of reference classes formed by mathematical roles of selection, the existence of a determinate procedure leading to a result is equated with the existence of a recursive function leading to that result. In the case of reference classes based on empirical predicates, the class of determinate procedures is the totality of experimental procedures extant in the scientific field in question. This line of argument is extended to explicate the concept of 'available evidence', which figures prominently in many theories of probability. Various objections to this explication of available evidence are considered and a more refined analysis eventually emerges. Chapter V considers Carnap's thesis that there exist two distinct concepts of probability and it is shown that this conclusion can be avoided by adopting the frequency theory for the singe case given in Chapter II in place of the standard frequency theory - the definition of probability expressed in our theory is shown to be an instance of the definition of probability as a quantitative relation between evidence and hypothesis and thus to have distinct affinities to Carnap's theory of probability. Chapter V concludes with a discussion of subjectivism in theories which make probability a relation between evidence and hypothesis and it is shown how both our theory of probability and the logical relation theory avoid such subjectivism. Chapter VI and VII are concerned with the principle of indifference. Chapter VI begins with a survey of a variety of opinions on the principle and then presents a new interpretation of it as a rudimentary form of semantic definition of probability. It is shown that the principle, in stating conditions under which alternatives are equally probable, actually fixes identity conditions for the concept of probability and, as widely understood since Frege, any statement which fixes identity conditions for a concept fixes a particular sense for that concept, a particular semantics. The definition of probability encapsulated in the principle is that of probability as a comparative relation of evidential support. It is further explained how such a comparative conception of probability will have led to the numerical assignments of probability traditionally arrived at by use of the principle of indifference - when, as was the case with the principle's usual employment, a set of mutually exclusive and exhaustive hypothses are (comparatively) judged to be equally likely, it follows directly from the axioms of probability that each hypothesis has the numerical probability 1-n. That the principle of indifference encapsulates the semantic definition of probability as the comparative concept of evidential support is found to accord with the classical theorists' definition of probability as the ratio of favourable cases to possible cases, where each case is equi-possible. If we take the expression 'equi-possible' to be an undefined primitive term, the classical definition of probability constitutes a natural (and fruitful) uninterpreted axiomatic definition of probability. In accordance with the argument of the earlier part of Chapter VI, the principle of indifference - which even in classical times was regarded as fixing a sense or interpretation for the expression 'equi-possible' - then can be understood as providing, in a rudimentary way, a semantic interpretation for the basic undefined term of this axiom system. Chapter VII which concludes the thesis, analyses a variety of objections that have been raised against the principle and the analysis there supports the over-all claim that the principle of indifference was the first, and therefore, not surprisingly, somewhat confused, attempt to state the highly plausible definition of probability as the comparative concept of evidential support.