Probability and Inductive Logic


[2. The Main Philosophical Tenets of Logical Positivism.]f. Probability and Inductive Logic.There were two different theories about probability proposed by the logical positivists:
According to Reichenbach, the meaning of a statement such as "the probability of P given Q is r" is that the limit of the relative frequency of objects P in the set of objects Q is r. That is, in a large sample of the set Q [say m objects from Q, where m is great] there are n objects P, and lim (n/m) = r, as m tends to infinity. Therefore Reichenbach asserted that a statement about the probability of a single event is meaningless. A statement about the probability of a single event  Reichenbach asserted  is an abbreviation: it refers to a whole series of events to which the event in question is assigned. A fundamental problem regarding probability is "Why are we justified in inferring that the observed relative frequency in a sequence of events will be preserved in a future continuation of the sequence?" (Reichenbach, Selected Writings, vol. 2, D. Reidel: Dordrecht, Holland, 1978, p. 339). This question  according to Reichenbach  is connected with the problem of induction. Suppose that the event B has happened m times among n events. Thus induction suggests that the probability of B is (m/n) ± δ, for a small value of δ. This conclusion is not necessarily true. Instead, induction supplies an asymptotic rule that, if the sequence has a limit of the frequency, eventually arrives at reliable predictions. Reichenbach admits that this inductive inference cannot be justified logically. However, this is not a problem, because in general probability statements are not meaningful in a twovalued logic, in which every statement is true or false. As Reichenbach said,
According to Reichenbach, the two truthvalues, true and false, must be replaced by a continuous scale of probability. Hume's problem of induction is thus "resolved on the grounds that the demand for a justification of probability in terms of deductive logic is unreasonable" (ibid. p. 344). Reichenbach's attitude towards induction and probability is in many respects dissimilar from the point of view of the other logical positivists, and frequently it seems in explicit opposition to Carnap's interpretation of probability. The following assertion, cited from Reichenbach's review of Carnap's Logical Structure of the World, is symptomatic of the differences between Carnap and Reichenbach.
Reichenbach's interpretation of probability seems very similar to Richard von Mises' frequency interpretation, and often these two interpretations are presented together. However, when Russell presented Reichenbach's theory as a development of von Mises' theory (Russell, Human Knowledge, 1948), Reichenbach wrote in a letter to Russell that his theory cannot be considered a development of that of von Mises. Reichenbach asserted that his first publication on probability had a frequency interpretation and was earlier than von Mises' publications. Moreover, Reichenbach said that von Mises' theory lacks of an application to the theory of induction and it is not connected with the logical symbolism.
Reichenbach used his frequency interpretation to
give a measure of the probability of a theory.
Suppose that T is a scientific theory; let Reichenbach defined also a secondform probability for a theory T: it is the probability of the statement "The firstform probability of T is q". This distinction is analogous to the distinction between the probability of the statement "Side six shows when the die is thrown" and the probability of the statement "The probability that side six will show when the die is thrown is 1/6". Carnap distinguished between two concepts of probability: statistical probability and logical probability. A statement about statistical probability belongs to the language of scientific theories and describes something about the facts of the nature. Hence it is an empirical statement (a synthetic statement), whose truth can be determined only by means of empirical procedures. On statistical probability Carnap agreed with Reichenbach's frequency interpretation. A statement about logical probability is an analytic statement, independent from the experience, whose truth is determined a priori. Carnap's works on probability were mainly dedicated to an explication of the concept of logical probability. Let P(h,e) be an abbreviation for "logical probability of h with respect to e". Carnap distinguished four aspects of the meaning of P(h,e).
Waismann proposed a logical interpretation of probability in his work "Logische Analyse des Wahrscheinlichkeitsbegriffs" in Erkenntnis, 1, 1930. His startingpoint is Wittgenstein's interpretation of probability. According to Waismann, we have to use the theory of probability when we do not know whether a proposition is true or false. In that circumstance, we can study the logical relationships between the statements that express our knowledge and we can determine their relative probability. Hence a probability is a mathematical measure of a logical relationship between propositions. What is the role of frequency in the logical interpretation? First of all, it is possible that we know so little about a physical condition that we can determine the probability only a posteriori by means of the frequency. Therefore, the relative frequency and the logical probability are obviously equal. In other circumstances, we can predict the probability through our knowledge of the relevant conditions and physical laws. In such situation, the frequency is used to verify the forecast.
