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[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:

• Frequency Interpretation: the probability is the limit of a frequency.
• Logical Interpretation: the probability is the degree of confirmation a statement receives from a given set of other statements.

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 two-valued logic, in which every statement is true or false. As Reichenbach said,

we do not infer from this fact that the justification of probability statements is impossible. We merely infer that the assumption of two-valued logic alone will not help. It is not possible to justify the system of scientific statements simply on the basis of deductive logic together with observational reports; this is our epistemological result. (ibid. p. 342).

According to Reichenbach, the two truth-values, 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.

It is a puzzle to me just how logical neo-positivism proposes to include assertions of probability in its system, and I am under the impression that this is not possible without an essential violation of its basic principles (Reichenbach, Selected Writings, vol. 1, p. 407).

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
    φ₁(x ₁₁) .... φ₁(x _1k)
    φ₂(x ₂₁) .... φ₂(x _2m)
    φ₃(x ₃₁) .... φ₃(x _3n)
    ....
    φ_r(x _r1) .... φ_r(x _rp)
be an enumeration of the testable propositions that are deducible from T. For example, φ₁(x₁₁) might stand for "The Geiger counter is deflected at moment x₁₁", φ₁(x₁₂) might stand for "The Geiger counter is deflected at moment x₁₂", and so on; φ₂(x₂₁) might stand for "A flash of light occurs at point x₂₁ of the screen", φ₂(x₂₂) might stand for "A flash of light occurs at point x₂₂ of the screen", and so on. Every line represents a propositional sequence; thus the first line represents the propositional sequence φ₁, the second line represents the propositional sequence φ₂, and so on. The probability of a propositional sequence is the ratio of the positive outcomes to the total outcomes, that is it is the ratio between the number of testable propositions that are true and the total number of testable propositions; in other words, this probability is the relative frequency of the true testable propositions. Let W(φ_i) be the probability of the propositional sequence φ_i; thus, according to Reichenbach, the first-form probability W(T) of the theory T is W(T) = W(φ₁)·W(φ₂)·....·W(φ _r).

Reichenbach defined also a second-form probability for a theory T: it is the probability of the statement "The first-form 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).

1. P(h,e) is the degree of the inductive support that the evidence e gives to a hypothesis h. It is determined by the semantic relation between the set of statements describing the evidence e and the set of statements describing the hypothesis h. In his late works, Carnap discarded this interpretation.
2. P(h,e) is a fair betting quotient for bets on h, where e is the total evidence. Carnap remarked that this interpretation is valid only if the risked amount is small compared to the fortune of the gambler.
3. P(h,e) is an estimate of a relative frequency. In this interpretation e describes the evidence, and h is a hypothesis assigning a property, say M, to an object which is not described by e. Thus P(h,e) is an estimate of the relative frequency of M in every class of objects that are not described by e. This interpretation gives a link between logical and statistical probability.
4. Let X be a rational agent and let U_X be the utility function of X [i.e., for every object O, the value of U_X (O) is a measure of the utility of O for X]. Consider the following offer: "X receives an object O if and only if the event h occurs". The value V of this offer for X is given by the formula V = P(h,e)·U_X(O), where e is the evidence. The rational agent X must accept the offer with the higher value V. In this interpretation logical probability is linked to the theory of decision.

Waismann proposed a logical interpretation of probability in his work "Logische Analyse des Wahrscheinlichkeitsbegriffs" in Erkenntnis, 1, 1930. His starting-point 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.

Synthetic and Analytic Statements

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