¹ Cf. Professor Ray H. Dotterer, “Ignorance and Equal Probability,” in Philosophy of Science, v. 8, no. 3, July, 1941, pp. 297-303.

² By “event”, I mean an individual single occurrence, i.e., an instance of a class of similar events, which is, in terms of Aristotelian logic, an infima species.

³ Dotterer formerly admitted this fact; cf. his Beginners’ Logic, p. 205. He now asserts that this statement is not quite accurate and declares that “statements about the fall of a coin or the death of a man are statements about the relation of an individual to a class.” On this point, cf. my n. 4.

⁴ The fallacy of division does not consist merely in the ambiguity of the word “all”, but in the fact that certain statements about groups do not apply to members of the group, so that inferences from such groups to their members are invalid.

If Professor Dotterer means literally that probability statements are merely “statements about the relation of an individual to a class,” he is correct. But the class to which John Jones belongs is strictly that of persons aged thirty; the probability frequency is, strictly speaking, a characteristic of the group aged thirty. A statement about the relation of an individual to a class can be put much more simply than by giving John Jones a certain probability frequency of death. It is quite sufficient to assert that he belongs to the class of those aged thirty (which indeed is the minor premise necessary to assign him this probability); this probability of death is only one of the various characteristics of the group aged thirty. If all that probability statements about individuals mean is the assigning of them to classes, why mention probabilities? It would seem enough merely to state the class to which they immediately belong, without adding the characteristics of that class. Since individual events moreover do not exhibit the frequencies of the class to which they belong, probability statements about such individuals are misleading, to say the least.

The real reason for assigning to John Jones or any other individual the probability of the class of persons aged thirty is to justify a certain psychological expectation about his death. But expectations and frequencies are different and should not be confounded.

⁵ This statement is taught by Pierce and Keynes and accepted by Dotterer; cf. op. cit. in Phil. of Science, p. 299.

⁶ Dotterer, ibid., 299.

⁷ A “fair coin” should not be defined circularly as one that obeys the mathematical laws of probability. A good definition of a fair coin is that it is one which is symmetrical (both externally in its shape and internally in its composition) about its two axes: about the line composing the center of the circles that is its circumference, and also about the plane midway between its two faces. Similarly, a fair die is one that is symmetrical about its three axes.

Few actual coins are completely “fair” in this exact sense; they nearly always have a little extra metal on the “head” side. Hence it should be expected (what is also an observed fact) that there is a slight tendency for more falls of heads than of tails. Since the departure of most actual coins from complete symmetry is so small, this deviation from the behavior of completely fair coins is quite slight and is observable only in long runs.

⁸ J. M. Keynes, A Treatise on Probability, p. 4.

⁹ Dotterer, ibid., p. 297.

¹⁰ Ibid., p. 303.

¹¹ The term, “degrees of knowledge,” is frequently another circumlocution for the fact that individuals belong to groups having differing frequencies. For example, we are said to have differing degrees of knowledge about the character of an individual being when we discover, successively, that he is (1) a human being, (2) a Caucasian, (3) a European, (4) British, and (5) an Englishman. These five are each groups possessing differing frequencies of occurrence for various characteristics. Placing him in each of these five groups may raise differing expectations in our mind concerning his honesty, corresponding to the frequency of honesty in each of these five groups, but in no case can we infer logically that he, as an individual, is either honest or dishonest, since either event would be quite consistent with the statements of frequency that are made. The case is precisely analogous when we discover that a certain putative truth (or hypothesis) denotes a type of event that (1) has occurred once, (2) has occurred many times, and (3) occurs constantly. If even constant occurrence is still consistent with the occurrence of exceptions (as even the law of the conservation of energy is now believed to be contradicted by the change of matter into energy), these three discoveries merely indicate that this putative truth belongs to groups having different (approximate) frequencies, and we ought not logically to infer anything (except an expectation) concerning the truth of this individual hypothesis from the fact that it belongs to any of these groups.

The assignment of a person or a putative truth to any one of these classes (except the widest one) does not presuppose the principle of insufficient reason; we have no right to assign any individual to a class (and consequently to entertain any justified expectation) except on quite adequate knowledge that he does belong to that class. The principle of insufficient reason is only useful in assigning an individual to the class of those events about whose frequency we are entirely ignorant, and hence in entertaining no expectation of one type of consequence rather than another. Ignorance justifies the assignment of an event to no class whatever, except the class of events about which we are ignorant.

¹² Ibid., p. 303.