Or any individual can be expressed in the form
(23) Let then any individual a be
Then by this definition
(24) Hence by def. 3
(25) Hence, by cond. 1 of def. 7,
(26) Hence, by cond. 4 of def. 7
Any Aa is a.
(27) In the same way if (Am,Bb) is any individual b
Any Bb is b.
(28) Then (Aa,Bb) belongs to the A which the individual a belongs to. For, by def. 3
Any (Aa,Bb) is Aa
And by (23) the individual a is Aa,Bn. Whence by def. 3, the individual a is Aa. Whence by cond. 1, of def. 7, the individual a belongs to no other A than Aa.
(29) Similarly (Aa,Bb) belongs to the B to which the individual b belongs.
(30) Moreover by def. 3 and (26) (Aa,Bb) is a.
(31) Similarly it is, also, b.
(32) Moreover by (19) it is an individual.
(33) And by cond. 6 of def. 7 it exists.
(34) In the same way, it could be shown that (Am,Bn) is an existent individual which belongs to the same A which the individual b belongs to, and the same B which the individual a belongs to.
Thus, for any individuals one a and the other b, there exists an individual which belongs to the same A as a and the same B as b, and an individual which belongs to the same A as b and the same B as a, provided a and b are independent.
This is the first part of our lemma.
[Venn’s The Logic of Chance]
P 21: North American Review 105(July 1867):317–21
The Logic of Chance. An Essay on the Foundations and Province of the Theory of Probability, with especial Reference to its Application to Moral and Social Science. By John Venn, M.A. London and Cambridge. 1866. 16mo. pp. 370.
Here is a book which should be read by every thinking man. Great changes have taken place of late years in the philosophy of chances. Mr. Venn remarks, with great ingenuity and penetration, that this doctrine has had its realistic, conceptualistic, and nominalistic stages. The logic of the Middle Ages is almost coextensive with demonstrative logic; but our age of science opened with a discussion of probable argument (in the Novum Organum), and this part of the subject has given the chief interest to modern studies of logic. What is called the doctrine of chances is, to be sure, but a small part of this field of inquiry; but it is a part where the varieties in the conceptions of probability have been most evident. When this doctrine was first studied, probability seems to have been regarded as something inhering in the singular events, so that it was possible for Bernouilli to enounce it as a theorem (and not merely as an identical proposition), that events happen with frequencies proportional to their probabilities. That was a realistic view. Afterwards it was said that probability does not exist in the singular events, but consists in the degree of credence which ought to be reposed in the occurrence of an event. This is conceptualistic. Finally, probability is regarded as the ratio of the number of events in a certain part of an aggregate of them to the number in the whole aggregate. This is the nominalistic view.
This last is the position of Mr. Venn and of the most advanced writers on the subject. The theory was perhaps first put forth by Mr. Stuart Mill; but his head became involved in clouds, and he relapsed into the conceptualistic opinion. Yet the arguments upon the modern side are overwhelming. The question is by no means one of words; but if we were to inquire into the manner in which the terms probable, likely, and so forth, have been used, we should find that they always refer to a determination of a genus of argument. See, for example, Locke on the Understanding, Book IV, ch. 15, §1. There we find it stated that a thing is probable when it is supported by reasons such as lead to a true conclusion. These words such as plainly refer to a genus of argument. Now, what constitutes the validity of a genus of argument? The necessity of thinking the conclusion, say the conceptualists. But a madman may be under a necessity of thinking fallaciously, and (as Bacon suggests) all mankind may be mad after one uniform fashion. Hence the nominalist answers the question thus: A genus of argument is valid when from true premises it will yield a true conclusion,—invariably if demonstrative, generally if probable. The conceptualist says, that probability is the degree of credence which ought to be placed in the occurrence of an event. Here is an allusion to an entry on the debtor side of man’s ledger. What is this entry? What is the meaning of this ought? Since probability is not an affair of morals, the ought must refer to an alternative to be avoided. Now the reasoner has nothing to fear but error. Probability will accordingly be the degree of credence which it is necessary to repose in a proposition in order to escape error. Conceptualists have not undertaken to say what is meant by “degree of credence.” They would probably pronounce it indefinable and indescribable. Their philosophy deals much with the indefinable and indescribable. But propositions are either absolutely true or absolutely false. There is nothing in the facts which corresponds at all to a degree of credence, except that a genus of argument may yield a certain proportion of true conclusions from true premises. Thus, the following form of argument would, in the long run, yield (from true premises) a true conclusion two-thirds of the time:—
A is taken at random from among the B’s;
∴ A is C
Truth being, then, the agreement of a representation with its object, and there being nothing in re answering to a degree of credence, a modification of a judgment in that respect cannot make it more true, although it may indicate the proportion of such judgments which are true in the long run. That is, indeed, the precise and only use or significance of these fractions termed probabilities: they give security in the long run. Now, in order that the degree of credence should correspond to any truth in the long run, it must be the representation of a general statistical fact,—a real, objective fact. And then, as it is the fact which is said to be probable, and not the belief, the introduction of “degree of credence” at all into the definition of probability is as superfluous as the introduction of a reflection upon a mental process into any other definition would be,—as though we were to define man as “that which (if the essence of the name is to be apprehended) ought to be conceived as a rational animal.”
To say that the conceptualistic and nominalistic theories are both true at once, is mere ignorance, because their numerical results conflict. A conceptualist might hesitate, perhaps, to say that the probability of a proposition of which he knows absolutely nothing is ½, although this would be, in one sense, justifiable for the nominalist, in as much as one half of all possible propositions (being contradictions of the other half) are true; but he does not hesitate to assume events to be equally probable when he does not know anything about their probabilities, and this is for the nominalist an utterly