So that
Boole gives (25), but not (26).
We pass now from the consideration of identities to that of equations.
Let every expression for a class have a second meaning, which is its meaning in an equation. Namely, let it denote the proportion of individuals of that class to be found among all the individuals examined in the long run.
Then we have
Let ba denote the frequency of b’s among the a’s. Then considered as a class, if a and b are events ba denotes the fact that if a happens b happens.
It will be convenient to set down some obvious and fundamental properties of the function ba.
The application of the system to probabilities may best be exhibited in a few simple examples, some of which I shall select from Boole’s work, in order that the solutions here given may be compared with his.
Example 1. Given the proportion of days upon which it hails, and the proportion of days upon which it thunders. Required the proportion of days upon which it does both.
Let 1
p
q
r
Answer. The required proportion is an unknown fraction of the least of the two proportions given, not less than p + q — 1.
By p might have been denoted the probability of the major, and by q that of the minor premise of a hypothetical syllogism of the following form:—
If a noise is heard, an explosion always takes place;
If a match is applied to a barrel of gunpowder, a noise is heard;
∴ If a match is applied to a barrel of gunpowder, an explosion always takes place.
In this case, the value given for r would have represented the probability of the conclusion. Now Boole (p. 284) solves this problem by his unmodified method, and obtains the following answer:—
where a is an arbitrary constant. Here, if q = 1 and p = 0, r = 0. That is, his answer implies that if the major premise be false and the minor be true, the conclusion must be false. That this is not really so is shown by the above example. Boole (p. 286) is forced to the conclusion that “propositions which, when true, are equivalent, are not necessarily equivalent when regarded only as probable.” This is absurd, because probability belongs to the events denoted, and not to forms of expression. The probability of an event is not altered by translation from one language to another.
Boole, in fact, puts the problem into equations wrongly (an error which it is the chief purpose of a calculus of logic to prevent), and proceeds as if the problem were as follows:—
It being known what would be the probability of Y, if X were to happen, and what would be the probability of Z, if Y were to happen; what would be the probability of Z, if X were to happen?
But even this problem has been wrongly solved by him. For, according to his solution, where
r must be at least as large as the product of p and q. But if X be the event that a certain man is a Negro, Y the event that he is born in Massachusetts, and Z the event that he is a white man, then neither p nor q is zero, and yet r vanishes.
This problem may be rightly solved as follows:—
Developing these expressions by (18) we have
The comparison of these two identities shows that
Then let
And we have
Ex. 2. (See Boole, p. 276.) Given r and q; to find p.
Ans. The required proportion lies somewhere between the proportion of days upon which it both hails and thunders, and that added to one minus the proportion of days when it thunders.
Ex. 3. (See Boole, p. 279.) Given, out of the number of questions put to two witnesses, and answered by yes or no, the proportion that each answers truly, and the proportion of those their answers to which disagree. Required, out of those wherein they agree, the proportion they answer truly and the proportion they answer falsely.
Let 1
p
q
r
w
w’
Now by (28.)
Substituting and transposing,
The differences of Boole’s system, as given by himself, from the modification of it given here, are