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On an Improvement in Boole’s Calculus of Logic

P 30: Presented 12 March 1867

The principal use of Boole’s Calculus of Logic lies in its application to problems concerning probability. It consists, essentially, in a system of signs to denote the logical relations of classes. The data of any problem may be expressed by means of these signs, if the letters of the alphabet are allowed to stand for the classes themselves. From such expressions, by means of certain rules for transformation, expressions can be obtained for the classes (of events or things) whose frequency is sought in terms of those whose frequency is known. Lastly, if certain relations are known between the logical relations and arithmetical operations, these expressions for events can be converted into expressions for their probability.

It is proposed, first, to exhibit Boole’s system in a modified form, and second, to examine the difference between this form and that given by Boole himself.

Let the letters of the alphabet denote classes whether of things or of occurrences. It is obvious that an event may either be singular, as “this sunrise,” or general, as “all sunrises.” Let the sign of equality with a comma beneath it express numerical identity. Thus is to mean that a and b denote the same class,—the same collection of individuals.

Let denote all the individuals contained under a and b together. The operation here performed will differ from arithmetical addition in two respects: 1st, that it has reference to identity, not to equality; and 2d, that what is common to a and b is not taken into account twice over, as it would be in arithmetic. The first of these differences, however, amounts to nothing, in as much as the sign of identity would indicate the distinction in which it is founded; and therefore we may say that

It is plain that

and also, that the process denoted by and which I shall call the process of logical addition, is both commutative and associative. That is to say

and

Let a, b denote the individuals contained at once under the classes a and b; those of which a and b are the common species. If a and b were independent events, a, b would denote the event whose probability is the product of the probabilities of each. On the strength of this analogy (to speak of no other), the operation indicated by the comma may be called logical multiplication. It is plain that

Logical multiplication is evidently a commutative and associative process. That is,

Logical addition and logical multiplication are doubly distributive, so that

and

where any of these letters may vanish. These formulæ comprehend every possible relation of a,b, and c; and it follows from them, that

But

So

But

Let be the sign of logical subtraction; so defined that

Here it will be observed that x is not completely determinate. It may vary from a to a with b taken away. This minimum may be denoted by a —b. It is also to be observed that if the sphere of b reaches at all beyond a, the expression is uninterpretable. If then we denote the contradictory negative of a class by the letter which denotes the class itself, with a line above it,¹ if we denote by v a wholly indeterminate class, and if we allow to be a wholly uninterpretable symbol, we have

which is uninterpretable unless

If we define zero by the following identities, in which x may be any class whatever,

then, zero denotes the class which does not go beyond any class, that is nothing or nonentity.

Let a ; b be read a logically divided by b, and be defined by the condition that

x is not fully determined by this condition. It will vary from a to and will be uninterpretable if a is not wholly contained under b. Hence, allowing [1;0] to be some uninterpretable symbol,

which is uninterpretable unless

Unity may be defined by the following identities in which x may be any class whatever.

Then unity denotes the class of which any class is a part; that is, what is or ens.

It is plain that if for the moment we allow a : b to denote the maximum value of a ; b, then

So that

The rules for the transformation of expressions involving logical subtraction and division would be very complicated. The following method is, therefore, resorted to.

It is plain that any operations consisting solely of logical addition and multiplication, being performed upon interpretable symbols, can result in nothing uninterpretable. Hence, if φ + × x signifies such an operation performed upon symbols of which x is one, we have

where a and b are interpretable.

It is plain, also, that all four operations being performed in any way upon any symbols, will, in general, give a result of which one term is interpretable and another not; although either of these terms may disappear. We have then

We have seen that if either of these coefficients i and j is uninterpretable, the other factor of the same term is nothing, or else the whole expression is uninterpretable. But

Hence

Developing by (18) , we have,

So that, by (11),

Developing x; y in the same way, we have²

So that, by (14),

Boole gives (20), but not (19).

In solving identities we must remember that

From the value of b cannot be obtained.

From a ; b the value of b cannot be determined.

Required to eliminate x.

Logically multiplying these identities, we get

For two terms disappear because of (17).

But we have, by (18),

Multiplying logically by x we get

Substituting these values above, we have

Required to eliminate x.

Now, developing as in (18), only in reference to φ(1) and φ(0) instead of to x and y,

But by (18) we have also,

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 days,

p days when it hails,

q days when it thunders,

r days when it hails and thunders.

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 the questions put to both witnesses,

p those which the first answers truly,

q those which the second answers truly,

r those wherein they disagree,

w those which both answer truly,

w’ those which both answer falsely.

Now by (28.)

Substituting and transposing,

The differences of Boole’s system, as given by himself, from the modification of it given here, are three.

1st. Boole does not make use of the operations here termed logical addition and subtraction. The advantages obtained by the introduction of them are three, viz. they give unity to the system; they greatly abbreviate the labor of working with it; and they enable us to express particular propositions. This last point requires illustration. Let i be a class only determined to be such that only some one individual of the class a comes under it. Then ,a is the expression for some a. Boole cannot properly express some a.

2d. Boole uses the ordinary sign of multiplication for logical multiplication. This debars him from converting every logical identity into an equality of probabilities. Before the transformation can be made the equation has to be brought into a particular form, and much labor is wasted in bringing it to that form.

3d. Boole has no such function as ab. This involves him in two difficulties. When the probability of such a function is required, he can only obtain it by a departure from the strictness of his system. And on account of the absence of that symbol, he is led to declare that, without adopting the principle that simple, unconditioned events whose probabilities are given are independent, a calculus of logic applicable to probabilities would be impossible.

The question as to the adoption of this principle is certainly not one of words merely. The manner in which it is answered, however, partly determines the sense in which the term “probability” is taken.

In the propriety of language, the probability of a fact either is, or solely depends upon, the strength of the argument in its favor, supposing all relevant relations of all known facts to constitute that argument. Now, the strength of an argument is only the frequency with which such an argument will yield a true conclusion when its premises are true. Hence probability depends solely upon the relative frequency of a specific event (namely, that a certain kind of argument yields a true conclusion from true premises) to a generic event (namely, that that kind of argument occurs with true premises). Thus, when an ordinary man says that it is highly probable that it will rain, he has reference to certain indications of rain,—that is, to a certain kind of argument that it will rain,—and means to say that there is an argument that it will rain, which is of a kind of which but a small proportion fail. “Probability,” in the untechnical sense, is therefore a vague word, in as much as it does not indicate what one, of the numerous subordinated and co-ordinated genera to which every argument belongs, is the one the relative frequency of the truth of which is expressed. It is usually the case, that there is a tacit understanding upon this point, based perhaps on the notion of an infima species of argument. But an infima species is a mere fiction in logic. And very often the reference is to a very wide genus.

The sense in which the term should be made a technical one is that which will best subserve the purposes of the calculus in question. Now, the only possible use of a calculation of a probability is security in the long run. But there can be no question that an insurance company, for example, which assumed that events were independent without any reason to think that they really were so, would be subjected to great hazard. Suppose, says Mr. Venn, that an insurance company knew that nine-tenths of the Englishmen who go to Madeira die, and that nine-tenths of the consumptives who go there get well. How should they treat a consumptive Englishman? Mr. Venn has made an error in answering the question, but the illustration puts in a clear light the advantage of ceasing to speak of probability, and of speaking only of the relative frequency of this event to that.³

1. So that, for example, ā denotes not-a.

2. a;b,c must always be taken as (a;b),c, not as a;(b,c).

3. See a notice, Venn’s Logic of Chance, in the North American Review for July, 1867.