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18

Boolian Algebra. First Lection

c. 1890

Houghton Library

§1. INTRODUCTORY

The algebra of logic (which must be reckoned among man’s precious possessions for that it illuminates the tangled paths of thought) was given to the world in 1842; and George Boole is the name, an honoured one upon other accounts in the mathematical world, of the mortal upon whom this inspiration descended. Although there had been some previous attempts in the same direction, Boole’s idea by no means grew from what other men had conceived, but, as truly as any mental product may, sprang from the brain of genius, motherless. You shall be told, before we leave this subject, precisely what Boole’s original algebra was; it has, however, been improved and extended by the labors of other logicians, not in England alone, but also in France, in Germany, and in our own borders; and it is to one of the modified systems which have so been produced that I shall first introduce you, and shall for the most part adhere. The whole apparatus of this algebra is somewhat extensive. You must not suppose that you are getting it all in the first, the second, or the third lection. But the subject-matter shall be so arranged that you may from the outset make some use of the notation described, and even apply it to the solution of problems.

A deficiency of pronouns makes itself felt in English, as in every tongue, whenever there is occasion to discourse concerning relations between more than two objects; so that, to supply the place of the wanting words, the designations, A, B, and C are resorted to, not only by geometricians for points, but also by lawyers and economists for persons and other parties. This device is already a long stride toward an algebraical notation; and in any mode of expression whose only elegance is to consist in absolute clearness and in the aid it affords to the mind in reasoning, the use of letters in place of words ought to be further extended.

Another serious imperfection of ordinary language, in its written form at least, belongs to our feeble marks of punctuation. The illustration of how a phrase may be ambiguous when written, from which the pauses of speech would remove all uncertainty, is now too stale a joke for the padding of a newspaper. But in algebra we find a method of punctuation which answers its purpose to perfection and is at the same time of the utmost simplicity. The plan is simply to enclose a phrase in parenthesis to show that it is to be treated as a unit in its combination with other phrases or single words. When one such parenthesis is included within another, the appearance of the ordinary curvilinear marks ( ) is varied, either by the use of square brackets [ ] or braces { }, or by making the lines heavier ( ), or larger. Sometimes, a vinculum or straight line drawn over the phrase or compound expression is used instead of the parenthesis. By this simple means, we readily distinguish between the black (lady’s veil) and the (black lady)’s veil; or between the following:—

The {(church of England)’s[(gunpowder plot) services]},

[The (church of England)’s][(gunpowder plot) services],

{(The church) of [England’s (gunpowder plot)]} services,

The {[(church of England)’s gun][(powder plot) services]},

etc. etc. etc.

Another fault of ordinary language as an instrument of reasoning is that it is more pictorial than diagrammatic. It serves the purposes of literature well, but not those of logic. The thought of the writer is encumbered with sensuous accessories. In striving to convey a clear conception of a complicated system of relations, the writer is driven to circumlocutions which distract the attention or to polysyllabic and unfamiliar words which are not very much better. Besides, almost every word signifies the most disparate and even contrary things in different connections, (for example, the “number of millimetres in an inch” is the same as “an inch in millimetres”), so that if the reader seizes the idea at all, he only does it by substituting for the signs in which it is expressed some mental diagram which embodies the same relations in a clearer form. Games of chess are described in old books after this fashion: “The white king’s pawn is advanced two squares. The black king’s pawn is advanced two squares. The white king’s knight is placed on the square in front of the king’s bishop’s pawn,” etc. In ancient writings arithmetical processes are performed in words with the same intolerable prolixity. To remedy this vice of language, what is required is a system of abbreviations of invariable significations and so chosen that the different relations upon which reasoning turns may find their analogues in the relations between the different parts of the expression. [Please to reflect on this last condition.] Among such abbreviations of quasi-diagrammatical power, we shall find the algebraical signs + and × of the greatest utility, owing to their being familiarly associated with the rules for using them.

§2. THE COPULA

In the special modification of the Boolian calculus now to be described, which I shall designate as Propositional Algebra, the letters of the alphabet are used to signify statements, the special statement signified by each letter depending on the convenience of the moment. The statement signified by a letter may be one that we believe or one that we disbelieve: it may be very simple or it may be indefinitely complex. We may, if we choose, employ a single letter to designate the whole contents of a book, or the sum of omniscience, or a falsehood as such. To use the consecrated term of logic which Appuleius, in the second century of our era, already speaks of as familiar, the letters of the alphabet are to be PROPOSITIONS. The final letters x, y, z, will be specially appropriated to the expression of formulae which hold good whatever statements these letters may represent; so that in such a formula each of these letters may be replaced throughout by any proposition whatever.

The idea is to express the degree of truth of propositions upon a quantitative scale, as temperatures are expressed by degrees of the thermometer scale. Only, since every proposition is either true or false, the scale of truth has but two points upon it, the true point and the false point. We shall conceive truth to be higher in the scale than falsity.

In that branch of the art of reasoning which this algebra immediately subserves, we are to study the modes of necessary inference. A proposition or propositions, called PREMISES, being taken for granted, the question is what other propositions, called CONCLUSIONS, these premises entitle us to affirm. The truth of the premises is not now to be examined, for that is assumed to have been satisfactorily determined, already; and any process of inference (i.e., formation of a conclusion from premises) will be satisfactory, provided it be such that the conclusion is certainly true unless the premises are false. That is to say, if P signifies the premises and C the conclusion, the condition of the validity of the inference is that either P is false or C is true. For human reason cannot undertake to guarantee that the conclusion shall be true if the premises on which it depends are false. It is true that we can imagine inferences which satisfy this condition and yet are illogical. Such are the following:—

P true, C true. The world is round; therefore, the sun is hot.

P false, C true. The world is square; therefore, the sun is hot.

P false, C false. The world is square; therefore, the sun is cold.

The reason why such inferences would be bad is that nobody could, in such cases, know that either P is false or C true, unless he knew already that P was false (when it would not properly be a premise), or else knew independently that C was true (when it would not properly be a conclusion drawn from P). But if the proverbial Angel Gabriel, who has been imagined as making so many extraordinary utterances, were to descend and tell me “Either the earth is not round, or the further side of the moon is blue,” it would be perfectly logical for me, from the known fact that the earth is round, to conclude that the other side of the moon is blue. It is true that the inference would not be what is called a complete or logical one; that is to say, the principle that either P is false or C true could not be known from the study of reasonings in general; but it would be a perfectly sound or valid inference.

From what has been said it is plain that that relation between two propositions which consists in our knowing that either the one is true or the other false is of prime importance as warranting an inference from the former to the latter. It is, therefore, desirable to have an abbreviation to express this relation. The sign is to be used in such a sense that x y means that x is at least as low on the scale of truth as y. The sign is to be called the COPULA, and for the sake of brevity it may be read “gives,” that is, warrants the inference of. A proposition like x y will be called a HYPOTHETICAL, the proposition x preceding the copula will be called the ANTECEDENT, and the proposition y following the copula will be called the CONSEQUENT. The meaning of may be more explicitly stated in the following propositions, which, for convenience of reference, I mark A, B, C.

A. If x is false, x y.

B. If y is true, x y.

C. If x y, either x is false or y is true.¹

Rules of the Copula

The sign is subject to three algebraical rules,² as follows:—

RULE I. If x y and y z, then x z. This is called the principle of the transitiveness of the copula.

RULE II. Either x y or y z.

RULE III. There are two propositions, u and v, such that v u is false.

To these is to be added the following:

RULE OF INTERPRETATION. If y is true, v y; and if y is false, y u.

Rule I can be proved from propositions, A, B, C. For by C, if x y either x is false or y is true. In the statement of A, substitute z for y. It then reads that if x is false, x z. Hence, if x y, either x z or y is true. Call this proposition P. In the statement of C, substitute y for x, and z for y. It then reads that if y z, either y is false or z is true. Combining this with P, we see that if x y and y z, either x z or z is true. Call this proposition Q. In the statement of B, substitute z for y. It then reads that if z is true, x z. Combining this with Q, we conclude that if x y and y z, x z. Q.E.D.

Rule II can be proved from propositions A and B alone. For in the statement of A, substitute y for x and z for y. It then reads that if y is false, y z. But by B, if y is true, x y. Hence, either x y or y z. Q.E.D.

Rule III can be proved from proposition C alone. For in the statement of C, substitute v for x and u for y; and it reads that if v u, either v is false or u is true. If, therefore, v is any true proposition, and u any false one, it is not true that v u. Thus, it is possible so to take u and v that v u shall be false. Q.E.D.

The rule of interpretation evidently follows from propositions A and B.

That these four rules fully represent propositions A, B, C, can be shown by deducing the latter from the former. It is left to the student to construct these proofs.

Let us consider the three algebraical rules by themselves, independently of the rule of interpretation. Rule I shows that x y expresses a relation between x and y analogous to that of numbers on a scale, the number x being at least as low on the scale as y. For if x is at least as low as y, and y at least as low as z, then x is at least as low as z. Rule II shows that this scale has not more than two places upon it. For if one number, y, could be lower on the scale than a second, x, and at the same time higher than a third, z, neither x y nor y z would be true. Rule III shows that the scale has at least two places. For if it had but one, any one number would be as low as any other and we should have x y for all values of x and y. Finally the rule of interpretation shows that the higher point on the scale represents the truth and the lower falsity. [Note A. In the ordinary logic, the fact that there are not more than two varieties of propositions in respect to truth is expressed by the so-called Principle of Excluded Middle, which is that every proposition is either true or false, (or A is either B or not-B); while the fact that there are at least two different varieties is expressed by the so-called Principle of Contradiction, which is that nothing is both true and false, (or A is not not-A).]

I now proceed to deduce a few useful formulae from Rules I, II, III. In Rule II, substitute x for z, and we have

(1). Either x y or y x.

In (1), substitute x for y, and we have

(2). x x.

By Rules II and III,

(3). x v.

(4). u x.

By Rule I,

(5). If x y, while it is false that x z, then it is false that y z.

(6). If y z, while it is false that x z, then it is false that x y.

(7). If it is false that x z, it is either false that x y or that y z.

By Rules I and III,

(8). It is either false that v x or false that x u.

By Rule II,

(9). Either v x or x u.

[Note B. An entire calculus of logic might be made with the sign alone. Some of the most important formulae would be, as follows:—

x x.

If x (y z), then y (x z).

(v x) x.

But such a calculus would be useless on account of its complexity.]

[Note C. The equation x = y means, of course, that x and y are at the same point of the scale. That is to say, the definition of x = y is contained in the following propositions:

A. If x = y, x y.

B. If x = y, y x.

C. If x y and y x, then x = y.

From these propositions, it follows that logical equality is subject to the following rules:—

i. x = x.

ii. If x = y, then y = x.

iii. If x = y and y = z, then x = z.

iv. Either x = y or y = z or z = x.

The proof of these from A, B, C, by means of Rules I, II, III, is left to the student.]

§3. LOGICAL ADDITION AND MULTIPLICATION

[Note. The sign as explained above, is, we may trust, free from every trace of ambiguity. But while it does not hesitate between two meanings, it does carry two meanings at one and the same time. The expression x y means that either x is false or y true; but it also means that x is at least as low as y upon a scale. In short, x y not only states something, but states it under a particular aspect; and though it is anything but a poetical or rhetorical expression, it conveys its purport by means of an arithmetical simile. Now, elegance requires that this simile, once adopted, should be adhered to; and elegance, as we shall find, is every whit as important a consideration in the art of reasoning as it is in the more sensuous modes to which the name of Art is commonly appropriated. Following out this analogy, then, we proceed to inquire what are to be the logical significations of addition, subtraction, multiplication, and division.]

Any two numbers whatever (say 5 and 2) might be chosen for u and v, the representatives of the false and the true; though there is some convenience in making v the larger. Then, the principle of contradiction is satisfied by these being different numbers; for a number, x, cannot at once be equal to 5 and to 2, and therefore the proposition represented by x cannot be at once true and false. But in order to satisfy the principle of excluded middle, that every proposition is either true or false, every letter, x, signifying a proposition must, considered as a number, be supposed subject to a quadratic equation whose roots are u and v. In short, we must have

(x − u)(v − x) = 0.

Since the product forming the left hand member of this equation vanishes, one of the factors must vanish. So that either x − u = 0 and x = u, or v − x = 0 and x = v. Another way of expressing the principle of excluded middle would be:

It will be found, however, that occasion seldom arises for taking explicit account of the principle of excluded middle.

The propositions

Either x is false or y is true,

and

Either y is false or z is true,

are expressed by the equations

(x − u)(v − y) = 0

(y − u)(v − z) = 0.

For, as before, to say that the product forming the first member of each equation vanishes, is equivalent to saying that one or other factor vanishes.

Let us now eliminate y from the above two equations. For this purpose, we multiply the first by (v − z) and the second by (x − u).

We, thus, get

(x − u)(v − y)(v − z) = 0

(x − u)(y − u)(v − z) = 0.

We now add these two equations and get

(x − u)(v − u)(v − z) = 0.

But the factor v − u does not vanish. We, therefore, divide by it, and so find

(x − u)(v − z) = 0.

The signification of this is,

Either x is false or z is true;

and this is the legitimate conclusion from the two propositions

Either x is false or y is true,

and

Either y is false or z is true.

Suppose, now, that we seek to find the expression of the precise denial of x [which in logical terminology is called the contradictory of x]. Call this X. Then it is necessary and sufficient that X should be true when x is false and false when x is true. We may therefore put

X = u + v − x

or

X = uv/x.

These two expressions are equal, by the equation of excluded middle.

The simplest expression of that proposition which is true if x, y, z are all true and is false if any of them are false is

The simplest expression for the proposition which is true if any of the propositions x, y, z, is true, but is false if all are false is

It is now easy to see that some values of u and v are much more convenient than others. For example, the proposition which asserts that some two at least of the three propositions, x, y, z, are true, is, if u = 2, v = 5,

but if u = − 1, v = + 1, the same statement is simply

x + y + z − xyz.

Perhaps the system which would most readily occur to a mathematician would be to take the true, v, as an odd number, and the false, u, as an even one,³ and not to discriminate between numbers except as odd or even. Thus, we should have

v = 1 = 3 = 5 = 7 = etc.

u = 0 = 2 = 4 = 6 = etc.

In other words, we should measure round a circle, having its circumference equal to 2; so that 2 would fall on 0, 3 on 1, etc. On this system, every possible algebraical expression formed by means of the addition and multiplication of propositions would have a meaning. Thus, x + y + z 1 etc. would mean that some odd number of the propositions, x, y, z, were true. While xyz etc. would mean that the propositions x, y, z, were all true. For these would be the conditions of the expressions representing odd numbers. Subtraction would have the same meaning as addition for we should have − x = x. A quotient, as x/y, would not properly signify a proposition, since it would not necessarily represent any possible whole number. Namely, if x were odd and y even, x/y would be a fraction.

1. In using the conjunctions “either … or,” I always intend to leave open the possibility that both alternatives may hold good. By “either x or y,” I mean “Either x or y or both.”

2. A “rule” in algebra differs from most other rules, in that it requires nothing to be done, but only permits us to make certain transformations.

3. The Pythagorean notion was that odd was good, even bad.