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17

Boolian Algebra

c. 1890

Houghton Library

The algebra of logic was invented by the celebrated English mathematician, George Boole, and has subsequently been improved by the labors of a number of writers in England, France, Germany, and America. The deficiency of pronouns in English, as in every other tongue, begins to be felt as soon as there is occasion to discourse of the relations of more than two objects, and forces the lawyer of today in speaking of parties, as it did Euclid of old in treating of the relative situations of many points, to designate them as A, B, C, etc. This device is already a long stride toward an algebraical notation. Two other kinds of signs, however, must be introduced at once. The first embraces the parentheses and brackets which are the punctuation marks of algebra. The imperfection of the ordinary system of punctuation is notorious; and it is too stale a joke to fill up the corner of a newspaper to show a phrase may be ambiguous when written from which the pause of speech would exclude all uncertainty. In our algebraical notation, we simply enclose an expression within a parenthesis to show that it is to be taken together as a unit. We thus easily distinguish the “black (lady’s veil),” from the “(black lady)’s veil.” In case it becomes necessary to enclose one parenthesis within another, we resort to square brackets [ ] for the outer one. The other signs of which we shall have immediate need are +, −, =, are of the nature of abbreviations. The sign +, now read plus, can be historically traced back, through successive insensible modifications, to the ancient word et, and. Without stopping to explain the origin of − and =, I merely remark that these signs are, virtually at least, mere phonograms of minus and est. Everybody knows how much abbreviations may lighten the labor of thought. In our ordinary Arabic notation for numbers, we have two kinds of signs, first, the ten figures, and second, the decimal places. Important as the figures are, they are not nearly so much so as the decimal places. Of the two conceivable imperfect systems of notation which should discard one and the other of these two classes of signs, we should find that one the more useful which should write for 123456 one, two, three, four, five, six, rather than that which should write 1 hundred and 2-ty 3 thousand 4 hundred and 5-ty 6. For what we need to aid our reasoning is a sign the parts of which stand to one another in relations analogous to those on which our reasoning is to hinge, so that we may just think of the signs themselves that are before our eyes, and not have to think of the things signified, which we could only do after all by calling up some mental image or sign which might answer the purpose of reasoning better than those that would be written down. Because the Arabic figures fulfill this condition to a certain extent, we are able to rattle off a long multiplication, thinking only of the figures and not of the numbers; and because we possess no notation for numbers which fulfill the condition perfectly, we find a great difficulty in reasoning about the divisibility of numbers and such like problems. A similar quasi-diagrammatical power is what gives the algebraical signs +, −, =, their great utility.

In that particular modification of the Boolian algebra to which I shall first introduce you, and which I shall chiefly use, the letters of the alphabet are used to signify statements. The special statement which each letter signifies will depend 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 like, use a simple letter to signify the entire contents of a book, or the sum total 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 signify. Of the special signs of invariable significance, the first consists in the writing down of a proposition by itself; and this has the effect of asserting it. This sign will receive a further development further on.

Equality and the cognate words, as well as the sign =, are used in such a sense that x = y (no matter what statements x and y may signify) means that x and y are equally true, that is, are either both true or both false. Thus, let D signify that the democrats will carry the next election and R that the republicans will lose it; then D = R means that either the democrats will carry the next election while the republicans will lose it, or the democrats will not carry it nor the republicans lose it. The exact meaning of the sign of equality, then, may be summed up in the following propositions, which I mark L, M, N, for convenience of future reference.

L. If x = y, then either x is true or y is false.¹

M. If x = y, then either x is false or y is true.

N. If x and y are either both true or false, then x = y.

From this definition of the sign of equality, it follows that in this algebra it is subject to precisely the same rules as in ordinary algebra.² These rules are as follows:

Rule 1. x = x.

Rule 2. If x = y, then y = x.

Rule 3. If x = y, and y = z, then x = z.

I proceed to give formal proofs of these rules; for though they are evidently true, it may not be quite evident that their truth follows necessarily, or how it does so, from the propositions L, M, N. At any rate the proofs will be valuable as examples of demonstration carried to the last pitch of formalism.

Rule 1. Any proposition, x, is either true or false. Call this statement E. In N, write x in place both of x and of y. From N, so stated, together with E, we conclude x = x.

Rule 2. Suppose x = y, which statement we may refer to as P. Then, all we have to prove is that y = x. From L and P, it follows that either x is false or y is true. Call these alternatives A and A′ respectively. We examine first the alternative A. By M and P, either x is true or y is false. Call this statement (having two alternatives) B. But no proposition, x, is both true and false. Call this statement C. From B and C, we conclude that y is false. Thus, the first alternative, A, is that x is false and y is false. Next, we examine the other alternative, A′. From M and P, we conclude B, as before. But no proposition, y, is both true and false. Call this statement C′. From B and C′, we conclude that x is true. Then the second alternative is that y is true and x is true. Thus, there are but two alternatives, either that x and y are both true or that they are both false. Call this compound statement D. In the statement of N, substitute x for y and y for x. Then, from N so stated, together with D, we conclude that y = x, which is all we had to prove.

Rule 3. Any proposition, y, is either true or false. Call these two alternatives A and A′. We first examine the alternative A. No proposition, y, is both true and false. Call this statement C. By M, A, and C, if x = y, then x is true. Call this conditional proposition B. In the statement of L substitute y for x and z for y. Then, from L, so stated, A, and C, we conclude that if y = z then z is true. Call this conditional proposition D. In the statement of N, substitute z for y. Then, from N, so stated, from A, D, it follows that if x = y and y = z, then x = z. Next, I examine the other alternative A′. From L, A′, and C, it follows that if x = y, then x is false. Call this statement B′. From M, stated as before, A′, and C, we conclude that if y = z, then z is false. Call this statement D′. Then from N, stated as before, A′, B′, and D′, we conclude that if x = y and y = z, then x = z. This being the case under both alternatives, we conclude it unconditionally.

Addition and multiplication, and their cognate words and algebraical signs, are used in such sense that x + y means that either x or y is true (without excluding the possibility of both being so), while xy means that both x and y are true. More explicitly, the meanings of the sum and product are summed up in the following propositions, which are lettered A, B, C, X, Y, Z, for convenience of reference.

A. Either x is false or x + y is true.	X. Either xy is false or x is true.
B. Either y is false or x + y is true.	Y. Either xy is false or y is true.
C. Either x + y is false or y is true.	Z. Either x or y is false or xy is true.

From these definitions it follows that in this algebra, all the ordinary rules of addition and multiplication hold good, together with some other rules besides. The rules common to logical and arithmetical algebra are the following.

Rule 4. The associative principle of addition.

(x + y) + z = x + (y + z).

Rule 5. The associative principle of multiplication. (xy)z = x(yz).

Rule 6. The commutative principle of addition. x + y = y + x.

Rule 7. The commutative principle of multiplication. xy = yx.

Rule 8. The distributive principle of multiplication with reference to addition. x(y + z) = xy + xz.

The rules peculiar to logical algebra may be stated as follows:

Rule 9. x + x = xx.

Rule 10. If x + y = xz, then either x + y = x or xz = x.

EXERCISE 1. Prove the above rules, from propositions L, M, N, A, B, C, X, Y, Z.

The above rules are made to conform as much as possible to those of ordinary algebra, and suppose that we are dealing with equations. But as a general rule, we shall not have any equations, but having written down a statement, the problem before us will be to ascertain what follows from it. In that case, it will be better to work by the following system of rules, which for the sake of distinction, I shall term principles.

Principle I. The commutative principle. The order of factors and additive terms is indifferent, that is, x + y = y + x and xy = yx.

Principle II. The principle of erasing parentheses. We always have a right to erase a parenthesis in any asserted proposition. This includes the associative principle, and also permits us to infer x + yz from (x + y)z.

Principle III. From any part of an asserted proposition, we have the right to erase any factor; and to any part we have a right to logically add anything we like. Thus, from xy we can infer x + z.

Principle IV. We have a right to repeat any factor, and to drop any additive term that is equal to another such term. Thus, from x we can infer xx, and from x + x we can infer x.

EXERCISE 2. Prove the above four principles from propositions L, M, and N, together with rules 4 to 10.

EXERCISE 3.

1. By means of the ten rules alone, prove that addition is distributive with respect to multiplication; that is, that

x + yz = (x + y)(x + z).

2. By means of the four principles alone, show that from x(y + z) we can infer xy + xz.

3. A chemist having a substance for examination, finds by one test that it contains either silver or lead, by a second test that it contains either silver or mercury, and by a third test that it contains either lead or mercury. Show by the four principles that it contains either silver and lead or silver and mercury or lead and mercury.

4. Show the same thing by means of the ten rules.

5. A substance known to be simple salt is shown by one test to be either a potassic or a sodic salt, by a second test to be either a potassic salt or a sulphate, by a third test to be either a sodic salt or a nitrate, and by a fourth test to be either a sulphate or a nitrate. Show by the four principles that it is either potassic nitrate or sodic sulphate. Show the same thing by the ten rules.

6. A simple salt is shown by one test to be either a salt of calcium, strontium, or barium; by a second test to be either a salt of calcium or strontium or an iodide; by a third test to be either a salt of strontium or barium or a chloride; by a fourth test to be either a salt of barium or calcium or a bromide; by a fifth test to be either a salt of calcium or a bromide or iodide; by a sixth test to be either a salt of strontium or a chloride or iodide; by a seventh to be either a salt of barium or a chloride or bromide; and by an eighth test to be either a chloride or bromide or iodide. Prove that it is either the chloride of calcium or the bromide of strontium or the iodide of barium.

1. I never use the locution “either … or …” to exclude the case of both members being true.

2. Note that the rules of algebra are “rules” in rather a peculiar sense. They do not compel us to do anything, but only permit us to perform certain transformations.