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′
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.
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