Figure 2 shows all these points and lines.
If L1 and L2 are any two lines in a plane, they have a point of intersection, and we may write
L3 = λL1 + (1 − λ)L2
for any third line passing through that point. Or if L1, L2, L3 have not a common intersection,
μ(λL1 + (1 − λ)L2) + (1 − μ)L3
may represent any line in the plane. Thus, the whole theory of lines is exactly like that of points.
1. In analytical geometry, a line always means a straight line.
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,