Writings of Charles S. Peirce: A Chronological Edition, Volume 8. Charles S. Peirce

Чтение книги онлайн.

       18

      Boolian Algebra. First Lection

      §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 СКАЧАТЬ