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
A. If x is false, x
B. If y is true, x
C. If x