The signs required in logic are of a very simple kind. As sameness or difference must exist between two things or notions, we need signs to indicate the things or notions compared, and other signs to denote the relations between them. We need, then, (1) symbols for terms, (2) a symbol for sameness, (3) a symbol for difference, and (4) one or two symbols to take the place of conjunctions.
Ordinary nouns substantive, such as Iron, Metal, Electricity, Undulation, might serve as terms, but, for the reasons explained above, it is better to adopt blank letters, devoid of special signification, such as A, B, C, &c. Each letter must be understood to represent a noun, and, so far as the conditions of the argument allow, any noun. Just as in Algebra, x, y, z, p, q, &c. are used for any quantities, undetermined or unknown, except when the special conditions of the problem are taken into account, so will our letters stand for undetermined or unknown things.
These letter-terms will be used indifferently for nouns substantive and adjective. Between these two kinds of nouns there may perhaps be differences in a metaphysical or grammatical point of view. But grammatical usage sanctions the conversion of adjectives into substantives, and vice versâ; we may avail ourselves of this latitude without in any way prejudging the metaphysical difficulties which may be involved. Here, as throughout this work, I shall devote my attention to truths which I can exhibit in a clear and formal manner, believing that in the present condition of logical science, this course will lead to greater advantage than discussion upon the metaphysical questions which may underlie any part of the subject.
Every noun or term denotes an object, and usually implies the possession by that object of certain qualities or circumstances common to all the objects denoted. There are certain terms, however, which imply the absence of qualities or circumstances attaching to other objects. It will be convenient to employ a special mode of indicating these negative terms, as they are called. If the general name A denotes an object or class of objects possessing certain defined qualities, then the term Not A will denote any object which does not possess the whole of those qualities; in short, Not A is the sign for anything which differs from A in regard to any one or more of the assigned qualities. If A denote “transparent object,” Not A will denote “not transparent object.” Brevity and facility of expression are of no slight importance in a system of notation, and it will therefore be desirable to substitute for the negative term Not A a briefer symbol. De Morgan represented negative terms by small Roman letters, or sometimes by small italic letters;[30] as the latter seem to be highly convenient, I shall use a, b, c, . . . p, q, &c., as the negative terms corresponding to A, B, C, . . . P, Q, &c. Thus if A means “fluid,” a will mean “not fluid.”
Expression of Identity and Difference.
To denote the relation of sameness or identity I unhesitatingly adopt the sign =, so long used by mathematicians to denote equality. This symbol was originally appropriated by Robert Recorde in his Whetstone of Wit, to avoid the tedious repetition of the words “is equal to;” and he chose a pair of parallel lines, because no two things can be more equal.[31] The meaning of the sign has however been gradually extended beyond that of equality of quantities; mathematicians have themselves used it to indicate equivalence of operations. The force of analogy has been so great that writers in most other branches of science have employed the same sign. The philologist uses it to indicate the equivalence of meaning of words: chemists adopt it to signify identity in kind and equality in weight of the elements which form two different compounds. Not a few logicians, for instance Lambert, Drobitsch, George Bentham,[32] Boole,[33] have employed it as the copula of propositions. De Morgan declined to use it for this purpose, but still further extended its meaning so as to include the equivalence of a proposition with the premises from which it can be inferred;[34] and Herbert Spencer has applied it in a like manner.[35]
Many persons may think that the choice of a symbol is a matter of slight importance or of mere convenience; but I hold that the common use of this sign = in so many different meanings is really founded upon a generalisation of the widest character and of the greatest importance—one indeed which it is a principal purpose of this work to explain. The employment of the same sign in different cases would be unphilosophical unless there were some real analogy between its diverse meanings. If such analogy exists, it is not only allowable, but highly desirable and even imperative, to use the symbol of equivalence with a generality of meaning corresponding to the generality of the principles involved. Accordingly De Morgan’s refusal to use the symbol in logical propositions indicated his opinion that there was a want of analogy between logical propositions and mathematical equations. I use the sign because I hold the contrary opinion.
I conceive that the sign = as commonly employed, always denotes some form or degree of sameness, and the particular form is usually indicated by the nature of the terms joined by it. Thus “6,720 pounds = 3 tons” is evidently an equation of quantities. The formula — × — = + expresses the equivalence of operations. “Exogens = Dicotyledons” is a logical identity expressing a profound truth concerning the character and origin of a most important group of plants.
We have great need in logic of a distinct sign for the copula, because the little verb is (or are), hitherto used both in logic and ordinary discourse, is thoroughly ambiguous. It sometimes denotes identity, as in “St. Paul’s is the chef-d’œuvre of Sir Christopher Wren;” but it more commonly indicates inclusion of class within class, or partial identity, as in “Bishops are members of the House of Lords.” This latter relation involves identity, but requires careful discrimination from simple identity, as will be shown further on.
When with this sign of equality we join two nouns or logical terms, as in