or grammatical point of view. But grammatical usage readily sanctions the free conversion of adjectives into substantives, and vice versû; 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 will lead to much 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 writing and reading 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 mode of expression. The late Prof. de Morgan represented negative terms by small Roman letters, or sometimes by small italic letters k, and as the latter seem to be highly convenient, I shall use a, b, c, d, e, ... p, q, r, &c., as the negative terms corresponding to A, B, C, D, E, P, Q, R, &c. Thus if A means 'fluid,' a will mean 'not-fluid,' and so on. ... k Formal Logic,' p. 38. C 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 equal1. The meaning of the sign has however been gradually extended beyond that of common equality; mathematicians have themselves used it to indicate equivalency of operations. The force of analogy has been so great that writers in all other branches of science have more or less employed the same sign. The philologist indicates by it equivalency of meaning of words: chemists adopt it to signify the identity in kind and equality in weight of the elements which form two different compounds. Not a few logicians, for instance Ploucquet, Condillac, George Bentham", Boole, have employed it as the copula of propositions. Prof. de Morgan declined to use it for this purpose, but still further extended its meaning so as to include the equivalency of a proposition with the premises from which it can be inferredo, and Herbert Spencer has applied it in a like manner P. 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 1 Hallam's 'Literature of Europe,' First Ed. vol. ii. p. 444. n Outline of a New System of Logic,' London, 1827, pp. 133, &c. o Formal Logic,' pp. 82, 106. In his later work, 'The Syllabus of a New System of Logic,' he discontinued the use of the sign. p 'Principles of Psychology,' Second Ed., vol. ii. pp. 54, 55. of the widest character and of the greatest importance— one indeed which it is a principal object of this work to endeavour to explain. The employment of the same sign in different cases would be wholly unphilosophical unless there were some real analogy between its diverse meanings. If such analogy exist, it is not only allowable, but highly desirable and even imperative, to use the symbol of equivalency with a generality of meaning corresponding to the generality of the principles involved. Accordingly Prof. 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 always denotes some form or degree of sameness or equivalency, and the particular form is usually indicated by the nature of the terms joined by it. Thus 6720 pounds = 3 tons' is evidently an equation of quantities. The formula presses the equivalency of operations. tyledons' is a logical identity expressing a profound truth concerning the character of vegetables. = + ex 'Exogens Dico We have great need in logic of a distinct sign for the copula, because the little verb is, 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 Hydrogen The least dense element, = we signify that the object or group of objects denoted by one term is identical with that denoted by the other in everything except the names. The general formula A =B must be taken to mean that A and B are symbols for the same object or group of objects. This identity may sometimes arise from the mere imposition of names, but it may also arise from the deepest laws of the constitution of nature; as when we say Gravitating matter = Matter possessing inertia, = Plagihedral quartz crystals Quartz crystals rotating the plane of polarisation of light. We shall need carefully to distinguish between relations of terms which can be modified at our own will and those which are fixed as expressing the laws of nature; but at present we are considering only the mode of expression. We may sometimes, but much less frequently, require a symbol to indicate difference or the absence of complete sameness. For this purpose we may generalise in like manner the symbol which was introduced by Wallis to signify difference between two numbers or quantities. The general formula ~ denotes that B and C are the names of some two objects. or groups of objects which are not identical with each other. Thus we may say Acrogens Flowering plants. ~ Snowdon The highest mountain in Great Britain. I shall also occasionally use the sign any to signify in the relation between most general manner the existence of the two terms connected by it. Thus might mean not only the relations of equality or inequality, sameness or difference, but any special relation of time, place, size, causation, &c. in which one thing may stand to another. By A B I mean, then, any two objects of thought related to each other in any matter whatsoever. General Formula of Logical Inference. The one supreme rule of inference consists, as I have said, in the direction to affirm of anything whatever is known of its like, equal or equivalent. The Substitution of Similars is a phrase which seems aptly to express the power of mutual replacement existing between any two objects which are to a sufficient degree like or equivalent. It is a matter for further investigation to point out when and for what purposes a degree of similarity less than complete identity is sufficient to warrant substitution. For the present we think only of the exact sameness expressed in the form A = B. Now if we take the letter C to denote any third conceivable object, and use the sign in its stated meaning of indefinite relation, then the general formula of all inference may be thus exhibited :— : or, in words-In whatever relation a thing stands to a second thing, in the same relation it stands to the like or equivalent of that second thing. The identity between A and B allows us indifferently to place A where B was or B where A was, and there is no limit to the variety of special meanings which we can bestow upon the signs used in this formula consistently with its truth. Thus if we first specify only the meaning of the sign, we may say that if C is the weight of B, then C is also the weight of A. Similarly If C is the father of B, C is the father of A; |