separate sound. In the case of substantial terms, certain qualities may be present equally in each minutest part as in the whole. The chemical nature of the largest mass of pure carbonate of lime is the same as the nature of the smallest particle. In the case of abstract terms, again, we cannot draw a distinction between whole and part; what is true of redness in any case is always true of redness, so far as it is merely red. Synthesis of Terms. We continually combine simple terms together so as to form new terms of more complex meaning. Thus, to increase the intension of meaning of a term we write it with an adjective or a phrase of adjectival nature. By joining "brittle" to "metal," we obtain a combined term, "brittle metal," which denotes a certain portion of the metals, namely, such as are selected on account of possessing the quality of brittleness. As we have already seen, brittle metal" possesses less extension and greater intension than metal. Nouns, prepositional phrases, participial phrases and subordinate propositions may also be added to terms so as to increase their intension and decrease their extension. " In our symbolic language we need some mode of indicating this junction of terms, and the most convenient device will be the juxtaposition of the letter-terms. Thus if A mean brittle, and B mean metal, then AB will mean brittle metal. Nor need there be any limit to the number of letters thus joined together, or the complexity of the notions which they may represent. Thus if we take the letters P = metal, Q = white, R = monovalent, = S of specific gravity 105. Tmelting above 1000° C., V = good conductor of heat and electricity, then we can form a combined term PQRSTV, which will denote "a white monovalent metal, of specific gravity 10'5, melting above 1000° C., and a good conductor of heat and electricity." There are many grammatical usages concerning the junction of words and phrases to which we need pay no attention in logic. We can never say in ordinary language "of wood table," meaning "table of wood;" but we may consider "of wood" as logically an exact equivalent of "wooden"; so that if X = of wood, there is no reason why, in our symbols, XY should not be just as correct an expression for "table of wood" as YX. In this case indeed we might substitute for "of wood" the corresponding adjective" wooden," but we should often fail to find any adjective answering exactly to a phrase. There is no single word by which we could express the notion "of specific gravity 10'5" but logically we may consider these words as forming an adjective; and denoting this by S and metal by P, we may say that SP means metal of specific gravity 105." It is one of many advantages in these blank letter-symbols that they enable us completely to neglect all grammatical peculiarities and to fix our attention solely on the purely logical relations involved. Investigation will probably show that the rules of grammar are mainly founded upon traditional usage and have little logical signification. This indeed is sufficiently proved by the wide grammatical differences which exist between languages, though the logical foundation must be the same. Symbolic Expression of the Law of Contradiction. The synthesis of terms is subject to the all-important Law of Thought, described in a previous section (p. 5) and called the Law of Contradiction. It is self-evident that no quality can be both present and absent at the same time and place. This fundamental condition of all thought and of all existence is expressed symbolically by a rule that a term and its negative shall never be allowed to come into combination. Such combined terms as Aa, Bb, Cc, &c., are self-contradictory and devoid of all intelligible meaning. If they could represent anything, it would be what cannot exist, and cannot even be imagined in the mind. They can therefore only enter into our consideration to suffer immediate exclusion. The criterion of false reasoning, as we shall find, is that it involves self-contradiction, the affirming and denying of the same statement. We might represent the object of all reasoning as the separation of the consistent and possible from the inconsistent and impossible; and we cannot make any statement except a truism without implying that certain combinations of terms are contradictory and excluded from thought. To assert that "all A's are B's" is equivalent to the assertion that "A's which are not B's cannot exist." It will be convenient to have the means of indicating the exclusion of the self-contradictory, and we may use the familiar sign for nothing, the cipher o. Thus the second law of thought may be symbolised in the forms = O Aa = O ABb = 0 ABCa We may variously describe the meaning of o in logic as the non-existent, the impossible, the self-inconsistent, the inconceivable. Close analogy exists between this meaning and its mathematical signification. Certain Special Conditions of Logical Symbols. In order that we may argue and infer truly we must treat our logical symbols according to the fundamental laws of Identity and Difference. But in thus using our symbols we shall frequently meet with combinations of which the meaning will not at first sight be apparent. If in one case we learn that an object is "yellow and round," and in another case that it is "round and yellow," there arises the question whether these two descriptions are identical in meaning or not. Again, if we proved that an object was "round round," the meaning of such an expression would be open to doubt. Accordingly we must take notice, before proceeding further, of certain special laws which govern the combination of logical terms. In the first place the combination of a logical term with itself is without effect, just as the repetition of a statement does not alter the meaning of the statement; "a round round object" is simply "a round object." What is yellow yellow is merely yellow; metallic metals cannot differ from metals, nor circular circles from circles. In our symbolic language we may similarly hold that AA is identical with A, or AAA = &c. A = AA = The late Professor Boole is the only logician in modern times who has drawn attention to this remarkable property of logical terms; but in place of the name which he gave to the law, I have proposed to call it The Law of Simplicity.2 Its high importance will only become apparent when we attempt to determine the relations of logical and mathematical science. Two symbols of quantity, and only two, seem to obey this law; we may say that I × I = I, and 0 x0 = 0 (taking o to mean absolute zero or I − 1); there is apparently no other number which combined with itself gives an unchanged result. I shall point out, however, in the chapter upon Number, that in reality all numerical symbols obey this logical principle. It is curious that this Law of Simplicity, though almost unnoticed in modern times, was known to Boëthius, who makes a singular remark in his treatise De Trinitate et Unitate Dei (p. 959). He says: "If I should say sun, sun, sun, I should not have made three suns, but I should have named one sun so many times." 3 Ancient discussions about the doctrine of the Trinity drew more attention to subtle questions concerning the nature of unity and plurality than has ever since been given to them. It is a second law of logical symbols that order of combination is a matter of indifference. Rich and rare gems' are the same as 66 rare and rich gems," or even as gems, rich and rare." Grammatical, rhetorical, or poetic usage may give considerable significance to order of expression. The limited power of our minds prevents our grasping many ideas at once, and thus the order of statement may produce some effect, but not in a simply logical manner. All life proceeds in the succession of time, and we are obliged to write, speak, or even think of things and their qualities one after the other; but between the things and their qualities there need be no such relation of order in 1 Mathematical Analysis of Logic, Cambridge, 1847, p. 17. An Investigation of the Laws of Thought, London, 1854, p. 31. 2 Pure Logic, p. 15. 3 "Velut si dicam, Sol, Sol, Sol, non tres soles effecerim, sed uno toties prædicaverim." D 1 time or space. The sweetness of sugar is neither before nor after its weight and solubility. The hardness of a metal, its colour, weight, opacity, malleability, electric and chemical properties, are all coexistent and coextensive, pervading the metal and every part of it in perfect community, none before nor after the others. In our words and symbols we cannot observe this natural condition; we must name one quality first and another second, just as some one must be the first to sign a petition, or to walk foremost in a procession. In nature there is no such precedence. I find that the opinion here stated, to the effect that relations of space and time do not apply to many of our ideas, is clearly adopted by Hume in his celebrated Treatise on Human Nature (vol. i. p. 410). He says :1—“ An object may be said to be no where, when its parts are not so situated with respect to each other, as to form any figure or quantity; nor the whole with respect to other bodies so as to answer to our notions of contiguity or distance. Now this is evidently the case with all our perceptions and objects, except those of sight and feeling. A moral reflection cannot be placed on the right hand or on the left hand of a passion, nor can a smell or sound be either of a circular or a square figure. These objects and perceptions, so far from requiring any particular place, are absolutely incompatible with it, and even the imagination cannot attribute it to them." A little reflection will show that knowledge in the highest perfection would consist in the simultaneous possession of a multitude of facts. To comprehend a science perfectly we should have every fact present with every other fact. We must write a book and we must read it successively word by word, but how infinitely higher would be our powers of thought if we could grasp the whole in one collective act of consciousness! Compared with the brutes we do possess some slight approximation to such power, and it is conceivable that in the indefinite future mind may acquire an increase of capacity, and be less restricted to the piecemeal examination of a subject. But I wish here to make plain that there is no logical foundation for the successive character of thought and reasoning unavoidable under our present mental conditions. Book i., Part iv., Section 5. |