CHAPTER II.
TERMS.
Every proposition expresses the resemblance or difference of the things denoted by its terms. As inference treats of the relation between two or more propositions, so a proposition expresses a relation between two or more terms. In the portion of this work which treats of deduction it will be convenient to follow the usual order of exposition. We will consider in succession the various kinds of terms, propositions, and arguments, and we commence in this chapter with terms.
The simplest and most palpable meaning which can belong to a term consists of some single material object, such as Westminster Abbey, Stonehenge, the Sun, Sirius, &c. It is probable that in early stages of intellect only concrete and palpable things are the objects of thought. The youngest child knows the difference between a hot and a cold body. The dog can recognise his master among a hundred other persons, and animals of much lower intelligence know and discriminate their haunts. In all such acts there is judgment concerning the likeness of physical objects, but there is little or no power of analysing each object and regarding it as a group of qualities.
The dignity of intellect begins with the power of separating points of agreement from those of difference. Comparison of two objects may lead us to perceive that they are at once like and unlike. Two fragments of rock may differ entirely in outward form, yet they may have the same colour, hardness, and texture. Flowers which agree in colour may differ in odour. The mind learns to regard each object as an aggregate of qualities, and acquires the power of dwelling at will upon one or other of those qualities to the exclusion of the rest. Logical abstraction, in short, comes into play, and the mind becomes capable of reasoning, not merely about objects which are physically complete and concrete, but about things which may be thought of separately in the mind though they exist not separately in nature. We can think of the hardness of a rock, or the colour of a flower, and thus produce abstract notions, denoted by abstract terms, which will form a subject for further consideration.
At the same time arise general notions and classes of objects. We cannot fail to observe that the quality *hardness* exists in many objects, for instance in many fragments of rock; mentally joining these together, we create the class *hard object*, which will include, not only the actual objects examined, but all others which may happen to agree with them, as they agree with each other. As our senses cannot possibly report to us all the contents of space, we cannot usually set any limits to the number of objects which may fall into any such class. At this point we begin to perceive the power and generality of thought, which enables us in a single act to treat of indefinitely or even infinitely numerous objects. We can safely assert that whatever is true of any one object coming under a class is true of any of the other objects so far as they possess the common qualities implied in their belonging to the class. We must not place a thing in a class unless we are prepared to believe of it all that is believed of the class in general; but it remains a matter of important consideration to decide how far and in what manner we can safely undertake thus to assign the place of objects in that general system of classification which constitutes the body of science.
*Twofold Meaning of General Names.*
Etymologically the *meaning* of a name is that which we are caused to think of when the name is used. Now every general name causes us to think of some one or more of the objects belonging to a class; it may also cause us to think of the common qualities possessed by those objects. A name is said to *denote* the object of thought to which it may be applied; it *implies* at the same time the possession of certain qualities or circumstances. The objects denoted form the *extent* of meaning of the term; the qualities implied form the *intent* of meaning. Crystal is the name of any substance of which the molecules are arranged in a regular geometrical manner. The substances or objects in question form the extent of meaning; the circumstance of having the molecules so arranged forms the intent of meaning.
When we compare general terms together, it may often be found that the meaning of one is included in the meaning of another. Thus all *crystals* are included among *material substances*, and all *opaque crystals* are included among *crystals*; here the inclusion is in extension. We may also have inclusion of meaning in regard to intension. For, as all crystals are material substances, the qualities implied by the term material substance must be among those implied by crystal. Again, it is obvious that while in extension of meaning opaque crystals are but a part of crystals, in intension of meaning crystal is but part of opaque crystal. We increase the intent of meaning of a term by joining to it adjectives, or phrases equivalent to adjectives, and the removal of such adjectives of course decreases the intensive meaning. Now, concerning such changes of meaning, the following all-important law holds universally true:--*When the intent of meaning of a term is increased the extent is decreased; and* vice versâ, *when the extent is increased the intent is decreased*. In short, as one is increased the other is decreased.
This law refers only to logical changes. The number of steam-engines in the world may be undergoing a rapid increase without the intensive meaning of the name being altered. The law will only be verified, again, when there is a real change in the intensive meaning, and an adjective may often be joined to a noun without making a change. *Elementary metal* is identical with *metal*; *mortal man* with *man*; it being a *property* of all metals to be elements, and of all men to be mortals.
There is no limit to the amount of meaning which a term may have. A term may denote one object, or many, or an infinite number; it may imply a single quality, if such there be, or a group of any number of qualities, and yet the law connecting the extension and intension will infallibly apply. Taking the general name *planet*, we increase its intension and decrease its extension by prefixing the adjective *exterior*; and if we further add *nearest to the earth*, there remains but one planet, *Mars*, to which the name can then be applied. Singular terms, which denote a single individual only, come under the same law of meaning as general names. They may be regarded as general names of which the meaning in extension is reduced to a minimum. Logicians have erroneously asserted, as it seems to me, that singular terms are devoid of meaning in intension, the fact being that they exceed all other terms in that kind of meaning, as I have elsewhere tried to show.[44]
[44] Jevons’ *Elementary Lessons in Logic*, pp. 41–43; *Pure Logic*, p. 6. See also J. S. Mill, *System of Logic*, Book I. chap. ii. section 5, and Shedden’s *Elements of Logic*, London, 1864, pp. 14, &c. Professor Robertson objects (*Mind*, vol. i. p. 210) that I confuse *singular* and *proper* names; if so, it is because I hold that the same remarks apply to proper names, which do not seem to me to differ logically from singular names.
*Abstract Terms.*
Comparison of objects, and analysis of the complex resemblances and differences which they present, lead us to the conception of *abstract qualities*. We learn to think of one object as not only different from another, but as differing in some particular point, such as colour, or weight, or size. We may then convert points of agreement or difference into separate objects of thought which we call qualities and denote by *abstract terms*. Thus the term *redness* means something in which a number of objects agree as to colour, and in virtue of which they are called red. Redness forms, in fact, the intensive meaning of the term red.
Abstract terms are strongly distinguished from general terms by possessing only one kind of meaning; for as they denote qualities there is nothing which they cannot in addition imply. The adjective “red” is the name of red objects, but it implies the possession by them of the quality *redness*; but this latter term has one single meaning--the quality alone. Thus it arises that abstract terms are incapable of plurality. Red objects are numerically distinct each from each, and there are multitudes of such objects; but redness is a single quality which runs through all those objects, and is the same in one as it is in another. It is true that we may speak of *rednesses*, meaning different kinds or tints of redness, just as we may speak of *colours*, meaning different kinds of colours. But in distinguishing kinds, degrees, or other differences, we render the terms so far concrete. In that they are merely red there is but a single nature in red objects, and so far as things are merely coloured, colour is a single indivisible quality. Redness, so far as it is redness merely, is one and the same everywhere, and possesses absolute oneness. In virtue of this unity we acquire the power of treating all instances of such quality as we may treat any one. We possess, in short, general knowledge.
*Substantial Terms.*
Logicians appear to have taken little notice of a class of terms which partake in certain respects of the character of abstract terms and yet are undoubtedly the names of concrete existing things. These terms are the names of substances, such as gold, carbonate of lime, nitrogen, &c. We cannot speak of two golds, twenty carbonates of lime, or a hundred nitrogens. There is no such distinction between the parts of a uniform substance as will allow of a discrimination of numerous individuals. The qualities of colour, lustre, malleability, density, &c., by which we recognise gold, extend through its substance irrespective of particular size or shape. So far as a substance is gold, it is one and the same everywhere; so that terms of this kind, which I propose to call *substantial terms*, possess the peculiar unity of abstract terms. Yet they are not abstract; for gold is of course a tangible visible body, entirely concrete, and existing independently of other bodies.
It is only when, by actual mechanical division, we break up the uniform whole which forms the meaning of a substantial term, that we introduce number. *Piece of gold* is a term capable of plurality; for there may be a great many pieces discriminated either by their various shapes and sizes, or, in the absence of such marks, by simultaneously occupying different parts of space. In substance they are one; as regards the properties of space they are many.[45] We need not further pursue this question, which involves the distinction between unity and plurality, until we consider the principles of number in a subsequent chapter.
[45] Professor Robertson has criticised my introduction of “Substantial Terms” (*Mind*, vol. i. p. 210), and objects, perhaps correctly, that the distinction if valid is extra-logical. I am inclined to think, however, that the doctrine of terms is, strictly speaking, for the most part extra-logical.
*Collective Terms.*
We must clearly distinguish between the *collective* and the *general meanings* of terms. The same name may be used to denote the whole body of existing objects of a certain kind, or any one of those objects taken separately. “Man” may mean the aggregate of existing men, which we sometimes describe as *mankind*; it is also the general name applying to any man. The vegetable kingdom is the name of the whole aggregate of *plants*, but “plant” itself is a general name applying to any one or other plant. Every material object may be conceived as divisible into parts, and is therefore collective as regards those parts. The animal body is made up of cells and fibres, a crystal of molecules; wherever physical division, or as it has been called *partition*, is possible, there we deal in reality with a collective whole. Thus the greater number of general terms are at the same time collective as regards each individual whole which they denote.
It need hardly be pointed out that we must not infer of a collective whole what we know only of the parts, nor of the parts what we know only of the whole. The relation of whole and part is not one of identity, and does not allow of substitution. There may nevertheless be qualities which are true alike of the whole and of its parts. A number of organ-pipes tuned in unison produce an aggregate of sound which is of exactly the same pitch as each 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 10·5, T = melting 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, Y = table,
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 10·5.” 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 A*a*, B*b*, C*c*, &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 0. Thus the second law of thought may be symbolised in the forms
A*a* = 0 AB*b* = 0 ABC*a* = 0
We may variously describe the meaning of 0 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
A = AA = AAA = &c.
The late Professor Boole is the only logician in modern times who has drawn attention to this remarkable property of logical terms;[46] but in place of the name which he gave to the law, I have proposed to call it The Law of Simplicity.[47] 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 1 × 1 = 1, and 0 × 0 = 0 (taking 0 to mean absolute zero or 1 – 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.
[46] *Mathematical Analysis of Logic*, Cambridge, 1847, p. 17. *An Investigation of the Laws of Thought*, London, 1854, p. 31.
[47] *Pure Logic*, p. 15.
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.”[48] 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.
[48] “Velut si dicam, Sol, Sol, Sol, non tres soles effecerim, sed uno toties prædicaverim.”
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 “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 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:[49]--“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.”
[49] Book i., Part iv., Section 5.
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. *We are logically weak and imperfect in respect of the fact that we are obliged to think of one thing after another.* We must describe metal as “hard and opaque,” or “opaque and hard,” but in the metal itself there is no such difference of order; the properties are simultaneous and coextensive in existence.
Setting aside all grammatical peculiarities which render a substantive less moveable than an adjective, and disregarding any meaning indicated by emphasis or marked order of words, we may state, as a general law of logic, that AB is identical with BA, or AB = BA. Similarly, ABC = ACB = BCA = &c.
Boole first drew attention in recent years to this property of logical terms, and he called it the property of Commutativeness.[50] He not only stated the law with the utmost clearness, but pointed out that it is a Law of Thought rather than a Law of Things. I shall have in various parts of this work to show how the necessary imperfection of our symbols expressed in this law clings to our modes of expression, and introduces complication into the whole body of mathematical formulæ, which are really founded on a logical basis.
[50] *Laws of Thought*, p. 29. It is pointed out in the preface to this Second Edition that Leibnitz was acquainted with the Laws of Simplicity and of Commutativeness.
It is of course apparent that the power of commutation belongs only to terms related in the simple logical mode of synthesis. No one can confuse “a house of bricks” with “bricks of a house,” “twelve square feet” with “twelve feet square,” “the water of crystallization” with “the crystallization of water.” All relations which involve differences of time and space are inconvertible; the higher must not be made to change places with the lower, nor the first with the last. For the parties concerned there is all the difference in the world between A killing B and B killing A. The law of commutativeness simply asserts that difference of order does not attach to the connection between the properties and circumstances of a thing--to what I call *simple logical relation*.