CHAPTER III.

PROPOSITIONS.

We now proceed to consider the variety of forms of propositions in which the truths of science must be expressed. I shall endeavour to show that, however diverse these forms may be, they all admit the application of the one same principle of inference that what is true of a thing is true of the like or same. This principle holds true whatever be the kind or manner of the likeness, provided proper regard be had to its nature. Propositions may assert an identity of time, space, manner, quantity, degree, or any other circumstance in which things may agree or differ.

We find an instance of a proposition concerning time in the following:--“The year in which Newton was born, was the year in which Galileo died.” This proposition expresses an approximate identity of time between two events; hence whatever is true of the year in which Galileo died is true of that in which Newton was born, and *vice versâ*. “Tower Hill is the place where Raleigh was executed” expresses an identity of place; and whatever is true of the one spot is true of the spot otherwise defined, but in reality the same. In ordinary language we have many propositions obscurely expressing identities of number, quantity, or degree. “So many men, so many minds,” is a proposition concerning number, that is to say, an equation; whatever is true of the number of men is true of the number of minds, and *vice versâ*. “The density of Mars is (nearly) the same as that of the Earth,” “The force of gravity is directly as the product of the masses, and inversely as the square of the distance,” are propositions concerning magnitude or degree. Logicians have not paid adequate attention to the great variety of propositions which can be stated by the use of the little conjunction *as*, together with *so*. “As the home so the people,” is a proposition expressing identity of manner; and a great number of similar propositions all indicating some kind of resemblance might be quoted. Whatever be the special kind of identity, all such expressions are subject to the great principle of inference; but as we shall in later parts of this work treat more particularly of inference in cases of number and magnitude, we will here confine our attention to logical propositions which involve only notions of quality.

*Simple Identities.*

The most important class of propositions consists of those which fall under the formula

A = B,

and may be called *simple identities*. I may instance, in the first place, those most elementary propositions which express the exact similarity of a quality encountered in two or more objects. I may compare the colour of the Pacific Ocean with that of the Atlantic, and declare them identical. I may assert that “the smell of a rotten egg is like that of hydrogen sulphide;” “the taste of silver hyposulphite is like that of cane sugar;” “the sound of an earthquake resembles that of distant artillery.” Such are propositions stating, accurately or otherwise, the identity of simple physical sensations. Judgments of this kind are necessarily pre-supposed in more complex judgments. If I declare that “this coin is made of gold,” I must base the judgment upon the exact likeness of the substance in several qualities to other pieces of substance which are undoubtedly gold. I must make judgments of the colour, the specific gravity, the hardness, and of other mechanical and chemical properties; each of these judgments is expressed in an elementary proposition, “the colour of this coin is the colour of gold,” and so on. Even when we establish the identity of a thing with itself under a different name or aspect, it is by distinct judgments concerning single circumstances. To prove that the Homeric χαλκός is copper we must show the identity of each quality recorded of χαλκός with a quality of copper. To establish Deal as the landing-place of Cæsar all material circumstances must be shown to agree. If the modern Wroxeter is the ancient Uriconium, there must be the like agreement of all features of the country not subject to alteration by time.

Such identities must be expressed in the form A = B. We may say

Colour of Pacific Ocean = Colour of Atlantic Ocean. Smell of rotten egg = Smell of hydrogen sulphide.

In these and similar propositions we assert identity of single qualities or causes of sensation. In the same form we may also express identity of any group of qualities, as in

χαλκός = Copper. Deal = Landing-place of Cæsar.

A multitude of propositions involving singular terms fall into the same form, as in

The Pole star = The slowest-moving star. Jupiter = The greatest of the planets. The ringed planet = The planet having seven satellites. The Queen of England = The Empress of India. The number two = The even prime number. Honesty = The best policy.

In mathematical and scientific theories we often meet with simple identities capable of expression in the same form. Thus in mechanical science “The process for finding the resultant of forces = the process for finding the resultant of simultaneous velocities.” Theorems in geometry often give results in this form, as

Equilateral triangles = Equiangular triangles. Circle = Finite plane curve of constant curvature. Circle = Curve of least perimeter.

The more profound and important laws of nature are often expressible in the form of simple identities; in addition to some instances which have already been given, I may suggest,

Crystals of cubical system = Crystals not possessing the power of double refraction.

All definitions are necessarily of this form, whether the objects defined be many, few, or singular. Thus we may say,

Common salt = Sodium chloride. Chlorophyl = Green colouring matter of leaves. Square = Equal-sided rectangle.

It is an extraordinary fact that propositions of this elementary form, all-important and very numerous as they are, had no recognised place in Aristotle’s system of Logic. Accordingly their importance was overlooked until very recent times, and logic was the most deformed of sciences. But it is impossible that Aristotle or any other person should avoid constantly using them; not a term could be defined without their use. In one place at least Aristotle actually notices a proposition of the kind. He observes: “We sometimes say that that white thing is Socrates, or that the object approaching is Callias.”[51] Here we certainly have simple identity of terms; but he considered such propositions purely accidental, and came to the unfortunate conclusion, that “Singulars cannot be predicated of other terms.”

[51] *Prior Analytics*, i. cap. xxvii. 3.

Propositions may also express the identity of extensive groups of objects taken collectively or in one connected whole; as when we say,

The Queen, Lords, and Commons = The Legislature of the United Kingdom.

When Blackstone asserts that “The only true and natural foundation of society are the wants and fears of individuals,” we must interpret him as meaning that the whole of the wants and fears of individuals in the aggregate form the foundation of society. But many propositions which might seem to be collective are but groups of singular propositions or identities. When we say “Potassium and sodium are the metallic bases of potash and soda,” we obviously mean,

Potassium = Metallic base of potash; Sodium = Metallic base of soda.

It is the work of grammatical analysis to separate the various propositions often combined into a single sentence. Logic cannot be properly required to interpret the forms and devices of language, but only to treat the meaning when clearly exhibited.

*Partial Identities.*

A second highly important kind of proposition is that which I propose to call *a partial identity*. When we say that “All mammalia are vertebrata,” we do not mean that mammalian animals are identical with vertebrate animals, but only that the mammalia form a *part of the class vertebrata*. Such a proposition was regarded in the old logic as asserting the inclusion of one class in another, or of an object in a class. It was called a universal affirmative proposition, because the attribute *vertebrate* was affirmed of the whole subject *mammalia*; but the attribute was said to be *undistributed*, because not all vertebrata were of necessity involved in the proposition. Aristotle, overlooking the importance of simple identities, and indeed almost denying their existence, unfortunately founded his system upon the notion of inclusion in a class, instead of adopting the basis of identity. He regarded inference as resting upon the rule that what is true of the containing class is true of the contained, in place of the vastly more general rule that what is true of a class or thing is true of the like. Thus he not only reduced logic to a fragment of its proper self, but destroyed the deep analogies which bind together logical and mathematical reasoning. Hence a crowd of defects, difficulties and errors which will long disfigure the first and simplest of the sciences.

It is surely evident that the relation of inclusion rests upon the relation of identity. Mammalian animals cannot be included among vertebrates unless they be identical with part of the vertebrates. Cabinet Ministers are included almost always in the class Members of Parliament, because they are identical with some who sit in Parliament. We may indicate this identity with a part of the larger class in various ways; as for instance,

Mammalia = part of the vertebrata. Diatomaceæ = a class of plants. Cabinet Ministers = some members of Parliament. Iron = a metal.

In ordinary language the verbs *is* and *are* express mere inclusion more often than not. *Men are mortals*, means that *men* form a part of the class *mortal*; but great confusion exists between this sense of the verb and that in which it expresses identity, as in “The sun is the centre of the planetary system.” The introduction of the indefinite article *a* often expresses partiality; when we say “Iron is a metal” we clearly mean that iron is *one only* of several metals.

Certain recent logicians have proposed to avoid the indefiniteness in question by what is called the Quantification of the Predicate, and they have generally used the little word *some* to show that only a part of the predicate is identical with the subject. *Some* is an *indeterminate adjective*; it implies unknown qualities by which we might select the part in question if the qualities were known, but it gives no hint as to their nature. I might make use of such an indeterminate sign to express partial identities in this work. Thus, taking the special symbol V = Some, the general form of a partial identity would be A = VB, and in Boole’s Logic expressions of the kind were much used. But I believe that indeterminate symbols only introduce complexity, and destroy the beauty and simple universality of the system which may be created without their use. A vague word like *some* is only used in ordinary language by *ellipsis*, and to avoid the trouble of attaining accuracy. We can always employ more definite expressions if we like; but when once the indefinite *some* is introduced we cannot replace it by the special description. We do not know whether *some* colour is red, yellow, blue, or what it is; but on the other hand *red* colour is certainly *some* colour.

Throughout this system of logic I shall dispense with such indefinite expressions; and this can readily be done by substituting one of the other terms. To express the proposition “All A’s are some B’s” I shall not use the form A = VB, but

A = AB.

This formula states that the class A is identical with the class AB; and as the latter must be a part at least of the class B, it implies the inclusion of the class A in that of B. We might represent our former example thus,

Mammalia = Mammalian vertebrata.

This proposition asserts identity between a part (or it may be the whole) of the vertebrata and the mammalia. If it is asked What part? the proposition affords no answer, except that it is the part which is mammalian; but the assertion “mammalia = some vertebrata” tells us no more.

It is quite likely that some readers will think this mode of representing the universal affirmative proposition artificial and complicated. I will not undertake to convince them of the opposite at this point of my exposition. Justification for it will be found, not so much in the immediate treatment of this proposition, as in the general harmony which it will enable us to disclose between all parts of reasoning. I have no doubt that this is the critical difficulty in the relation of logical to other forms of reasoning. Grant this mode of denoting that “all A’s are B’s,” and I fear no further difficulties; refuse it, and we find want of analogy and endless anomaly in every direction. It is on general grounds that I hope to show overwhelming reasons for seeking to reduce every kind of proposition to the form of an identity.

I may add that not a few logicians have accepted this view of the universal affirmative proposition. Leibnitz, in his *Difficultates Quædam Logicæ*, adopts it, saying, “Omne A est B; id est æquivalent AB et A, seu A non B est nonens.” Boole employed the logical equation *x* = *xy* concurrently with *x* = *vy*; and Spalding[52] distinctly says that the proposition “all metals are minerals” might be described as an assertion of *partial identity* between the two classes. Hence the name which I have adopted for the proposition.

[52] *Encyclopædia Britannica*, Eighth Ed. art. Logic, sect. 37, note. 8vo. reprint, p. 79.

*Limited Identities.*

An important class of propositions have the form

AB = AC,

expressing the identity of the class AB with the class AC. In other words, “Within the sphere of the class A, all the B’s are all the C’s;” or again, “The B’s and C’s, which are A’s, are identical.” But it will be observed that nothing is asserted concerning things which are outside of the class A; and thus the identity is of limited extent. It is the proposition B = C limited to the sphere of things called A. Thus we may say, with some approximation to truth, that “Large plants are plants devoid of locomotive power.”

A barrister may make numbers of most general statements concerning the relations of persons and things in the course of an argument, but it is of course to be understood that he speaks only of persons and things under the English Law. Even mathematicians make statements which are not true with absolute generality. They say that imaginary roots enter into equations by pairs; but this is only true under the tacit condition that the equations in question shall not have imaginary coefficients.[53] The universe, in short, within which they habitually discourse is that of equations with real coefficients. These implied limitations form part of that great mass of tacit knowledge which accompanies all special arguments.

[53] De Morgan, *On the Root of any Function*. Cambridge Philosophical Transactions, 1867, vol. xi. p. 25.

To De Morgan is due the remark, that we do usually think and argue in a limited universe or sphere of notions, even when it is not expressly stated.[54]

[54] *Syllabus of a proposed System of Logic*, §§ 122, 123.

It is worthy of inquiry whether all identities are not really limited to an implied sphere of meaning. When we make such a plain statement as “Gold is malleable” we obviously speak of gold only in its solid state; when we say that “Mercury is a liquid metal” we must be understood to exclude the frozen condition to which it may be reduced in the Arctic regions. Even when we take such a fundamental law of nature as “All substances gravitate,” we must mean by substance, material substance, not including that basis of heat, light, and electrical undulations which occupies space and possesses many wonderful mechanical properties, but not gravity. The proposition then is really of the form

Material substance = Material gravitating substance.

*Negative Propositions.*

In every act of intellect we are engaged with a certain identity or difference between things or sensations compared together. Hitherto I have treated only of identities; and yet it might seem that the relation of difference must be infinitely more common than that of likeness. One thing may resemble a great many other things, but then it differs from all remaining things in the world. Diversity may almost be said to constitute life, being to thought what motion is to a river. The perception of an object involves its discrimination from all other objects. But we may nevertheless be said to detect resemblance as often as we detect difference. We cannot, in fact, assert the existence of a difference, without at the same time implying the existence of an agreement.

If I compare mercury, for instance, with other metals, and decide that it is *not solid*, here is a difference between mercury and solid things, expressed in a negative proposition; but there must be implied, at the same time, an agreement between mercury and the other substances which are not solid. As it is impossible to separate the vowels of the alphabet from the consonants without at the same time separating the consonants from the vowels, so I cannot select as the object of thought *solid things*, without thereby throwing together into another class all things which are *not solid*. The very fact of not possessing a quality, constitutes a new quality which may be the ground of judgment and classification. In this point of view, agreement and difference are ever the two sides of the same act of intellect, and it becomes equally possible to express the same judgment in the one or other aspect.

Between affirmation and negation there is accordingly a perfect equilibrium. Every affirmative proposition implies a negative one, and *vice versâ*. It is even a matter of indifference, in a logical point of view, whether a positive or negative term be used to denote a given quality and the class of things possessing it. If the ordinary state of a man’s body be called *good health*, then in other circumstances he is said *not to be in good health*; but we might equally describe him in the latter state as *sickly*, and in his normal condition he would be *not sickly*. Animal and vegetable substances are now called *organic*, so that the other substances, forming an immensely greater part of the globe, are described negatively as *inorganic*. But we might, with at least equal logical correctness, have described the preponderating class of substances as *mineral*, and then vegetable and animal substances would have been *non-mineral*.

It is plain that any positive term and its corresponding negative divide between them the whole universe of thought: whatever does not fall into one must fall into the other, by the third fundamental Law of Thought, the Law of Duality. It follows at once that there are two modes of representing a difference. Supposing that the things represented by A and B are found to differ, we may indicate (see p. 17) the result of the judgment by the notation

A ~ B.

We may now represent the same judgment by the assertion that A agrees with those things which differ from B, or that A agrees with the not-B’s. Using our notation for negative terms (see p. 14), we obtain

A = A*b*

as the expression of the ordinary negative proposition. Thus if we take A to mean quicksilver, and B solid, then we have the following proposition:--

Quicksilver = Quicksilver not-solid.

There may also be several other classes of negative propositions, of which no notice was taken in the old logic. We may have cases where all A’s are not-B’s, and at the same time all not-B’s are A’s; there may, in short, be a simple identity between A and not-B, which may be expressed in the form

A = *b*.

An example of this form would be

Conductors of electricity = non-electrics.

We shall also frequently have to deal as results of deduction, with simple, partial, or limited identities between negative terms, as in the forms

*a* = *b*, *a* = *a**b*, *a*C = *b*C, etc.

It would be possible to represent affirmative propositions in the negative form. Thus “Iron is solid,” might be expressed as “Iron is not not-solid,” or “Iron is not fluid;” or, taking A and *b* for the terms “iron,” and “not-solid,” the form would be A ~ *b*.

But there are very strong reasons why we should employ all propositions in their affirmative form. All inference proceeds by the substitution of equivalents, and a proposition expressed in the form of an identity is ready to yield all its consequences in the most direct manner. As will be more fully shown, we can infer *in* a negative proposition, but not *by* it. Difference is incapable of becoming the ground of inference; it is only the implied agreement with other differing objects which admits of deductive reasoning; and it will always be found advantageous to employ propositions in the form which exhibits clearly the implied agreements.

*Conversion of Propositions.*

The old books of logic contain many rules concerning the conversion of propositions, that is, the transposition of the subject and predicate in such a way as to obtain a new proposition which will be true when the original proposition is true. The reduction of every proposition to the form of an identity renders all such rules and processes needless. Identity is essentially reciprocal. If the colour of the Atlantic Ocean is the same as that of the Pacific Ocean, that of the Pacific must be the same as that of the Atlantic. Sodium chloride being identical with common salt, common salt must be identical with sodium chloride. If the number of windows in Salisbury Cathedral equals the number of days in the year, the number of days in the year must equal the number of the windows. Lord Chesterfield was not wrong when he said, “I will give anybody their choice of these two truths, which amount to the same thing; He who loves himself best is the honestest man; or, The honestest man loves himself best.” Scotus Erigena exactly expresses this reciprocal character of identity in saying, “There are not two studies, one of philosophy and the other of religion; true philosophy is true religion, and true religion is true philosophy.”

A mathematician would not think it worth while to mention that if *x* = *y* then also *y* = *x*. He would not consider these to be two equations at all, but one equation accidentally written in two different manners. In written symbols one of two names must come first, and the other second, and a like succession must perhaps be observed in our thoughts: but in the relation of identity there is no need for succession in order (see p. 33), each is simultaneously equal and identical to the other. These remarks will hold true both of logical and mathematical identity; so that I shall consider the two forms

A = B and B = A

to express exactly the same identity differently written. All need for rules of conversion disappears, and there will be no single proposition in the system which may not be written with either end foremost. Thus A = AB is the same as AB = A, *a*C = *b*C is the same as *b*C = *a*C, and so forth.

The same remarks are partially true of differences and inequalities, which are also reciprocal to the extent that one thing cannot differ from a second without the second differing from the first. Mars differs in colour from Venus, and Venus must differ from Mars. The Earth differs from Jupiter in density; therefore Jupiter must differ from the Earth. Speaking generally, if A ~ B we shall also have B ~ A, and these two forms may be considered expressions of the same difference. But the relation of differing things is not wholly reciprocal. The density of Jupiter does not differ from that of the Earth in the same way that that of the Earth differs from that of Jupiter. The change of sensation which we experience in passing from Venus to Mars is not the same as what we experience in passing back to Venus, but just the opposite in nature. The colour of the sky is lighter than that of the ocean; therefore that of the ocean cannot be lighter than that of the sky, but darker. In these and all similar cases we gain a notion of *direction* or character of change, and results of immense importance may be shown to rest on this notion. For the present we shall be concerned with the mere fact of identity existing or not existing.

*Twofold Interpretation of Propositions.*

Terms, as we have seen (p. 25), may have a meaning either in extension or intension; and according as one or the other meaning is attributed to the terms of a proposition, so may a different interpretation be assigned to the proposition itself. When the terms are abstract we must read them in intension, and a proposition connecting such terms must denote the identity or non-identity of the qualities respectively denoted by the terms. Thus if we say

Equality = Identity of magnitude,

the assertion means that the circumstance of being equal exactly corresponds with the circumstance of being identical in magnitude. Similarly in

Opacity = Incapability of transmitting light,

the quality of being incapable of transmitting light is declared to be the same as the intended meaning of the word opacity.

When general names form the terms of a proposition we may apply a double interpretation. Thus

Exogens = Dicotyledons

means either that the qualities which belong to all exogens are the same as those which belong to all dicotyledons, or else that every individual falling under one name falls equally under the other. Hence it may be said that there are two distinct fields of logical thought. We may argue either by the qualitative meaning of names or by the quantitative, that is, the extensive meaning. Every argument involving concrete plural terms might be converted into one involving only abstract singular terms, and *vice versâ*. But there are reasons for believing that the intensive or qualitative form of reasoning is the primary and fundamental one. It is sufficient to point out that the extensive meaning of a name is a changeable and fleeting thing, while the intensive meaning may nevertheless remain fixed. Very numerous additions have been lately made to the extensive meanings both of planet and element. Every iron steam-ship which is made or destroyed adds to or subtracts from the extensive meaning of the name steam-ship, without necessarily affecting the intensive meaning. Stage coach means as much as ever in one way, but in extension the class is nearly extinct. Chinese railway, on the other hand, is a term represented only by a single instance; in twenty years it may be the name of a large class.