The late Professor Boole first drew attention, so far as I know, to this property of logical terms, and he called it the property of Commutativeness. 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 the following pages 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.

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 place with the lower, or 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 shall call simple logical relations.

e 'Laws of Thought,' p. 29.

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 influence, that what is true of one thing or circumstance 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 degree. 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 versa. '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 or an equation; whatever is true of the number of men is true of the number of minds, and vice versa. 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 or form of identity, all such expressions of identity 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 the 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 by memory or otherwise 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 that of hydrogen sulphide,' 'the taste of silver hyposulphite is that of cane sugar,' the sound of an earthquake is that of distant artillery. Such are propositions stating, accurately or otherwise, the identity or non-identity of simple physical sensations. Judgments of this kind are necessarily presupposed 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 of gold. I must make a judgment of the colour, the specific gravity, the hardness, sound, and chemical properties; and each of these judgments might be 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 xaλkós is copper we must show the identity of each quality recorded of xaλkós with a quality of copper. To establish Deal as the landing-place of Cæsar, every circumstance 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.

χαλκός

All such identities may 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 sensations. But in the same form we may express identity of any group of qualities, as in Xaλkós = 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 Queen 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 a.' 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 identities; in addition to some instances which have already been given I may suggest

Crystals of cubical system = Crystals incapable of double refraction.

All definitions are necessarily of this form of simple identity, 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 quite 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 ".' Here we certainly have simple identity of terms; but he

a Thomson and Tait, Treatise on Natural Philosophy,' vol. i. P. 182. b Prior Analytics,' I. cap. xxvii. 3.