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 = &c. = BCA Boole first drew attention in recent years 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 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. 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. 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. 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 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, that is to say, 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 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 propositious 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 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, 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 Xaλrós = 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. = 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. 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. = 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." 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." 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; 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. 1 Prior Analytics, i. cap. xxvii. 3. |