Parallel to the three operations of the mind, we have the three products, (1) terms, (2) propositions, and (3) syllogisms.
§3. Words are either (1) categorematic, or (2) syncategorems. Categorematic words are those which can be subjects or predicates. Syncategorems are such words as all, some, not. That is, there are words having independent meaning, and words whose meaning depends on their connections with other words.
A name is either (1) singular, or (2) common. That is, it names something having an independent existence or a class existing only in the connection of resemblance between singulars.
A name is either (1) finite, or (2) infinite, that is, formed by the negative particle prefixed to another name, as non-man. This follows the division of propositions into affirmative and negative.
A name is either (1) positive, (2) privative, or (3) negative. A positive name is an independent one, a negative name is one which is connected with a quality not possessed; a privative name is one in which this connection is suggested by a really analogous case where the quality denied is possessed. Thus, when a stone is said to be non-seeing, this is a negative predicate; when a man is said to be blind, this is a privative predicate.
A name is either (1) univocal, (2) equivocal, or (3) analogous. A univocal name has but one meaning, unconnected with any other. An equivocal name has two or more meanings which it reunites in a blind, irrational way. An analogous name has two or more meanings which are connected by a mediating reason.
Anybody who chooses to do so can easily go through the whole treatise in this way, and convince himself that these concepts,—or independency, blind connection, mediation,—enter into every distinction of logic. Independency or arbitrariness is the idea of something being a first, and no second to anything. Blind connection is the idea of something being second to something else, without any third to bring the connection about. Mediation is the idea of thirdness to a first and second that are thus brought into relation.
Of course, many maxims of logic require attention in the prolegomena of philosophy. I will illustrate one of these. The sole justification that any hypothesis can have is that it explains facts and renders them intelligible. If then a philosopher like Herbert Spencer tells us that the principle of the conservation of energy or anything else of the sort is a primitive law of nature, or if another tells me that feeling is an ultimate property of protoplasm, I ask them how they know this. To say that a law is primitive or a fact ultimate is to say that this law or this fact is absolutely inexplicable and unintelligible. But this is a hypothesis. And what does this hypothesis serve to explain? Nothing, except that it is difficult to show from what the law or fact in question is derived; a fact sufficiently explicable by the vera causa of our ignorance. There are, I grant, some facts of which it is absurd to ask for an explanation; as, for example, that things are many and not all alike, for it is of the nature of a hypothesis to explain likenesses or definite relations of unlikeness. The mere absence of definite relations is not a thing requiring any explanation at all. To ask what determines anything to indeterminacy is a senseless and meaningless question. But when a fact or principle is determinate and peculiar, to silence inquiry as to its derivation by the theory that no theory can explain it, this is an affront to reason, and a plain inconsistency. A philosopher who confesses his inability to explain why space has three dimensions, no more and no less, or anything else of like difficulty, may be pardoned, so long as he offers no other excuse than his ignorance. But if he sets up a theory of space which represents its three-dimensional character to be something absolutely inexplicable, now and forever, that theory is to be condemned as involving the hypothesis that no hypothesis is possible. It is the old sophism of the Cretan.
The science which, next after logic, may be expected to throw the most light upon philosophy, is mathematics. It is historical fact, I believe, that it was the mathematicians Thales, Pythagoras, and Plato who created metaphysics, and that metaphysics has always been the ape of mathematics. Seeing how the propositions of geometry flowed demonstratively from a few postulates, men got the notion that the same must be true in philosophy. But of late mathematicians have fully agreed that the axioms of geometry (as they are wrongly called) are not by any means evidently true. Euclid, be it observed, never pretended they were evident; he does not reckon them among his κοιναὶ ἔννοιαι, or things everybody knows,1 but among the αἴτηματα, postulates, or things the author must beg you to admit, because he is unable to prove them. At any rate, it is now agreed that there is no reason whatever to think the sum of the three angles of a triangle precisely equal to 180°. It is generally admitted that the evidence is that the departure from 180° (if there is any) will be greater the larger the triangle, and in the case of a triangle having for its base the diameter of the earth’s orbit and for its apex the furthest star, the sum hardly can differ, according to observation, so much as 0″.1. It is probable the discrepancy is far less. Nevertheless, there is an infinite number of different possible values, of which precisely 180° is only one; so that the probability is as 1 to ∞, or 0 to 1, that the value is just 180°. In other words, it seems for the present impossible to suppose the postulates of geometry precisely true. The matter is reduced to one of evidence; and as absolute precision is beyond the reach of direct observation, so it can never be rendered probable by evidence, which is indirect observation.
Thus, the postulates of geometry must go into the number of things approximately true. It may be thousands of years before men find out whether the sum of the three angles of a triangle is greater or less than 180°; but the presumption is it is one or the other.
Now what is metaphysics, which has always formed itself after the model of mathematics, to say to this state of things? The mathematical axioms being discredited, are the metaphysical ones to remain unquestioned? I trow not. There is one proposition, now held to be very certain, though denied throughout antiquity, namely that every event is precisely determined by general laws, which evidently never can be rendered probable by observation, and which, if admitted, must, therefore, stand as self-evident. This is a metaphysical postulate closely analogous to the postulates of geometry. Its fate is sealed. The geometrical axioms being exploded, this is for the future untenable. Whenever we attempt to verify a physical law, we find discrepancies between observation and theory, which we rightly set down as errors of observation. But now it appears we have no reason to deny that there are similar, though no doubt far smaller, discrepancies between the law and the real facts. As Lucretius says, the atoms swerve from the paths to which the laws of mechanics would confine them. I do not now inquire whether there is or not any positive evidence that this is so. What I am at present urging is that this arbitrariness is a conception occurring in logic, encouraged by mathematics, and ought to be regarded as a possible material to be used in the construction of a philosophical theory, should we find that it would suit the facts. We observe that phenomena approach very closely to satisfying general laws; but we have not the smallest reason for supposing that they satisfy them precisely.
Philosophy can draw from mathematics many other valuable ideas, at which I can here only hint.
Certain other conceptions of modern mathematics are indispensible to a philosophy which is to be upon the intellectual level of our age. In the first place, there is the conception of a space of more than three dimensions, ordinarily regarded as highly mysterious, but really easy enough. We can have no visual image of a space of three dimensions; we can only see its projections upon surfaces. A perspective view, or picture on a plane surface, represents all we can see at any one time. It would, therefore, be unreasonable to ask how a space of four or five dimensions would look, in any other than a projective sense. Take one of those glass paper-weights cut into the form of a polyhedron. Photographs of such a body of three dimensions taken from one point of view before and after giving it one turn are sufficient to determine how it must look however it be turned. Now, if it had four dimensions instead of three, the difference would be that by a certain peculiar effort we could turn it so as to give a different perspective form from any that the first two pictures would account for. Now, the reader does not know all about the geometry of three or even that of two dimensions; therefore, he cannot