CHAPTER III.

THE PRIMARY AXIOMS OF PURE THOUGHT.

HAVIN

AVING defined Logic to be the Science of the Necessary Laws of Pure Thought, our first object must be to ascertain what are the Fundamental and Universal Laws, here called Primary Axioms, to which all Thought, as such, is subject. In the separate consideration, which will come afterwards, of the three classes of Thoughts, namely, Concepts, Judgments, and Reasonings, we may expect

to find Special Laws or Rules which are applicable only to one or two of these divisions. Such Special Rules may or may not be derivative in character; that is, they may be either immediate inferences from the Primary Axioms which govern all the products of the Thinking Faculty, or they may be independent, as resting upon their own evidence. Of this hereafter. But our first inquiry must be, whether there are any Axioms of universal applicability, which underlie and govern every act and product of the human Understanding; and, if there are such, to determine their character and significance.

If there are such Axioms, they must be few, meagre in import, not susceptible of proof, and recognizable by all as familiar truisms, which have always implicitly directed their thoughts, though perhaps, on account of their very obviousness, they have never been explicitly stated or drawn out into distinct consciousness. They must have these characteristics, because they concern only the Forms of Thought, or the manner of thinking irrespective of what

we are thinking about; and as these Forms themselves are necessarily limited in number and narrow in significance, the Axioms which underlie them all, and constitute their common features, must be still fewer and poorer in import. They cannot admit of proof, as their truth is presupposed in every act of reasoning, and therefore no argument or proof is possible unless their veracity is taken for granted. They must be recognized by all as mere truisms, because they are thus self-evident, and because their truth has been acknowledged and acted upon in every Form of Thought which we have ever experienced. The First Principles of all the sciences are avowedly thus few and meagre, as is seen to be the case with the introductory axioms of Geometry and Physics. With still more reason do we expect the First Principles of all Thought to possess this character, as they stand in the same relation to the axioms of the special sciences, that these axioms do to the most advanced theorems which have been built upon them, or which have been constructed by taking them for granted.

After this explanation, we need not be surprised to find that all the Primary Axioms of Pure Thought are perhaps reducible to this single principle:- All Thought must be consistent with itself. If it be inconsistent, if, directly or indirectly, it contradicts itself, it is self-destructive, and the Thought is null. Thus stated, the principle is coincident with that which is usually called the Law of Contradiction, though, as Hamilton remarks, it ought rather to be termed the Law of Non-Contradiction. Practically speaking, every Thought which must be rejected as formally invalid — that is, which is radically vicious in Form, whatever be its Matter-offends against this principle. By logicians generally, however, this principle has been explicated into three general Axioms, called the Law of Identity, the Law of Contradiction, and the Law of Excluded Middle. The ground of this explication may be thus set forth.

The primary element of all Thought is a Judgment, which arises from a Comparison. Hence, all Thought must proceed either by affirmation or denial, as these are the only two possible forms of Judgment. Having compared any two Concepts with each other, we either perceive their identity, similarity, congruence, or some other relation whereby we affirm their union in one act of Thought; or we perceive the opposite relation between them, such as difference, unlikeness, or incompatibility, whereby we deny one of the other. As any Concept can be compared with any other, and as the Judgment which follows such comparison must either affirm or deny one of the other, there being no third form of Judgment conceivable, we have the Axiom which is usually called the Law of Excluded Third or Excluded Middle, Lex Exclusi Tertii aut Medii. Either A is B, or A is not B: if we make any Judgment, that is, if we think at all, one of these two must be true; for no third form is conceivable. It has been enounced in various forms: Of two contradictory judgments, one must be true; Every predicate may be affirmed or denied of every subject; Every conceivable thing is either A or not-A. Of course, A and not-A, taken together, include the universe, the universe not only of all that is actual, but of all that is conceivable; for as not-A excludes A only and nothing else, it includes the universe excepting A only.

Still further: Not only are affirmation and negation the only conceivable forms of Judgment, but, as contradictory opposites, they are absolutely incompatible or mutually destructive. The admission of one is tantamount to a rejection of the other. If taken together, they destroy each other, and the Thought is rendered null. To express this truth algebraically, A not-A=0. Here we have the well-known Law of Contradiction, more properly of Non-Contradiction, of which the formula is, A is not not-A

Evidently this Law is the principle of all logical negation and discrimination. It has been variously expressed: -Contradictory attributes cannot be affirmed of the same subject; What is contradictory is inconceivable. It is less correctly expressed in the adage, "It is impossible for the same thing to be and not to be." This is a maxim which concerns the Matter of Thought, and therefore we must add to it the material limitations, in the same place, at the same time, in the same respect, &c. It is a mistake, then, to. maintain that the Axiom, "Contradictory attributes cannot be affirmed of the same subject," is not universally true, because we can form such assertions as this: A man can be both young and not-young, though not at the same time. In Logic, where we consider only the Form of the Thought, a Judgment must be expressed by the present tense of the verb to be; for what we affirm is not the past or future union of two real phenomena, but the present coexistence and agreement of two Concepts in the mind. Hence, the logical Judgment, this man is NOT young, is absolutely incompatible with the assertion, this man is young, though it is compatible with the very different assertion, this man HAS BEEN young.

Once more: The formula, A is not not-A, proves, on reduction, to be the exact equivalent or consequence of this, A is A. Here we have the principle of affirmation and agreement, as the former was that of negation and difference. If an object cannot be thought under contradictory attributes, it is because it has a definite character of its own, excluding one of the contradictories through including the other. "The universe of conceivable objects," to adopt Mr. Mansel's language, "embraces both A and not-A; it is only when definitely conceived as the one, that an object cannot be conceived as the other. Every object of thought, as such, is thus conceived by limitation and difference; as having definite characteristics by which it is

marked off and distinguished from all others; as being, in short, itself, and nothing else." Here, then, we have a third Primary Axiom, expressed as the Law of Identity : Every A is A; Every object of thought is conceived as itself; Every thing is equal to itself or agrees with itself; Every whole is the sum of all its parts.

Thus we have three Primary Axioms of Pure Thought, -the Law of Identity, the Law of Contradiction, and the Law of Excluded Middle, all of which may be regarded as explications of the single rule, that all Thought must be consistent with itself, or as corollaries from this one principle, that Judgment, which is the basis of all Thought, proceeds only by affirmation and denial. The mutual dependence and correlation of these three Axioms may be further illustrated thus.

I can think any object only by placing it under a Concept, or Class-notion expressed by a General Term; and I can do this only by recognizing that it possesses the attributes which belong to this Concept and are common to all the members of this Class (Law of Identity, affirmation of similarity or agreement); by discriminating it from other objects which have different attributes (Law of Contradiction, negation of agreement); and both this affirmation and denial proceed by the Law of Excluded Middle, which declares, for each given attribute, that the one or the other is absolutely necessary. Either it does, or does not, belong to the object, and the object does or does not belong to the Class. In respect to the Laws of Identity and Contradiction, says Sir William Hamilton, "each infers the other, but only through the principle of Excluded Middle; and the principle of Excluded Middle only exists through the supposition of the two others. Thus, the principles of Identity and Contradiction cannot move, cannot be applied,—except through supposing the principle of Excluded Middle; and this last cannot be conceived existent except