and 'some A is not B'; and of these, both can be neither true (Law of Contradiction), nor false (Law of Excluded Middle); one must be false, and the other true. If all the things belonging to the class A are, however, individually considered, that is, if 'A' be taken as standing, at the same time, for a single individual only, then, of that individual, either 'B' or 'not-B' must be true. Thus 'wise' or 'not-wise' must be true of a single individual man, that is, of every man considered as an individual thing, one or other of these two contradictory terms must be true, though, on the whole, some individuals may belong to the class of wise, and others to the class of not-wise.

§ 4. (4) The next principle that we shall give here is a postulate of Logic. It is thus stated by Hamilton :- "The only postulate of Logic which requires an articulate enouncement is the demand, that before dealing with a judgment or reasoning expressed in language, the import of its terms should be fully understood; in other words, Logic postulates to be allowed to state explicitly in language all that is implicitly contained in the thought1:" that is, given a term, proposition, or argument, the thought expressed by it, or its meaning and import may be stated in any other form of words, which expresses the same thing. Thus, in describing the logical characters of a term or of a proposition, it is allowable to make any verbal changes we like, in order to reduce it to the logical form, provided the meaning remains the same. In testing an argument we may state it in any form of words we please, provided the thought contained in the constituent propositions or in the argument as a whole remains unaltered.

§ 5. Mill regards all the four principles given above as postulates. "Whatever is true in one form of words is true also in every other form of words which conveys the same meaning 2." He gives this for the Principle of Identity, regards it as the most universal postulate of Logic, and calls it a first Principle of

1 Hamilton's Lectures, Vol. I. p. 114.

2 An Examination of Hamilton's Philosophy, p. 482.

Thought. According to him the postulate we have given above is included in this. For the Principle of Contradiction, Mill gives the following postulate: "The affirmation of any assertion and the denial of its contradictory are logical equivalents, which it is allowable and indispensable to make use of as mutually convertible1." For the affirmation of the assertion "A is B," we may substitute the denial of its contradictory "A is not B"; or for the affirmation of the assertion "A is not B" we may substitute the denial of its contradictory ‘A is B': that is, the denial of ‘A is B' and the assertion of its contradictory ‘A is not B' are logically the same. For the Principle of Excluded Middle, Mill gives the postulate that it is allowable "to substitute for the denial of either of two contradictory propositions, the assertion of the other 2." That is, of the two propositions 'A is B' and 'A is not B,' we may substitute the assertion of one for the denial of the other: for the denial of 'A is B' we may substitute the assertion of 'A is not B'; and for that of the latter the assertion of the former.

Mill calls his three postulates the 'universal postulates of reasoning,' which ought to be placed, at the earliest, in the second part of Logic—the Theory of Judgments; since they essentially involve the ideas of truth and falsity, which are attributes of judgments only, not of names or concepts. This remark seems not applicable to his first postulate (that for the Law of Identity: "Whatever is true in one form of words is true also in every other form of words, which conveys the same meaning") as we require it for making verbal alterations, and for stating in logical form the meaning of a term, before describing its logical characters. Still less is the remark applicable to the postulate which we have given above. We require the aid of that postulate in order to state explicitly the thought that is implicitly contained in a term, and, in the case of an ambiguous term, to recognize its different meanings and treat them as such. It is hardly necessary to say that it is impossible to describe the logical characters 2 Ibid. p. 490.

1 Ibid. p. 488.

of a term without fully understanding and explicitly stating its meaning or meanings, the thought or thoughts, the attribute or thing, signified by it. For this reason, all the principles are here placed in the Introduction before the first part of Logic treating of Terms or Concepts.

Hamilton calls the first three principles the 'fundamental laws of thought,' and prefers to call the second the 'Law of Noncontradiction,' "as it enjoins the absence of contradiction as an indispensable condition of thought1."

ences.

Ueberweg calls them the Principles or Axioms of Inference, and places them at the beginning of the part treating of InferTo these three he adds a fourth, namely, the Axiom of the (determining or sufficient) Reason. The statement of this Principle or Axiom by Leibnitz seems to be the best, and is as follows:-" In virtue of this principle we know that no fact can be found real, no proposition true, without a sufficient reason, why it is in this way rather than in another."

According to Ueberweg the Axiom of Contradiction and the Axiom of Excluded Middle may be comprehended in a general principle, namely, the Principle of Contradictory Disjunction. The formula of this is :-'A is either B or is not-B,' which means that 'A' cannot be both 'B' and 'not-B' (Law of Contradiction), and that it must be one or the other (Law of Excluded Middle).

Ueberweg gives also another axiom which he calls the Axiom of Consistency. He states it as follows :-'A which is B is B, i. e., every attribute which belongs to the subject notion may serve as a predicate to the same.' He regards this axiom as allied with the Axiom of Identity 2.

§ 6. To the principles given above should be added the following:

(5) Aristotle's Dictum de omni et nullo 3. "Whatever is affirmed or denied of a class distributively may be affirmed or

1 Hamilton's Lectures, Vol. III. p. 82.

2 Ueberweg's Logic, English Translation, pp. 231, 275, 281, 283, &c.

3 See below, Part III. Chapter IV.

denied of every thing belonging to that class"; or, "what belongs to a higher class belongs to a lower." Some logicians maintain that it can be deduced from the three Laws of Thought, while others regard it as an independent axiom incapable of deduction from those laws.

(6) The fundamental axioms or canons of Syllogism as given by different logicians (Mill, Martineau, Thompson, Lambert, Whately, &c.1).

(7) The Mathematical Axioms :-(1) that of Argumentum à fortiori, namely, that "a thing which is greater than a second, which is greater than a third, is greater than the third"; (2) the axiom that "two things equal to the same thing are equal to each other"; and other axioms of a similar nature.

1 See below, Appendix A.

PART I-TERMS.

CHAPTER I.

THE VARIOUS DIVISIONS OF TERMS.

§ 1. A name may be defined as a sign for a thing or things. More accurately, it is a word, or a combination of words, signifying some object of thought, or something real or imaginary, mental or material, substantive or attributive, phenomenal or nöumenal. For example, the words 'animal,' 'plant,' 'flower,' 'table,' 'paper,' 'chair' are names of real things, while the words ‘centaur,' ‘golden mountain,' &c., are names standing for imaginary objects; the words 'mind,' 'soul,' 'spirit,' 'self,' &c., are names signifying mental things or substances, while the words 'gold,' 'silver,' 'mineral,' 'copper,' &c., are names standing for material things; the words 'sensation,' 'pleasure,' 'pain,' 'perception,' 'imagination,' 'memory,' &c., are names expressing attributes of mind, while 'solidity,' 'colour,' 'figure,' 'hardness,' &c., are words signifying attributes of matter; the words 'thinking,' 'perceiving,' 'feeling,' 'wishing,' 'hoping,' &c., are names expressing acts or phenomena of mind, while the words 'moving,' 'melting,' 'expanding,' 'cooling,' &c., are words signifying phenomena or changes of bodies; the words 'thing-in-itself,' 'matter-in-itself,' 'mind-in-itself,' are names expressing nöumena or realities which are believed to underlie all phenomena; and the