be false, at the same time, of one and the same individual thing. If the term B be not true of the individual thing A, then the term not-B must be true of it; if the term not-B be not true of it, then B must be true of it. In other words, two contradictory propositions cannot both be false; taking A as before to mean one and the same individual thing, and using the term B in the same sense in both, the two propositions 'A is B' and 'A is not-B' are contradictory and cannot both be false; if one be false, the other must be true; that is, if the proposition 'A is B' be false, then the proposition 'A is not-B' must be true, and if ‘A is not-B' be false, then ‘A is B' must be true. For example, the two propositions, 'a leaf is green,' and 'a leaf is not-green,' cannot both be false; a leaf is either 'green' or 'not-green': if the term 'green' be not true of a leaf, then its contradictory 'not-green' must be true of it; that is, two contradictory terms cannot both be false of one and the same thing. Similarly, 'yellow' and 'not-yellow,' 'liquid' and 'notliquid,' 'good and not-good' cannot both be false of one and the same thing, such as a piece of gold, a sample of water, or any other individual thing: if one of them be false of any one of these things, then the other must be true of it. In other words, of the two contradictory propositions "a leaf is green” and “a leaf is not-green," both cannot be false; if one be false, the other must be true; similarly, of the contradictory propositions "this sample of water is "cold," and "this sample of water is not-cold," "this piece of gold is yellow," and "this piece of gold is notyellow," "this piece of chalk is solid," and "this piece of chalk is not-solid,” both cannot be false: if one be false, the other must be true. According to the Principle of Contradiction, two contradictory propositions cannot both be true, that is, one must be false; and, according to the Principle of Excluded Middle, both of them cannot be false, that is, one must be true. Of the two contradictory propositions, 'A is B' and 'A is not-B' (taking A to mean an individual thing, and using A and B in the same sense in both), one must be false according to the former, and one must be true according to the latter; that is, if the proposition 'A is B' be true, then the proposition 'A is not-B' must be false; if 'A is not-B' be true, then 'A is B' must be false; and if the proposition 'A is B' be false, then 'A is not-B' must be true; if ‘A is not-B' be false, then ‘A is B' must be true. According to the two principles, therefore, the truth of one contradictory proposition implies the falsity of the other, and the falsity of one implies the truth of the other; that is, of two contradictory propositions one must be true by the Principle of Excluded Middle, and the other must be false by the Principle of Contradiction. We have taken above A to mean an individual thing, one and the same thing; and, in that case, two contradictory terms B and not-B cannot both be either true or false of A; or, in other words, the two propositions 'A is B' and 'A is not-B' are contradictory, and cannot both be either true or false. But if A signifies a class of things, that is, if A be a general term or a name for each individual of a number of things, then the two contradictory terms B and not-B might both be true or false of A. 'B' might be true of some individuals and false of others, all belonging to 'A,' so that the two propositions 'A is B' and 'A is not-B' would both be false in one sense, and true in another— false if 'A' is taken universally, that is, if A stands for all the individuals of the class, and true if 'A' is taken partially, that is, if A stands for a part, or at least one individual, of the class. Let us take, for example, the common name 'man' and the two contradictory terms 'wise' and 'not-wise.' Now, man as a class is not either 'wise' or 'not-wise'; in other words, the two propositions 'man is wise' and 'man is not-wise' are both false, if the term 'man' be taken universally to denote all men, while they are both true if the term 'man' be taken partially to denote some men or at least one man. Hence two contradictory terms may be both false of a class; that is, the two propositions 'A is B' and 'A is not-B' may be both false, if 'A' be a general term or common name. In other words, the two contradictory propositions are then not 'A is B' and 'A is not-B,' but 'all A is B,' 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. PRAPY T Thought. According to him the postulate we have given above 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 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 Identity2. § 6. To the principles given above should be added the following: (5)/Aristotle's Dictum de omni et nullo3. "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. |