follows a progressive route from principles to conclusions, even when discovery (supposing discovery foregone) was made by analysis or regression to principles, of which expository method no better illustration could be given than the practice of Euclid in the demonstration of his 'Elements.' On the other hand, it may be said that the line of discovery is itself the line upon which the truth about any question can best be expounded or understood for the same reason that was found successful in discovery, namely, that the mind (now of the learner) has before it something quite definite and specific to start from; upon which view, the method of exposition should be analytic or regressive to principles, at least wherever the discovery took that route. The blending of both methods, when possible, is doubtless most effective; otherwise it depends upon circumstances-chiefly the character of the learner, but also the nature of the subject in respect of complexity, which should be preferred,-when one alone is followed1." § 12. By some logicians Deductive Logic is regarded as identical with Formal Logic; by others as a part of Material Logic. According to all, it does not directly concern itself with the real truth or falsity of its data, but with their formal correctness or freedom from inconsistency, and with the legitimacy of the results from them. In this work it is proposed to treat of the following subjects:-The fundamental principles; the name, the concept, the term and its divisions; denotation, connotation, extension, comprehension; the proposition, the judgment, and their divisions; the predicables; the theory of predication and the import of propositions; definition, division; inference, reasoning and their divisions; immediate inference and its divisions; the syllogism, its divisions, its canons, its rules, its figures, its moods, its function and value; reduction; fallacies; probable reasoning and probability. 1 Encyclopædia Britannica, 9th edition, Vol. 1. p. 797. CHAPTER II. THE FUNDAMENTAL PRINCIPLES OF DEDUCTIVE LOGIC. § 1. THERE is great difference of opinion among logicians as to the nature, number, name, origin, and place in a Treatise on Logic, of what we have here called the fundamental principles of Deductive Logic. They may be stated as follows ::(1) "A is A." "A thing is what it is." "Every thing is equal to itself." "Every thing is what it is." This is called the Principle or Axiom of Identity. It really means that the data, with which we start in Deductive Logic, must remain unaltered; that, by them we must abide in all our deductions and reasonings. If we have granted or assumed that a certain thing possesses a certain attribute, we must always admit that; if we have used a term in a certain meaning, we must always use it in that meaning, or give notice when any change is made. In Deductive Logic things and their attributes, or thoughts, are supposed to be unalterably fixed; and the same thing must always be regarded as possessed of the same attributes. In nature, no doubt, a thing may change and have attributes which it did not originally possess; but Deductive Logic takes no cognizance of such changes. It assumes, on the contrary, that all things and their relations are as absolutely fixed and permanent as are the properties and relations of Geometrical Figures. And the principle or axiom of identity expresses this unalterable or absolutely fixed nature of things, postulated in Deductive Logic, by stating that "Every thing is what it is," that is, it cannot change and be other than what it is, nor can it lose any of its properties or attributes. In other words, the element of time or change has no place in Deductive Logic. "The same § 2. (2) "A cannot be both B and not-B." thing cannot be both B and not-B." "This paper cannot be both white and not-white." This is called the Principle or Axiom of Contradiction. It means that two contradictory terms B and not-B cannot both be true, at the same time, of one and the same individual thing A. If the term B be true of the individual thing A, then the term not-B cannot, at the same time, be true of it; or if the term not-B be true of it, then B cannot, at the same time, be true of it. In other words, two contradictory propositions cannot both be true; taking A to mean an individual thing, one and the same thing, and using B in the same sense in both, the two propositions 'A is B' and 'A is not-B' are contradictory, and cannot both be true: if one be true, the other must be false; that is, if 'A is B' be true, then 'A is not-B' must be false; and if 'A is not-B' be true, then 'A is B' must be false. For example, a leaf cannot, at the same time, be 'green' and 'not-green'; if it is 'green,' it cannot, at the same time, be 'not-green' (see p. 10); a piece of gold cannot, at the same time, be 'yellow' and 'not-yellow'; if it is 'yellow,' it cannot, at the same time, be 'not-yellow'; a sample of water cannot, at the same time, be 'liquid' and 'not-liquid,' 'cold' and 'not-cold,' 'hot' and 'not-hot'; if it has one quality, it cannot, at the same time, have the contradictory quality; 'cold' and 'not-cold,' 'liquid' and 'not-liquid' are contradictory qualities, and cannot be possessed, at the same time, by the same thing. Similarly, a thing cannot at the same time be 'mortal' and 'notmortal,' 'extended' and 'not-extended,' 'organized' and 'notorganized,' 'existent' and 'not-existent,' 'good' and 'not-good'; if it has one of these contradictory attributes, it cannot, at the same time, have the other. , § 3. (3) "A is either B or not-B." "The same thing is either B or not-B." "This paper is either white or not-white." This is called the Principle or Axiom of Excluded Middle. It means that two contradictory terms, B and not-B, cannot both 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,' |