formal when the object, whether actually existing or not, is simply free from any self-contradiction. The latter is the end of what is called Formal Logic, and the former of what is called Material Logic. In Formal Logic, the concepts, judgments, and reasonings need not be really true. It is sufficient if they conform to the fundamental principles of consistency or laws of thought, as they are called, and be free from any inner contradiction or inconsistency. In Material Logic, also called by Mill the Logic of Truth, they must be true or right, and correspond to the realities actually existing; they must be valid not only formally, but also really; they must be free not only from any self-contradiction, but also from any inconsistency with reality, that is, a concept must be an attribute or a collection of attributes actually existing in things, a judgment, a relation between two true concepts, and a reasoning must lead to a conclusion that agrees with fact. The end of Material Logic is thus the attainment of truth in the stricter and proper sense, that is, of real truth, while the end of Formal Logic is merely consistency or freedom from selfcontradiction. Formal Logic is often called Pure Logic, and also the Logic of Consistency. Hamilton's definition of Logic, as given above, is a definition of Formal Logic, while Mill's and Spencer's are definitions of Material Logic. In the latter we are concerned with terms, propositions, and arguments that have reference to actual existences, while in the former we are concerned not with what is actual, but with what is possible, not with what is real in Nature, but with what may be realized in Thought. Formal Logic includes in its sphere all possible notions, judgments, and reasonings, or all possible attributes, and their relations, and does not confine itself to what is actual or real in Nature. The definition which we have given at the beginning of this chapter is that of Formal or of Material Logic according as the word valid is taken to mean mere conformity to the principles of consistency, or agreement with reality, that is, according as it means merely formally valid or really valid and true. If the products of comparison, namely, concepts, judgments, and reasonings, are required to agree with the actually existing things and phenomena, then our definition becomes the definition of Material Logic. If, on the contrary, they are required simply to be free from self-contradiction, then our definition becomes the definition of Formal Logic. § 11. Logic is usually regarded as consisting of three parts,— the first part treating of the process and products of conception; the second, of judgment; and the third, of reasoning or inference. To these three parts may be added a fourth, namely, Method, treating of the arrangement or disposing of a series of reasonings in an essay or discourse. Method has been defined as "the art of disposing well a series of many thoughts, either for discovering truth when we are ignorant of it, or for proving it to others when it is already known." "Thus there are two kinds of Method, one for discovering truth, which is called analysis, or the method of resolution, and which may also be termed the method of invention; and the other for explaining it to others when we have found it, which is called synthesis, or the method of composition, and which may be also called the method of doctrine1." "Without stepping," says Professor Robertson, "beyond the bounds of Logic conceived as a formal doctrine, a fourth department under the name of method or disposing may be added to the three departments regularly assigned-conceiving (simple apprehension), judging, reasoning; and this would consider how reasonings, when employed continuously upon any matter whatever, should be set forth to produce their combined effect upon the mind. The question is formal, being one of mere exposition, and concerns the teacher in relation to the learner. How should results, attained by continuous reasoning, be set before the mind of a learner? Upon a line representing the course by which they were actually wrought out, or always in the fixed order of following from express principles to which preliminary assent is required? If the latter, all teaching becomes synthetic, and 1 Professor Baynes' Port Royal Logic, pp. 308-9. 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 "The same it lose any of its properties or attributes. In other words, the element of time or change has no place in Deductive Logic. § 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 |