Logic may be most briefly defined, in accordance with the etymology of the word, as the science of reasoning or 'the art of thinking.' It is a scientific account of the laws which regulate the passage in thought from one statement to another, and which must be observed if the thinking process is to be valid. The theory of every operation is later than its performance, and men were accustomed to think correctly long before they began to reflect upon their thinking faculties and the processes by which their results were reached. The attention which Socrates devoted to the meaning and justification of general names is signalised by Aristotle as the beginning of logical theory. It was Aristotle himself, however, who first elaborated the idea of the science, and defined its sphere by separating it from the metaphysical questions with which logical discussions are always associated in his predecessors. The six treatises afterwards collected under the name of the Organon contain the gist of what is still taught as formal logic; but the term logic was probably first used by the Stoics in the wide sense with which we are familiar. Aristotle himself possessed no single name for the science of which he was the founder.
The independence which Aristotle conferred upon the new science has enabled it to survive to the present day almost without change, and with very few additions of importance. But, while the edifice of Aristotle remains architectonically complete upon its own basis, it has become customary to add to this science of logic proper a second part, called Mixed, Material, or Inductive Logic, embracing an account of the methods of science and the conditions of scientific proof. The modern version of the Aristotelian Logic is then called, by way of distinction, Pure or Formal Logic. The meaning of this designation is that logic, as such, takes no account of the matter of our reasonings—i.e. of the things reasoned about: it deals solely with the form or skeleton of the reasoning process itself. Thus, if we say, 'Englishmen are white-skinned,' logic has no occasion to consider the truth of this statement as a matter of fact or science; it deals only with the form of the proposition or judgment as a general logical mould into which any pair of notions may be fitted. It treats the proposition, in short, only so far as it is expressible in the form, 'X is Y.' To this abstraction from all questions regarding the adequacy of our notions, and the material truth of our assertions, formal logic owes its completeness as a science. It looks upon thought, not as the expression of the truth of things, but as a series of mechanical operations, and its aim is to lay down the general or symbolic forms which these operations must assume in order to insure that the end shall be consistent with the beginning. It is apparent, then, that in any reasoning process formal logic only guarantees that the conclusion is true if the premises from which we started were true. It has accordingly been called the logic of consistency, as opposed to induction, which seeks to be a logic of truth. Pure logic takes its material, as it were, ready-made from the hands of observation, and merely watches over its correct manipulation. Reasoning in the strict logical sense is, in fact, merely analytic; the conclusion only brings to explicit consciousness what was implied or involved in the premises. Formal logic is thus, in its most general aspect, an application, by means of many subordinate rules, of the laws of identity and non-contradiction. Practically, however, it is of great service in clarifying the thought of the individual, though, in a sense, merely teaching him what he knows already.
Formal logic is usually treated under the three heads of Notions, Judgments, and Reasonings; or, if regard be had to the verbal expression of thought, the Notion, Judgment, and Reasoning appear respectively as Term, Proposition, and Syllogism. Though pure logic has strictly nothing to say about the formation of general names or the adequacy of our notions, it is customary for logical writers to expound under the first head the nature of generalisation and definition—the processes by which our notions are formed and tested. The Judgment, however, may be taken as the unit in logic, for it is only in their relation as subject and predicate of a judgment that notions become susceptible of logical treatment. The combination of two judgments (involving three notions), in such a form that a third judgment is deduced from them, constitutes a Syllogism—e.g. ‘All fishes are cold-blooded. The whale is not cold-blooded. Therefore the whale is not a fish.’ The variations of this fundamental type of reasoning constitute the scholastic doctrine of the moods and figures of the Syllogism. As an appendix to this exposition of the normal forms of inference there follows a discussion of the different classes of fallacies to which any deviation from them may give rise. It is in this aspect that logic vindicates its claim to be ‘a cathartic of the human mind.’ For, like ethics, logic is a normative science; that is to say, it does not, like the physical sciences, or like psychology, simply generalise facts. Its laws are not statements of what always happens, but rules of what ought to be done. This distinction contains the answer to the question, once much debated, whether logic is a science or an art. The question is essentially a dispute about words.
The perception that pure logic treats thought simply as a process of comparison and classification has induced a number of recent logicians (chiefly English) to attempt an extension of Aristotle’s scheme by a thorough-going application of the notion of logical quantity. Thus, Sir W. Hamilton maintained that the relation between subject and predicate in a proposition is that of logical equation. The proposition, ‘All men are mortal,’ means, when fully expressed, ‘All men are some mortals.’ If the predicate be thus explicitly quantified, it is evident that we may substitute for the copula the algebraical symbol of equation. This doctrine, which is known as the Quantification of the Predicate, was expounded by Archbishop Thomson, Spencer Baynes, and others. It leads to a multiplication of the old propositional and syllogistic forms, but in its Hamiltonian form it has been shown by Venn to rest on a confusion of views. A similar line of thought has been worked out by Jevons, who defines inference as ‘the substitution of similars.’ He would make the proposition run—‘All men are mortal men’ (All is ). De Morgan’s formula for the proposition resembles this; but his innovations, as well as Boole’s development of logic into a branch of mathematics, are rather specimens of the ingenuity of their authors than transcripts of actual thought-processes. They show no signs of taking their place as a permanent addition to logical doctrine. The same may be said of Jevons’ Method of Indirect Inference, by which he claims to have reached the same results as Boole without the use of mathematics. The Method consists in ‘developing’ all the possible combinations of the terms mentioned in the premises, and then proceeding, by elimination of those which violate the conditions there laid down, to reach those combinations which are consistent with our data. Jevons applied his principle in the invention of a logical machine which effects this process of counting out with unerring accuracy; but where the terms are multiplied to any extent the operation is, of course, cumbrous in the extreme.
Bacon is commonly regarded as the founder of Inductive Logic. In his Novum Organum he put himself at the head of the revolt against the scholastic logic which marked the men of the Renaissance, and, though his own apprehension of scientific method was gravely defective, his eloquence and his position made him the most influential prophet of the scientific movement which Galileo and others had initiated. In point of fact he came to supplement the old, not to supersede it; but he allowed his dislike of the abuses of the Aristotelian logic to carry him away into indiscriminate denunciation. Bacon’s animus is perhaps excusable as the zeal of the reformer; and it may be granted that in the Aristotelian logic, as in Greek philosophy generally, there is a tendency to let the study of words usurp the place of the investigation of facts. The middle ages had exaggerated this tendency by habitually assuming the distinctions existing among things to be correctly and adequately rendered by traditional names. Beyond this, Bacon’s diatribes against ‘syllogism’ betray a misapprehension of the real function of formal logic, which, as has been seen, makes no pretensions to be an instrument of scientific discovery. Inductive theory has received many developments since the time of Bacon, notably at the hands of J. S. Mill. The progress of science has made it easier to formulate its methods and to determine the conditions of valid scientific proof. It is sufficient here to point out that, whereas in formal or deductive logic, reasoning proceeds from a whole to the particulars included under that whole, we seem in inductive logic to rise, in reliance on the uniformity of nature, from observation of particulars to the enunciation of a universal proposition. The nature of the certainty which belongs to such scientific generalisations is one of the subjects which the philosophy of induction has to deal with. The profound interest and value of these investigations, when compared with the rigid framework of symbols with which pure logic presents us, may well lead men to overestimate the former at the expense of the latter. But the two disciplines are essentially distinct; and the exactness and scientific completeness of pure or formal logic will always constitute it a valuable educational instrument.
BIBLIOGRAPHY.—The handiest elementary manuals of logic are those by Jevons and Fowler—Jevons’ Elementary Lessons in Logic, Fowler’s Deductive and Inductive Logic—to which may be added Whately’s Logic, an older book, and Keynes’s Formal Logic, which is somewhat more advanced. Among larger treatises in English of comparatively recent date may be mentioned Mill’s Logic, Hamilton’s Lectures on Logic, Ueberweg’s Logic (translated), Bradley’s Principles of Logic, Bosanquet’s Logic, Venn’s Empirical Logic, Jevons’ Principles of Science, Lotze’s Logic (translated). The German works of Sigwart and Wundt should also be named. Thomson’s Out- lines of the Laws of Thought, Baynes's New Analytic of Logical Forms, Jevons' Pure Logic and Other Papers, Venn's Symbolic Logic, and the works of De Morgan and Boole deal with proposed developments of logic on algebraic lines. There is an elaborate history of logic by Prantl in German; and the works of Trendelenburg in German and of Hamilton and Mansel in English are also valuable in this connection.