Logic (Gr. nryos, reason), the science of reasoning. More strictly and properly, it is the science of deducing ideas or conceptions one from another, and of constructing them into propositions, arguments, and systems. A wide range and great diversity of topics have, however, been included in the various treatises written under the name. Some have understood by it an account of the whole mental activity, and defined it as the art of thinking. Others have made it comprise only a knowledge of the first principles, or axioms, from which we reason. Others appear to have held it responsible for the truthfulness of all professedly logical reasonings and processes. Others again have regarded it as chiefly or exclusively an instrument of invention and discovery, and worthless except for the attainment of some new truth. It is now generally held that logic assumes certain first principles or axioms, from which as premises to reason; that it is concerned with the form only of reasoning or argument, and not at all with the subject matter; that it is and of necessity must be a purely a priori science, and moreover a hypothetical science, since it neither assumes nor proves as such the reality of anything, does not assert that any objects corresponding to our conceptions do really exist, but only gives results and conclusions based on premises, which are true provided the premises are true.

Logic is thus limited to the method of reasoning. Though commonly regarded as consisting of two parts, analytics and method, it is essentially a constructive science; it explains the way in which theories and systems are constructed from our primary ideas of objects, and it proves and tests, not their truth, but their legitimacy as deductions. In this view it presupposes psychology, which is a sort of natural history of thought, and it is preliminary and prerequisite to ontology, the science of being. - Logic begins with ideas. Our ideas of objects are complex wholes, and may be analyzed into conceptions of the known properties of objects. Thus, snow is represented by its properties of whiteness, coldness, etc, and an orange by its color, shape, etc. These properties, or rather the terms describing them, become predicates which we may affirm of the object. Thus, having analyzed our idea of an orange, we obtain the properties of roundness, etc, and hence may say, "The orange is round," etc. Or, forming a generic conception, we may say, "An orange is a fruit;" "Men are animals." We may thus predicate P of M, and M of S, and then, dropping the common or middle term M, may predicate P of S, a proposition derived by deduction from the two premises or primary judgments.

The formula, "M is P, S is M, therefore S is P," is called a syllogism, a term which includes any possible combination of two propositions from which is deduced a third, which is hence called a conclusion. The conclusions of preceding syllogisms may become the premises of others ad infinitum. The premises may be negative as well as affirmative - S are not P, as well as S are P; they may also include only a part of the subject, as some S are P, some S are not P. Hence there are four cardinal propositions:

Universal affirmative: All S are P.

" negative: No 8 are P. Particular affirmative: Some S are P.

" negative: Some S are not P.

For convenience these propositions are designated by the first four vowels; thus: A, universal affirmative; E, universal negative; I, particular affirmative; O, particular negative. Combining these four propositions in all possible ways of three in a set, we obtain 64 sets, which are called moods. Of these moods, however, only 11 are found to give valid conclusions, viz.: AAA, AAI, AEE, AEO, All, AOO, EAE, EAO, EIO, IAI, and OAO. It is found also that the position of the middle term is of essential importance; for let the mood AAA be written thus: "All M are P; all S are M; therefore all S are P;" and it is evident at once that if M is included in the class P, and S is included in the class M, then S must be included in P also. But if the same mood be written, "All P are M; all S are M," then it does not follow that S is included in P; for men are animals, and horses are animals, but men are not therefore horses. Every mood of the syllogism thus has what are termed figures, of which there are four.

In the first figure, the middle term is the subject of the major premise and the predicate of the minor; in the second, the middle term is the predicate of both premises; in the third, it is the subject of both premises; and in the fourth, it is the predicate of the major premise and the subject of the minor. The 11 moods each having 4 figures would give 44 syllogisms, of which, however, only 19 are found by examination to be distinct and valid. These are designated by the capital vowels in the following mnemonic hexameters:

BArbArA, cElArEnt, DArII. fErIOque, prioris: CExArE, cAmEstrEs, fEstInO, bArOkO, secundce: Tertia dArAptI. dIsArnIs, dAtIsI. fElAptOn, BOkArdO, fErIsOn, habet: quarta insuper addit, BrAmAntIp, cAmEnsEs, dImArIs, fEsApO, frEslsOn.

When one of the premises is understood, but not expressed, in the statement, the syllogism is called an enthymeme. When several premises are employed for the same conclusion, several syllogisms are in fact abridged into one formula, which is called a sorites. When one premise is assumed as hypothetically true, and the conclusion is stated as depending upon the truth of the other alone, we have what is called a conditional judgment; and if the conclusion is stated as depending upon the falsity of the other, we have a disjunctive judgment. A conditional or disjunctive proposition may be made the major premise, and then the syllogism be completed as follows: "If A is B, C is D; but A is B; therefore C is D." In this case the syllogism is called a conditional syllogism, or sometimes a hypothetical syllogism. "Either A is B or C is D; but A is not B; therefore C is D." In this case the syllogism is called disjunctive. The major premise may affirm only a comparison or relation between the terms, as: " Where the boy is, there the father is; but the boy is at home; therefore, the father is at home." - Besides the fulfilment of all the conditions of the formulas in syllogisms, there are found to be also certain conditions and laws in regard to the use of words, which are necessary to the validity of the reasoning.

The violation of these laws gives rise to fallacies, of which there are reckoned 13, 6 in dictione and 7 extra dictionem. 1. Equivocation occurs when a word is used in the same formula in two different senses. 2. Amphibology when a word is so used as to leave it doubtful whether it be a subject or predicate, or when the reference of a pronoun is ambiguous. 3 and 4. Composition and division are caused by using the same term both collectively and distributively in the same formula, thus: " 3 and 2 are two numbers; but 5 is 3 and 2; therefore, 5 is two numbers." Here 3 and 2 are used distributively in the major and collectively in the minor premise. The reverse is true of the word Romans in the following: "The Romans conquered Carthage; Brutus and Caesar were Romans; therefore Brutus and Caesar conquered Carthage." 5. Accent may occasion a fallacy by varying the meaning of a proposition. Thus the purport of the question, "Do you ride to town to-day?" may be changed five times by changing the accented word, or omitting the emphatic accent. 6. The form of the expression (figura dictionis) may lead to a fallacy, as when we infer from the fact that one word ending in a, as mensa, is of the feminine gender, that therefore another word with a like termination, as poeta, is feminine also. 7. The fallacy of accidents arises when we affirm of something described by some accidental property or circumstance what is true only of its substance, as: " We buy raw meat in the market; what we buy in the market, we eat; therefore, we eat raw meat." Here we do not buy meat because it is raw, but because it is meat, for its essence and not for its accidents, and only its essential quality is common to the different members of the argument. 8. Mistaken application consists in giving to a statement a universal application when it was intended for only a limited one. 9. The ignoratio elenchi occurs when we either fail to give for any particular conclusion the premises required, or draw from given premises a conclusion not legitimately following from them, or employ a legitimate syllogism which does not give the conclusion that the occasion demanded. 10. The a non causa, pro causa, occurs when we reason from a premise that is true, but not a premise to the conclusion which we profess to draw from it. 11. The fallacy of consequences consists in employing a conclusion not derived from the premises. 12. The petitio principii, or begging the question, assumes as true that which should be proved. 13. The fallacy of many questions occurs when several interrogatories are either expressly or implicitly so combined into one that they must all receive the same answer, though truth requires that some be answered affirmatively and others negatively. - Aristotle was the creator of the science of logic (though he says that Zeno the Eleatic was the founder of dialectics), and his writings have been the basis of most of the treatises on logic that have since appeared.

Six separate works constitute his Organon. In his "Categories" he treats of the highest generic ideas, which he reduces to ten, and of the nature of terms. In his " Prior Analytics " he examines the nature of propositions and the theory of conclusions; in his " Posterior Analytics," of demonstrable knowledge and the methods of reasoning. His " Topics " embrace dialectics and the discussion of first principles; his Sophistica are devoted to fallacies; and he also wrote a work on the art of expression. The whole system of Aristotle is crude and perplexed, as is usually the case with the first draft or statement of anything that lies far beyond the ordinary thought of men. There has, however, until a late period been little done in the department of logic more than to simplify and rearrange the materials furnished by the Stagirite. He recognized and discussed only the first three figures, and the discovery of the fourth is ascribed to Galen. Moreover, he scarcely regards the hypothetical syllogisms as modes of reasoning at all; the discovery of these is ascribed to Theophrastus. It was clearly seen by Aristotle that reasoning depends in some way on the relations of the logical wholes (individual, species, and genus) to one another.

Porphyry in his " Introduction to Aristotle" explained more fully and clearly than his master had done the predica-bles, as they were called, namely, genus, species, differentia, property, and accident. Logic was extensively studied during the middle ages, though no important advance was made in its development. Its use gave rise to the scholastic method, which consists in applying the formulas of reasoning to terms, or to general principles deduced by definition or otherwise from terms. This method is of course legitimate, and the only one that is at all legitimate, in mathematics, and in all a priori or demonstrative sciences. But in the natural sciences the first principles or topics are the facts of nature; and a careful observation, analysis, and classification of them, together with an induction from them, must precede any useful deduction. The discovery of this great principle led to a disregard of the proper sphere and use of formal logic, and brought the whole subject into neglect and contempt; and the inductive was generally proclaimed to be of vastly more use than the scholastic method.

Induction, however, had not wholly escaped the attention of Aristotle, who defined it as "the method by which we pass from particular instances to general truths." The natural sciences all begin with induction. The philosophy of the method has not, however, been explained to universal satisfaction. The Novum Organum of Bacon was designed to show its necessity and practical application, rather than the philosophic grounds on which its validity rests. The works of Herschel, "Preliminary Discourse on the Study of Natu-ral Philosophy," and Whewell, " History and Philosophy of the Inductive Sciences," have greatly improved the matter since Bacon's time. But the "System of Logic" by John Stuart Mill is regarded as the best exposition of the inductive method that has yet been produced. During the general neglect of logic, one of the most important works produced in its interest was La logique, ou l'Art de penser (16G2), usually called the Port-Royal logic, by several authors, among whom Arnauld, Nicole, and Sacy were most prominent. It was really in the interest of the scholastic method, though intended otherwise, and though the scholastic rules and formulas were illustrated by new and well chosen examples, which constitute the great merit of the work.

It was widely read, and gave a new impulse to the study. At the beginning of the next century Wolf published his great treatise on logic, in which he attempted to incorporate the peculiarities of the Leibnitzian philosophy, and which gave the direction to speculations on this subject in Germany, leading the German writers to regard the fundamental laws of thought which underlie and give validity to logical formulas, rather than their practical value or application. In 1816 Hegel completed the publication of his "Logic," in which the term is used with a breadth of meaning peculiar to his philosophical system. The Hegelian logic is the law of absolute being, the scientific exposition of the pure conceptions of reason, of the absolute idea; its domain is the absolute truth as it is in itself, apart from its manifestations; it represents God as he is in his eternal being, before the creation of the world or of any finite mind; it is the analysis of the successive stages of history in their abstract form.

It thus constitutes the first and highest part of the Hegelian scheme of absolute idealism, and since the time of Hegel the German writings that have appeared under the name of logic have followed very much in the same direction, discussing questions which we are accustomed to regard as belonging to ontology under the title logic, rather than what we expect to find in books on this subject. Archbishop Whately published his " Elements of Logic " in 1826, when this branch of study was at its lowest ebb in the English universities. This work has had probably a wider circulation and more extensive use than any other ever written on the subject, and had the effect of recalling public attention to its importance. He maintained that induction as well as deduction should be regarded as a branch of logic, and consequently attempted to explain the philosophy of induction and to show its accordance with the deductive formulas; and while the writers of the German schools treated logic as chiefly or exclusively concerned with thought, Whately regarded it as chiefly concerned with words.

His work gave rise to many other efforts in the same department, prominent among which was the " System of Logic, Ratiocinative and Inductive," by John Stuart Mill (1843), in which the author treats the grounds and fundamental principles rather than the formulas of reasoning. Being an eminent thinker of the sensational school, he does not make logic an a priori science, but aims to systematize the inductive method and reduce it to strict rules. The work abounds in valuable practical hints and reflections, and the concluding portion endeavors to solve the question whether from moral and social phenomena the instrument of logic may not derive a body of truths empirically acquired and universally assented to, like many of the laws of the physical world. In 1847 Prof. De Morgan published his treatise on "Formal Logic," an attempt to construct the science on a new basis. A mathematician of high repute, his work is difficult of comprehension to all except scholars in his own department. The peculiarity of its fundamental principle is that it ignores the distinction between a unit and an individual. Units, however, are not, and individuals are distinguishable from one another.

Six men, for example, are not distinguished as mere units from any other six objects of thought; but it is obvious that we may predicate of six men what would not be true of six individuals in any other species of objects; and logic does not deal with its objects as mere units, but as individuals making up species and genera. If the subject in any affirmative proposition denote an individual, the predicate will denote the species in which it is comprehended; and if the subject denote a species, the predicate will denote the comprehending genus; but the argument neither establishes nor affirms any numerical relation between them. Sir William Hamilton dissented from the views of Whately and his followers, who considered logic as chiefly concerned with language and as including the department of dialectics. He maintained that it is exclusively occupied with the forms of reasoning, that it takes no notice of the subject matter, and has no connection with psychological processes. The peculiarity of his system results from what he calls the quantification of the predicate, a fact which in his view had hitherto been overlooked. Besides the four kinds of propositions designated by A, E, I, and O, he distinguishes four others.

It had previously been held that all universal propositions as such and of necessity distributed the subject, and negative propositions the predicate. Thus in the universal affirmative, "All men are animals," the subject only is taken into the scope of the proposition as a logical whole. We here speak of " all men " as a class, but not of "all animals," and we say or imply nothing concerning the latter except that some of them are men. The universal negative distributes both terms, and in like manner it has been held that the particular affirmative takes neither of its terms as a whole, and that the particular negative distributes the predicate only. But Sir William Hamilton holds that we may have affirmative propositions with the subject distributed, and negatives with or without the predicate distributed; and he proposes to designate the eight propositions which result as A, U, I, Y, E, n, 0, w. The scheme, presenting the quantity of the predicate, is as follows:

U. Toto-total: All S is all P. A. Toto-partial: All S is some P. T. Parti-total: Some S is all P. I. Parti-partial: Some S is some P.

E. Toto-total: All S is not all P.

n. Toto-partial: All S is not some P.

O. Parti-total: Some S is not all P.

w. Parti-partial: Some S is not some P.

This view, if it be accepted, revolutionizes the theory of the syllogism, and the whole system of logic as commenced by Aristotle and elaborated by his followers down to the time of Hamilton. De Morgan asserted that this theory of quantification was substantially the same as his own. George Boole, for many years mathematical professor in Queen's college, Cork, published "Mathematical Analysis of Logic" (1847), and " Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities" (1854). An elementary treatise on logic by Dr. W. D. Wilson, then professor in Geneva college, N. Y. (since 1868 in Cornell university), was published in 1856. In 1872 he reissued his " Logic," in a form which is rather a new treatise than a new edition of the former work. In this latter he takes the ground distinctly that logic deals, not with ideas or conceptions at all, but with things, using words only as representing things under the various aspects in which they are contemplated by the mind.

He holds that all the laws and formulas of reasoning are derived from one or another of the four relations of things, namely: 1, individual to species, species to germs, etc.; 2, comparison of quantity, time, place, etc.; 3, cause and effect; 4, the relation of parts to their wholes, and vice versa. In his "Introduction to the Study of Metaphysics" Dr. Wilson has pointed out a new form of logic, applicable to the investigation and criticism of metaphysical facts and phenomena. In this he starts with the principle that language consists of nouns which denote things, and that all other words are used as subsidiary to the nouns in making sentences; thus, while the nouns denote the things we are speaking of, the other words indicate the relation which we suppose to exist between the things denoted by the nouns. Again, as language is but an expression of the facts and states of consciousness, we may use the words that constitute any sentence as a means of developing what was contained or even implied in the thought or mental state which gave rise to the sentence, and thus obtain a more rigorous and exact analysis of the phenomena of consciousness than we have been accustomed to.

On the other hand, Dr. McCosh, president of Princeton college, has published a work, "The Laws of Discursive Thought" (New York, 1870), in which he holds that logic is based upon and deals chiefly with "the notion." His first part treats of " the notion," and occupies nearly half of the entire work. In this the author treats with great clearness and precision of the nature and relation of terms as the foundation of all reasoning. Thomson's " Outline of the Necessary Laws of Thought" is based on Sir William Hamilton's theory of what is called "the quantification of predicate." It has been extensively used, and is a very valuable treatise. Prof. Francis Bowen, of Harvard university, published in 1804 a "Treatise on Logic," in which the two sys-terns, which may perhaps be best designated as the Aristotelian and the Hamiltonian, are both given with great clearness and impartiality, in such a way as to enable the learner to compare them easily and judge of their respective merits. Other important works produced in this country on the subject are: "The Elements of Logic," by Prof. Levi Hedge (1816), founded on the Scotch philosophy, and therefore omitting all metaphysical discussions of formulas and a priori conditions of thought; "The Elements of Logic," by Prof. Henry P. Tappan (1844), founded on the philosophy of Kant, and occupied rather with the conditions and laws of thought than with the application of logical formulas; "The Science of Logic," by Prof. A. Mahan (1857); and "System of Logic," by P. McGregor (New York, 1862). Prof. Bain, of the university of Aberdeen, published in 1870 (new ed., 1874) "Logic, Deductive and Inductive," a work which aims at embracing a full course of the science as it is usually taught, and also in the wider sense in which it is conceived and treated by Mill. The author also treats of the principles of psychology so far as he considers a knowledge of them necessary to a right understanding of logic.

A peculiar and useful feature of the book is its treatment of the great generalization of the 19th century, variously designated as the correlation, conservation, persistence, or indestructibility of force or energy. It also contains an account of the various modifications of the science and additions to it recommended by Hamilton, De Morgan, Boole, and others, and examples of its application to the other sciences. The work is valuable not only as a treatise on logic, but as explaining and illustrating the methods employed in modern scientific investigations. In 1872 Prof. Jevons, of Owens college, Manchester, England, published a small work, " Elementary Lessons in Logic," in which he gave a plain and fair statement of the theories of Hamilton, De Morgan, and Boole, although he regarded them as too recent to be generally adopted, and still preferred the old or Aristotelian system. Sub-sequently Prof. Jevons issued a larger and more comprehensive work, "The Principles of Science: a Treatise on Logic and Scientfic Method" (London, 1874), in which he proposes a new system of representing the logical formulas.

It is in a measure based upon the systems of Hamilton, De Morgan, and Boole, though different from them, supplying their deficiencies and correcting some errors which had been found to be involved in each of them. He assumes what is called the entire quantification of the terms, and treats the "some" which has been regarded as the sign of a particular or partial proposition, as in the statement "Some men are wise," as an adjective differentiating a class as completely as any other adjective, "as "good men." Then, representing each noun and each adjective by a letter, supposing S to stand for " some," M for " men," and W for " wise," we should have S M = M W, "Some men are men wise;" or in the ordinary form of expression, where the recurring noun in the predicate is omitted by ellipsis, we have " Some men are wise." Or if we take the following syllogism in Barbara, " All metals are elements, and all elements are incapable of decomposition," and, considering " incapable of decomposition " as a single word, "indecomposable," use the initials of each word for symbols, we have M = M E, and M E = I. Replacing the subject of the last proposition by M, which is shown by the first to be equivalent (logically) to the subject of the last, we have M = I, namely, "Metal is indecomposable," for our conclusion.

The author then proceeds at great length to discuss the two methods of reasoning, deductive and inductive. He holds that deduction is first in order, and that even induction is but a method and form of deduction, inasmuch as induction always presupposes an assumed principle or hypothesis as its major premise, although he does not call it by that name. He dissents entirely from Bacon's view of induction, and agrees, as he affirms, with the method pursued by Copernicus, Kepler, Galileo, Newton, etc, rather than that which was taught as a theory by Bacon. Prof. Jevons illustrates his system of notation and his theories of deduction by a wide range and most ample citation of examples. He discusses very elaborately the various methods of measurement and observation, with cautions against the errors to which the student is liable; and presents, on the whole, the most complete survey of the whole field of knowledge and inquiry that has yet been given to the public. In his psychology he is evidently a sensationalist, in that he does not believe either in any a priori element of knowledge or in any insight into the notion of things by which we can obtain necessary truths and axioms, which, while they may have been obtained on the occasion of an act of sense-perception, do nevertheless transcend the truths of sense-perception, and assert what can never be proved as absolute truths by any of the a posteriori processes, or by any processes that are based on sense-perception alone.

Though thus a sensationalist in his psychology, Prof. Jevons is not, as one would naturally expect, a materialist in his ontology. He thinks that the modern doctrines of evolution, development, etc, as held by Spencer, Darwin, and those of their schools, are not only not proved, but cannot be proved on any premises within the sphere of knowledge, as distinguished from mere conjecture and hypothesis, by the use of any of the methods or modes of reasoning known to logic or admissible within the domain of science. And even as a hypothesis designed to explain observed and known phenomena, he thinks that the system of the modern materialists, when attempting to explain the phenomena of the universe without the recognition of a personal creator, introduces more mysteries or insoluble problems than it solves. - In Germany, from the time of Hegel, logic has followed mostly in the direction he gave it (already described), "as," in the words of Ueberweg, " that part of philosophy which considers reason itself as the prius of nature and spirit." Among the writers in this school may be mentioned, as most worthy of note, Kuno Fischer, Logik und Meta-physik (Heidelberg, 1852; 2d ed., 1865); Ha-nusch, Handbuch der wissenschaftlichen Denk-lehre (Lemberg, 1843; 2d ed., Prague, 1850); Rosenkranz, Wissenschaft der logischen Idee (1858-'9; together with Epilegomena, 1862); and Karl Werder, Logik ah Commentar und Erganzung zu Hegel's Wissenschaftslehre der Logik (Berlin, 1841). But as early as 1832 Beneke published his Lehrouch der Logik als Kunstlehre, which was a protest in some sense against the direction Hegel had given to speculations under the name of logic, and was based upon a more practical view of the nature of the science.

As a follower of Beneke we have Dressier, Die Grundlehren der Psychologie und Logik (Leipsic, 1867; 2d ed., 1870). We have also as specially worth noticing, and not in the Hegelian line, Trendelenburg's Elementa Logices Aristotelicce (Berlin, 1836; 6th ed., 1868; with supplementary Erlauterungen, 2d ed., 1861), and Ueberweg's System der Logik und Geschichte der logischen Lehren (Bonn, 1857; 3d ed., 1868). In 1872 Robert Grass-mann, a younger brother of the mathematician (see Grassmann), published Die Begriffslehre oder Logik, zweites Buch der Formenlehre oder Mathematik, in which he treats the whole science as a branch of mathematics. The system is analogous to that of Boole. The logical doctrines in regard to ideas, judgments, and inference are expressed in the form of definitions and equations between arbitrary symbols, and are treated like the theorems of algebra according to fixed rules of operation.