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Lachelier, J.: (1831-1918) A French philosopher who, though he wrote little, exerted a considerable direct personal influence on his students at the Ecole Normale Superieure; he was the teacher of both E. Boutroux and H. Bergson. His philosophical position was a Kantian idealism modified by the French "spiritualism" of Maine de Biran and Ravaisson.
Note sur le pari de Pascal. -- L.W.
Lamaism: (from Tibetan b La-ma, honorable title of a monk) The religious beliefs and institutions of Tibet, derived from Mahayana Buddhism (q.v.) which was first introduced in the 7th century by the chieftain Sron-tsan-gampo, superimposed on the native Shamaistic Bon religion, resuscitated and mixed with Tantric (q.v.) elements by the mythic Hindu Padmasambhava, and reformed by the Bengalese Atisa in the 11th and Tsong-kha-pa at the turn of the 14th century. The strong admixture of elements of the exorcismal, highly magically charged and priest-ridden original Bon, has given Buddhism a turn away from its philosophic orientation and produced in Lamaism a form that places great emphasis on mantras (q.v.) -- the most famous one being om mani padme hum) -- elaborate ritual, and the worship of subsidiary tutelary deities, high dignitaries, and living incarnations of the Buddha. This worship is institutionalized, with a semblance of the papacy, in the double incarnation of the Bodhisattva (q.v.) in the Dalai-Lama who resides with political powers at the capital Lhasa, and the more spiritual head Tashi-Lama who rules at Tashi-Ihum-po. Contacts with Indian and Chinese traditions have been maintained for centuries and the two canons of Lamaism, the Kan-jur of 108 books and the Tan-jur of 225 books represent many translations as well as original works, some of great philosophical value. -- K.F.L.
Lmbert is known also for important contributions to mathematics, and astronomy; also for his work in logic, in particular his (unsuccessful, but historically significant) attempts at construction of a mathematical or symbolic logic. Cf. C. I. Lewis, Survey of Symbolic Logic. -- A.C.
Lamennais, R.: (1782-1854) Leader of a Platonic-Christian movement in the Catholic clergy of France. He advanced the idea of "inspired mankind." He attacked the eighteenth century for its principles and its method. In finding dissolution and destruction as its aftermath, he advocated a return to the Catholic Church as the solution.
squisse d'une philosophic, 1841-46. -- L.E.D.
Lamettrie, Julien Offroy de: (1709-1751) A French materialist and author of L'homme machine, in which he expresses his belief that the soul is a product of bodily growth; he maintains that the brain has its "thought muscles" just as the leg has its "walk muscles."
Le systeme d' Epicure, 1750. -- R.B.W.
Lange, Friedrich Albert: (1828-1875) Celebrated for his History of Materialism, based upon a qualified Kantian point of view, he demonstrated the philosophical limitations of metaphysical materialism, and his appreciation of the value of materialism as a stimulus to critical thinking. He worked for a greater understanding of Kant's work and anticipated fictionalism. -- H.H.
Language, Functions of: Some utterances (a) are produced by a speaker, (b) induce effects in an interpreter, (c) are related to a certain subject-matter (which may, but in general will not, include either the speaker or interpreter). According as one or other of the relations in which the utterance stands to the several factors of such speech-situations is selected for attention, the (token) utterance may be said to have expressive, evocative and referential functions. The utterance expresses thoughts, desires, attitudes of the speaker; evokes reactions (thoughts, evaluations, tendencies to action) in the hearer; designates or refers to its reference.
While all three functions are normally distinguishable in any given utterance, type-sentences (and, derivatively, words) may be classified according as one or the other of the functions normally predominates in the occurrence of the corresponding tokens. (Thus exclamations are predominantly expressive, commands evocative, scientific generalizations referential.) Such distinctions may, in turn, be made the basis for distinguishing different types of linguistic systems.
While most writers on language agree us to the value of making some such distinctions, there is little agreement as to the number and kinds of functions which may usefully be recognised. There is even less agreement about nomenclature. The account given follows that of Kretschmer (Sprache, 61 ff. in Gercke and Norden, Einleitung in die Altertumszvissenschaft, I) and Bühlcr (Sprachtheorie, passim). Ogden and Richards distinguish five functions (Meaning of Meaning, 357 fF.). The broad distinction between "referential" and "emotive" uses of language, due to the same authors, has been widely accepted. -- M.B.
Language, Philosophy of: Any philosophical investigation arising from study of concrete, actualized, languages, whether "living" or "dead". By "language" is here to be understood a system of signs (whether words or ideograms) used in regular modes of combination, in accordance with conventionally established rules, for the purpose of communication.
Philosophers have in the past been concerned with two questions covered by our definition, though attempts to organize the subject as an autonomous department of philosophy are of recent date.
Enquiries into the origin of language (e.g. in Plato's Kratylos) once a favorite subject for speculation, are now out of fashion, both with philosophers and linguists.
Enquiries as to the nature of language (as in Descartes, Leibniz, and many others) are, however, still central to all philosophical interest in language. Such questions as "What are the most general characters of symbolism?", "How is 'Language' to be defined?", "What is the essence of language?", "How is communication possible?", "What would be the nature of a perfect language?", are indicative of the varying modulations which this theme receives in the works of contemporaries.
The analysis of meaning. Problems arising here involve attention to those under the next heading.
The formulation of general descriptive schemata. Topics of importance here include the identification and analysis of different ways in which language is used, and the definition of men crucial notions as "symbol'', "grammar", "form", "convention", "metaphor", etc.
The study of fully formalized language systems or "calculi". An increasingly important and highly technical division which seeks to extend and adapt to all languages the methods first developed in "metamathematics" for the study of mathematical symbolism.
Applications to problems in general philosophy. Notably the attempt made to show that necessary propositions are really verbal; or again, the study of the nature of the religious symbol. Advance here awaits more generally acceptable doctrine in the other divisions.
The Theory of Speech and Language.
See also Communication, Meaning, Referent, Semiotic, Sign, Symbol, Functions of Language, Scientific Empiricism.- -- M.B.
Language of Science: See Scientific Empiricism II B 1.
Lao Tzu: Whether the founder of Taoism (tao chia) was the same as Li Erh and Li An, whether he lived before or after Confucius, and whether the Tao Te Ching (Eng. trans.: The Canon of Reason and Virtue by P. Carus, The Way and Its Power by A. Waley, etc.) contains his teachings are controversial. According to the Shih Chi (Historical Records), he was a native of Chu (in present Honan), land of romanticism in the south, and a custodian of documents whom Confucius went to consult on rituals. Thus he might have been a priest-teacher who, by advocating the doctrine of "inaction", attempted to preserve the declining culture of his people, the suppressed people of Yin, while Confucius worked hard to promote the culture of the ruling people of Chou. -- W.T.C.
Lassalle, Ferdinand: (1825-1864) Was influenced in his thought by Fichte and Hegel but soon assumed a distinctly materialist position. His main interest and activity lay in the field of political and economic philosophy; he advocated, and worked for, the formation of trade unions in Germany and adhered to socialism.
Main works: Die Philosophie Herakleitos d. Dunklen, 1858; System d. Eruorbenen Rechte, 1861; and political speeches and pamphlets in Collected Works (Leipzig, 1899-1901). -- R.B.W.
Latency: (Lat. latere, to be hidden) (a) In metaphysics, the term latency is equivalent to potency or potentiality. See Potentiality.
(b) In epistemology and psychology, the term is applied to knowledge, e.g. memory, which lies dormant in the mind but is capable of becoming actual and explicit (see W. Hamilton, Lectures on Metaphysics, xviii, cited by J. M. Baldwin, Dictionary of Philosophy and Psychology, Vol. I, p. 628). Latency in this restricted sense, designates phenomena now embraced by the term subconscious. See Subconscious. -- L.W.
Latin-American Philosophy: Philosophy in Latin America may be divided into three periods.
(1) The scholastic period begins with the Recognitio summularum of Alonso de la Veracruz (1554) and continues to the dawn of the nineteenth century. According to Ueberweg, the influence of Duns Scotus during this period was greater than that of Thomas Aquinas.
(2) The predominantly naturalistic and positivistic period coincides roughly with the nineteenth century. The wars of independence were accompanied by revolt from scholasticism. In the early part of the century, liberal eclectics like Cousin and P. Janet were popular in South America, but French eighteenth century materialism exerted an increasing influence. Later, the thought of Auguste Comte and of Herbert Spencer came to be dominant especially in Mexico, Brazil, Argentina, and Chile. Even an idealistically inclined social and educational philosopher like Eugenio Maria de Hostos (1839-1903), although rejecting naturalistic ethics, maintains a positivistic attitude toward metaphysics.
(3) The predominantly idealistic period of the twentieth century was initiated by the work of the Argentine Alejandro Korn (1860-1936), who introduced modern German philosophy to his fellow-countrymen. Francisco Romero, also an Argentine, has brought about the translation of many European philosophical classics into Spanish. Leibniz, Kant, Hegel, and the more recent neo-Kantians and phenomcnologists have exerted wide influence in Latin America. North American personalism has also attracted attention. In Mexico, Jose Vasconcelos and Pedro Gringoire reflect in their own syntheses the main streams of idealistic metaphysics, ethics, esthetics. Puerto Rico, with its recent publication of the writings of Hostes, is also a center of philosophic activity. There are signs of growing philosophical independence throughout Latin America. -- J.F., E.S.B.
Latitudinarianism: (1) A party in the Church of England (middle of the 17th century) aiming to reconcile contending parties by seeking a broad basis in common doctrines. (2) A term applied to a liberal opinion which allows diversity in unity. (3) A term used derisively as meaning indifference to religious doctrines. -- V.F.
Law: (in Kant) "Every formula which expresses the necessity of an action is called a law" (Kant). -- P. A.S.
Law, Chinese School of: See Fa chia and Chinese philosophy.
Law of Population: In economics, the tendency of population to encroach upon the means of subsistence. First announced by Malthus (1766-1834), the Law asserts that the increase of unchecked population is in geometric ratio while the increase of the means of subsistence is in arithmetic ratio, so that population must always press upon the limits of the means of subsistence. -- J.K.F.
Laws of thought: See Logic, traditional.
Leading principle: The general statement of the validity of some particular form of valid inference (see Logic, formal) may be called its leading principle.
Or the name may be applied to a proposition or sentence of logic corresponding to a certain form of valid inference. E.g., the law of exportation (q. v.) may be called the leading principle of the form of valid inference known as exportation. -- A.C.
Legal Philosophy: Deals with the philosophic principles of law and justice. The origin is to be found in ancient philosophy. The Greek Sophists criticized existing laws and customs by questioning their validity: All human rules are artificial, created by enactment or convention, as opposed to natural law, based on nature. The theory of a law of nature was further developed by Aristotle and the Stoics. According to the Stoics the natural law is based upon the eternal law of the universe; this itself is an outgrowth of universal reason, as man's mind is an offshoot of the latter. The idea of a law of nature as being innate in man was particularly stressed and popularized by Cicero who identified it with "right reason" and already contrasted it with written law that might be unjust or even tyrannical. Through Saint Augustine these ideas were transmitted to medieval philosophy and by Thomas Aquinas built into his philosophical system. Thomas considers the eternal law the reason existing in the divine mind and controlling the universe. Natural law, innate in man participates in that eternal law. A new impetus was given to Legal Philosophy by the Renaissance. Natural Jurisprudence, properly so-called, originated in the XVII. century. Hugo Grotius, Thomas Hobbes, Benedictus Spinoza, John Locke, Samuel Pufendorf were the most important representatives of that line of thought. Grotius, continuing the Scholastic tradition, particularly stressed the absoluteness of natural hw (it would exist even if God did not exist) and, following Jean Bodin, the sovereignty of the people. The idea of the social contract traced all political bodies back to a voluntary compact by which every individual gave up his right to self-government, or rather transferred it to the government, abandoning a state of nature which according to Hobbes must have been a state of perpetual war. The theory of the social compact more and more accepts the character of a "fiction" or of a regulative idea (Kant). In this sense the theory means that we ought to judge acts of government by their correspondence to the general will (Rousseau) and to the interests of the individuals who by transferring their rights to the commonwealth intended to establish their real liberty. Natural law by putting the emphasis on natural rights, takes on a revolutionary character. It played a part in shaping the bills of rights, the constitutions of the American colonies and of the Union, as well as of the French declaration of the rights of men and of citizens. Natural jurisprudence in the teachings of Christian Wolff and Thomasius undergoes a kind of petrification in the vain attempt to outline an elaborate system of natural law not only in the field of international or public law, but also in the detailed regulations of the law of property, of contract, etc. This sort of dogmatic approach towards the problems of law evoked the opposition of the Historic School (Gustav Hugo and Savigny) which stressed the natural growth of laws ind customs, originating from the mysterious "spirit of the people". On the other hand Immanuel Kant tried to overcome the old natural law by the idea of a "law of reason", meaning an a priori element in all existing or positive law. In his definition of law ("the ensemble of conditions according to which everyone's will may coexist with the will of every other in accordance with a general rule of liberty"), however, as in his legal philosophy in general, he still shares the attitude of the natural law doctrine, confusing positive law with the idea of just law. This is also true of Hegel whose panlogism seemed to lead in this very direction. Under the influence of epistemological positivism (Comte, Mill) in the later half of the nineteenth century, legal philosophy, especially in Germany, confined itself to a "general theory of law". Similarily John Austin in England considered philosophy of law concerned only with positive law, "as it necessarily is", not as it ought to be. Its main task was to analyze certain notions which pervade the science of law (Analytical Jurisprudence). In recent times the same tendency to reduce legal philosophy to logical or at least methodological tasks was further developed in attempting a pure science of law (Kelsen, Roguin). Owing to the influence of Darwinism and natural science in general the evolutionist and biological viewpoint was accepted in legal philosophy: comparative jurisprudence, sociology of law, the Freirecht movement in Germany, the study of the living law, "Realism" in American legal philosophy, all represent a tendency against rationalism. On the other hand there is a revival of older tendencies: Hegelianism, natural law -- especially in Catholic philosophy -- and Kantianism (beginning with Rudolf Stammler). From here other trends arose: the critical attitude leads to relativism (f.i. Gustav Radbruch); the antimetaphysical tendency towards positivism -- though different from epistemological positivism -- and to a pure theory of law. Different schools of recent philosophy have found their applications or repercussions in legal philosophy: Phenomenology, for example, tried to intuit the essences of legal institutions, thus coming back to a formalist position, not too far from the real meaning of analytical jurisprudence. Neo-positivism, though so far not yet explicitly applied to legal philosophy, seems to lead in the same direction. -- W.E.
Legalism, ethical: The insistence on a strict literal or overt observance of certain rules of conduct, or the belief that there are rules which must be so obeyed. Opposed on the one hand by the view which emphasizes the spirit over the letter of the law, and on the other by the view which emphasizes a consideration of the	value of the consequences of actions and rules of action. Deontological ethics is often said to be legalistic. Cf. F. Cohen, Legal Ideals and Ethical Systems. -- W.K.F.
(b) Generic relationship or partial relationship. See Mo che. -- W.T.C.
Leibniz, Gottfried Withelm: (1646-1716) Born in Leipzig, where his father was a professor in the university, he was educated at Leipzig, Jena, and Altdorf University, where he obtained his doctorate. Jurist, mathematician, diplomat, historian, theologian of no mean proportions, he was Germany's greatest 17th century philosopher and one of the most universal minds of all times. In Paris, then the centre of intellectual civilization (Moliere was still alive, Racine at the height of his glory), where he had been sent on an official mission of state, he met Arnauld, a disciple of Descartes who acquainted him with his master's ideas, and Huygens who taught him as to the higher forms of mathematics and their application to physical phenomena. He visited London, where he met Newton, Boyle, and others. At the Hague he came face to face with the other great philosopher of the time, Spinoza. One of Leibniz's cherished ideas was the creation of a society of scholars for the investigation of all branches of scientific truth to combine them into one great system of truth. His philosophy, the work "of odd moments", bears, in content and form, the impress of its haphazard origin and its author's cosmopolitan mode of large number of letters, essays, memoranda, etc., published in various scientific journals. Universality and individuality characterize him both as a man and philosopher.
Leibniz's philosophy was the dawning consciousness of the modern world (Dewey). So gradual and continuous, like the development of a monad, so all-inclusive was the growth of his mind, that his philosophy, as he himself says, "connects Plato with Democritus, Aristotle with Descartes, the Scholastics with the moderns, theology and morals with reason." The reform (if all science was to be effected by the use of two instruments, a universal scientific language and a calculus of reasoning. He advocated a universal language of ideographic symbols in which complex concepts would be expressed by combinations of symbols representing simple concepts or by new symbols defined as equivalent to such a complex. He believed that analysis would enable us to limit the number of undefined concepts to a few simple primitives in terms of which all other concepts could be defined. This is the essential notion back of modern logistic treatments.
In contributing some elements of a "universal calculus" he may be said to have been the first serious student of symbolic logic. He devised a symbolism for such concepts and relations as "and", "or", implication between concepts, class inclusion, class and conceptual equivalence, etc. One of his sets of symbolic representations for the four standard propositions of traditional logic coincides with the usage of modern logic He anticipated in the principles of his calculus many of the important rules of modern symbolic systems. His treatment, since it was primarily intensional, neglected important extensional features of recent developments, but, on the other hand, called attention to certain intensional distinctions now commonly neglected.
Leibniz is best known in the history of philosophy as the author of the Monadology and the theory of the Pre-established Harmony both of which see.
Monadologie, 1714 (letter to Prince Eugene of Savoy).
No complete edition of L. exists, but the Prussian Academy of Sciences began one and issued 4 vols. to date. Cf. Gerhardt's edition of L's philosophical works (7 vols., 1875-90) and mathematical works (1849-63), Foucher de Careil's edition, 7 vols. (1859-75), O. Klopp's edition of L.'s historico-political works, 10 vols. (1864-77), L. Couturat's Opuscules et fragments inedits de L., 1903. -- K.F.L.
Lemma: (Gr. lemma) In Aristotle's logic a premiss of a syllogism. -- G.R.M.
In mathematics, a theorem proved for the sake of its use in proving another theorem. The name is applied especially in cases where the lemma ceases to be of interest in itself after proof of the theorem for the sake of which it was introduced. -- A.C.
Lenin, V. I.: (Ulianov, Vladimir Ilyich) Lenin is generally regarded as the chief exponent of dialectical materialism (q.v.) after Marx and Engels. He was born April 22, 1870, in Simbirsk, Russia, and received the professional training of a lawyer. A Marxist from his student days onward, he lived many years outside of Russia as a political refugee, and read widely in the social sciences and philosophy. In the latter field his "Philosophical Note Books" (as yet untranslated into English) containing detailed critical comments on the works of many leading philosophers, ancient and modern, and in particular on Hegel, indicate his close study of texts. In 1909, Lenin published his best known philosophic work "Materialism and Empirio-Cnticism" which was directed against "a number of writers, would-be Marxists" including Bazarov, Bogdanov, Lunacharsky, Berman, Helfond, Yushkevich, Suvorov and Valentinov, and especially against a symposium of this group published under the title, "Studies in the Philosophy of Marxism" which in general adopted the "positivistic" position of Mach and Avenanus.
In his economic and political writings, Lenin extended and developed the doctrines of Marx and Engels especially in their application to a phase of capitalism which emerged fully only	after their death -- imperialism. In the same fashion Lenin built upon and further extended the Marxist doctrine of the state in his "State and Revolution", written just before the revolution of 1917. In this work Lenin develops a concept like the dictatorship of the proletariat which Marx treated only briefly and generally, elaborates a distinction like that between socialism and communism, only implicit in Marx's work, and asserts a thesis like the possibility of socialism in one country, towards which Marx was negative in the light of conditions as he knew them. After the Bolsheviks came to power, Lenin headed the government until his death on January 21, 1924. In Russian, Lenin's "Collected Works" comprise thirty volumes, with about thirty additional volumes of miscellaneous writings ("Leninskie Sborniki"). The principal English translations are the "Collected Works", to comprise thirty volumes (of which five in eight books have been published to date), the "Selected Works" comprising twelve volumes (for philosophical materials, see especially Volume XI, "Theoretical Principles of Marxism"), and the Little Lenin Library, made up mostly of shorter works, comprising 27 volumes to date. -- J.M.S.
Lessing, Gotthold Ephraim: (1729-1781) German dramatist and critic. He is best known in the philosophic field for his treatise on the limitations of poetry and the plastic arts in the famous "Laokoon." In the drama, "Nathan the Wise," he has added to the world's literature a profound plea for religious toleration. -- L.E.D.
Leucippus: (a. 450 B.C.) A contemporary of Empedocles and Anaxagoras and founder of the School of Abdera, developed the fruitful principle that all qualitative differences in nature may be reduced to quantitative ones. Thus Leucippus breaks up the homogeneous "Being" of Parmenides into an infinity of equally homogeneous parts or atoms and he distributes these, in an infinite variety of forms, through infinite space. These small particles of "Being" are separated from one another by that which is not-Being, i.e. by empty space. "Becoming", or the coming into being of things, is essentially the result of the motion of these atoms in space and their accidental coming together. -- M.F.
Level: A grade or type of existence or being which entails a special type of relatedness or of organization, with distinctive laws. The term has been used primarily in connection with theories of emergent evolution where certain so-called higher levels, e.g. life, or mind, are supposed to have emerged from the lower levels, e.g. matter, and are considered to exhibit features of novelty not predictable from the lower levels. -- A.C.B.
Levy-Bruhl, Lucien: (1857-1939) Professor of Philosophy at the Sorbonne 1899-1939, represents a sociological and anthropological approach to philosophy; his chief contribution is an anthropological study of primitive religion which emphasizes the "prelogical" or mystical character of the thinking of primitive peoples. La Mentalite primitive (1922), Eng. trans., 1923; L'Ame Primitive (1927). His other writings include: History of Modern Philosophy in France (Eng. trans., 1899); The Philosophy of Auguste Comte (1900, Eng. trans., 1903). -- L.W.
Lewis, Clarence Irving: (1883-) Professor of Philosophy at Harvard. In Logic, Lewis has originated and defended strict implication (q.v.) in contrast to material implication, urging that formal inference should be based on a relation which can be known to hold without knowing what is true or false of this particular universe. See his Survey of Symbolic Logic, and his and C. H. Langford's Symbolic Logic, esp. Ch. VIII. Lewis has argued also for "queer logics", that is, abstract systems somewhat different from the abstract system usually interpreted as logic. Lewis raises the question how "queer" a system can be and still be interpretable properly as a system of logic.
"A priori truth is definitive in nature and rises exclusively from the analysis of concepts."
"The choice of conceptual systems for . . . application [to particular given experiences] is . . . pragmatic."
"That experience in general is such as to be capable of conceptual interpretation . . . could not conceivably be otherwise." -- C.A.B.
Li: Reason; Law; the Rational Principle. This is the basic concept of modern Chinese philosophy. To the Neo-Confucians, especially Ch'eng I-ch'uan (1033-1107), Ch'eng Ming-tao (1032-1086) and Chu Hsi (1130-1200), Reason is the rational principle of existence whereas the vital force (ch'i) is the material principle. All things have the same Reason in them, making them one reality. By virtue of their Reason, Heaven and Earth and all things are not isolated. The Reason of a thing is one with the Reason of all things. A thing can function easily if it follows its own Reason. Everything can be understood by its Reason. This Reason of a thing is the same as its nature (hsingj. Subjectively it is the nature, objectively it is Reason. Lu Hsiang-shan (1139-1193) said that there is only one mind and there is only one Reason, which are identical. It fills the universe, manifesting itself everywhere. To Wang Yang-ming (1473-1529), the mind itself is the embodiment of Reason. To say that there is nothing existing independent of Reason is to say that there is nothing apart from the mind. See Li hsueh, Chinese philosophy, and ch'i. -- W.T.C.
Li: Propriety; code of proper conduct; rules of social contact; good manners; etiquett; mores; rituals; rites; ceremonials. In Confucius, it aims at true manhood (jen) through self-mastery, and central harmony (ho). "Propriety regulates and refines human feelings, giving them due allowance, so as to keep the people within bounds." It is "to determine human relationships, to settle suspicions and doubts, to distinguish similarity and difference, and to ascertain right and wrong." "The rules of propriety	are rooted in Heaven, have their correspondences in Earth, and are applicable to spiritual beings." "Music unites, while rituals differentiate. . . . Music comes from the inside, while rituals come from the outside. Because music comes from the inside, it is characterized by quiet and calm. And because rituals come from the outside, they are characterized by formalism. . . . Truly great music shares the principles of harmony with the universe, and truly great ritualism shares the principles of distinction with the universe. Through the principles of harmony, order is restored in the physical world, and through the principles of distinction, we are enabled to offer sacrifices to Heaven and Earth. . . . Music expresses the harmony of the universe, while rituals express the order of the universe. Through harmony all things are influenced, and through order all things have a proper place. Music rises from Heaven, while rituals are patterned on Earth. . . ." (Early Confucianism.) "The code of propriety has three sources: Heaven and Earth gave birth to it -- this is a source; our ancestors made it fit the situation -- this is a source; the princes and teachers formed it -- this is a source." (Hsun Tzu, c 335-c 238 B.C.) -- W.T.C.
(b) Benefit, "that which, when obtained, gives pleasure," or the largest amount of happiness for the greatest number of people, as a result of Universal Love (chien ai). Righteousness, loyalty, filial piety, and accomplishment are forms of li. (Mohism and Neo-Mohism.) -- W.T.C.
Libertarianism: (Lat. libertas, freedom) Theory of the freedom of the will. See Free-Will. -- L.W.
Liberty: (in Scholasticism) Of exercise: Is the same as liberty of contradiction: a potentiality for either one of two contradictories, as to do good or not to do good, to act or not to act.
Of specification; is the same as liberty of contrariety: a potentiality for either one of two contraries, as to do good or to do evil. -- H.G.
Liberum Arbitrium: The freedom of indifference (liberum arbitrium indifferentiae) is the ability of the will to choose independently of antecedent determination. See Free-Will. -- L.W.
Lichtenberg, Georg Christoph: (1742-1799) Influential German satirist. Made discoveries in physics. He leaned towards theoretical materialism, and yet had a strong religious (Spinozistic) element. -- H.H.
Main works: Briefe aus England, 1776-8; editor of Göttingisches Mag. d. Literatur u. Wissensch., 1780-2; Ausführlkhe Erklärung d. Hogarthschen Kupferstiche, 1794-9.
Lieh Tzu: Nothing is known of Lieh Tzu (Lieh Yu-k'ou, c. 450-375 B.C.) except that he was a Taoist. The book Lieh Tzu (partial Eng. tr. by L. Giles: Taoist Teachings from the Book of Lieh Tzu) which bears his name is a work of the third century A.D. -- W.T.C.
Li hsueh: The Rational Philosophy or the Reason School of the Sung dynasty (960-1279) which insisted on Reason or Law (li) as the basis of reality, including such philosophers as Chou Lien-hsi (1017-1073), Shao K'ang-chieh (1011-1077), Chang Heng-ch'u (1020-1077), Ch'eng I-ch'uan (1033-1107), Ch'eng Ming-tao (1032-1086), Chu Hsi (1130-1200), and Lu Hsiang-shan (1139-1193). It is also called Hsing-li Hsueh (Philosophy of the Nature and Reason) and Sung Hsueh (Philosophy of the Sung Dynasty). Often the term includes the idealistic philosophy of the Ming dynasty (1368-1644), including Wang Yang-ming (1473-1529), sometimes called Hsin Hsueh (Philosophy of Mind). Often it also includes the philosophy of the Ch'ing dynasty (1644-1911), called Tao Hsueh, including such philosophers as Yen Hsi-chai (1635-1704) and Tai Tung-yuan (1723-1777). For a summary of the Rational Philosophy, see Chinese philosophy. For its philosophy of Reason (li), vital force (ch'i), the Great Ultimate (T'ai Chi), the passive and active principles (yin yang), the nature of man and things (hsing), the investigation of things to the utmost (ch'iung li), the extension of knowledge (chih chih), and its ethics of true manhood or love (jen), seriousness (ching) and sincerity (ch'eng), see articles on these topics. -- W.T.C.
Let f be a monadic function for which the range of the independent variable and the range of the dependent variable both consist of real numbers; let b and c be real numbers; and let g be the monadic function so determined that g(c)=b, and g(x)=f(x) if x is different from c. (The range of the independent variable for g is thus the same as that for f, with the addition of the real number c if not already included.) The limit of f(x) as x approaches c is said to be b if g is continuous at c. -- More briefly but less accurately, the limit of f(x) as x approaches c is the value which rrust be assigned to f for the argument c in order to make it continuous at c.
The limit of f(x) as x approaches infinity is said to be b, if the limit of h(x) as x approaches 0 is b, where h is the function so determined that h(x)=f( 1/x).
In connection with the infinite sequence of real numbers a1, a2, a3 . . ., a monadic function a may be introduced for which the range of the independent variable consists of the positive integers 1, 2, 3, . . . and a(1)=a1, a(2)=a2, a(3)=a3, . . . . It can then be shown that the limit of the infinite sequence as above defined is	the same as the limit of a(x) as x approaches infinity.
(Of course it is not meant to be implied in the preceding that the limit of an infinite sequence or of a function always exists. In particular cases it may happen that there is no limit of an infinite sequence, or no limit of f(x) as x approaches c, etc.) -- A.C.
Limitative: Tending to restrict; pertaining to the limit-value. In logic, an affirmative infinitated judgment, often employed as a third quality added to affirmative and negative. More specifically used by Kant to denote judgments of the type, "Every A is a not-B", and since Kant, applied to the judgments known to the older logicians as indefinite. -- J.K.F.
Limiting Notion: The notion of the extreme applicability of an universal principle considered as a limit. Employed by Kant (1724-1804) in his Kritik der Reinen Vernunft, A 255, to indicate the theory that experience cannot attain to the noumenon. -- J.K.F.
Limits of Sensation: The two limiting sensations in the sensory continuum of any given sense: (a) the lower limit is the just noticeable sensation which if the stimulus producing it were diminished, would vanish altogether or -- in the view of some psychologists -- would pass into the unconscious. See Threshold of Consciousness, (b) The upper limit is the maximum sensation such that if the producing stimulus were increased the resultant sensation would again vanish. -- L.W.
Line of Beauty: Title given by Wm. Hogarth to an undulating line supposedly containing the essence of the graphically beautiful, and so regarded as both the cause and the criterion of beauty; particular lines and paintings become beautiful as and because they exhibit this line. According to Hogarth, such lines must express "symmetry, variety, uniformity, simplicity, intricacy, and quantity". (Analysis of Beauty, London, 1753, p. 47.) -- I.J.
Lipps, Theodor: (1851-1914) Eminent German philosopher and psychologist. The study of optical illusions led him to his theory of empathy. Starts with the presupposition that every aesthetic object represents a living being, and calls the psychic state which we experience when we project ourselves into the life of such an object, an empathy (Einfühlung) or "fellow-feeling". He applied this principle consistently to all the arts. The empathic act is not simply kinaesthetic inference but has exclusively objective reference. Being a peculiar source of knowledge about other egos, it is a blend of inference and intuition. Main works: Psychol. Studien, 1885; Grundzüge d. Logik, 1893; Die ethische Grundfragen, 1899; Aesthetik, 2 vols., 1903-06; Philos. u. Wirklichkeit, 1908; Psychol. Untersuch., 2 vols., 1907-12. -- H.H.
Localization, Cerebral: (Lat. locus, place) The supposed correlation of mental processes, sensory and motor, with definite areas of the brain. The theory of definite and exact brain localization has been largely disproven by recent physiological investigations of Franz, Lashley and others. -- L.W.
Locke, John: (1632-1714) The first great British empiricist, denied the existence of innate ideas, categories, and moral principles. The mind at birth is a tabula rasa. Its whole content is derived from sense-experience, and constructed by reflection upon sensible data. Reflection is effected through memory and its attendant activities of contemplation, distinction, comparison in point of likeness and difference, and imaginative recompositon. Even the most abstract notions and ideas, like infinity, power, cause and effect, substance and identity, which seemingly are not given by experience, are no exceptions to the rule. Thus "infinity" confesses our inability to limit in fact or imagination the spatial and temporal extension of sense-experience; "substance," to perceive or understand why qualities congregate in separate clumps; "power" and "cause and effect," to perceive or understand why and how these clumps follow, and seemingly produce one another as they do, or for that matter, how our volitions "produce" the movements that put them into effect. Incidentally, Locke defines freedom as liberty, not of choice, which is always sufficiently motivated, but of action in accordance with choice. "Identity" of things, Locke derives from spatial and temporal continuity of the content of clumps of sensations; of structure, from continuity of arrangement in changing content; of person, from continuity of consciousness through memory, which, incidentally, permits of alternating personalities in the same body or of the transference of the same personality from one body to another.
In these circumstances real knowledge is very limited. "Universals" register superficial resemblances, not the real essences of things. Experience directly "intuits" identity and diversity, relations, coexistences and necessary connections in its content, and, aided by memory, "knows" the agreements and disagreements of ideas in these respects. We also feel directly (sensitive knowledge) that our experience comes from without. Moreover, though taste, smell, colour, sound, etc. are internal to ourselves (secondary qualities) extension, shape, rest, motion, unity and plurality (primary qualities) seem to inhere in the external world independently of our perception of it. Finally, we have "demonstrative knowledge" of the existence of God. But of anything other than God, we have no knowledge except such as is derived from and limited by the senses.
Locke also was a political, economic and religious thinker of note. A "latitudinarian" and broad churchman in theology and a liberal in politics, he argued against the divine right of kings and the authority of the Bible and the Church, and maintained that political sovereignty rests upon the consent of the governed, and ecclesiastical authority upon the consent of reason. He was also an ardent defender of freedom of thought and speech. Main works: Two Treatises on Gov't, 1689; Reasonableness in Christianity, 1695; Some Thoughts on Education, 1693; An Essay on Human Understanding, 1690. -- B.A.G.F.
Logic, formal: Investigates the structure of propositions and of deductive reasoning by a method which abstracts from the content of propositions which come under consideration and deals only with their logical form. The distinction between form and content can be made definite with the aid of a particular language or symbolism in which propositions are expressed, and the formal method can then be characterized by the fact that it deals with the objective form of sentences which express propositions and provides in these concrete terms criteria of meaningfulness and validity of inference. This formulation of the matter presupposes the selection of a particular language which is to be regarded as logically exact and free from the ambiguities and irregularities of structure which appear in English (or other languages of everyday use) -- i.e., it makes the distinction between form and content relative to the choice of a language. Many logicians prefer to postulate an abstract form for propositions themselves, and to characterize the logical exactness of a language by the uniformity with which the concrete form of its sentences reproduces or parallels the form of the propositions which they express. At all events it is practically necessary to introduce a special logical language, or symbolic notation, more exact than ordinary English usage, if topics beyond the most elementary are to be dealt with (see logistic system, and semiotic).
Concerning the distinction between form and content see further the articles formal, and syntax, logical.
1. THE PROPOSITIONAL CALCULUS formalizes the use of the sentential connectives and, or, not, if . . . then. Various systems of notation are current, of which we here adopt a particular one for purposes of exposition. We use juxtaposition to denote conjunction ("pq" to mean "p and q"), the sign ∨ to denote inclusive disjunction ("p ∨ q" to mean ("p or q or both"), the sign + to denote exclusive disiunction ("p + q" to mean "p or q but not both"), the sign ∼ to denote negation ("∼p" to mean "not p"), the sign ⊃ to denote the conditional ("p ⊃ q" to mean "if p then q," or "not both p and not-q"), the sign ≡ to denote the biconditional ("p ≡ q" to mean "p if and only if q," or "either p and q or not-p and not-q"), and the sign | to denote alternative denial ("p | q" to mean "not both p and q"). -- The word or is ambiguous in ordinary English usage between inclusive disjunction and exclusive disjunction, and distinct notations are accordingly provided for the two meanings of the word, The notations "p ⊃ q" and "p ≡ q" are sometimes read as "p implies q" and "p is equivalent to q" respectively. These readings must, however, be used with caution, since the terms implication and equivalence are often used in a sense which involves some relationship between the logical forms of the propositions (or the sentences) which they connect, whereas the validity of p ⊃ q and of p ≡ q requires no such relationship. The connective ⊃ is also said to stand for "material implication," distinguished from formal implication (§ 3 below) and strict implication (q. v.). Similarly the connective ≡ is said to stand for "material equivalence."
A | B → ∼A ∨ ∼B.
A ⊃ B → ∼A ∨ B.
A ≡ B → [B ⊃ A][A ⊃ B].
A + B → [∼B]A ∨ [∼A]B.
if A and B are formulas [A] ∨ [B] is a formula.
The formulas of the propositional calculus as thus defined will in general contain more brackets than are necessary for clarity or freedom from ambiguity; in practice we omit superfluous brackets and regard the shortened expressions as abbreviations for the full formulas. It will be noted also that, if A and B are formulas, we regard [A] | [B], [A] ⊃ [B], [A] ≡ [B], and [A] + [B], not as formulas, but as abbreviations for certain formulas in accordance with the above given definitions.
In order to complete the setting up of the propositional calculus as a logistic system (q. v.) it is necessary to state primitive formulas and primitive rules of inference. Of the many possible ways of doing this we select the following.
[A ∨ A] ⊃ A.
A ⊃ [B ⊃ AB].
A ⊃ [A ∨ B].
[A ∨ B] ⊃ [B ∨ A].
[A ⊃ B] ⊃ [[C ∨ A] ⊃ [C ∨ B]].
(The complete list of primitive formulas is thus infinite, but there are just seven possible forms of primitive formulas as above.) There is one primitive rule of inference, as follows: Given A and A ⊃ B to infer B. This is the inference known as modus ponens (see below, §2).
if A and A ⊃ B are theorems then B is a theorem.
if the inference from A1, A2, . . . , An to C and the inference from A1, A2, . . . , An to C ⊃ B are both valid inferences then the inference from A1, A2, . . . , An to B is a valid inference.
It can be proved that the inference from A1, A2, . . . , An to B is a valid inference of the propositional calculus if (obviously), and only if (the deduction theorem), [A1 ⊃ [A2 ⊃ . . . [An ⊃ B] . . . ]] is a theorem of the propositional calculus.
The reader should distinguish between theorems about the propositional calculus -- the deduction theorem, the principles of duality (below), etc. -- and theorems of the propositional calculus in the sense just defined. It is convenient to use such words as theorem, premiss, conclusion both for propositions (in whatever language expressed) and for formulas representing propositions in some fixed system or calculus.
In the foregoing the list of fundamental propositional symbols has been left unspecified. A case of special importance is the case that the fundamental propositional symbols are an infinite list of variables, p, q, r, . . ., which may be taken as representing ambiguously any proposition whatever -- or any proposition of a certain class fixed in advance (the class should be closed under the operations of negation, conjunction, and inclusive disjunction). In this case we speak of the pure propositional calculus, and refer to the other cases as applied propositional calculus (although the application may be to something as abstract in character as the pure propositional calculus itself, as, e.g., in the case of the pure functional calculus of first order (§3), which contains an applied propositional calculus).
In formulating the pure propositional calculus the primitive formulas may (if desired) be reduced to a finite number, e.g., to the seven listed above with A, B, C taken to be the particular variables p, q, r. A second primitive rule of inference, the rule of substitution is then required, allowing the inference from a formula A to the formula obtained from A by substituting a formula B for a particular variable in A (the same formula B must be substituted for all occurrences of that variable in A). The definition of a theorem is then given in the same way as before, allowing for the additional primitive rule, the definition of a valid inference must, however, be modified.
In what follows (to the end of §1) we shall, for convenience of statement, confine attention to the case of the pure propositional calculus. Similar statements hold, with minor modifications, for the general case.
The formulation which we have given provides a means of proving theorems of the propositional calculus, the proof consisting of an explicit finite sequence of formulas, the last of which is the theorem proved, and each of which is either a primitive formula or inferable from preceding formulas by a single application of the rule of inference (or one of the rules of inference, if the alternative formulation of the pure propositional calculus employing the rule of substitution is adopted). The test whether a given finite sequence of formulas is a proof of the last formula of the sequence is effective -- we have the means of always determining of a given formula whether it is a primitive formula, and the means of always determining of a given formula whether it is inferable from a given finite list of formulas by a single application of modus ponens (or substitution). Indeed our formulation would not be satisfactory otherwise. For in the contrary case a proof would not necessarily carry conviction, the proposer of a proof could fairly be asked to give a proof that it was a proof -- in short the formal analysis of what constitutes a proof (in the sense of a cogent demonstration) would be incomplete.
However, the test whether a given formula is a theorem, by the criterion that it is a theorem if a proof of it exists, is not effective -- since failure to find a proof upon search might mean lack of ingenuity rather than non-existence of a proof. The problem to give an effective test by means of which it can always be determined whether a given formula is a theorem is the decision problem, of the propositional calculus. This problem can be solved either by the process of reduction of a formula to disjunctive normal form, or by the truth-table decision procedure. We state the latter in detail.
p ∨ q is false if p and q are both false, true otherwise.
Thus, given a formula of the (pure) propositional calculus and an assignment of a truth-value to each of the variables appearing, we can reckon out by a mechanical process the truth-value to be assigned to the entire formula. If, for all possible assignments of truth-values to the variables appearing, the calculated truth-value corresponding to the entire formula is truth, the formula is said to be a tautology.The test whether a formula is a tautology is effective, since in any particular case the total number of different assignments of truth-values to the variables is finite, and the calculation of the truth-value corresponding to the entire formula can be carried out separately for each possible assignment of truth-values to the variables.
Now it is readily verified that all the primitive formulas are tautologies, and that for the rule of modus ponens (and the rule of substitution) the property holds that if the premisses of the inference are tautologies the conclusion must be a tautology. It follows that every theorem of the propositional calculus is a tautology. By a more difficult argument it can be shown also that every tautology is a theorem. Hence the test whether a formula is a tautology provides a solution of the decision problem of the propositional calculus.
As corollaries of this we have proofs of the consistency of the propositional calculus (if A is any formula, A and ∼A cannot both be tautologies and hence cannot both be theorems) and of the completeness of the propositional calculus (it can be shown that if any formula not already a theorem, and hence not a tautology, is added to the list of primitive formulas, the calculus becomes inconsistent on the basis of the two rules, substitution and modus ponens).
If A ≡ B is a theorem, and D is the result of replacing a particular occurrence of A by B in the formula C, then the inference from C to D is a valid inference.
The dual of a formula C of the propositional calculus is obtained by interchanging conjunction and disjunction throughout the formula, i.e., by replacing AB everywhere by A ∨ B, and A ∨ B by AB. Thus, e.g., the dual of the formula ∼[pq ∨ ∼r] is the formula ∼[[p ∨ q] ∼r]. In forming the dual of a formula which is expressed with the aid of the defined connectives, |, ⊃, ≡, +, it is convenient to remember that the effect of interchanging conjunction and (inclusive) disjunction is to replace A|B by ∼A∼B, to replace A ⊃ B by ∼A B; and to interchange ≡ and +.
if A ≡ B is a theorem, then A* ≡ B* is a theorem.
Names given to particular theorems of the propositional calculus are usually thought of as applying to laws embodied in the theorems rather than to the theorems as formulas; hence, in particular, the same name is applied to theorems differing only by alphabetical changes of the variables appearing; and frequently the name used for a theorem is used also for one or more forms of valid inference associated with the theorem. Similar remarks apply to names given to particular theorems of the functional calculus of first order, etc.
Princibia Mathematica. 2nd edn., vol. 1. Cambridge. England, 1925.
Introduction to a general theory of elementary propositions, American Journal of Mathematics, vol. 43 (1921), pp. 163-181.
Elementary Logic, Boston and New York. 1941.
Recherches sur la Theorie de la Demonstration, Warsaw, 1930.
Grundzuge der theoretischen Logik, 2nd edn., Berlin, 1938.
Grundlagen der Mathematik, vol. 1, Berlin, 1934; also Supplement III to vol. 2. Berlin. 1939.
2. HYPOTHETICAL SYLLOGISM, DISJUNCTIVE SYLLOGISM, DILEMMA are names traditionally given to certain, forms of inference, which may be identified as follows with certain particular forms of valid inference of the propositional calculus (see § 1).
The hypothetical syllogism has two kinds or moods. Modus ponens is the inference from a major premiss A ⊃ B and a minor premiss A to the conclusion B. Modus tollens is the inference from a major premiss A ⊃ B and a minor premiss ∼B to the conclusion ∼A.
from A + B and ∼A to B.
from A + B and B to ∼A.
In each case the first premiss named is the major premiss and the second one the minor premiss.
The inferences from A ⊃ B and C ⊃ A to C ⊃ B, and from A ⊃ B and C ⊃ ∼B to C ⊃ ∼A are called pure hypothetical syllogisms, and the above simpler forms of the hypothetical syllogism are then distinguished as mixed hypothetical. Some recent writers apply the names, modus ponens and modus tollens respectively, also to these two forms of the pure hypothetical syllogism. Other variations of usage or additional forms arc also found. Some writers include under these heads forms of inference which belong to the functional calculus of first order rather than to the propositional calculus.
System der Logik, 4th edn., Bonn, 1874.
An Introduction to Logic, 2nd edn., Oxford, 1916.
General Logic, New York, 1931.
An Introduction to Symbolic Logic, 1937. Appendix A.
For the interpretation of the calculus we must presuppose a certain domain of individuals. This may be any well-defined non-empty domain, within very wide limits. Different posslhle choices of the domain of individuals lead to different interpretations of the calculus.
a list of functional constants, which denote particular propositional functions of individuals. There shall be an effective notational criterion associating with each functional constant a positive integer n, the functional constant denoting an n-adic propositional function of individuals.
One or more of the lists (2), (3), (4) may be empty, but not both (3) and (4) shall be empty. The list (1) is required to be infinite, and the remaining lists may, some or all of them, be infinite. Finally, no symbol shall be duplicated either by appearing twice in the same list or by appearing in two different lists; and no functional constant shall contain braces [ ] (or either a left brace or a right brace) as a constituent part of the symbol.
When (2) and (3) are complete -- i.e., contain all the variables indicated above (an infinite number of propositional variables and for each positive integer n an infinite number of functional variables with subscript n) -- and (4) is empty, we shall speak of the pure functional calculus of first order. When (2) and (3) are empty and (4) is not empty, we shall speak of a simple applied functional calculus of first order.
if A is a formula and X is an individual variable, (EX)[A] is a formula.
[A] ⊃x [B] → (X)[[A] ⊃ [B]].
[A] ≡x [B] → (X)[[A] ≡ [B]].
[A] ∧x [B] → (EX)[[A][B]].
(Here A and B are any formulas, and X is any individual variable. Brackets may be omitted when superfluous.) If F and G denote monadic propositional functions, we say that F(X) ⊃x G(X) expresses formal implication of the function G by the function F, and F(X) ≡x G(X) expresses formal equivalence of the two functions (the adjective formal is perhaps not well chosen here but has become established in use).
[A ⊃x B] ⊃ [A⊃ (X)B].
[B ⊃x A] ⊃ [(EX)B ⊃ A].
Given A and A ⊃ B to infer B (the rule of modus ponens).
Given A to infer (X)A, where X is any individual variable (the rule of generalization). In applying the rule of generalization, we say that the variable X is generalized upon.
The theorems of the functional calculus of first order are the formulas which can he derived from the primitive formulas by a succession of applications of the primitive rules of inference. An inference from premisses A1, A2, . . . , An to a conclusion B is a valid inference of the functional calculus of first order if B becomes a theorem upon adding A1, A2, . . . , An to the list of primitive formulas and at the same time restricting the rule of generalization by requiring that the variable generalized upon shall not be any one of the free individual variables of A1, A2, . . . . , An. It can be proved that the inference from A1, A2, . . . , An to B is a valid inference of the functional calculus of first order if (obviously), and only if (the deduction threorem), [A2 ⊃ [A2⊃ . . . [An ⊃ B] . . . ]] is a theorem of the functional calculus of first order.
It can be proved that if A ≡ B is a theorem, and D is the result of replacing a particular occurrence of A by B in the formula C, then the inference from C to D is a valid inference.
The consistency of the functional calculus of first order can also be proved without great difficulty.
A formula is said to be in prenex normal form if all the quantifiers which it contains stand together at the beginning, unseparated by negations (or other sentential connectives), and the scope of each quantifier (i.e., the extent of the bracket [ ] following the quantifier) is to the end of the entire formula. In the case of a formula in prenex normal form, the succession of quantifiers at the beginning is called the prefix; the remaining portion contains no quantifiers and is the matrix of the formula. It can be proved that for every formula A there is a formula B in prenex normal form such that A ≡ B is a theorem; and B is then called a prenex normal form of A.
A formula of the pure functional calculus of first order which contains no free individual variables is said to be satisfiable if it is possible to determine the underlying non-empty domain of individuals and to give meanings to the propositional and functional variables contained -- namely to each propositional variable a meaning as a particular proposition and to each n-adic functional variable a meaning as an n-adic propositional function of individuals (of the domain in question) -- in such a way that (under the accepted meanings of the sentential connectives, the quantifiers, and application of function to argument) the formula becomes true. The meaning of the last word, even for abstract, not excluding infinite, domains, must be presupposed -- a respect in which this definition differs sharply from most others made in this article.
which is satisfiable in an infinite domain of individuals but not in any finite domain -- the negation is satisfiable in any non-empty domain.
It can be shown that all theorems A of the pure functional calculus of first order which contain no free individual variables have the property that ∼A is not satisfiable. Hence the pure functional calculus of first order is not complete in the strong sense in which the pure	propositional calculus is complete. Gödel has shown that the pure functional calculus of first order is complete in the weaker sense that if a formula A contains no free individual variables and ∼A is not satisfiable then A is a theorem.
the so-called set-theoretic decision problem, to find an effective test by means of which it can always be determined whether a given formula containing no free individual variables is satisfiable.
It follows from Gödel's completeness theorem that these two forms of the decision problem are equivalent: a solution of either would lead immediately to a solution of the other.
Church has proved that the decision problem of the pure functional calculus of first order is unsolvable. Solutions exist, however, for several important special cases. In particular a decision procedure is known for the case of formulas containing only monadic function variables (this would seem to cover substantially everything considered in traditional formal logic prior to the introduction of the modern logic of relations).
if X and Y are any individual variables, and B results from the substitution of Y for a particular free occurrence of X in A, which is not in a sub-formula of A of the form (Y)[C], then [X = Y] ⊃ [A ⊃ B] is a primitive formula.
We speak of the pure functional calculus of first order with equality when the lists of propositional variables and functional variables are complete and the only functional constant is =; we speak of a simple applied functional calculus of first order with equality when the lists of propositional variables and functional variables are empty.
The addition to the functional calculus of first order of individual constants (denoting particular individuals) is not often made -- unless symbols for functions from individuals to individuals (so-called "mathematical" or "descriptive" functions) are to be added at the same time. Such an addition is, however, employed in the two following sections as a means of representing certain forms of inference of traditional logic. The addition is really non-essential, and requires only minor changes in the definition of a formula and the list of primitive formulas (allowing the alternative of individual constants at certain places where the above given formulation calls for free individual variables).
Principia Matbematica, 2nd edn., vol. 1, Cambridge, England, 1925.
Die Vollständigkeit der Axiome des logiscben Funktionen-kalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349-360.
Grundzüge der theoretischen Logik, 2nd edn., Berlin, 1938.
Hilbert and Bernays, Grundlagen der Mathematik, vol. 1, Berlin, 1934, and vol. 2, Berlin, 1939.
The traditional account includes under A and E respectively, also (propositions denoted by) P(A) and ∼P(A), where A is an individual constant. These singular propositions are ignored in our account of opposition and immediate inference, but will appear in § 5 as giving variant forms of certain syllogisms.
respectively. The question concerning the choice between these two interpretations is known as the problem of existential import of propositions. We prefer to introduce (Ex)S(x) as a separate premiss at those places where it is required.
Given a fixed subject S and a fixed predicate P, we have, according to the square of opposition, that A and O are contradictory, E and I are contradictory, A and E are contrary, I and O are subcontrary, A and I are subaltern, E and O are subaltern. The two propositions in a contradictory pair cannot be both true and cannot be both false (one is the exact negation of the other). The two propositions in a subaltern pair are so related that the first one, together with the premiss (Ex)S(x), implies the second (subalternation). Under the premiss (Ex)S(x), the contrary pair, A, E, cannot be both true, and the subcontrary pair, I, O, cannot be both false.
Simple conversion of a proposition, A, E, I, or O, consists in interchanging S and P without other change. Thus the converse of S(x) ⊃x P(x) is P(x) ⊃x S(x), and the converse of S(x) ⊃x ∼P(x) is P(x) ⊃x ∼S(x). In mathematics the term converse is used primarily for the simple converse of a proposition A; loosely also for any one of a number of transformations similar to this (e.g., F(x)G(x) ⊃x H (x) may be said to have the converse F(x)H(x) ⊃x G(x)). Simple conversion of a proposition is a valid inference, in general, only in the case of E and I.
Conversion per accidens of a proposition A. i.e., of S(x) ⊃x P(x), yields P(x) ∧x S(x).
Obversion of a proposition A, E, I, or O consists in replacing P by a functional constant p which denotes the negation of the propositional function (property) denoted by P, and at the same time inserting ∼ if not already present or deleting it if present. Thus the obverse of S(x) ⊃x P(x) is S(x) ⊃x ∼p(x) (the obverse of "all men are mortal" is "no men are immortal"). The obverse of S(x) ⊃x ∼P(x) is S(x) ⊃x p(x); the obverse of S(x) ∧x P(x) is S(x) ∧x ∼p(x); the obverse of S(x) ∧x ∼P(x) is S(x) ∧x p(x).
The name immediate inference is given to certain inferences involving propositions A, E, I, O. These include obversion of A, E, I, or O, simple conversion of E or I, conversion per accidens of A, subalternation of A, E. The three last require the additional premiss (Ex)S(x). Other immediate inferences (for which the terminology is not wholly uniform among different writers) may be obtained by means of sequences of these: e.g., given that all men are mortal we may take the obverse of the converse of the obverse and so infer that all immortals are non-men (called by some the contrapositive, by others the obverted contrapositive).
Formal Logic, New York, 1939.
5. CATEGORICAL SYLLOGISM is the name given to certain forms of valid inference (of the functional calculus of first order) which involve as premisses two (formulas representing) categorical propositions, having a term in common -- the middle term. Using S, M, P as minor term, middle term, and major term, respectively, we give the traditional classification into figures and moods. In each case we give the major premiss first, the minor premiss immediately after it, and the conclusion last; in some cases we give a third (existential) premiss which is suppressed in the traditional account. Because of the admission of singular propositions under the heads A, E, two different forms of valid inference appear in some cases under the same figure and mood -- these singular forms are separately listed.
The five moods of the fourth figure are sometimes characterized instead as indirect moods of the first figure, the two premisses (major and minor) being interchanged, and the names being then given respectively as Baralipton, Celantes, Dabitis, Fapesmo, Frisesomorum. (Some add the five "weakened" moods, Barbari, Celaront, Cesaro, Camestros, Calemos, to be obtained respectively from Barbara, Celarent, Cesare, Camestres, Calemes, by subalternation of the conclusion.) Other variations in the names of the moods are also found. These names have a mnemonic significance, the first three vowels indicating whether the major premiss, minor premiss, and conclusion, in order, are A, E, I, or O; and some of the consonants indicating the traditional reductions of the other moods to the four direct moods of the first figure.
The Port-Royal Logic, translated by T. S. Baynes, 2nd edn., London, 1851.
F. Ueberweg, System der Logik, 4th edn., Bonn, 1874.
M. Eaton, General Logic, New York, 1931.
A mathematical treatment of the rules of the syllogism, Mind, vol. 45 (1936), pp. 209-216, 416.
6. THEORY OF TYPES. In the functional calculus of first order, variables which appear as arguments of propositional functions or which are bound by quantifiers must be variables which are restricted to a certain limited range, the kinds of propositions about propositional functions which cannot be expressed in the calculus. The uncritical attempt to remove this restriction, by introducing variables of unlimited range (the range covering both non-functions and functions of whatever kind) and modifying accordingly the definition of a formula and the lists of primitive formulas and primitive rules of inference, leads to a system which is formally inconsistent through the possibility of deriving in it certain of the logical paradoxes (q. v.). The functional calculus of first order may, however, be extended in another way, which involves separating propositional functions into a certain array of categories (the hierarchy of types), excluding. propositional functions which do not fall into one of these categories, and -- besides propositional and individual variables -- admitting only variables having a particular one of these categories as range.
For convenience of statement, we confine attention to the pure functional calculus of first order. The first step in the extension consists in introducing quantifiers such as (F1), (EF1), (F2), (EF3), etc., binding n-adic functional variables. Corresponding changes are made in the definition of a formula and in the lists of primitive formulas and primitive rules of inference, allowing for these new kinds of bound variables. The resulting system is the functional calculus of second order. Then the next step consists in introducing new kinds of functional variables; namely for every finite ordered set k, l, m, . . . , p of i non-negative integers (i = 1, 2, 3, . . .) an infinite list of functional variables Fklm . . .p, Gklm . . .p, . . . , each of which denotes ambiguously any i-adic propositional function for which the first argument may be any (k-1)-adic propositional function of individuals, the second argument any (l-1)-adic propositional function of individuals, etc. (if one of the integers k, l, m, . . . , p is 1 the corresponding argument is a proposition -- if 0, an individual). Then quantifiers are introduced binding these new kinds of functional variables; and so on. The process of alternately introducing new kinds of functional variables (denoting propositional functions which take as arguments propositional functions of kinds for which variables have already been introduced) and quantifiers binding the new kinds of functional variables, with appropriate extension at each stage of the definition of a formula and the lists of primitive formulas and primitive rules of inference, may be continued to infinity. This leads to what we may call the functional calculus of order omega, embodying the (so-called simple) theory of types.
where X and Y are any two variables of the same type and F is a monadic functional variable of appropriate type. The notation X = Y may then be interpreted as denoting equality or identity.
The functional calculus of order omega (as just described) can be proved to be consistent by a straightforward generalization of the method employed by Hilbert and Ackermann to prove the consistency of the functional calculus of first order.
For many purposes, however, it is necessary to add to the functional calculus of order omega the axiom of infinity, requiring the domain of individuals to be infinite. -- This is most conveniently done by adding a single additional primitive formula, which may be described by referring to § 3 above, taking the formula, which is there given as an example of a formula satisfiable in an infinite domain of individuals but not in any finite domain, and prefixing the quantifier (EF) with scope extending to the end of the formula. This form of the axiom of infinity, however, is considerably stronger (in the absence of the axiom of choice) than the "Infin ax" of Whitehead and Russell.
Other primitive formulas (possibly involving new primitive notations) which may be added correspond to the axiom of choice (q. v.) or are designed to introduce classes (q. v.) or descriptions (q. v.). Functional abstraction (q. v.) may also be Introduced by means of additional primitive formulas or primitive rules of inference, or it may be defined with the aid of descriptions. Whitehead and Russell employ the axiom of infinity and the axiom of choice but avoid the necessity of special primitive formulas in connection with classes and descriptions by introducing classes and descriptions as incomplete symbols.
With the aid of the axiom of infinity and a method of dealing with classes and descriptions, the non-negative integers may be introduced in any one of various ways (e.g., following Frege and Russell, as finite cardinal numbers), and arithmetic (elementary number theory) derived formally within the system. With the further addition of the axiom of choice, analysis (real number theory) may be likewise derived.
No proof of consistency of the functional calculus of order omega (or even of lower order) wiih the axiom of infinity added is known, except by methods involving assumptions so strong as to destroy any major significance.
According to an important theorem of Gödel, the functional calculus of order omega with the axiom of infinity added, if consistent, is incomplete in the sense that there are formulas A containing no free variables, such that neither A nor ∼A is a theorem. The same thing holds of any logistic system obtained by adding new primitive formulas and primitive rules of inference, provided only that the effective (recursive) character of the formal construction of the system is retained. Thus the system is not only incomplete but, in the indicated sense, incompletable. The same thing holds also of a large variety of logistic systems which could be considered as acceptable substitutes for the functional calculus of order omega with axiom of infinity; in particular the Zermelo set theory (§ 9 below) is in the same sense incomplete and incompletable.
The formalization as a logistic system of the functional calculus of order omega with axiom of infinity leads, by a method which cannot be given here, to a (definite but quite complicated) proposition of arithmetic which is equivalent to -- in a certain sense, expresses -- the consistency of the system. This proposition of arithmetic can be represented within the system by a formula A containing no free variables, and the following second form of gödel's incompleteness theorem can then be proved: If the system is consistent, then the formula A, although its meaning is a true proposition of arithmetic, is not a theorem of the system. We might, of course, add A to the system as a new primitive formula -- we would then have a new system, whose consistency would correspond to a new proposition of arithmetic, represented by a new formula B (containing no free variables), and we would still have in the new system, if consistent, that B was not a theorem.
Principia Mathematica, 3 vols., Cambridge England, 1st edn. 1910-13, 2nd edn. 1925-27.
Abriss der Logistik, Vienna, 1929.
A System of Logistic, Cambridge, Mass., 1934.
A formulation of the simple theory of types, The Journal of Symbolic Logic, vol. 5 (1940), pp. 56-68.
Einige Betrachtungen uber die Begriffe der ω-Widerspruchsfreiheit und der ω-Vollstandigkeit, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 97-112.
Die Widerspruchsfreiheit der Stufenlogik, Mathematische Zeitschrift, vol. 4l (1936), pp. 357-366.
K. Gödel, Uber formal unenscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173-198.
Extensions of some theorems of Gödel and Church, The Journal of Symbolic Logic, vol. 1 (1936), pp. 87-91.
An informal exposition of proofs of Gödel's theorems and Church's theorem, The Journal of Symbolic Logic, vol. 4 (1939), pp. 53-60.
Grundlagen der mathematik, vol. 2, Berlin, 1939.
While the algebra of classes can be set up as an independent logistic system, we shall here describe it instead by reference to the functional calculus of first order (§ 3), using two monadic functional constants, ∨ and ∧ and an infinite list of monadic functional variables F1, G1, H1, . . . corresponding in order to the class variables a, b, c, . . . respectively. Given any expression A of the algebra of classes, the corresponding formula A‡ of the functional calculus is obtained by replacing complementation, logical sum, and logical product respectively by negation, inclusive disjunction, and conjunction, and at the same time replacing ∨, ∧, a, b, c, . . . respectively by ∨(x), ∧(x), F1(x), G1(x), H1(x), . . . . Given any formula A = B of the algebra of classes the corresponding formula of the functional calculus is A‡ ≡x B‡. Given any formula A ≠ B of the algebra of classes, the corresponding formula of the functional calculus is ∼[A‡ ≡x B‡]. A formula C is a theorem of the algebra of classes if and only if the inference from (x)∨(x) and (x)∼∧(x) to C‡ (where C‡ corresponds to C) is a valid inference of the functional calculus. The inference from premisses D1, D2, . . ., Dn to a conclusion C is a valid inference of the algebra of classes if and only if the inference from (x)∨(x), (x)∼∧(x), D1‡, D2‡, . . . , Dn‡ to C‡ is a valid inference of the functional calculus.
This isomorphism between the algebra of classes and the indicated part of the functional calculus of first order can be taken as representing a parallelism of meaning. In fact, the meanings become identical if we wish to construe the functional calculus in extension, (see the article propositional function); or, inversely, if we wish to construe the algebra of classes in intension, instead of the usual construction.
If we deal only with formulas of the algebra of classes which are equations (i.e., which have the form A = B), the above description by reference to the functional calculus may be replaced by a simpler description using the applied propositional calculus (§ 1) whose fundamental proposittonal symbols are xε∨, xε∧, xεa, xεb, xεc, . . . . Given an equation, C of the algebra of classes, the corresponding formula C† of the propositional calculus is obtained by replacing equality ( = ) by the biconditional ( ≡ ), replacing complementation, logical sum, and logical product respectively by negation, inclusive disjunction, and conjunction, and at the same time replacing ∨, ∧, a, b, c, . . . respectively by xε∨, xε∧, xεa, xεb, xεc, . . . . An equation C is a theorem of the algbera of classes if and only if the inference from xε∨, ∼xε∧ to C† is a valid inference of the propositional calculus; analogously for valid inferences of the algebra of classes in which the formulas involved are equations.
As a corollary of this, every theorem of the pure propositional calculus (§ 1) of the form A ≡ B has a corresponding theorem of the algebra of classes obtained by replacing the principal occurrence of ≡ by =, elsewhere replacing negation, inclusive disjunction, and conjunction respectively by complementation, logical sum, and logical product, and at the same time replacing propositional variables by class variables. Likewise, every theorem A of the pure propositional calculus has a corresponding theorem B = ∨ of the algebra of classes, where B is obtained from A by replacing negation, inclusive disjunction, and conjunction respectively by complementation, logical sum, and logical product, and replacing propositional variables by class variables.
The dual of a formula or an expression of the algebra of classes is obtained by interchanging logical sum and logical product, and at the same time interchanging ∨ and ∧. The principle of duality holds, that the dual of every theorem is also a theorem.
A ⊂ B → A ? −B = A.
Instead of algebra of classes, the term Boolean algebra is used primarily when it is intended that the formal system shall remain uninterpreted or that interpretations other than that	described above shall be admitted. For the related idea of a Boolean ring see the paper of Stone cited below.
Algebra der Logik, vol. 1, vol. 2 part 1, and vol. 2 part 2, Leipzig, 1890, 1891, 1905.
Sets of independent postulates for the algebra of logic, Transactions ot the American Mathematical Societv. vol. 5 (1904), pp. 288-309.
An Introduction to Symbolic Logic. 1937.
M. H. Stone, The representation of Boolean algebras, Bulletin of the American Mathematical Society. vol. 44 (1938), pp. 807-816.
8. ALGEBRA OF RELATIONS or algebra of relatives deals with relations (q. v.) in extension whose domains and converse domains are each contained in a fixed universe of discourse (which must be a class having at least two members), in a way similar to that in which the algebra of classes deals with classes. Fundamental ideas involved are those of the universal relation and the null relation; the relations of identity and diversity; the contrary and the converse of a relation; the logical sum, the logical product, the relative sum, and the relative product of two relations.
The universal relation ∨ can be described by saying that x∨y holds for every x and y in the universe of discourse. The null relation ∧ is such that x∧y holds for no x and y in the universe of discourse. The relation of identity I is such that xIx holds for every x in the universe of discourse and xIy fails if x is not identical with y. The relation of diversity J is such that xJx fails for every x in the universe of discourse and xJy holds if x is not identical with y.
The contrary −R of a relation R is the relation such that x −R y holds if and only if xRy fails (x and y being in the universe of discourse). In particular, the universal relation and the null relation are each the contrary of the other; and the relations of identity and diversity are each the contrary of the other.
The converse of a relation R is the relation S such that xSy if and only if yRx. The usual notation for the converse of a relation is obtained by placing a breve ? over the letter denoting the relation. A relation is said to be symmetric if it is the same as its converse. In particular, the universal relation, the null relation, and the relations of identity and diversity are symmetric.
The logical sum R ∪ S of two relations R and S is the relation such that x R∪S y holds if and only if at least one of xRy and xSy holds. The logical product R∩S of two relations R and S is the relation such that x R∩S y if and only if both xRy and xSy. The relative product R|S of two relations R and S is a relation such that x R|S y if and only if there is a z (in the universe of discourse) such that xRz and zSy. The relative sum of two relations R and S is the relation which holds between x and y if and only if for every z (in the universe of discourse) at least one of xRz and zSy holds. The square of a relation R is R|R.
The signs = and ≠ are used to form equations and inequations in the same way as in the algebra of classes. An isomorphism between	the algebra of relations and an appropriately chosen part of the functional calculus of first order can also be exhibited in the same way as was done in § 7 above for the algebra of classes. A principle of duality also holds, where duality consists in interchanging logical sum and logical product, relative sum and relative product, ∨ and ∧, | and J.
The portion of the algebra of relations which involves, besides relation variables, only the universal relation and the null relation, and the operations of contrary, logical sum, and logical product, and =, ≠, is isomorphic with (formally indistinguishable from) the algebra of classes.
R ⊂ S → R ∩ −S = ∧.
When R ⊂ S, we say that R is contained in S, or that S contains R.
A relation is transitive if it contains its square, reflexive if it contains I, irreflexive if it is contained in J, asymmetric if its square is contained in J.
Algebra der Logik, vol. 3, Leipzig, 1895.
On the calculus of relations, The Journal of Symbolic Logic, vol. 6 (1941), pp. 73-.
9. ZERMELO SET THEORY. The attempt to devise a system which deals with the logic of classes in a more comprehensive way than is done by the algebra of classes (§ 7), and which, in particular, takes account of the relation e between classes (see the article class), must be carried out with caution in order to avoid the Russell paradox and similar logical paradoxes (q. v.).
There are two methods of devising such a system which (so at least it is widely held or conjectured) do not lead to any inconsistency. One of these involves the theory of types, which was set forth in § 6 above, explicitly for propositional functions, and by implication for classes (classes being divided into types according to the types of the monadic propositional functions which determine them). The other method is the Zermelo set theory, which avoids this preliminary division of classes into types, but imposes restrictions in another direction.
Z=Y → ZεX ⊃x YεX.
Z⊂Y → XεZ ⊃x XεY.
Here X, Y, and Z are to be taken as individual variables ("individual" in the technical sense of § 3), and X is to be determined according to an explicit rule so as to be different from Y and Z.
In the axiom of subset formation, A is any formula not containing t as a free varibale (in general, A will contain x as a free variable). In the axiom of replacement, A is any formula which contains neither t nor x' as a free variable (in general, A will contain x and y as free variables). These two axioms are thus represented each by an infinite list of primitive formulas -- the remaining axioms each by one primitive formula.
The axiom of extensionality as above stated has (incidentally to its principal purpose) the effect of excluding non-classes entirely and assuming that everything is a class. This assumption can be avoided if desired, at the cost of complicating the axioms somewhat -- one method would be to introduce an additional functional constant, expressing the property to be a class (or set), and to modify the axioms accordingly, the domain of individuals being thought of as possibly containing other things besides sets.
The restriction which is imposed in order to avoid paradox can be seen in connection with the axiom of subset formation. Instead of this axiom, an uncritical formulation of axioms for set theory might well have included (Et)[xεt ≡x A], asserting the existence of a set t whose members are the sets x satisfying an arbitrary condition A expressible in the notation of the system. This, however, would lead at once to the Russell paradox by taking A to be ∼ xεx and then going through a process of inference which can be described briefly by saying that x is put equal to t. As actually proposed, however, the axiom of subset formation allows the use of the condition A only to obtain a set t whose members are the sets x which are members of a previously given set z and satisfy A. This is not known to lead to paradox.
The notion of an ordered pair can be introduced into the theory by definition, in a way	which amounts to identifying the ordered pair (x, y) with the set a which has two and only two members, x' and y', x' being the set which has x as its only member, and y' being the set which has x and y as its only two members. (This is one of various similar possible methods.) Relations in extension may then be treated as sets of ordered pairs.
The Zermelo set theory has an adequacy to the logical development of mathematics comparable to that of the functional calculus of order omega (§ 6). Indeed, as here actually formulated, its adequacy for mathematics apparently exceeds that of the functional calculus; however, this should not be taken as an essential difference, since both systems are incomplete, in accordance with Gödel'a theorem (§ 6), but are capable of extension.
Besides the Zermelo set theory and the functional calculus (theory of types), there is a third method of obtaining a system adequate for mathematics and at the same time -- it is hoped -- consistent, proposed by Quine in his book cited below (1940). -- The last word on these matters has almost certainly not yet been said.
Einleitung in die Mengenlehre, 3rd edn., Berlin, 1928.
Set-theoretic foundations for logic, The Journal of Symbolic Logic, vol. 1 (1936), pp. 45-57.
Mengentheorettsche Begründung der Logik, Mathematische Annalen, vol. 115 (1937), pp. 1 -22.
A system of axiomatic set theory, The Journal of Symbolic Logic, vol. 2 (1937), pp. 65-77, and vol. 6 (1941), pp. 1-17.
Mathematical Logic, New York, 1940.
Logic, symbolic, or mathematical logic, or logistic, is the name given to the treatment of formal logic by means of a formalized logical language or calculus whose purpose is to avoid the ambiguities and logical inadequacy of ordinary language. It is best characterized, not as a separate subject, but as a new and powerful method in formal logic. Foreshadowed by ideas of Leibniz, J. H. Lambert, and others, it had its substantial historical beginning in the Nineteenth Century algebra of logic (q. v.), and received its contemporary form at the hands of Frege, Peano, Russell, Hilbert, and others. Advantages of the symbolic method are greater exactness of formulation, and power to deal with formally more complex material. See also logistic system. -- A. C.
A Survey of Symbolic Logic, Berkeley, Cal., 1918.
Symbolic Logic, New York and London, 1932.
An Introduction to Symbolic Logic, Boston and New York, or London, 1937.
Introduction to Logic and to the Methodology of Deductive Sciences, New York, 1941.
Elementary Logic, Boston and New York, 1941.
A bibliography of symbolic logic, The Journal of Symbolic Logic, vol. 1 (1936) pp 121-218, and vol. 3 (1938), pp. 178-212.
Nove Lezioni di Logica Simbolica. Rome, 1938.
Geschichte der Logik, Berlin, 1931.
Logic, traditional: the name given to those parts and that method of treatment of formal logic which have come down substantially unchanged from classical and medieval times. Traditional logic emphasizes the analysis of propositions into subject and predicate and the associated classification into the four forms, A, E, I, O; and it is concerned chiefly with topics immediately related to these, including opposition, immediate inference, and the syllogism (see logic, formal). Associated with traditional logic are also the three so-called laws of thought -- the laws of identity (q. v.), contradiction (q. v.) -- and excluded middle (q. v.) -- and the doctrine that these laws are in a special sense fundamental presuppositions of reasoning, or even (by some) that all other principles of logic can be derived from them or are mere elaborations of them. Induction (q. v.) has been added in comparatively modern times (dating from Bacon's Novum Organum) to the subject matter of traditional logic. -- A. C.
La Logique ou l'Art de Penser, better known as the Port-Royal Logic, 1st edn., Paris, 1662 ; reprinted, Paris, 1878; English translation by T. S. Baynes, 2nd edn., London, 1851.
System der Logik und Geschichte der logischen Lehren, 1st edn., Bonn, 1857; 4th edn., Bonn, 1874.
Gescbichte der Logik im Abendlande, 4 vols., Leipzig, 1855-1870; reprinted, Leipzig, 1927.
Per la Storia della Logica, Bologna, 1922 ; English translation by J. Rosenthal, New York, 1929.
Logical Empiricism: See Scientific Empiricism I.
Logical meaning: See meaning, kinds of, 3.
Logical truth: See Meaning, kinds of, 3; and Truth, semantical.
Logistic: The old use of the word logistic to mean the art of calculation, or common arithmetic, is now nearly obsolete. In Seventeenth Century English the corresponding adjective was also sometimes used to mean simply logical. Leibniz occasionally employed logistica (as also logica mathematica) as one of various alternative names for his calculus ratiocinator. The modern use of logistic (French logistique) as a synonym for symbolic logic (q. v.) dates from the International Congress of Philosophy of 1904, where it was proposed independently by Itelson, Lalande, and Couturat. The word logistic has been employed by some with special reference to the Frege-Russell doctrine that mathematics is reducible to logic, but it would seem that the better usage makes it simply a synonym of symbolic logic. -- A. C.
Ilme Congres de Philosophie, Revue de Metaphysique et de Morale, vol. 12 (1904), see p. 1042.
a determination of the circumstances under which a finite sequence of formulas is a proof of the last formula of the sequence as a consequence of a certain set of formulas (when there is a proof of a formula B as a consequence of the set of formulas A1, A2, . . . , An, we say that the inference from the premisses A1, A2, . . . , An to the conclusion B is a valid inference of the logistic system).
It is not excluded that the class of proofs in the sense of (3) should be empty. But every proof of a formula B as a consequence of an empty set of formulas, in the sense of (4), must also be a proof of B in the sense of (3), and conversely. Moreover, if to the proof of a formula B as a consequence of A1, A2, . . . , An are prefixed in any order proofs of A1, A2, . . . , An, the entire resulting sequence of formulas must be a proof of B; more generally, if to the proof of a formula B as a consequence of A1, A2, . . . , An are prefixed in any order proofs of a subset of A1, A2, . . . , An as consequences of the remainder of A1, A2, . . . , An, the entire resulting sequence must be a proof of B as a consequence of this remainder.
a list of primitive rules of inference each of which prescribes that under certain circumstances a formula B shall be an immediate consequence of a set of formulas A1, A2, . . . , An.
The list of primitive formulas may be empty -- this is not excluded. Or the primitive formulas may be included under the head of primitive rules of inference by allowing the case n=0 in (6). A proof is then defined as a finite sequence of formulas each of which is either a primitive formula or an immediate consequence of preceding formulas by one of the primitive rules of inference. A proof as a consequence of a set of formulas A1, A2, . . . , An is in some cases defined as a finite sequence of formulas each of which is either a primitive formula, or one of A1, A2, . . . , An, or an immediate consequence of preceding formulas by one of the primitive rules of inference; in other cases it may be desirable to impose certain restrictions upon the application of the primitive rules of inference (e.g., in the case of the functional calculus of first order -- logic, formal, § 3 -- that no free variable of A1, A2, . . . , An shall be generalized upon).
A logistic system need not be given any meaning or interpretation, but may be put forward merely as a formal discipline of interest for its own sake; and in this case the words proof, theorem, valid inference, etc., are to be dissociated from their every-day meanings and taken purely as technical terms. Even when an interpretation of the system is intended, it is a requirement of rigor that no use shall be made of the interpretation (as such) in the determination whether a sequence of symbols is a formula, whether a sequence of formulas is a proof, etc.
The kind of an interpretation, or assignment of meaning, which is normally intended for a logistic system is indicated by the technical terminology employed. This is namely such an interpretation that the formulas, some or all of them, mean or express propositions; the theorems express true propositions; and the proofs and valid inferences represent proofs and valid inferences in the ordinary sense. (Formulas which do not mean propositions may be interpreted as names of things other than propositions, or may be interpreted as containing free variables and having only an ambiguous denotation -- see variable.) A logistic system may thus be regarded as a device for obtaining -- or, rather stating -- an objective, external criterion for the validity of proofs and inferences (which are expressible in a given notation).
A logistic system which has an interpretation of the kind in question may be expected, in general, to have more than one such interpretation.
It is usually to be required that a logistic system shall provide an effective criterion for recognizing formulas, proofs, and proofs as a consequence of a set of formulas; i.e., it shall be a matter of direct observation, and of following a fixed set of directions for concrete operations with symbols, to determine whether a given finite sequence of primitive symbols is a formula, or whether a given finite sequence of formulas is a proof, or is a proof as a conseqence of a given set of formulas. If this requirement is not satisfied, it may be necessary -- e.g. -- given a particular finite sequence of formulas, to seek by some argument adapted to the special case to prove or disprove that it satisfies the conditions to be a proof (in the technical sense); i.e., the criterion for formal recognition of proofs then presupposes, in actual application, that we already know what a valid deduction is (in a sense which is stronger than that merely of the ability to follow concrete directions in a particular case). See further on this point logic, formal, § 1.
The requirement of effectiveness does not compel the lists of primitive symbols, primitive formulas, and primitive rules of inference to be finite. It is sufficient if there are effective criteria for recognizing formulas, for recognizing primitive formulas, for recognizing applications of primitive rules of inference, and (if separately needed) for recognizing such restricted applications of the primitive rules of inference as are admitted in proofs as a consequence of a given set of formulas.
With the aid of Gödel's device of representing sequences of primitive symbols and sequences of formulas by means of numbers, it is possible to give a more exact definition of the notion of effectiveness by making it correspond to that of recursiveness (q. v.) of numerical functions. E.g., a criterion for recognizing primitive formulas is effective if it determines a general recursive monadic function of natural numbers whose value is 0 when the argument is the number of a primitive formula, 1 for any other natural number as argument. The adequacy of this technical definition to represent the intuitive notion of effectiveness as described above is not immediately clear, but is placed beyond any real doubt by developments for details of which the reader is referred to Hilbert-Bernays and Turing (see references below).
The requirement of effectiveness plays an important role in connection with logistic systems, but the necessity of the requirement depends on the purpose in hand and it may for some purposes be abandoned. Various writers have proposed non-effective, or non-constructive, logistic systems; in some of these the requirement of finiteness of length of formulas is also abandoned and certain infinite sequences of primitive symbols are admitted as formulas.
For particular examples of logistic systems (all of which satisfy the requirement of effectiveness) see the article logic, formal, especially §§1, 3, 9.
The Logical Syntax of Language, New York and London, 1937.
Was ist ein Kalkül und was hat Frege für eine pünktliche Beantwortung dieser Frage geleistet?, Semester-Berichte, Münster i. W., summer 1935, pp. 16-54.
A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, ser. 2 vol. 42 (1937), pp. 230-265, and Correction, ibid., ser. 2 vol. 43 (1937), pp. 544-546.
A. M. Turing, Computability and λ-definability, The Journal of Symbolic Logic, vol. 2 (1937), pp. 153-163.
Logomachy: (Gr. logos, word + mache, battle) A contention in which words are involved without their references. A contention which lacks the real grounds of difference, or one in which allegedly opposed views are actually not on the same level of discourse. A battle of words alone, which ignores their symbolic character. -- J. K. F.
Logos: (Gr. logos) A term denoting either reason or one of the expressions of reason or order in words or things; such as word, discourse, definition, formula, principle, mathematical ratio. In its most important sense in philosophy it refers to a cosmic reason which gives order and intelligibility to the world. In this sense the doctrine first appears in Heraclitus, who affirms the reality of a Logos analogous to the reason in man that regulates all physical processes and is the source of all human law. The conception is developed more fully by the Stoics, who conceive of the world as a living unity, perfect in the adaptation of its parts to one another and to the whole, and animated by an immanent and purposive reason. As the creative source of this cosmic unity and perfection the world-reason is called the seminal reason (logos spermatikos), and is conceived as containing within itself a multitude of logoi spermatikoi, or intelligible and purposive forms operating in the world. As regulating all things, the Logos is identified with Fate (heimarmene); as directing all things toward the good, with Providence (pronoia); and as the ordered course of events, with Nature (physis). In Philo of Alexandria, in whom Hebrew modes of thought mingle with Greek concepts, the Logos becomes the immaterial instrument, and even at times the personal agency, through which the creative activity of the transcendent God is exerted upon the world. In Christian philosophy the Logos becomes the second person of the Trinity and its functions are identified with the creative, illuminating and redemptive work of Jesus Christ. Finally the Logos plays an important role in the system of Plotinus, where it appears as the creative and form-giving aspect of Intelligence (Nous), the second of the three Hypostases. -- G. R. M.
Lombard, Peter: (c. 1100-c. 1160) Was the author of the Four Books of Sentences, i.e. a compilation of the opinions of the Fathers and early teachers of the Catholic Church concerning various points in theology. He was born at Lumello in Lombardy, studied at Bologna, Rheims and the School of St. Victor in Paris. He was made Bishop of Paris in 1159. The Libri IV Sententiarum was used as a textbook in Catholic theology for more than two centuries, hence it has been commented by all the great theologians of the 13th and I4th centuries. The Franciscans of Quaracchi have published a critical edition in 2 vols. (Quaracchi, 1916). -- V. J. B.
Metaphysik, 1879 (Eng. tr. 1884). -- K. F. L.
Love: (in Max Scheler) Giving one's self to a "total being" (Gesamtwesen); it therefore discloses the essence of that being; for this reason love is, for Scheler, an aspect of phenomonelogical knowledge. -- P. A. S.
Lovejoy, Arthur O.: (1873-) Emeritus Professor of Philosophy of Johns Hopkins University. He was one of the contributors to "Critical Realism." He wrote the famous article on the thirteen pragmatisms (Jour. Philos. Jan. 16, 1908). Also critical of the behavioristic approach. His best known works are The Revolt against Dualism and his recent, The Great Chain of Being, 1936. The latter exemplified L's method of tracing the history of a "unit-idea." A. O. L. is the first editor of the Journal of the History of Ideas (1940-). He is an authority on Primitivism (q.v.) and Romanticism (q.v.). -- L.E.D.
Löwenheim's theorem: The theorem, first proved by Löwenheim, that if a formula of the pure functional calculus of first order (see Logic, formal § 3), containing no free individual variables, is satisfiable (see ibid.) at all, it is satisfiable in a domain of individuals which is at most enumerable. Other, simpler, proofs of the theorem were afterwards given by Skolem, who also obtained the generalization that, if an enumerable set of such formulas are simultaneously satisfiable, they are simultaneously satisfiable in a domain of individuals at most enumerable.
There follows the existence of an interpretation of the Zermelo set theory (see Logic, formal, § 9) -- consistency of the theory assumed -- according to which the domain of sets is only enumerable; although there are theorems of the Zermelo set theory which, under the usual interpretation, assert the existence of the non-enumerable infinite.
A like result may be obtained for the functional calculus of order omega (theory of types) by utilizing a representation of it within the Zermelo set theory. It is thus in a certain sense impossible to postulate the non-enumerable infinite: any set of postulates designed to do so will have an unintended interpretation within the enumerable. Usual sets of mathematical postulates for the real number system (see number) have an appearance to the contrary only because they are incompletely formalized (i.e., the mathematical concepts are formalized, while the underlying logic remains unformalized and indefinite).
The situation described in the preceding paragraph is sometimes called Skolem's paradox, although it is not a paradox in the sense of formal self-contradiction, but only in the sense of being unexpected or at variance with preconceived ideas. -- A.C.
Sur la portee du theoreme de Löwenheim-Skolem, Les Entretiens de Zurich sur les Fondements et la Methode des Sciences Mathematiques, Zurich 1941, pp. 25-52.
Lucretius, Carus: (98-54 B.C.) Noted Roman poet, author of the famous didactic poem De Natura Rerum, in six books, which forms an interesting exposition of the philosophy of Epicureanism. -- M..F.
Lu Hsiang-shan: (Lu Chiu-yiian, Lu Tzu-ching, 1139-1192) Questioned Ch'eng I-ch'uan's interpretation of Confucianism when very young, later often argued with Chu Hsi, and claimed that "the six (Confucian) classics are my footnotes." This official-scholar served as transition from the Reason School (li hsueh) of Neo-Confucianism to the Mind School (hsin hsueh) of Neo-Confucianism. His complete works, Lu Hsiang-shan Ch'uan-chi, number 36 chuans in four volumes. -- W.T.C.
Lullic art: The Ars Magna or Generalis of Raymond Lully (1235-1315), a science of the highest and most general principles, even above metaphysics and logic, in which the basic postulates of all the sciences are included, and from which he hoped to derive these fundamental assumptions with the aid of an ingenious mechanical contrivance, a sort of logical or thinking machine. -- J.J.R.
Lumen naturale: Natural light, equivalent to lumen naturalis rationis, in medieval philosophy and theology denoted the ordinary cognitive powers of human reason unaided by the supernatural light of grace, lumen gratiae, or divine revelation, lumen fidei. -- J.J.R.
The phrase "natural light of reason" occurs also in the scientific writings of Galileo (q.v.) and Descartes (q.v.) -- P.P.W.
Lutheranism: An ecclesiastical school of thought claiming Martin Luther (1483-1546) as its source and inspiration. See Reformation. The Protestant doctrine of salvation by faith, the free grace of God, wholly without earned merit and institutional sanctions, is emphasized. The essence of the church-community is held to revolve about the pure, revealed Word of God and the sacraments of baptism and communion. Varieties of Lutheranism range from a liberal acknowledgment of the Augsburg Confession of 1530 to a more strict adherence to the several Lutheran documents collectively known as the Book of Concord. -- V.F.
b. Identified with art in general because it symbolizes expression of sentiment (Croce). -- L.V.

References: V. 
 §2
 §1
 § 1
 § 5
 § 3
 § 7
 § 6
 § 3
 § 3
 § 1
 V. 
 § 3
 § 9