C

In mathematics, the word calculus has many specific applications, all conforming more or less closely to the above statement. Sometimes, however, the simple phrase "the calculus" is used in referring to those branches of mathematical analysis (q.v.) which are known more explicitly as the differential calculus and the integral calculus. -- A.C.

Newton was influenced by Henry More, e.g. in viewing space as the sensorium of God. See Cudworth, Deism. Cf. M.H. Nicolson, Conway Papers.

All nature is suffused with a love of God and a desire to return to him, witnessed by the laws of motion governing inanimate bodies, the law of self-preservation in organic life, and by man's conscious search for the divine.

Campanella was a political philosopher. In his City of the Sun he conceived a Utopia built on Platonic lines. He was also an ardent champion of the temporal power of the Papacy and of its political as well as its religious sovereignty through the world. -- B.A.G.F.

Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, edited by E. Zermelo, and with a life by A. Fraenkel, Berlin, 1932.

If two classes a and b have no members in common, the cardinal number of the logical sum of a and b is uniquely determined by the cardinal numbers of a and b, and is called the sum of the cardinal number of a and the cardinal number of b.

0 is the cardinal number of the null class. 1 is the cardinal number of a unit class (all unit classes have the same cardinal number).

A cardinal number is inductive if it is a member of every class t of cardinal numbers which has the two properties, (1) 0∈ t, and (2) for all x, if x∈ t and y is the sum of x and 1, then y∈ t. In other (less exact) words, the inductive cardinal numbers are those which can be reached from 0 by successive additions of 1. A class b is infinite if there is a class a, different from b, such that a ⊂ b and a is equivalent to b. In the contrary case b is finite. The cardinal number of an infinite class is said to be infinite, and of a finite class, finite. It can be proved that every inductive cardinal number is finite, and, with the aid of the axiom of choice, that every finite cardinal number is inductive.

The most important infinite cardinal number is the cardinal number of the class of inductive cardinal numbers (0, 1, 2, . . .); it is called aleph-zero and symbolized by a Hebrew letter aleph followed by an inferior 0.

For brevity and simplicity in the preceding account we have ignored complications introduced by the theory of types, which are considerable and troublesome. Modifications are also required if the account is to be incorporated into the Zermelo set theory. --A.C. G. Cantor, Contributions to the Founding of the Theory of Trasfinite Numbers, translated and with an introduction bv P.E.B. Jourdain, Chicago and London, 1915. Whitehead and Russell, Principia Mathematica, vol. 2.

Carnap's work has been devoted especially to formal logic and its applications to problems of epistemology and the philosophy of science. His writings in formal logic include a textbook of mathematical logic and a comprehensive monograph devoted to logical syntax, a new branch of logical research to whose development Carnap has greatly contributed.

In his logical work, he has been specially interested in the nature of mathematics and its relation to logic. He has treated these topics in a number of special articles and in a monograph. The latter also includes an introduction to the youngest field of modern logic, semantics.

Carnap's contributions to the study of epistemological and philosophical problems may be characterized as applications of the methods of logical analysis to the languages of everyday life and of science. His books contain applications to the fundamental problems of epistemology, expound the principles of physicalism (q.v.) which was developed by Carnap and Neurath and which offers, amongst others, a basis for a more cautious version of the ideas of older behaviorism and for the construction of one common unified language for all branches of empirical science (see Unity of Science). Main works: Logische Aufblou der Welt; Abriss der Logistik; Logische Syntax der Sprache "Testability and Meaning," Phil. of Sci. (1916). -- C.G.H.

Discours de la methode, (preface to his Geometric, Meteores, Dieptrique) Meditationes de prima philosophia,

Principia philosophiae,

Passions de l'ame,

Regulae ad directionem ingenii,

Le monde.

1. innate, by which he seems to mean little more than the mental power to think things or thoughts;
2. adventitious, which come to him from without;
3. factitious, produced within his own mind.

1. Created spirits, i.e. the finite soul-substance of each man: these are immaterial agencies capable of performing spiritual operations, loosely united with bodies, but not extended since thought is their very essence.
2. Uncreated Spirit, i.e. God, confined neither to space nor time, All-Good and All-Powerful, though his Existence can be known clearly, his Nature cannot be known adequately by men on earth, He is the God of Christianity, Creator, Providence and Final Cause of the universe.
3. Bodies, i.e. created, physical substances existing independently of human thought and having as their chief attribute, extension. Cartesian physics regards bodies as the result of the introduction of "vortices", i.e. whorls of motion, into extension. Divisibility, figurability and mobility, are the notes of extension, which appears to be little more thin what Descartes' Scholastic teachers called geometrical space.

Apart from philosophy, Descartes' contribution to the development of analytical geometry, the theory of music and the science of optics, are noteworthy achievements.

Descartes is one of the fathers of modern philosophy; his general influence is too extensive to be detailed. Leibniz, Spinoza, Malebranche, Clauberg, De La Forge, Geulincx, Placentius, Chouet, Legrand, Corneio -- these and many others spread Cartesianism throughout Europe. (See Boutroux, "Descartes and Cartesianism," Camb. Mod. Hist., IV, ch. 27.) At present, German Phenomenology, French Spiritualism and Positivism, Bergsonism, and certain forms of Catholic thought represented by J. Geyser in Germany and M. Blondel in France, are offshoots of Cartesianism.

Oeuvres completes, ed. C. Adam et P. Tannery, 13 vols. (Paris, 1896-1911). The Philos. Works of Descartes, transl. by Haldane and Ross, 2 vols. (Cambridge, 1911-12). Fischer, K., Descartes and his School (London, 1887). Gilson, E., Le role de la pensee medievale dans la formation du systeme cartesien (Paris, 1930). Maritain, J., Le songe de Descartes (Paris, 1932). Gemelli, A. (ed.), Cartesio (symposium) (Milan, 1937). -- V.J.B.

(Ger. kategorial) In Husserl: Of or pertaining to the function or the result of ego-spontaneity as conffering logical form on substrates and producing syntactical objects. -- D.C.

(In Kant) A judgment comprising two concepts related by a copula, typically an attribute (predicate) asserted of a substance or thing (subject). Kant denied that hypothetical and disjunctive propositions can be reduced to categorical ones and insisted that each of the forms of judgment denotes a distinct function of the understanding. See Logik, § 24. -- O.F.K.

(in Kant) Any of twelve forms or relating principles of the understanding, constituting necessary conditions of experience. Kant sought to deriv e an exhaustive list of pure forms of the understanding from the forms of judgment in the traditional logic. His list of categories comprises three each of quantity, quality, relation, and modality. See Kantianism. -- O.F.K.

In aesthetics: Purification of and liberation from passions in art (Aristotle). First idea of the distinction between form and sentiment. -- L.V.

1. a relation between events, processes, or entities in the same time series, such that
   1. when one occurs, the other necessarily follows (sufficient condition),
   2. when the latter occurs, the former must have preceded (necessary condition),
   3. both conditions a and b prevail (necessary and sufficient condition),
   4. when one occurs under certain conditions, the other necessarily follows (contributory, but not sufficient, condition) ("multiple causality" would be a case involving several causes which are severally contributory and jointly sufficient); the necessity in these cases is neither that of logical implication nor that of coercion;
2. a relation between events, processes, or entities in the same time series such that when one occurs the other invariably follows (invariable antecedence),
3. a relation between events, processes, or entities such that one has the efficacy to produce or alter the other;
4. a relation between events, processes, or entities such that without one the other could not occur, as in the relation between
   1. the material out of which a product is made and the finished product (material cause),
   2. structure or form and the individual embodying it (formal cause),
   3. a goal or purpose (whether supposed to exist in the future as a special kind of entity, outside a time series, or merely as an idea of the pur-poser) and the work fulfilling it (final cause),
   4. a moving force and the process or result of its action (efficient cause);
5. a relation between experienced events, processes, or entities and extra-experiential but either temporal or non-temporal events, processes, or entities upon whose existence the former depend;
6. a relation between a thing and itself when it is dependent upon nothing else for its existence (self-causality);
7. a relation between an event, process, or entity and the reason or explanation for its being;
8. a relation between an idea and an experience whose expectation the idea arouses because of customary association of the two in this sequence;
9. a principle or category introducing into experience one of the aforesaid types of order; this principle may be inherent in the mind, invented by the mind, or derived from experience; it may be an explanatory hypothesis, a postulate, a convenient fiction, or a necessary form of thought. Causality has been conceived to prevail between processes, parts of a continuous process, changing parts of an unchanging whole, objects, events, ideas, or something of one of these types and something of another. When an entity, event, or process is said to follow from another, it may be meant that it must succeed but can be neither contemporaneous with nor prior to the other, that it must either succeed or be contemporaneous with and dependent upon but cannot precede the other, or that one is dependent upon the other but they either are not in the same time series or one is in no time series at all.

1. the material cause, or that out of which something arises,
2. the formal cause, that is, the pattern or essence determining the creation of a thing,
3. the efficient cause, or the force or agent producing an effect; and
4. the final cause, or purpose. Many thinkers spoke also of
5. the first cause, usually conceived as God.

In scholasticism: Four causes are distinguished, in accordance with Aristotle. Efficient cause, by which any change is brought about in the order new being arises -- prime matter in regard to substantial second matter in regard to accidental forms (Cf. Form, Matter) -- formal cause, the act by which a material substratum is determined towards a new being -- substantial or accidental -- final cause, that because of which something is or becomes. All things tend towards an end by a "natural appetite". -- R.A.

Chance events, according to Aristotle, are occurrences purposive in appearance but not actually the result of either conscious or unconscious teleology. -- G.R.M.

In Cournot, following Aristotle, the co-incidence of two causally determined series of events. In Peirce (q.v.), a vera causa and metaphysically grounded category. See Tychism.

L. Couturat, La Logique de Leibniz (1901).

1. Honesty; sincerity; absence of fault; actuality.
2. Reverence; seriousness.
3. Being one's true self; absolute true self; truth, in the sense of "fulfillment of the self," which "is the beginning and end of material existence," and "without which there is no material existence." "Being true to oneself (or sincerity) is the law of Heaven. To try to be true to oneself is the law of man." "Only those who are their absolute true selves in the world can fulfill their own nature," "the nature of others," "the nature of things," "help Nature in growing and sustaining," and "become equals of Heaven and Earth." (Early Confucianism, Neo-Confucianism.)

Being true to the nature of being (of man and things), which is "the character of the sage," "the basis of the five cardinal moral principles and the source of the moral life." It is "the state of tranquillity without movement." (Chou Lien-hsi, 1017-1073.) "Sincerity (ch'eng) is the way of Heaven, whereas seriousness (ching) is the essence of human affairs. When there is seriousness, there is sincerity." "Sincerity means 'to have no depraved thought'." (Ch'eng I-ch'uan, 1033-1107 and Ch'eng Ming-tao, 1032-1086.) "It may also be expressed as the principle of reality." (Chu Hsi, 1130-1200.) -- W.T.C.

(b) The knowledge that rises above distinctions, attainable only when the human mind, completely comprehends Heaven, nature and the moral law. (Chang Heng-Ch'u, 1020-1077). -- W.T.C.

1. Breath; the vital fluid.
2. Force; spirit.
3. The vital force, as expressed in the operation and succession of the active principle (yang) and the passive principle (yin) and the Five Agents or Elements (wu hsing). To Chou Lien-hsi (1017-1073), this material principle is identical with yin yang and the Five Elements. To Chang Heng-ch'u (1020-1077) it is the reality of the Ultimate Vacuity, having the two aspects of yin and yang. It is to the Ultimate Vacuity (Tai Hsu) as ice is to water. Ch'eng I-ch'uan (1033-1107) and Ch'eng Ming-tao (1032-1086) considered all that has physical form to be identical with the vital force. It is the principle of differentiation and individuation. When a thing disintegrates, the vital force is at an end, not to appear again in the creative process. A new entity is constituted of new vital force. Thus it is also the principle of novelty in creation. It is produced by Reason (li). But to the Neo-Confucians, especially Chi Hsi (1130-1200), Reason has no control over it. The two can never be separated; without it, Reason would having nothing to be embodied in.
4. In aesthetics: Rhythmic vitality; vitalizing spirit; strength of expression or brush stioke.

(b) Religion, especially used in tsung chiao. K'ung Chiao (Confucianism) and Tao Chiao (Taoism) may either mean (a) the ethical, political, and philosophical teachings of Confucius and Lao Tzu respectively and their followers, or (b) the state cult of the worship of Heaven and ancestors and the folk religion of nature and spirit worship, respectively. -- W.T.C.

1. Wisdom, one of the three Universally Recognized Moral qualities of man (ta te), the Three Moral Qualities of the superior man (san te), the Four Fundamentals of the moral life (ssu tuan), and the Five Constant Virtues (wu ch'ang). (Confucianism.)
2. Knowledge; intelligence.
3. Discriminate knowledge; small knowledge, which is incapable of understanding Tao.
4. Intuitive knowledge (liang chih). (Wang Yang-ming, 1473-1529.) -- W.T.C

(a) Marks or qualities of a thing. All things are marks or predicates.

(b) That which is designated by a name.

(c) An idea or concept which Kung-sun Lung used to designate the universal. -- W.T.C.

1. Personal experience, or knowledge obtained through the contact of one's knowing faculty and the object to be known. (Neo-Mohists.)
2. Parents.
3. Kinship, as distinguished from the more remote relatives and strangers, such distinction being upheld by Confucians as essential to the social structure but severely attacked by the Mohists and Legalists as untenable in the face of the equality of men.
4. Affection, love, which it is important for a ruler to have toward his people and for children toward parents. (Confucianism.)

Confucianism (ju chia), on the other hand, advocated true manhood (jen) as the highest good, the superior man (chun tzu) as the ideal being, and cultivation of life (hsiu shen) as the supreme duty of man. It was toward this moralism and humanism that Confucius (551-479 B.C.) taught the doctrines of "chung," or being true to the principles of one's nature, and "shu," or the application of those principles in relation to others, as well as the doctrine of the Golden Mean (chung yung), i.e., "to find the central clue of our moral being and to be harmonious with the universe." Humanism was further strengthened by Mencius (371-289 B.C.) who insisted that man must develop his nature fully because benevolence (jen) and righteousness (i) are natural to his nature which is originally good, and again reinforced by Hsun Tzu (c. 335-286 BC) who, contending that human nature is evil, advocated the control of nature. Amid this antagonism between naturalism and humanism, however, both schools conceived reality as unceasing change (i) and incessant transformation, perpetually in progress due to the interaction of the active (yang) and passive (yin) cosmic principles.

Taoism, however, became too mystical, and Confucianism too formalistic. "Hundred schools" grew and flourished, many in direct opposition to Taoism and Confucianism. There was Mohism (Mo, founded by Mo Tzu, between 500 and 396 B.C.) which rejected formalism in favor of "benefit" and "utility" which are to be promoted through universal love (chien ai), practical observation and application, and obedience to the will of Heaven. There was Neo-Mohism (Mo che, 300 B.C.) which, in trying to prove the thesis of Mohism, developed an intricate system of logic. There was Sophism (ming chia, 400 B.C.) which displayed much sophistry about terms and concepts, particularly about the relationship between substance and quality (chien pai). There was Legalism (fa chia, 500-200 B.C.) which advocated law, statecraft, and authority as effective instruments of government. finally, there was the Yin Yang school (400-200 B.C.) which emphasized yin and yang as the two fundamental principles, always contrasting but complementary, and underlying all conceivable objects, qualities, situations, and relationships. It was this school that provided a common ground for the fusion of ancient divergent philosophical tendencies in medieval China.

Medieval Chinese philosophy was essentially a story of the synthesis of indigenous philosophies and the development of Buddhism. In the second century B.C., the Yin Yang movement identified itself with the common and powerful movement under the names of the Yellow Emperor and Lao Tzu (Huang Lao). This, in turn, became interfused with Confucianism and produced the mixture which was the Eclectic Sinisticism lasting till the tenth century A.D. In both Huai-nan Tzu (d. 122 B.C.), the semi-Taoist, and Tung Chung-shu (177-104- B.C.), the Confucian, Taoist metaphysics and Confucian ethics mingled with each other, with yin and yang as the connecting links. As the cosmic order results from the harmony of yin and yang in nature, namely, Heaven and Earth, so the moral order results from the harmony of yang and yin in man, such as husband and wife, human nature and passions, and love and hate. The Five Agents (wu hsing), through which the yin yang principles operate, have direct correspondence not only with the five directions, the five metals, etc., in nature, but also with the five Constant Virtues, the five senses, etc., in man, thus binding nature and man in a neat macrocosm-microcosm relationship. Ultimately this led to superstition, which Wang Ch'ung (27-c. 100 A.D.) vigorously attacked. He reinstated naturalism on a rational ground by accepting only reason and experience, and thus promoted the critical spirit to such an extent that it gave rise to a strong movement of textual criticism and an equally strong movement of free political thought in the few centuries after him.

In the meantime, Taoism degenerated and identified itself with the lowest forms of religious worship. Its naturalistic philosophy was carried to the point of fatalistic mechanism in Lieh Tzu (c. 300 A.D.) and was made the theoretical basis for alchemy and the search for longevity in Ko Hung (c. 268-c. 334 A.D.). In Kuo Hsiang (c. 312 A.D.), however, the true spirit of Taoism revived. He restored and developed the Taoist doctrines of naturalism and spontaneous transformation to a position of dignity.

Parallel with these developments was the growth of Buddhism in China, a story too long to relate here. Many Buddhist doctrines, latent in India, were developed in China. The nihilism of Madhyamika (Sun-lan, c. 450-c. 1000) to the effect that reality is Void in the sense of being "devoid" of any specific character, was brought to fullness, while the idealism of Vijnaptimatravada (Yogacara, Fahsiang, 563-c. 1000), which claimed that reality in its imaginary, dependent and absolute aspects is "representation-only," was pushed to the extreme. But these philosophies failed because their extreme positions were not consonant with the Chinese Ideal of the golden mean. In the meantime, China developed her own Buddhist philosophy consistent with her general philosophical outlook. We need only mention the Hua-yen school (Avatamisaka, 508) which offered a totalistic philosophy of "all in one" and "one in all," the T'ien-t'ai school (c. 550) which believes in the identity of the Void, Transitoriness, and the Mean, and in the "immanence of 3,000 worlds in one moment of thought," and the Chin-t'u school (Pure Land, c. 500) which bases its doctrine of salvation by faith and salvation for all on the philosophy of the universality of Buddha-nature. These schools have persisted because they accepted both noumenon and phenomenon, both ens and non-ens, and this "both-and" spirit is predominantly characteristic of Chinese philosophy.

The most strange development was Ch'an (Meditation, Zen, c. 500). It is basically a method of "direct intuition into the heart to find Buddha-nature," a method based, on the one hand, on the eightfold negation of production and extinction, annihilation and permanence, unity and diversity, and coming and departing, and, on the other hand, on the affirmation of the reality of the Buddha-nature in all things. Its sole reliance on meditation was most un-Chinese, but it imposed on the Chinese mind a severe mental and spiritual discipline which was invigorating as well as fascinating. For this reason, it exerted tremendous influence not only on Taoism which had much in common with it and imitated it in every way, but also on Neo-Confucianism, which stood in diametrical opposition to it.

Neo-Confucianism developed in three phases, namely the Reason school in the Sung period (960-1279), the Mind school in the Ming period (1388-1644) and the Moral-Law school in the Ch-ing period (1644-1911). The central idea of the movement is focused on the Great Ultimate (T'ai Chi) and Reason (li). The Great Ultimate moves and generates the active principle, yang, when its activity reaches its limit, and engenders the passive principle, yin, when it becomes tranquil. The eternal oscillation of yin and yang gives rise to the material universe through their Five Agents of Water, Fire, Wood, Metal and Earth. Thus, reality is a progressively evolved and a well-coordinated system.

This dynamic and orderly character of the universe is due to Reason and the vital force. As the Ch'eng brothers (I-ch'uan, 1033-1077, and Min-tao, 1032-1086) said, "All things have the same Reason in them." Thus, Reason combines the Many into One, while the vital force differentiates the One into the Many, each with its own "determinate nature." The two principles, however, are not to be sharply contrasted, for neither is independent of the other. Reason operates through, and is embodied in, the vital force. It is this cooperative functioning of theirs that makes the universe a cosmos, a harmonious system of order and sequence. "Centrality is the order of the universe and harmony is its unalterable law." As such the cosmos is a moral order. This is the main reason why the greatest of the Neo-Confucians, Chu Hsi (1130-1200) said that "the Great Ultimate is nothing but the Reason of ultimate goodness."

Furthermore, the universe is a social order, and nothing can stand by itself. At the same time, everything has its opposite. "No two of the productions of creation are alike," and the Taoist doctrine of the equality of things must be rejected. In the eternal sequence of appearance and disappearance every creation is new, and the Buddhist doctrine of transmigration must be rejected.

In order to appreciate fully the meaning of the universe, man must comprehend Reason. This can be done by "investigating things to the utmost" (ko wu), that is, by "investigating the Reason of things to the utmost (ch'iung li)." When sufficient effort is made, and understanding naturally comes, one's nature will be realized and his destiny will be fulfilled, since "the exhaustive investigation of Reason, the full realization of one's nature, and the fulfillment of destiny are simultaneous." When one understands Reason, he will find that "All people are brothers and sisters, and all things are my companions," because all men have the same Reason in them. Consequently one should not entertain any distinction between things and the ego. This is the foundation of the Neo-Confucian ethics of jen, true manhood, benevolence or love. Both the understanding of Reason and the practice of jen require sincerity (ch'eng) and seriousness (ching) which to the Neo-Confucians almost assumed religious significance. As a matter of fact these have a certain correspondence with the Buddhist dhyana and prajna or meditation and insight. Gradually the Neo-Confucian movement became an inward movement, the mind assuming more and more importance.

When it came to the Ming period especially in Wang Yang-ming (1473-1529), Reason became identified with Mind. Mencius' doctrine of intuitive knowledge (liang chih) was revived and made the basis of his theory of the identity of knowledge and conduct and the sacred duty of man to "fully exercise his mind" and to "manifest his illustrious virtues."

Wang Yang-ming considered desire as an obstacle to the mind. The Neo-Confucians of the Ch'ing period, especially Tai Tung-yuan (1723-1777), however, argued that since desire is part of our nature, it has its rightful place, just as the vital force has its rightful place beside Reason. The main problem then would be to attain the harmony of human passion (ch'ing) and the originally good human nature (hsing). Thus Neo-Confucianism reasserted the principle of central harmony (chung yung), and central harmony is the Moral Law (tao). This Law finds expression in constant and orderly transformation, the realization of which is Reason. It will be seen that Neo-Confucianism is essentially compatible with western philosophy and science. It is to be expected, therefore, that both Neo-Confucianism and western thought will play a great role in any future philosophy in China. -- W.T.C

References:

Alfred Forke,

Geschichte der neueren chinesischen Philosophie, De Gruyter & Co., Hamburg, 1938;

Fung Yu-lan,

A History of Chinese Philosophy (ancient period), tr. by D. Bodde, Henri Vetch, Peiping, 1937;

Hu Shih,

The Development of the Logical Method in Ancient China, The Oriental Book Co., Shanghai, 1922.

(b) Cardinal standards or directions in Confucian ethics and government. -- W.T.C.

(b) Seriousness, the inner state of respect or politeness (kung). With respect to daily affairs, it is expressed in care, vigilance, attention, etc., and with respect to the laws of the universe, it is expressed in sincerity (ch'eng), especially toward the Reason (li) of things. "Seriousness is the basis of moral cultivation, the essence of human affairs, just as sincerity is the way of Heaven." It is "to straighten one's internal life and righteousness (i) is to square one's external life." It means "unity of mind and absolute equanimity and absolute steadfastness." (Neo-Confucianism.) -- W.T.C.

1. Essence. "Essence and vital force (ch'i) constitute things."
2. Purity; the pure nature.
3. Spirit; intelligence.
4. Concentration; unity of thought.

(b) Quietude; quiescence; interpreted by the Taoist as absence of desire and unity of thought, by Confucians in general as the original state of human nature, and by Hsun Tzu (c. 335-c. 288 B.C.) as the mind not being disturbed by such things as dreams. -- W.T.C.

(a) Human nature (which is originally tranquil) when moved and awakened and expressed in the seven feelings (joy, anger, sorrow, fear, love, hatred, and desire), like and dislike, and the sense of advantage and disadvantage.

(b) The impure side of man, born of the passive (yin) vital force (ch'i) as contrasted with the pure, the nature (hsing) born of the active (yang) vital force and expressed in the Five Constant Virtues (wu ch'ang).

(c) Human nature, or feelings original in or proper to man. -- W.T.C.

See Cause, Determinism, Will.

An equivalent assumption, called by Russell the multiplicative axiom and afterwards adopted by Zermelo as a statement of his Auswahl-prinzip, is as follows: Given a class K whose members are non-empty classes no two of which have a member in common, there exists a class A (the Auswahlmenge) all of whose members are members of members of K and which has one and only one member in common with each member of K. Proof of equivalence of the multiplicative axiom to the axiom of choice is due to Zermelo. -- A.C.

E. Zermelo,

Beweis, dass jede Menge wohlgeord net werden kann, Mathematische Annalen, vol. 59 (1904), pp. 514-516.

B. Russell,

On some difficulties in the theory of transfinite numbers and order types. Proceedings of the London Mathematical Society, ser 2. vol. 4 (1906), pp 29-53.

E. Zermelo,

Neuer Beweis fur die Moglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65 (1908), pp. 107-128.

K. Gödel,

The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton, N.J., 1940.

(b) Power; authority; force.

(c) Expediency, a Confucian ethical concept which justifies deviation from cardinal standards or directions (ching) in ethics and government under certain abnormal circumstances. -- W.T.C.

(b) Completeness of life, that is, "all desires reach a proper harmony." (Taoism.) -- W.T.C.

1. The Mean. See Chung yung. (Confucius.)
2. The central self or moral being, in which "the passions such as joy, anger, grief, and pleasure have not awakened," and which exists "in a state of absolute tranquillity without being moved." See ho. (Early Confucianism; Neo-Confucianism.)
3. The central or the proper principle; the Moral Law (tao); the "ultimate principle" of the universe; "the great basis of existence"; "the beginning and the end of the universe."
4. The principle of centrality, which is observable in everything, that everything should have the proper balance of activity and tranquillity. (Tung Chung-shu, 177-104 B.C., Ch'eng Ming-tao, 1032-1086.)
5. Impartiality; the principle of neutrality which is present in every human heart.
6. The inner life; the inner principle. (Lao Tzu.)

1. Being true to the principle of the self; being true to the originally good nature of the self; being one's true self; the Confucian "central thread or principle" (i kuan) with respect to the self, as reciprocity (shu) is that principle with respect to others. See i kuan.
2. Exerting one's pure heart to the utmost, to the extent of "not a single thought not having been exhausted", honesty, sincerity; devotion of soul, conscientiousness. (Confucianism.) "Honesty (chung) is complete realization of one's nature" whereas truthfulness (hsin) is "complete realization of the nature of things." "Honesty (chung) is the subjective side of truthfulness (hsin) whereas truthfulness is the objective side of honesty." (Ch'eng Ming-tao, 1032-1086.)

   "Honesty is exerting one's heart to the utmost whereas truthfulness is the observance of the Reason of things." (Chu Hsi, 1230-1300.)

3. Impartiality, especially in love and profit,
4. Loyalty, especially to one's superiors, faithfulness.

1. The Golden Mean. See Chung and i kuan.
2. Centrality and harmony, a law "from whose operation we cannot for one instant in our existence escape", the central clue to man's moral being which unites him to the universal order (or to attain central harmony). (Early Confucianism.)
3. The Universal and the Changeless which is the true principle of things and the eternal law of the universe. (Neo-Confucianism.)

1. The superior man, the perfect man, the moral man, the noble man. "There may have been a superior man who is not a true man (jen), but there has never been an inferior man (hsiao jen) who is a true man." The superior man "makes upward progress," "understands profit," and "despises the ordinances of Heaven, great men, and the words of the sages." (Confucius.)

   "The superior man's moral order is on the increase, while the inferior man's moral order is on the decrease." "The superior man abides by what is internal, whereas the inferior man abides by what is external." (Ancient Confucianism )

   "The superior man makes advance in the moral law, whereas the inferior man makes advance in profit." "The superior man enjoys in the fulfillment of the moral law, whereas the inferior man enjoys in the fulfillment of his desires." (Medieval Confucianism.) The superior man "sees what is great and far" and is interested in "helping things to perfection," whereas the inferior man "sees what is small and near" and is interested in destroying things." (Neo-Confucianism.)

2. A ruler.
3. Husband (as in the Odes).

Cicero: De Natura deorum; De officiis; Disputationes: Tuscalanae; De finibus bonorum et malorum.

Whitehead and Russell, by introducing classes into their system only as incomplete symbols, "avoid the assumption that there are such things as classes." Their method (roughly) is to reinterpret a proposition about a class determined by a propositional function A as being instead an existential proposition, about some propositional function formally equivalent to A.

See also Logic, formal, §§ 7, 9. -- A.C.

B. Greek and Roman art in which perfect balance between body and spirit is achieved (Hegel). Contrasted with Modern and Romantic. -- L.V.

2. Process of grouping species into genera, genera into still larger genera, and so on to the summum genus (q.v.). -- A.C.B.

1. non-propositional apprehension (perception, memory, introspection, etc.) as well as
2. propositions or judgments expressive of such apprehension.

In Scholasticism: Whatever is known is, as known, an accident of the knowing soul and therefore caused by an informing agent. All knowledge ultimately is due to an affection of the senses which are informed by the agency of the objects through a medium. The immutation of the sense organ and the corresponding accidental change of the soul are called species sensibilis impressa. The conscious percept is the species expressa. Intellectual knowledge stems from the phantasm out of which the active intellect disengages the universal nature which as species intelligibilis impressa informs the passive intellect and there becomes, as conscious concept. the species expressa or verbum mentis. Sensory cognition is a material process, but it is not the matter of the particular thing which enters into the sensory faculties; rather they supply the material foundation for the sensible form to become existent within the mind. Cognition is, therefore, "assimilation" of the mind to its object. The cognitive mental state as well as the species by which it originates are "images" of the object, in a metaphorical or analogical sense, not to be taken as anything like a copy or a reduplication of the thing. The senses, depending directly on the physical influence exercised by the object, cannot err; error is of the judging reason which may be misled by imagination and neglects to use the necessary critique. -- R.A.

Abstractive: That meaning of cognition which lacks one of the two requisites for intuitive knowledge: for in abstrictive cogniti n either we know things through other things, and not through their proper images -- or we know th'ngs that are not present: e.g., the knowledge we now have of God, through creatures -- or the knowledge we have of Adam, a being not present to us.

Comprehensive: Strictly speaking, that which is adequate to or fully commensurate with the object, -- a knowledge in which the whole object is known completely and in every way in which it can be known -- even to all the effects and consequences with which it has an intrinsic connection. This knowledge must be clear, certain, evident, and quidditative, because it is the most perfect type of knowledge corresponding to the object. E.g., God's complete knowledge of Himself.

Intuitive: Requires two things: (1) that it result from the proper species, or the proper image of the object itself, impressed upon the mind by the object or by God, and (2) that it bear upon an object that is really present with the greatest clearness and certitude. Our knowledge of the sun is intuitive while we are looking at the sun, and that knowledge which the blessed have of God is intuitive.

Quidditative: In the strict sense, is that which arises from the proper image of an object, like intuitive knowledge, and besides, penetrates distinctly, with a clear, proper, and positive concept, the essential predicates of a thing even to the last difference. The knowledge which God has of Himself is of this kind. But quidditative knowledge in the wide sense is any knowledge of the quiddity or essence of an object, or any definition explaining what a thing is. -- H.G.

1. real and existent e.g. in veridical perception and memory;
2. abstract and ideal e.g. in conception and valuation;
3. fictitious, e.g. in imagination and hallucination.

1. Composition is a union of parts such that the component parts are discernible in the compound. Thus the visual and factual data which combine to form a total percept are recognizable in the resultant percept.
2. Fusion is a union of parts into a whole in which the identity of the parts is obliterated. Thus the amalgamation of two sense images to form a new quality would, if this phenomenon were psychologically possible, be an instance of psychic fusion.

M. Schönfinkel,

Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92 (1924), pp. 305-316.

H. B. Curry,

Grundlagen der kombmatorischen Logik, American Journal of Mathematics, vol. 52 (1930), pp. 509-536, 789-834.

H. B. Curry,

The universal quantifier in combinatory logic, Annals of Mathematics, ser. 2, vol. 32 (1931), pp. 154-180.

H. B. Curry,

Apparent variables from the standpoint of combinatory logic, Annals of Mathematics, ser. 2, vol 34 (1933), pp. 381-404.

H. B. Curry,

Functionality in combinatory logic, Proceedings of the National Academy of Sciences, vol. 20 (1934), pp. 584-590.

J. B. Rosser,

A mathematical logic without variables, Annals of Mathematics, ser. 2, vol. 36 (1935), pp 127-150, and Duke Mathematical Journal, vol. 1 (1935), pp. 328-355.

H. B. Curry,

A revision of the fundamental rules of combinatory logic, The Journal of Symbolic Logic, vol. 6 (1941), pp. 41-53.

H. B. Curry,

Consistency and completeness of the theory of combinators, ibid , pp. 54-61.

By a familiar ambiguity the term is used indiscriminately to refer either to the process by which such accordance is brought about, or that with regard to which accord between the speaker and interpreter is achieved.

The definition is intended to cover the communication of attitudes, evaluations, desires, etc., as well as of judgments or assertions. See Functions of Language, Speech Situation. -- M.B.

1. ordinary sense-mediated communication by means of speech, writing, gesture, facial expression and bodily attitude and
2. allegedly direct contact between minds by mental telepathy and other occult means.

Given the concept of semantical truth (q.v.), we may also define a logistic system as complete if every true formula of the system is a theorem. This sense of completeness is not, in general, equivalent to the other; and may be the weaker one if formulas containing free variables occur. See Logic, formal, §§ 3, 6. -- A.C.

[p ⊃ q][p ⊃ r] ⊃ [p ⊃ qr].

1. unifying and relating manifold facts or ideas;
2. deducing something from premises;
3. accommodating new facts or ideas to established knowledge;
4. seeing a thing or idea in its proper or significant context;
5. relating a fact or idea to something known, universal and subject to law.

Logic: The sum of characteristics which connote a class notion symbolized by a general term. Also, the features common to a number of in stances or objects. Thus, the connotation (q.v.) or intension (q.v.) of a concept. See Intension. -- O.F.K.

In scholasticism: the "word of the mind" (verbum mentis) by which the possible intellect expresses (therefore also in later writers species expressa) the universal nature disengaged by the active intellect from the phantasm and transmitted as species intelligibilis to the possible intellect. -- R.A.

In Kant: In the strict sense, any generic or class term, exclusive or relational terms or categories. Sometimes, loosely, any general or abstract representation. -- O.F.K.

In Husserl: 1. An expressible sense. 2. An eidos as intended. -- D.C.

1. the generic concept refers to a class of resembling particulars,
2. the object of a concept is a universal essence pervading the particulars, but having; no reality apart from them,
3. concepts refer to abstracta, that is to say, to ideal objects envisaged by the mind but having no metaphysical status.

See Scholasticism.

Precise conjunction or accompaniment, spatial or temporal. -- C.A.B.

2. Neo-idealism (q.v.) in Italy introduce a second type of concrete universal whose element^ lack the character of dialectical opposition and logical abstractness. -- W.L.

Santayana calls universals "concretions of discourse." (Life of Reason, vol. I (Reason in Common Sense)).

E. Condillac, Traite des systemes, 1846; Traite des sensations, 1854; Langue des calculs, 1858.

A sentence of the form A ⊃ B (or a proposition expressed by such a sentence) -- verbally, "if A then B" -- may be called a conditional sentence (or proposition). -- A. C.

(b) In psychology. Behavior of a living organism reacting to environmental stimuli. See Behaviorism. -- A.J.B.

In 496 B.C., he began 14 years of travelling from state to state, offering his service. He was politely consulted by princes and dukes, but no one would put his moral doctrines into practice. He was even sent away from Ch'i, threatened in Sung, driven out of Sung and Wei, and surrounded between Ch'en and Ts'ai. When in difficulty, he exclaimed, "Heaven has endowed me with a moral destiny. What can Huan Tuei (who threatened him) do to me?" Eventually he retired to Lu to study, teach and write.

He lived in the time when the moral and cultural traditions of Chou were in rapid decline. Attempting to uphold the Chou culture, he taught poetry, history, ceremonies and music to 3,000 pupils, becoming the first Chinese educator to offer education to any who cared to come with or without tuition. He taught literature, human conduct, being one's true self and honesty in social relationships. He wrote the chronicles called Spring and Autumn. His tacit judgments on social and political events were such that "unruly ministers and villainous sons were afraid" to repeat their evil deeds.

He severely disciplined himself and practiced what he taught. He loved poetry, ceremonies and music. He was serious, honest, polite, filially pious towards his mother, stern toward his son, and friendly to his pupils. His most reliable teachings are found in the Lun Yu (Analects), aphorisms recorded by his followers. -- W.T.C.

1. Grammar: The inflections of a verb.
2. Biology: The union of male and female plant or animal.
3. Logic: Joining the extreme terms of a syllogism by the middle term; joining dissimilar things by their common characteristics or by analogy.
4. Ethics: Conjugations or pairings of the passions: love and hate, desire and avoidance, pleasure and sadness, etc. Synonymous with connexio.
5. Metaphysics: In Aristotle, De Gen. et Corr., the pairings of opposites in the simple bodies: dry and hot (fire), hot and moist (air), moist and cold (water), cold and dry (earth).

1. The sum of the constitutive notes of the essence of a concept as it is in itself and not as it is for us. This logical property is thus measured by the sum of the notes of the concept, of the higher genera it implies, of the various essential attributes of its nature as such. This term is synonymous with intension and comprehension; yet, the distinctions between them have been the object of controversies.
2. J. S. Mill identifies connotation with signification and meaning, and includes in it much less than under comprehension or intension. The connotation of a general term (singular terms except descriptions are non-connotative) is the aggregate of all the other general terms necessarily implied by it is an abstract possibility and apart from exemplification in the actual world. It cannot be determined by denotation because necessity does not always refer to singular facts. Logicians who adopt this view distinguish connotation from comprehension by including in the latter contingent characters which do not enter in the former. Comprehension is thus the intensional reference of the concept, or the reference to universals of both general and singular terms. The determination of the comprehension of a concept is helped by its denotation, considering that reference is made also to singular, contingent, or particular objects exhibiting certain characteristics. In short, the connotation of a concept is its intensional reference determined intensionally; while its comprehension is its intensional reference extensionally determined.
3. It may be observed that such a distinction and the view that the connotation of a concept contains only the notes which serve to define it, involves the nominalist principle that a concept may be reduced to what we are actually and explicitely thinking about the several notes we use to define it. Thus the connotation of a concept is much poorer than its actual content. Though the value of the concept seems to be saved by the recognition of its comprehension, it may be argued that the artificial introduction into the comprehension of both necessary and contingent notes, that is of actual and potential characteristics, confuses and perverts the notion of connotation as a logical property of our ideas.

1. Broadest sense: noematically intentional, conscious of something. A process may be "conscious" in this sense even if it is not "conscious"' in the following sense.

2. Narrower sense: "Actual", belonging to the cogito. As living in a process that is "conscious" in this second sense, the ego is also said to be "conscious", "awake", and "conscious of" (awake to) the intentional object of the process. As objects of processes that are conscious (in either of the first two senses), objects are occasionally referred to as "conscious". -- D.C.

The analysis of conscioisness proceeds in two principal directions:

1. a distinction may be drawn between the act of consciousness and the content of consciousness and the two may even be considered as separable ingredients of consciousness, and
2. consciousness is analyzed into its three principal functions: cognition, affection and conation.

Locke, Reid and others restricted consciousness to the reflective apprehension of the mind of its own processes but this usage has been abandoned in favor of the wider definition indicated above and the term introspection is used to designate this special kind of consciousness. See Behaviorism. -- L.W.

(Ger. Bewusstsein) In Husserl:

1. Noematic intentionality in general. The intentional constituting of the temporal stream-of-consciousness itself is an instance of "consciousness" in this broad sense, though it is intrinsically prior to the constituted stream.
2. The stream of subjective process, or any part of it, as having the characteristic of noematic intentionality.
3. The stream of "actual" subjective process, or any part of it; the "ego cogito".

Since this definition of consistency is relative to the choice if a particular notation as representing negation, the following definition is sometimes used instead: (2) A logistic system is consistent if not every formula (not every sentence) is a theorem. In the case of many familiar systems, under the usual choice as to which notation represents negation, the equivalence of this sense of consistency to the previous one is immediate.

Closely related to (2), and applicable to logistic systems containing the pure propositional calculus (see Logic, formal, § 1) or an appropriate part of it, is the notion of consistency in the sense of E. L. Post, according to which a system is consistent if a formula composed of a single propositional variable (say the formula p) is not a theorem. -- A.C.

Thus in ordinary numerical algebra and in real number theory, the symbols x, y, z are variables, while 0, 1, 3, -- 1/2, π, e are constants. In such mathematical contexts the term constant is often restricted to unambiguous (non-variable) names of numbeis. But such symbols as +, =, < may also be called constants, as denoting particular functions and relations.

In various mathematical contexts, the term constant will be found applied to letters which should properly be called variables (according to our account here), but which are thought of as constant relatively to other variables appearing. The actual distinction in such cases, as revealed by logistic formalization, either is between free and bound variables, or concerns the order and manner in which the variables are bound by quantifiers, abstraction operators, etc.

In mathematics, the word constant may also be employed to mean simply a number ("Eulers constant"), or, in the physical sciences, to mean a physical quantity ("the gravitational constant," "Planck's constant"). -- A.C.

In Kant: Whatever enters into the structure of actual experience. Thus, the categories are constitutive of knowledge of nature because they are necessary conditions of any experience or knowledge whatever. In contrast, the transcendent Ideas (God, the total Cosmos, and the immortal Soul) are not constitutive of anything, since they do not serve to define or compose real objects, and must be restricted to a regulative and speculative use. See Crit. of Pure Reason, Transc. Dialectic, Bk. II, ch. II, Sec. 8. -- O.F.K.

(b) In recent epistemology: Contemplation is knowledge of an object in contrast to enjoyment which is the minds' direct self-awareness. (Cf. S. Alexander, Space, Time and Deity, Vol I, p. 12.) -- L.W.

In this broad sense contingency appears always to imply a reference to some basis in relation to which a given thing may be said to be contingent, and in view of the two referents most commonly employed it is possible to distinguish two chief types: (1) logical contingency, and (2) physical contingency. The first is contingency with respect to the laws of logic, the second contingency with respect to the laws of nature. A given state of affairs, e.g., the existence of a snowflake with a given shape, is logically contingent in that the laws of logic do not suffice to establish that such a thing does or does not exist. This same state of affairs would not ordinarily be held to be physically contingent, however, for, although the laws of nature alone do not suffice to determine that there is such a snowflake, still it would be held on the general hypothesis of determinism that, given the specific conditions under which the water was frozen, it was determined by physical laws that a snowflake would exist and that it would have this shape and no other.

A narrower, less philosophical employment of "contingent" emphasizes the aspect of dependence of one state of affairs upon another state of affairs in accordance with the laws of nature. In this usage an event A is said to be contingent upon B when the occurrence of A depends upon the occurrence of B, and it is usually implied that the occurrence of B is itself uncertain. -- F.L.W.

In metaphysics: The opposite of determinism, which holds that free activity may enter causally into natural processes. See Boutroux. -- R.T.F.

Leibniz distinguished contingent truths (verites de fait) from necessarv truths of reason (verites de raison), Hume (q.v.) regarded all causal assertions as contingent upon certain habits of the mind. See Cause, Probability.

A "substantive", or "existent" is defined by Johnson as anything manifested in space or time. The substantives divide into two subclasses, continuants and occurrents: those which continue to exist, and those which cease to exist. Every occurrent is referable to one or more continuant.

While continuants are collections or sets of occurrents, every collection of occurrents does not constitute a continuant, but only those possessing a certain type of unity. This unity is not an "unknown somewhat" supporting the observable properties; nor does it imply the permanence of any given property. Rather it is a "causal unity of connection between its temporarally or spatially separated manifestations" (Ibid., III, p. 99).

Johnson recognizes two fundamentally distinct types of continuant: physical and psychical, -- the "occupant" (of space), and the "experient". -- F.L.W.

If a class b ii ordered by a relation R, and a ⊂ b, we say that z is an upper bound of a if, for all x, x∈a implies xRz; and that z is a least upper bound of a if z is an upper bound of a and there is no upper bound y of a, different from z, such that yRz.

A class b ordered by a relation R is said to have continuous order (Dedekindian continuity) if it is compactly ordered by R and every non-empty class a, for which a ⊂ b, and which has an upper bound, has a least upper bound.

An important mathematical example of continuous order is afforded by the real numbers, ordered by the relation not greater than. According to usual geometric postulates, the points on a straight line also have continuous order, and, indeed, have the same order type as the real numbers.

The term continuity is also employed in mathematics in connection with functions of various kinds. We shall state the definition for the case of a monadic function f for which the range of the independent variable and the range of the dependent variable both consist of real numbers (see the article Function).

Let us use R for the relation not greater than among real numbers. A neighborhood of a real number c is determined by two real numbers m and n -- both different from c and such that mRc and cRn -- and is the class of real numbers x, other than m and n, such that mRx and xRn. The function f is said to be continuous at the real number c if the three following conditions are satisfied:

1. c belongs to the range of the independent variable;
2. in every neighborhood of c there are numbers other than c belonging to the range of the independent variable;
3. corresponding to every neighborhood b of f(c) there is a neighborhood a of c such that, for every real number x belonging to the range of the independent variable, x∈a implies f(x) ∈ b.

E. V. Huntington, The Continuum, Cambridge, Mass., 1917.

S(s) ⊃_x P(x) is ∼P(x) ⊃_x ∼S(x).

Strictly speaking, there are no contraries in the category of substance, since substances are the subject of contraries, nor in the category of quantity, since these are relative. Two contrary states cannot obtain in one and the same individual at the same time and in the same respect; cf. contradiction. Some contraries, e.g. good-bad, black-white, have intermediaries; while others do not, e.g. odd-even. (ii) Propositions: Two universal propositions, having opposite quality (i.e. one affirmative and one negative) are contrary; De Interpretattone, 17b-4, See Logic, formal § 4, 8.

(b) Physics: In Greek philosophy, the ultimate principles of nature and change were contraries: e.g. love-strife; motion-rest; potentiality-actuality. All motion is between contraries. See Heraclitus, Empedodes, Aristotle. -- L.M.H.

In mathematics, any system of designating points by means of ordered sets of n numbers may be called an n-dimensional coordinate system, and the n numbers so associated with any point are then called its coordinates. Coordinates may also be used in like fashion for various other things besides points. -- A.C.

If the relation R is such that when C and D so correspond, C must always have exactly k members and D exactly 1 member, the correspondence is termed a k-1 correspondence. By an obvious extension it is customary to speak also of many-one, one-many, and many-many correspondences.

Thus the heads and tails of coins are said to be in one-one correspondence; the square roots of positive integers in two-one correspondence with the positive integers. See One-one. -- M.B.

Contrasted with the Coherence Theory of Truth. Cf. B. Russell, An Inquiry into Meaning and Truth, 1941, for defence, and F. H. Bradley, Essays on Truth and Reality, for criticisms of the theory. -- M.B.

In more general epistemology: Theory of knowledge which maintains that truth attaches to a proposition by virtue of its capacity to represent or portray fact. -- A.C.B.