Dagobert D. Runes, Dictionary of Philosophy, 1942.
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Na chia: The coordination and interlocking of the Ten Celestial Stems with the Eight Elements (pa kua), to the end that the first Stem, which is the embodiment of the active or male cosmic force, and the second Stem, which is the reservoir of the passive or female cosmic force, gather in the center and the highest point in the universe. Taoist religion. -- W.T.C.
Naive Realism: The view of the man in the street. This view is an uncritical belief in an external world and the ability to know it. -- V.F.
Name: A word or symbol which denotes (designates) a particular thing is called a proper name of that particular thing.

In English and other natural languages there occur also common names (common nouns), such a common name being thought of as if it could serve as a name of anything belonging to a specified class or having specified characteristics. Under usual translations into symbolic notation, common names are replaced by proper names of classes or of class concepts; and this would seem to provide the best logical analysis. In actual English usage, however, a common noun is often more nearly like a variable (q. v.) having a specified range. -- A.C.

Name relation or meaning relation: The relation between a symbol (formula, word, phrase) and that which it denotes or of which it is the name.

Where a particular (Interpreted) system does not contain symbols for formulas, it may be desirable to employ Gödel's device for associating (positive integral) numbers with formulas, and to consider the relation between a number and that which the associated formula denotes. This we shall call the numerical name relation and distinguish it from the relation between a formula and that which it denotes by calling the latter the semantical name relation.

In many (interpreted) logistic systems -- including such as contain, with their usual interpretations, the Zermelo set theory, or the simple theory of types with axiom of infinity, or the functional calculus of second order with addition of Peano's postulates for arithmetic -- it is impossible without contradiction to introduce the numerical name relation with its natural properties, because Grelling's paradox or similar paradoxes would result (see paradoxes, logical). The same can be said of the semantical name relation in cases where symbols for formulas are present.

Such systems may, however, contain partial name relations which function as name relations in the case of some but not all of the formulas of the system (or of their associated Gödel numbers).

In particular, it is normally possible -- at least it does not obviously lead to contradiction in the case of such systems as the Zermelo set theory or the simple theory of types (functional calculus of order omega) with axiom of infinity -- to extend a system L1 into a system L2 (the semantics of L1 in the sense of Tarski), so that L2 shall contain symbols for the formulas of L1, and for the essential syntactical relations between formulas of L1, and for a relation which functions as a name relation as regards all the formulas of L1 (or, in the case of the theory of types, one such relation for each type), together with appropriate new primitive formulas. Then L2 may be similarly extended into L3, and so on through a hierarchy of systems each including the preceding one as a part.

Or, if L1 contains symbols for positive integers, we may extend L1 into L2 by merely adding a symbol for a relation which functions as a numerical name relation as regards all numbers of formulas of L1 (or one such relation for each type) together with appropriate new primitive formulas; and so on through a hierarchy of systems L1, L2, L3, . . . .

See further Semantics; Semtotic 2; Truth, Semantical. -- A.C.

Nama-rupa: (Skr.) "Name and form", a stereotyped formula for the phenomenal world, or its conceptual and material aspects; also: "word and beauty", as forms of manifestation. See Rupa. -- K.F.L.
Nascency: (Lat. nascens, ppr. of nasci, to be born) A potency which is in process of actualization. See Potency; Potentiality. The term may be applied generally to anything in process of coming into being but it is particularly appropriate to psychological states and feelings. -- L.W.
Nascent: A term applied to a thing or a state of mind at an early stage of its development when it is as yet scarcely recognizable. See Nascency. The term, as applied by H. Spencer (Psychology, § 195) to psychological states, foreshadowed the later theory of the subconscious. See Subconscious; Latency. -- L.W.
Nastika: (Skr.) "Not orthodox", not acknowledging the authority of the Veda (s.v.) -- K.F.L.
Nativism: Theory that mind has elements of knowledge not derived from sensation. Similar to the common sense theory of T. Reid (1710-1796) and the Scotch School. Introduced as a term by Helmholtz (1821-1894) for the doctrine that there are inherited items in human knowledge which are, therefore, in each and every individual independently of his experience. The doctrine of innate ideas. Opposed to: radical empiricism. See Transcendentalism. -- J.K.F.
Natorp, Paul: (1854-1924) Collaborating with Cohen, Natorp applied the transcendental method to an interpretation of Plato, to psychology and to the methodology of the exact sciences. Like Cohen, Natorp really did not contribute to the scientific development of critical philosophy but prepared the way for philosophical mysticism. Cf. Platos Ideenlehre, 1903; Kant u. d. Marburger Schule, 1915. -- J. K.
Natural: (in Scholasticism) As opposed to supernatural, is that which belongs to (or is due to) a thing according to its nature, as it is natural to man to know; as opposed to voluntary and free, it is that which is done without the command and the advertence of the will, but of nature's own accord, e.g. to sleep, as opposed to chance, it is that which happens through natural causes, as the falling of a stone. Sometimes it is used to refer to a physical body composed of matter and form. -- H.G.
Natural election: The inherent desire of all things for all other things in a certain order. First employed by Francis Bacon (1561-1626) in a passage quoted by A. N. Whitehead (1861-) from the Silva Silvarum "there is a kind of election to embrace that which is agreeable and to exclude or expel that which is ingrate". First erected into a philosophical principle by John Laird (1887-) in The Idea of Value, following a suggestion m Montaigne's Essays. Value, considered as a larger category than human value, an ingredient of the natural world but regarded without its affective content. Syn. with objective value, as independent of the cognitive process. -- J.K.F.
Naturalism: Naturalism, challenging the cogency of the cosmological, teleological, and moral arguments, holds that the universe requires no supernatural cause and government, but is self-existent, self-explanatory, self-operating, and self-directing, that the world-process is not teleological and anthropocentric, but purposeless, deterministic (except for possible tychistic events), and only incidentally productive of man; that human life, physical, mental, moral and spiritual, is an ordinary natural event attributable in all respects to the ordinary operations of nature; and that man's ethical values, compulsions, activities, and restraints can be justified on natural grounds, without recourse to supernatural sanctions, and his highest good pursued and attained under natural conditions, without expectation of a supernatural destiny. -- B.A.G.F.

The general philosophical position which has as its fundamental tenet the proposition that the natural world is the whole of reality. "Nature" and "natural world" are certainly ambiguous terms, but this much is clear in thus restricting reality, naturalism means to assert that there is but one system or level of reality, that this system is the totality of objects and events in space and time; and that the behavior of this system is determined only by its own character and is reducible to a set of causal laws. Nature is thus conceived as self-contained and self-dependent, and from this view spring certain negations that define to a great extent the influence of naturalism. First, it is denied that nature is derived from or dependent upon any transcendent, supernatural entities. From this follows the denial that the order of natural events can be intruded upon. And this in turn entails the denial of freedom, purpose, and transcendent destiny.

Within the context of these views there is evidently allowance for divergent doctrines, but certain general tendencies can be noticed. The metaphysics of naturalism is always monistic and if any teleological element is introduced it is emergent. Man is viewed as coordinate with other parts of nature, and naturalistic psychology emphasizes the physical basis of human behavior; ideas and ideals are largely treated as artifacts, though there is disagreement as to the validity to be assigned them. The axiology of naturalism can seek its values only within the context of human character and experience, and must ground these values on individual self-realization or social utility; though again there is disagreement as to both the content and the final validity of the values there discovered. Naturalistic epistemologies have varied between the extremes of rationalism and positivism, but they consistently limit knowledge to natural events and the relationships holding between them, and so direct inquiry to a description and systematization of what happens in nature. The beneficent task that naturalism recurrently performs is that of recalling attention from a blind absorption in theory to a fresh consideration of the facts and values exhibited in nature and life.

In aesthetics: The general doctrine that the proper study of art is nature. In this broad sense, artistic naturalism is simply the thesis that the artist's sole concern and function should be to observe closely and report clearly the character and behavior of his physical environment. Similarly to philosophical naturalism, aesthetic naturalism derives much of its importance from its denials and from the manner in which it consequently restricts and directs art. The artist should not seek any "hidden" reality or essence; he should not attempt to correct or complete nature by either idealizing or generalizing; he should not impose value judgments upon nature; and he should not concern himself with the selection of "beautiful" subjects that will yield "aesthetic pleasure". He is simply to dissect and describe what he finds around him. Here, it is important to notice explicitly a distinction between naturalism and romanticism (q.v.): romanticism emphasizes the felt quality of things, and the romanticist is primarily interested in the experiences that nature will yield, naturalism emphasizes the objective character of things, and is interested in nature as an independent entity. Thus, romanticism stresses the intervention of the artist upon nature, while naturalism seeks to reduce this to a minimum.

Specifically, naturalism usually refers to the doctrines and practices of the 19th century school of realism which arose as the literary analogue of positivism, and whose great masters were Flaubert, Zola, and de Maupassant. The fundamental dogma of the movement, as expressed by Zola in "Le Roman experimental" and "Les Romanciers naturalistes", states that naturalism is "the scientific mdhod applied to literature". Zola maintains that the task of the artist is to report and explain what happens in nature, art must aim at a literal transcript of reality, and the artist attains this by making an analytic study of character, motives, and behavior. Naturalism argues that all judgments of good and bad are conventional, with no real basis in nature, so art should seek to understand, not to approve or condemn. Human behavior is regarded as largely a function of environment and circumstances, and the novelist should exhibit these in detail, with no false idealizing of character, no glossing over of the ugly, and no appeal to supposed hidden forces. -- I.J.

Naturalistic ethics: Any view according to which ethics is an empirical science, natural or social, ethical notions being reduced to those of of the natural sciences and ethical questions being answered wholly on basis of the findings of those sciences. -- W.K.F.
Naturalistic fallacy, the: The procedure involved in metaphysical and naturalistic systems of ethics, and said by G. E. Moore and his followers to be a fallacy, of deriving ethical conclusions from non-ethical premises or of defining ethical notions in non-ethical terms. See Naturalistic ethics, Metaphysical ethics. -- W.K.F.
Natural law: (in legal philosophy) A "higher law" as opposed to the positive law of a state. The rules of natural law were supposed to be universally valid and therefore natural. They are discoverable by reason alone (rationalism). Natural law theories originated in ancient Greek philosophy. From the Renaissance on they were used as an argument for liberal political doctrines. There is a marked tendency in recent legal philosophy to revive the natural law doctrine. -- W.E.
Nature Philosophers: Name given to pre-Socratic "physiologers" and to Renaissance philosophers who revived the study of physical processes. Early in the 16th century, as a result of the discovery of new lands, the revival of maritime trade, and the Reformation, there appeared in Europe a renewed interest in nature. Rationalism grown around the authorities of the Bible and Aristotle was challenged and the right to investigate phenomena was claimed. Interest in nature was directed at first toward the starry heaven and resulted in important discoveries of Copernicus, Galileo and Kepler. The scientific spirit of observation and research had not yet matured, however, and the philosophers of that time blended their interest in facts with much loose speculation. Among the nature philosophers of that period three deserve to be mentioned specifically, Telesio, Bruno and Carnpanella, all natives of Southern Italy. Despite his assertions that thought should be guided by the observation of the external world, Bernardino Telesio (1508-1588) confined his works to reflections on the nature of things. Particularly significant are two of his doctrines, first, that the universe must be described in terms of matter and force, the latter classified as heat and cold, and second, that mind is akin to matter. Giordano Bruno (1548-1600), a Dominican monk and a victim of the Inquisition, was greatly influenced by the Copernican conception of the universe regarded by him as a harmonious unity of which the earth was but a small and not too important part. The concept of unity was not a condition of human search for truth but a real principle underlying all things and expressing the harmonious order of Divine wisdom. Deity, in his view, was the soul of nature, operating both in the human minds and in the motion of bodies. Consequently, both living beings and material objects must be regarded as animated. Tomaso Campanella (1568-1639), another Dominican monk, was also persecuted for his teachings and spent 27 years in prison. He contended that observations of nature were not dependent on the authority of reason and can be refuted only by other observations. His interests lay largely along the lines previously suggested by Telesio, and much of his thought was devoted to problems of mind, consciousness and knowledge. He believed that all nature was permeated by latent awareness, and may therefore be regarded as an animist or perhaps pantheist. Today, he is best known for his City of the Sun, an account of an imaginary ideal state in which existed neither property nor nobility and in which all affair were administered scientifically. -- R.B.W.
Natural Realism: In epistemology, the doctrine that sensation and perception can be relied upon to give indubitable evidence of the real existence of the external world. Theory that realism is part of the inherent common sense of mankind. First advanced by T. Reid (1710-1796) and held by his followers of the Scotch school. Also known as the comrnon-sense philosophy. See Realism. -- J.K.F.
Natural Selection: This is the corner stone of the evolutionary hypothesis of Charles Darwin. He found great variation in and among types as a result of his extensive biological investigations and accounted for the modifications, not bvysome act of special creation or supernatural intervention, but by the descent, generation after generation, of modified species selected to survive and reproduce the more useful and the more successfully adapted to the environmental struggle for existence. He elaborated a corollary to this general theory in his idea of sexual selection. See Evolutionism, Charles Darwin, Herbert Spencer. -- L.E.D.
Natural Theology: In general, natural theology is a term used to distinguish any theology based upon the fundamental premise of the ability of man to construct his theory of God and of the world out of the framework of his own reason and of reasonable probability from the so-called "revealed theology" which presupposes that God and divine purposes are not open to unaided human understanding but rest upon a supernatural and not wholly understandable basis. See Deism; Renaissance. During the 17th and 18th centuries there were attempts to set up a "natural religion" to which men might easily give their assent and to offset the extravagant claims of the supernaturalists and their harsh charges against doubters. The classical attempt to make out a case for the sweet reasonableness of a divine purpose at work in the world of nature was given by Paley in his Natural Theology (1802). Traditional Catholicism, especially that of the late middle Ages developed a kind of natural theology based upon the metaphysics of Aristotle. Descartes, Spinoza and Leibniz developed a more definite type of natural theology in their several constructions of what now may well be called philosophical theology wherein reason is made the guide. Natural theology has raised its head in recent times in attempts to combat the extravagant declarations of theologians of human pessimism. The term, however, is unfortunate because it is being widely acknowledged that so-called "revealed theology" is natural (recent psychological and social studies) and that natural theology need not deny to reason its possible character as the bearer of an immanent divine revelation. -- V.F.
Nature: A highly ambiguous term, of which the following meanings are distinguished by A. O. Lovejoy:
  1. The objective as opposed to the subjective.
  2. An objective standard for values as opposed to custom, law, convention.
  3. The general cosmic order, usually conceived as divinely ordained, in contrast to human deviations from this.
  4. That which exists apart from and uninfluenced by man, in contrast with art.
  5. The instinctive or spontaneous behavior of man as opposed to the intellective.
Various normative meanings are read into these, with the result that the "natural" is held to be better than the "artificial", the "unnatural", the "conventional" or customary, the intellectual or deliberate, the subjective. -- G.B.

In Aristotle's philosophy: (1) the internal source of change or rest in an object as such, in distinction from art, which is an external source of change. Natural beings are those that have such an internal source of change. Though both matter and form are involved in the changes of a natural being, its nature is ordinarily identified with the form, as the active and intelligible factor. (2) The sum total of all natural beings. See Aristotelianism. -- G.R.M.

Nature "naturing": (Natura naturans, in Scholasticism) God. Nature "natured" (Natura naturata) is the complexus of all created things. Sometimes nature is used for the essence of a thing or for natural causes, and in this sense it is said nature does nothing in vain, for the generation and birth of living beings, for substantial form, and for the effective or passive principle of motion and rest. -- H.G.
Necessary: According to distinctions of modality (q. v.), a proposition is necessary if its truth is certifiable on a priori grounds, or on purely logical grounds. Necessity is thus, as it were a stronger kind of truth, to be distinguished from the contingent truth of a proposition which might have been otherwise. (As thus described, the notion is of course vague, but it may in various ways be given an exact counterpart in one logistic system or another.)

A proposition may also be said to be necessary if it is a consequence of some accepted set of propositions (indicated by the context), even if this accepted set of propositions is not held to be a priori. See Necessity.

That a propositional function F is necessary may mean simply (x)F(x), or it may mean that (x)F(x) is necessary in one of the preceding senses. -- A.C.

Necessary condition: F is a necessary condition of G if G(x) ⊃x F(x). F is a necessary and sufficient condition of G if G(x) ≡ F(x). -- A.C.
Necessitarianism: (Lat. necessitas, necessity) Theory that every event in the universe is determined by logical or causal necessity. The theory excludes both physical indeterminacy (chance) and psychical indeterminacy (freedom). Necessitarianism, as a theory of cosmic necessity, becomes in its special application to the human will, determinism. See Determinism. -- LW.
Necessity: A state of affairs is said to be necessary if it cannot be otherwise than it is. Inasmuch as the grounds of an assertion of this kind may in general be one of three very distinct kinds, it is customary and valuable to distinguish the three types of necessity affirmed as
  1. logical or mathematical necessity,
  2. physical necessity, and
  3. moral necessity.
The distinction between these three was first worked out with precision by Leibniz in his Theodicee.

Logical, physical, and moral necessity are founded in logical, physical, and moral laws respectively. Anything is logically necessary the denial of which would violate a law of logic. Thus in ordinary commutative algebra the implication from the postulates to ab-ba is logically necessary, since its denial would violate a logical law (viz. the commutative rule) of this system.

Similarly, physically necessary things are those whose denial would violate a physical or natural law. The orbits of the planets are said to be physically necessary. Circular orbits for the planets are logically possible, but not physically possible, so long as certain physical laws of motion remain true. Physical necessity is also referred to as "causal" necessity.

As moral laws differ widely from logical and physical laws, the type of necessity which they generate is considerably different from the two types previous defined. Moral necessity is illustrated in the necessity of an obligation. Fulfillment of the obligation is morally necessary in the sense that the failure to fulfill it would violate a moral law, where this law is regarded as embodying some recognized value. If it is admitted that values are relative to individuals and societies, then the laws embodying these values will be similarly relative, and likewise the type of thing which these laws will render morally necessary.

While these three types of necessity are generally recognized by philosophers, the weighting of the distinctions is a matter of considerable divergence of view. Those who hold that the distinctions are all radical, sharply distinguish between logical statements, statements of fact, and so-called ethical or value statements. On the other hand, the attempt to establish an a priori ethics may be regarded as an attempt to reduce moral necessity to logical necessity; while the attempt to derive ethical evaluations from the statements of science, e.g. from biology, is an attempt to reduce moral necessity to physical or causal necessity. -- F.L.W.

Negation: The act of denying a proposition as contrasted with the act of affirming it. The affirmation of a proposition p, justifies the negation of its contradictory, p', and the negation of p justifies the affirmation of p'. Contrariwise the affirmation of p' justifies the negation of p and the negation of p' justifies the affirmation of p. -- C.A.B.

The negation of a proposition p is the proposition ∼p (see Logic, formal, § 1). The negation of a monadic propositional function F is the monadic propositional function λ[∼F(x)]; similarly for dyadic propositional functions, etc.

Or the word negation may be used in a syntactical sense, so that the negation of a sentence (formula) A is the sentence ∼A. -- A. C.

Negative proposition: See affirmative proposition.
Negative Sensation: Term used by Wundt to designate sensations produced by stimuli below the threshold of positive sensation. See Limits of Sensation. The term has largely been discarded because the existence of such sensations is now generally denied. -- L.W.
Nei sheng: Often used as referring to the man who attained to complete self-cultivation, sage-hood. (Confucius.) -- H.H.
Nei tan: Internal alchemy, as a means of nourishing life, attaining Tao and immortality, including an elaborate system of breathing technique, diet, and the art of preserving unity of thought (tsun i, tsun hsiang, tsun ssu). Also called t'ai hsi. For external alchemy, see Wai tan. (Taoist religion.) -- W.T.C.
Neo-Confucianism: See Li hsueh and Chinese philosophy.
Neo-Criticism: The designation of his philosophy used by Cournot, and in the early stage of his thought by Renouvier, who later changed to Personalism as the more fitting title. See also Monadology, The New. -- R.T.F.
Neo-Hegelianism: The name given to the revival of the Hegelian philosophy which began in Scotland and England about the middle of the nineteenth century and a little later extended to America. Outstanding representatives of the movement in England and Scotland are J. H. Stirling, John and Edward Caird, T. H. Green (perhaps more under the influence of Kant), F. H. Bradley, B. Bosanquet, R. B. Haldane, J. E. McTaggart and, in America, W. T. Harris and Josiah Royce. Throughout, the representatives remained indifferent to the formal aspects of Hegel's dialectic and subscribed only to its spirit -- what Hegel himself described as "the power of negation" and what Bosanquet named the argumentum a contingentia mundi. -- G.W.C.
Neo-Idealism: Primarily a name given unofficially to the Italian school of neo-Hegelianism headed by Benedetto Croce and Giovanni Gentile, founded on a basic distinction that it proposes between two kinds of "concrete universals" (s.v.). In addition to the Hegelian concrete universal, conceived as a dialectical synthesis of two abstract opposltes, is posited a second type in which the component elements are "concretes" rather than dialectical abstracts, i.e. possess relative mutual independence and lack the characteristic of logical opposition. The living forms of Mind, both theoretical and practical, are universal in this latter sense. This implies that fine art, utility, and ethics do not comprise a dialectical series with philosophy at their head, i.e. they are not inferior forms of metaphysics. Thus neo-Idealism rejects Hegel's panlogism. It also repudiates his doctrine of the relative independence of Nature, the timeless transcendence of the Absolute with respect to the historical process, and the view that at any point of history a logically final embodiment of the Absolute Idea is achieved. -- W.L.
Neo-intuitionism: See Intuitionism.
Neo-Kantianism: A group of Kantian followers who regard the thing-in-itself or noumenal world as a limiting concept rather than, as did Kant, an existent, though unknowable realm. Reality is for the Neo-Kantians a construct of mind, not another realm. Even Kant's noumenal world is a construct of mind. The phenomenal world is the real and it is the realm of ideas. Hence Neo-Kantianism is a form of idealism. Hermann Cohen, a Neo-Kantian, spoke of the world as the creative act of thought. This idealism is sometimes termed "positivistic." -- V.F.
Neology: Literally, the introduction of new words or new meanings. In theology the neologist is the heretic who introduces a new doctrine. In the latter sense, the rationalist was called a neologist by the traditional theologian. -- V.F.
Neo-Mohism: Sec Mo che and Chinese philosophy.
Neo-Platonism: New Platonism, i.e. a school of philosophy established perhaps by Ammonius Saccus in the second century A.D., in Alexandria, ending as a formal school with Proclus in the fifth century. See Plotinism. -- V.J.B.
Neo-Pythagoreanism: A school of thought initiated in Alexandria, according to Cicero, by Nigidius Figulus, a Roman philosopher who died in 45 B.C. It was compounded of traditional Pythagorean teachings, various Platonic, Aristotelian and Stoic doctrines, including some mystical and theosophical elements. -- J.J.R.
Neo-Scholasticism: See Scholasticism.
Nescience: (Lat. nesciens, ignorant) A state of ignorance such as is professed by the agnostic. See Agnosticism. -- L.W.
Nestorians: A Christian sect dating from the 5th century. Nestorius, a patriarch of Constantinople 428-431) opposed the designation "Mother of God" (a declaration of Origen's) applied to Mary, the mother of Jesus. He said that Christ had two distinct natures and that Mary, a human being, could not have delivered anyone but a human. The emphasis is upon the genuine human nature and the exemplary value of Christ. Nestorianism was not only a Christo-logical viewpoint and the only cause for much theological dispute; it was also a part of a political and ecclesiastical feud between bishops east and west. The council of Ephesus in 431 declared the view heretical. Nevertheless the Nestorian churches spread widely and continues until our present time in Asiatic Turkey and Persia. -- V.F.
Neti, neti: (Skr.) "Not this, not that", famous passage in the Brhadaranyaka Upanishad 2.3.6 et al. loc., giving answer to questions as to the nature of brahman (q. v.), thus hinting its indefinability. -- K.F.L.
Neutralism: A type of monism which holds that reality is neither mind nor matter but a single kind of stuff of which mind and matter are but appearances of aspects. Spinoza is the classical representative. -- H.H.
Neutral Monism: Theory of American New Realism, derived from W. James essay "Does Consciousness Exist?", Journal of Philosophy, 1904, which reduces the mental as well as physical to relations among neutral entities (i.e. entities which are in themselves neither mental nor physical). The theory is qualitatively monistic in its admission of only one kind of ultimate reality viz. neutral or subsistent entities but is numerically pluralistic in acknowledging a multiplicity of independent reals. -- L.W,
New Academy: Name commonly given to what is also called the Third Academy, started by Carneades (214-129 B.C.) who substituted a theory of probability for the principle of doubt which had been introduced into Plato's School by Arcesilaus, the originator of the Second or Middle Academy. The Academy later veered toward eclecticism and eventually was merged with Neo-Platonism. -- J.J.R.
New Realism: A school of thought which dates from the beginning of the twentieth century. It began as a movement of reaction against the wide influence of idealistic metaphysics. Whereas the idealists reduce everything to mind, this school reduced mind to everything. For the New Realists Nature is basic and mind is part and parcel of it. How nature was conceived (whether materialistic, neutralistic, etc.) was not the important factor. New Realists differed here among themselves. Their theory of knowledge was strictly monistic, the subject and object are one since there is no fundamental dualism. Two schools of New Realists are recognized:

(a) English New Realists: Less radical in that mind was given a status of its own character although a part of its objective environment. Among distinguished representatives were: G. E. Moore, Bertrand Russell, S. Alexander, T. P. Nunn, A. Wolf, G. F. Stout,

(b) American New Realists: More radical in that mind tended to lose its special status in the order of things. In psychology this school moved toward behaviorism. In philosophy they were extreme pan-objectivists. Distinguished representatives: F. J. E. Woodbridge, G. S. Fullerton, E. B. McGilvary and six platformists (so-called because of their collaboration in a volume The New Realism, published 1912): E. B. Holt, W. T. Marvin, W. P. Montague, R. B. Perry, W. B. Pitkin, E. G. Spaulding. The American New Realists agreed on a general platform but differed greatly among themselves as to theories of reality and particular questions. -- V.F.

Newton's Method: The method of procedure in natural philosophy as formulated by Sir Isaac Newton, especially in his Rules of Reasoning in Philosophy (Mathematical Principles of Natural Philosophy, Book III). These rules are as follows:
  1. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
  2. Therefore to the same natural effects we must, as far as possible, assign the same causes.
  3. The qualities of bodies, which admit neither intension nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.
  4. In experimental philosophy we are to look upon propositions collected by general induction from phaenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phaenomena occur, by which they may either be made more accurate, or liable to exceptions.
To this passage should be appended another statement from the closing pages of the same work. "I do not make hypotheses; for whatever is not deduced from the phaenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy." -- A.C.S.
Nicht-Ich: (Ger. non-ego) Anything which is not the subjective self. Fichte accounted for the not-self in terms of the ontologically posited subjective self. The not-self is the external, outer world opposed to the ego. -- H.H.
Nicolai, Friedrich: (1733-1811) Was one of the followers of Leibniz-Wolffian school which developed an eclectic reconciliation of rationalism and empiricism in a popular form that served to lay a foundation for the Kantian critical philosophy. -- L.E.D
Nicomachus: Of Gerasa in Arabia, a Neo-Py-thagorean (q.v.) philosopher of the second century. -- M.F.
Nidra: (Skr.) Sleep. In Indian philosophy, particularly the Yoga (s.v.), not considered void of mental activity. -- K.F.L.
Nietzsche, Friedrich: (1844-1900) Nietzsche's discovery and description of "resentment", to mention only one of his major achievements, stamps him as one of the philosophical psychologists of the last century. His critique of the antiquated and false values of the educated middle class led pre-war generations to the pursuit of anethics of more realistic ideals. See Superman.

He was the first to recognize a fundamental critical difference between the philosopher and the scientist. He found those genuine ideals in the pre-Socratic period of Greek culture which he regarded as essential standards for the deepening of individuality and real culture in the deepest sense, towards which the special and natural sciences, and professional or academic philosophers failed to contribute. Nietzsche wanted the philosopher to be prophetic, originally forward-looking in the clarification of the problem of existence. Based on a comprehensive critique of the history of Western civilization, that the highest values in religion, morals and philosophy have begun to lose their power, his philosophy gradually assumed the will to power, self-aggrandizement, as the all-embracing principle in inorganic and organic nature, in the development of the mind, in the individual and in society. More interested in developing a philosophy of life than a system of academic philosophy, his view is that only that life is worth living which develops the strength and integrity to withstand the unavoidable sufferings and misfortunes of existence without flying into an imaginary world.

His major works are: Thus Spake Zarathustra; Beyond Good and Evil, and Genealogy of Morals. -- H.H.

Nihil est in intellectu quod non prius fuerit in sensu: (Lat.) Nothing is in the intellect which was not first in sense. All the materials, or content, of higher, intellectual cognition are derived from the activity of lower, sense cognition. A principle subscribed to by Aristotle, St. Thomas and Locke; opposed by Plato, St. Augustine and Leibniz (who qualified the proposition by adding: nisi intellectus ipse, i.e. except for what is already present as part of the innate nature of the intellect, thus making it possible for Kant to suggest that certain forms of sensibility and reason are prior to sense experience). -- V.J.B.
Nihil ex nihilo: (Lat.) Nothing comes from nothing; a negative statement of the principle of sufficient reason. -- V.J.B.
Nihilism: The doctrine that nothing, or nothing of a specified and very general class, exists, or is knowable, or is valuable. Thus Gorgias held that
  1. Nothing exists;
  2. Even if something did exist it could not be known;
  3. Even if it were known this knowledge could not be communicated.
Schopenhauer's pessimism and denial of the Will expresses a nihilistic attitude toward the so-called values of the world. As a social doctrine Nihilism is the belief that progress is possible only through the destruction of all social and political organizations. See Anarchism. -- C.A.B.
Nihilism, ethical: The denial of the validity of all distinctions of moral value. As this position involves in effect the denial of possibility of all ethical philosophy, it has seldom been taken by philosophers. In the history of thought, however, a less pure ethical nihilism sometimes appears as an intermediate stage in a philosophy which wishes to deny the validity of all previous systems of value as a preliminary to substituting a new one in their places. -- F.L.W.
Nimbarka: An Indian thinker and theologian of the 12th century A.D., of Vedantic (q.v.), Vishnuite persuasion, who assumed the world and the human soul to be essentially and eternally different from Vishnu, yet constituting a certain unity with him because of: complete dependence. -- K.F.L.
Nirguna: (Skr.) "Devoid of qualities" (cf. guna), predicated as early as the Upanishads (q.v.) of the Absolute as its in-it-self aspect (cf. saguna). The highest reality is conceived to be of such fulness, such transcendence that it has no part in the manifold of the phenomenal which is mere maya (q.v.) in Sankara's (q.v.) philosophy in so far as it is esoteric. -- K.F.L.
Nirvana: (Skr. blown out) The complete extinction of individuality, without loss of consciousness, in the beatific rejoining of the liberated with the metaphysical world-ground. A term used principally by Buddhists though denoting a state the attainment of which has been counselled from the Upanishads (q.v.) on as the summum bonum. It is invariably defined as a condition in which all pain, suffering, mental anguish and, above all, samsara (q.v.) have ceased. It is doubtful that complete extinction of life and consciousness or absolute annihilation is meant. -- K.F.L.
Nisus: The creative principle of emergent evolution. See Emergent Evolution. -- R.B.W.
Nitya-vada: (Skr.) The Vedantic (q.v.) theory (vada) which asserts that reality is eternal (nitya), change being unreal. -- K.F.L.
Niyama: (Skr.) The imposing on oneself of good and kind habits, including bathing, eating clean food, steeling the body, contentedness, cheerfulness, study, and piety. -- K.F.L.
Noema: (Ger. Noema) In Husserl: The objective sense of a noesis, together with the character of the sense as posited in a certain manner, as given or emptily intended in a certain manner, etc. For every dimension of the noesis there is a corresponding dimension of the noema. See note under noesis. -- D.C.
Noematic: (Ger. noematisch) In Husserl: Of or pertaining to noema. Noematic sense, see Sense. -- D.C.
Noesis: (Gr. Noesis) In Husserl: 1. That current in the stream of consciousness which is intrinsically intentional in that it points to an object as beyond itself. The noesis animates the intrinsically non -intentional hyletic current in the stream. (See Hyle). 2. A particular instance of the ego cogito. Note: In Husserl's usage, noesis and noema are very rarely restricted to the sphere of "thinking" or "intellect" (however defined) but are rather extended to all kinds of consciousness. -- D.C.

In Greek philosophers The exercise of nous, or reason, the activity of intellectual apprehension and intuitive thought. See Nous; Aristotelianism. -- G.RM.

Noetic: Ihe character some entities have due to their resulting from the activity of nous or reason. Thus those concepts which are non-sensuous and non-empirical but are conceived by reason alone are noetic, the noetic aspects of reality are those which are knowable by reason. In a more general sense, "noetic" is equivalent to "cognitive". -- C.A.B.

(Ger. noetisch) In Husserl: Of or pertaining to noesis. See note under noesis. -- D.C.

Nolition: (Lat. nolo, I am unwilling) The state or act of negative volition. -- V.J.B.
Nominal: Hnving to do with names, nouns, words, or symbols rather than with that which would ordinarily be regarded as symbolized by these verbal forms. See Nominalism. -- C.A.B.
Nominalism: (Lat. nominalis, belonging to a name) In scholastic philosophy, the theory that abstract or general terms, or universals, represent no objective real existents, but are mere words or names, mere vocal utterances, "flatus vocis". Reality is admitted only to actual physical particulars. Universals exist only post res. Opposite of Realism (q.v.) which maintains that universals exist ante res. First suggested by Boethius in his 6th century Latin translation of the Introduction to the Categories (of Aristotle) by Porphyry (A.D. 233-304). Porphyry had raised the question of how Aristotle was to be interpreted on this score, and had decided the question in favor of what was later called nominalism. The doctrine did not receive any prominence until applied to the Sacrament of the Eucharist by Berengar in the 11th century. Berengar was the first scholastic to insist upon the evidence of his senses when examining the nature of the Eucharist. Shortly after, Roscellinus, who had broadened the doctrine to the denial of the reality of all universals and the assertion of the sole reality of physical particulars, was forced by the Council of Soissons to recant. Thereafter, despite Abelard's unsuccessful attempt to reconcile the doctrine with realism by finding a half-way position between the two, nominalism was not again explicitly held until William of Occam (1280-1349) revived it and attempted to defend it within the limits allowed by Church dogma. In the first frankly nominalistic system Occam distinguished between the real and the grammatical meanings of terms or universal. He assigned a real status to universals in the mind, and thus was the first to see that nominalism can have a subjective as well as an objective aspect. He maintained that to our intellects, however, everything real must be some particular individual thing. After Occam, nominalism as an explicitly held doctrine disappeared until recently, when it has been restated in certain branches of Logical Positivism. -- J.K.F.
  1. Non-existence or the non-existent; absence or piivation of existence or the existent;
  2. absence of determinateness or what is thus indeterminate;
  3. unreality of the unreal -- either lack of any reality or what is so lacking (absence, negation, or privation of reality), or lack of a particular kind of reality or what is so lacking; otherness or existents of another order of reality than a specified type; failure to fulfill the defining criteria of some category, or what so fails;
  4. a category encompassing any of the above.
Confusion of non-existence and unreality renders paradoxical the question whether non-being is. -- M.T.K.
Non causa pro causa, or false cause, is the fallacy, incident to the method of proof by reductio ad absurdum (q. v.), when a contradiction has been deduced from a number of assumptions, of inferring the negation of one of the assumptions, say M, where actually it is one or more of the other assumptions which are false and the contradiction could have been deduced without use of M. This fallacy was committed, e.g., by Burali-Forti in his paper of 1897 (see Paradoxes, logical) when he inferred the existence of ordinal numbers a, b such that a is neither less than, equal to, nor greater than b, upon having deduced what is now known as Burali-Forti's paradox from the contrary assumption he had used without question the assumption that there is a class of all ordinal numbers. -- A.C.
Non-centre theory: Ascribes the unity of mind to the fact a number of contemporary mental events are directly interrelated in certain characteristic ways. (Broad). -- H .H.
Non-contradiction, law of: Same as Contradiction, law of (q.v.). -- A.C.
Non-ego: The outer world that has no independent self-existence. (Fichte). -- H.H.
Non-Euclidean geometry: Euclid's postulates for geometry included one, the parallel postulate, which was regarded from earliest times (perhaps even by Euclid himself) as less satisfactory than the others. This may be stated as follows (not Euclid's original form but an equivalent one) Through a given point P not on a given line l there passes at most one line, in the plane of P and l, which does not intersect l. Here "line" means a straight line extended infinitely in both directions (not a line segment).

Attempts to prove the parallel postulate from the other postulates of Euclidean geometry were unsuccessful. The undertaking of Saccheri (1733) to make a proof by reductio ad absurdum of the parallel postulate by deducing consequences of its negation did, however, lead to his developing many of the theorems of what is now known as hyperbolic geometry. The proposal that this hyperbolic geometry, in which Euclid's parallel postulate is replaced by its negation, is a system equally valid with the Euclidean originated with Bolyai and Lobachevsky (independently, c 1825). Proof of the self-consistency of hyperbolic geometry, and thus of the impossibility of Saccheri's undertaking, is contained in results of Cayley (1859) and was made explicit by Klein in 1871; for the two-dimensional case another proof was given by Beltrami in 1868.

The name non-Euclidean geometry is applied to hyperbolic geometry and generally to any system in which one or more postulates of Euclidean geometry are replaced by contrary assumptions. (But geometries of more than three dimensions, if they otherwise follow the postulates of Euclid, are not ordinarily called non-Euclidean.)

Closely related to the hyperbolic geometry is the elliptic geometry, which was introduced by Klein on the basis of ideas of Riemann. In this geometry lines are of finite total length and closed, and every two coplanar lines intersect in a unique point.

Still other non-Euclidean geometries are given an actual application to physical space -- or rather, space-time -- in the General Theory of Relativity.

Contemporary ideas concerning the abstract nature of mathematics (q. v.) and the status of applied geometry have important historical roots in the discovery of non-Euclidean geometries. -- A.C.

G. Saccheri,
Euclides Vindicatus, translated into English by G. B Halsted, Chicago and London, 1920.
H. P. Manning,
Non-Euclidean Geometry, 1901.
J. L. Coolidge,
The Elements of Non-Euclidean Geometry, Oxford. 1909.

Non-Naturalistic ethics: Any ethical theory which holds that ethical properties or relations are non-natural. See Non-natural properties, Intuitionism. -- W.K.F.
Non-Natural Properties: A notion which plays an important part in recent intuitionistic ethics. A non-natural property is one which is neither natural, as yellow and pleasantness are, nor metaphysical, as absoluteness and being commanded by God are. It is, then, a property which is apprehended, not by sensation or by introspection, but in some other way, and which is somehow non-descriptive, non-expository, or non-existential. It is also said sometimes, e.g. by G. E. Moore and W. D. Ross, to be a consequential property, i.e. a property which a thing has in virtue of its having another property, as when an experience is good in virtue of being pleasant. See Intuitionism. -- W.K.F.
Non sequitur is any fallacy which has not even the deceptive appearance of valid reasoning, or in which there is a complete lack of connection between the premisses advanced and the conclusion drawn. By some, however, non sequitur is identified with Aristotle's fallacy of the consequent, which includes the two fallacies of denial of the antecedent (q. v.) and affirmation of the consequent (q. v.). -- A.C.
Noology: (Gr. nous, Mind; logos, Science) A term variously used, but without common acceptance, for the science of mind or of its noetic function. According to several 17th century German writers (Colovius, Mejerus, Wagnerus, Zeidlerus) it is the science of the first principles of knowledge. Crusius identified it with psychology. According to Kant it is the rationalistic theory of innate ideas. For Bentham "noological" is a synonym of logical. Noology is the field of mental science in which the will does not function in the production of mental events, that branch of psychology concerned with the field of purely mental change. For Hamilton it is the science of the noetic, i.e. the function and content of intellectual intuition or pure reason. Eucken distinguished noological method from the psychological and cosmological. Its object is the Spiritual Life, i.e. the source of Reality, and the self-contained goal in which man participates. For H. Gomperz it is the science that mediates between logic and psychology. -- W.L.
Norm: (Lat. norma, rule)
  1. General: Standard for measure. Pattern. Type.
  2. In ethics: Standard for proper conduct. Rule for right action.
  3. In axiology: Standard for judging value or evaluation.
  4. In aesthetics: Standard for judging beauty or art. Basis for criticism,
  5. In logic: Rule for valid inference.
  6. In psychology: Class average test score.
-- A.J.B.
Normative: (Lat. normatus, pp. of normo, square) Constituting a standard; regulative. Having to do with an established ideal. In scientific method: concerning those sciences which have subject-matters containing values, and which set up norms or rules of conduct, such as ethics, aesthetics, politics. The ideal formulation of any science. Opposite of empirical. -- J.K.F.
Nota notae est nota rei ipsius: (Lat) That which falls within the comprehension of a "note", i.e. a known component of a thing, also falls within the comprehension of the thing, an attempted formulation of the supreme principle of syllogistic reasoning on the basis of comprehension rather than extension; Kant is said to have offered this principle in place of the famous extensivist rule, the dictum de omni et nullo (q.v.). -- V.J.B.
Notations, logical: There follows a list of some of the logical symbols and notations found in contemporary usage. In each case the notation employed in articles in this dictionary is given first, afterwards alternative notations, if any.

PROPOSITIONAL CALCULUS (see Logic, formal, § 1, and strict implication)

pq, the conjunction of p and q, "p and q." Instead of simple juxtaposition of the propositional symbols, a dot is sometimes written between, as p·q. Or the common abbreviation for and may be employed as a logical symbol, p & q. Or an inverted letter ∨, usually from a gothic font, may be used. In the Lukasiewicz notation for the propositional calculus, which avoids necessity for parentheses, the conjunction of p and q id Kpq.

p ∨ q, the inclusive disjunction of p and q, "p or q." Frequently the letter ∨ is from a gothic font. In the Lukasiewicz notation, Apq is employed.

∼p, the negation of p, "not p." Instead of ∼, a dash − may be used, written either before the propositional symbol or above it. Heyting adds a short downward stroke at the right end of the dash (a notation which has come to be associated particularly with the intuitionistic propositional calculus and the intuitionistic concept of negation). Also employed is an accent after the propositional symbol (but this more usual as a notation for the complement of a class). In the Lukasiewicz notation, the negation of p is Np.

p ⊃ q, the material implication of q by p, "if p then q." Also employed is a horizontal arrow, p → q. The Lukasiewicz notation is Cpq.

p ≡ q, the material equivalence of p and q, "p if and only if q." Another notation which has sometimes been employed is p ⊃⊂ q. Other notations are a double horizontal arrow, with point at both ends, and two horizontal arrows, one above the other, one pointing forward and the other back. The Lukasiewicz notation is Epq.

p + q, the exclusive disjunction of p and q, "p or q but not both." Also sometimes used is the sign of material equivalence ≡ with a vertical or slanting line across it (non-equivalence). In connection with the Lukasiewicz notation, Rpq has been employed.

p|q, the alternative denial of p and q, "not both p and q." -- For the dual connective, joint denial ("neither p nor q"), a downward arrow has been used.

p 3 q, the strict implication of q by p, "p strictly implies q."

p = q, the strict equivalence of p and q, "p strictly implies q and q strictly implies p." Some recent writers employ, for strict equivalence, instead of Lewis's =, a sign similar to the sign of material equivalence, ≡, but with four lines instead of three.

Mp, "p is possible." This is Lukasiewicz's notation and has been used especially in connection with his three-valued propositional calculus. For the different notion of possibility which is appropriate to the calculus of strict implication, Lewis employs a diamond.

CLASSES (see class, and logic, formal, §§ 7, 9):

x∈a, "x is a member of the class a," or, "x is an a." For the negation of this, sometimes a vertical line across the letter epsilon is employed, or a ∼ above it.

a ⊂ b, the inclusion of the class a in the class b, "a is a subclass of b." This notation is usually employed in such a way that a ⊂ b does not exclude the possibility that a = b. Sometimes, however, the usage is that a ⊂ b ("a is a proper subclass of b") does exclude that a = b; and in that case another notation is used when it is not meant that a = b is excluded, the sign = being either surcharged upon the sign ⊂ or written below it (or a single horizontal line below the ⊂ may take the place of =).

∃!a, "the class a is not empty [has at least one member]," or, "a's exist."

ιx, or ι‘x, or [x] -- the unit class of x, i.e., the class whose single member is x.

V, the universal class. Where the algebra of classes is treated in isolation, the digit 1 is often used for the universal class.

Λ, the null or empty class. Where the algebra of classes is treated in isolation, the digit 0 is often used.

−a, the complement of a, or class of non-members of the class a. An alternative notation is a a'.

a ∪ b, the logical sum, or union, of the classes a and b. Alternative notation, a + b.

a ∩ b, the logical product, or intersection, or common part, of the classes a and b. Alternative notation, ab.

RELATIONS (see Relation, and Logic, formal, § 8; (where a notation used in connection with relations is here given as identical with a corresponding notation for classes, the relational notation will also often be found with a dot added to distinguish it from the one for classes):

xRy, "x has [or stands in, or bears] the relation R to y."

R ⊂ S, "the relation R is contained in [implies] the relation S."

&esixt;!R, "the relation R is not null [holds in at least one instance]."

A downward arrow placed between (e.g.) x and y denotes the relation which holds between x and y (in that order) and in no other case.

V, the universal relation. Schröder uses 1.

Λ, the null relation. Schröder uses 0.

−R, the contrary, or negation, of the relation R. The dash may also be placed over the letter R (or other symbol denoting a relation) instead of before it.

R ∪ S, the logical sum of the relations R and S, "R or S." Schröder uses R+S.

R ∩ S, the logical product of the relations R and S, "R and S." Schröder uses R·S.

I, the relation of identity -- so that xIy is the same as x = y. Schröder uses 1'.

J, the relation of diversity -- so that xJy is the same as x ≠ y. Schröder uses 0'.

A breve is placed over the symbol for a relation to denote the converse relation. An alternative notation for the converse of R is Cnv‘R.

R + S, the relative sum of R and S. Schröder adds a leftward hook at the bottom of the vertical line in the sign +.

R|S, the relative product of R and S. Schröder uses a semicolon to symbolize the relative product, but the vertical bar, or sometimes a slanted bar, is now the usual notation.

R2, the square of the relation R, i.e., R|R.

Similarly for higher powers of a relation, as R3, etc.

R‘y, the (unique) x such that xRy, "the R of y." Frequently the inverted comma is of a bold square (bold gothic) style.

R‘‘b, the class of x's which bear the relation R to at least one member of the class b, "the R's of the b's." Then R‘‘ιy, or R‘‘ι‘y, is the class of x's such that xRy, "the R's of y."

A forward pointing arrow is placed over (e. g.) R to denote the relation of R‘&slquo;ιy to y. Similarly a backward pointing arrow placed over R denotes the relation of the class of y's such that xRy to x.

An upward arrow placed between (e.g.) a and b denotes the relation which holds between x and yŽ if and only if x∈a and y∈b.

The left half of an upward arrow placed between (e.g.) a and R denotes the relation which holds between x and y if and only if x∈a and xRy, in other words, the relation R with its domain limited to the class a.

The right half of an upward arrow placed between (e.g.) R and b denotes the relation which holds between x and y if and only if xRy and y∈b; in other words the relation R with its converse domain limited to b.

The right half of a double -- upward and downward -- arrow placed between (e.g.) R and a denotes the relation which holds between x and y if and only if xRy and both x and y are members of the class a; in other words, the relation R with its field limited to a.

D‘R, the domain of R.

(|‘R, the converse domain of R.

C‘R, the field of R.

Rpo, the proper ancestral of R -- i.e., the relation which holds between x and y if and only if x bears the first or some higher power of the relation R to y (where the first power of R is R).

R* the ancestral of R -- i.e., the relation which holds between x and y if and only if x bears the zero or some higher power of the relation R to y (where the zero power of R is taken to be, either I, or I with its field limited to the field of R).

QUANTIFIERS (see Quantifiers, and Logic, formal, §§3, 6)

(x), universal quantification with respect to x -- so that (x)M may be read "for every x, M." An alternative notation occasionally met with, instead of (x), is (∀x), usually with the inverted A from a gothic or other special font. Another notation is composed of a Greek capital pi with the x placed either after it, or before it, or as a subscript. -- Negation of the universal quantifier is sometimes expressed by means of a dash, or horizontal line, over it.

(Ex), existential quantification with respect to x -- so that (Ex)M may be read "there exists an x such that M." The E which forms part of the notation may also be inverted; and, whether inverted or not, the E is frequently taken from a gothic or other special font. An alternative notation employs a Greek capital sigma with x either after it or as a subscript. -- Negation of the existential quantifier is sometimes expressed by means of a dash over it.

x, formal implication with respect to x. See definition in the article logic, formal, § 3.

x, formal equivalence with respect to x. See definition in logic, formal, § 3.

x. See definition in logic, formal, § 3.

xy or ⊃x,y -- formal implication with respect to x and y. Similarly for formal implication with respect to three or more variables.

xy, or ≡x,y -- formal equivalence with respect to x and y. Similarly for formal equivalence with respect to three or more variables.

ABSTRACTION, DESCRIPTIONS (see articles of those titles):

λx, functional abstraction with respect to x -- so that λxM may be read "the (monadic) function whose value for the argument x is M."

x, class abstraction with respect to x -- so that x M may be read "the class of x's such that M." An alternative notation, instead of x , is x3.

xy, relation abstraction with respect to x and y -- so that xyM may be read "the relation which holds between x and y if and only if M."

(i x), description with respect to x -- so that (i x)M may be read "the x such that M."

E! is employed in connection with descriptions to denote existence, so that E!(ix)M may be read "there exists a unique x such that M."


F(x), the result of application of the (monadic, propositional or other) function F to the argument x -- the value of the function F for the argument x -- ftF of x." Sometimes the parentheses are omitted, so that the notation is Fx. -- See the articles function, and propostttonal function.

F(x, y), the result of application of the (dyadic) function F to the arguments x and y. Similarly for larger numbers of arguments.

x = y, the identity or equality of x and y, "x equals y." See logic, formal, §§ 3, 6, 9.

x ≠ y, negation of x = y.

|- is the assertion sign. See assertion, logical.

Dots (frequently printed as bold, or bold square, dots) are used in the punctuation of logical formulas, to avoid or replace parentheses. There are varying conventions for this purpose.

→ is used to express definitions, the definiendum being placed to the left and the definiens to the right. An alternative notation is the sign = (or, in connection with the propositional calculus, ≡) with the letters Df, or df, written above it, or as a subscript, or separately after the definiens.

Quotation marks, usually single quotes, are employed as a means of distinguishing the name of a symbol or formula from the symbol or formula itself (see syntax, logical). A symbol or formula between quotation marks is employed as a name of that particular symbol or formula. E.g., 'p' is a name of the sixteenth letter of the English alphabet in small italic type.

The reader will observe that this use of quotation marks has not been followed in the present article, and in fact that there are frequent inaccuracies from the point of view of strict preservation of the distinction between a symbol and its name. These inaccuracies are of too involved a character to be removed by merely supplying quotation marks at appropriate places. But it is thought that there is no point at which real doubt will arise as to the meaning intended. -- Alonzo Church

Nothing: Literally, not a thing. According to Kant, emptiness of concept, object or intuition. According to Hegel, the immediate, indeterminate notion of being. According to Peirce, that which possesses contrary attributes. -- J.K.F.

In translation into logical notation, the word nothing is usually to be represented by the negation of an existential quantifier. Thus "nothing has the property F" becomes "∼(Ex)F(x)." -- A.C.

Notion: (Ger. Begriff) This is a technical term in the writings of Hegel, and as there used it has a dual reference. On one side, it refers to the essence or nature of the object of thought; on the other side, it refers to the true thought of that essence or nature. These two aspects of the Notion are emphasized at length in the third part of the Logic (The Doctrine of the Notion), where it is dialectically defined as the synthesis of Being and Essence under the form of the Idea (Die Idee). -- G.W.C.
Notiones communes: Cicero's translation of the phrase koinai ennoiai, by which the Stoics designated such notions as good, evil, and the existence of God, which they regarded as common to all men, and as, in some sense, natural (physikai) or implanted (emphytai), though not, perhaps, in the sense of being literally innate. -- W.K.F.
Noumenal World: The real world as opposed to the appearance world. Kant said of the noumenal realm that it cannot be known. -- V.F.
Noumenon: (Gr. noumenon) In Kant: An object or power transcending experience whose existence is theoretically problematic but must be postulated by practical reason. In theoretical terms Kant defined the noumenon positively as "the object of a non-sensuous intuition," negatively as "not an object of the sensuous intuition;" but since he denied the existence of any but sensuous intuitions, the noumenon remained an unknowable "x". In his practical philosophy, however, the postulation of a noumenal realm is necessary in order to explain the possibility of freedom. See Kantianism. -- O.F.K.
Noun: In English and other natural languages, a word serving as a proper or common name (q.v.). -- A.C.
Nous: (Gr. nous) Mind, especially the highest part of mind, viz. reason; the faculty of intellectual (as distinct from sensible) apprehension and of intuitive thought. In its restricted sense nous denotes the faculty of apprehending the first principles of science, the forms, and the eternal intelligible substances, and is thus distinguished from discursive thought. In this sense nous is regarded as the essence of the divine being. In man Aristotle distinguishes between the nous pathetikos, or passive reason, and a higher active reason, called by the commentators nous poietikos, which alone is truly divine and eternal, and which is related to the nous pathetikos as form to matter. See Aristotelianism. -- G.RM.
Null class: See Logic, formal, §7.
Number: The number system of mathematical analysis may be described as follows -- with reference, not to historical, but to one possible logical order.

First are the non-negative integers 0, 1, 2, 3, . . . , for which the operations of addition and multiplication are determined. They are ordered by a relation not greater than -- which we shall denote by R -- so that, e.g., 0R0, 0R3, 2R3, 3R3, 57R218, etc.

These are extended by introducing, for every pair of non-negative integers a, b, with b different from 0, the fraction a/b, subject to the following conditions (which can be shown to be consistent):

  1. a/1 = a;
  2. a/b = c/d if and only if ad = bc;
  3. a/b R c/d if and only if ad R bc;
  4. a/b + c/d = (ad + bc)/bd;
  5. (a/b)(c/d) = ac/bd.
The resulting system is that of the non-negative rational numbers, which are compactly ordered but not continuously ordered (see continuity) by the relation R (as extended).

Then the next step is to introduce, for every non-negative rational number r, a corresponding negative rational number −r, subject to the conditions:

  1. −r = −s if and only if r = s;
  2. −r = s if and only if r = 0 and s = 0;
  3. −rR−s if and only if sRr;
  4. −rRs;
  5. sR−r if and only if r = 0 and s = 0;
  6. −r + s = s + −r = either t, where r + t = s, or −t where s + t =r;
  7. −r + −s = −(r + s);
  8. (−r)s = s(−r) = −(rs);
  9. (−r)(−s) = rs.
Here r, s, t are variables whose range is the non-negative rational numbers. The extended system, comprising both non-negative rational numbers and negative rational numbers is the system of rational numbers -- which are compactly ordered but not continuously ordered by the relation R (as extended).

If we make the minimum extension of the system of rational numbers which will render the order continuous, the system of real numbers results. Addition and multiplication of real numbers are uniquely determined by the meanings already given to addition and multiplication of rational numbers and the requirement that addition of, or multiplication by, a fixed real number (on right or left) shall be a continuous function (see continuity). Subtraction and division may be introduced as inverses of addition and multiplication respectively.

Finally, the complex numbers are introduced as numbers a+bi, where a and b are real numbers. There is no ordering relation, but addition and multiplication are determined as follows:

(a+bi) + (c+di) = (a + c) + (b + d)i.
(a+bi)(c+di) = (ac − bd) + (ad + bc)i.
In particular i (i.e., 0+1i) multiplied by itself is −1. A number of the form a+0i may be identified with the real number a; other complex numbers are called imaginary numbers, and those of the form 0+bi are called pure imaginaries.

(It is, of course, not possible to define i as "the square root of −l." The foregoing statement corresponds to taking i as a new, undefined, symbol. But there is an alternative method, of logical construction, in which the complex numbers are defined as ordered pairs (a, b) of real numbers, and i is then defined as (0, 1).)

In a mathematical development of the real number system or the complex number system, an appropriate set of postulates may be the starting point. Or the non-negative integers may first be introduced (by postulates or otherwise -- see arithmetic, foundations of) and from these the above outlined extensions may be provided for by successive logical constructions, in any one of several alternative ways.

The important matter is not the definition of number (or of particular numbers), which may be made in various ways more or less indifferently, but the internal structure of the number system.

For the notions of cardinal number, relation-number, and ordinal number, see the articles of these titles. -- Alonzo Church

R. Dedekind,
Essays on the Theory of Numbers, translated by W. W. Beman, Chicago, 1901.
E. V. Huntington,
A set of postulates for real algebra, Transactions of the American Mathematical Society, vol. 6 (1905), pp. 17-41.
E. V. Huntington,
A set of postulates for ordinary complex algebra, ibid., pp. 209-229.
E. Landau,
Grundlagen der Analysis, Leipzig, 1930.

Numinous: A word coined from the Latin "numen" by Rudolf Otto to signify the absolutely unique state of mind of the genuinely religious person who feels or is aware of something mysterious, terrible, awe-inspiring, holy and sacred. This feeling or awareness is a mysterium tremendum, beyond reason, beyond the good or the beautiful. This numinous is an a priori category and is the basis of man's cognition of the Divine. See his book The Idea of the Holy (rev. ed., 1925). -- V.F.
Nuñez Regüeiro, Manuel: Born in Uruguay, March 21, 1883. Professor of Philosophy at the National University of the Litoral in Argentine. Author of about twenty-five books, among which the following are the most important from a philosophical point of view:
Fundamentos de la Anterosofia, 1925;
Anterosofia Racional, 1926;
De Nuevo Hablo Jesus, 1928;
Filosofia Integral, 1932;
Del Conocimiento y Progreso de Si Mismo, 1934;
Tratado de Metalogica, o Fundamentos de Una Nueva Metodologia, 1936;
Suma Contra Una Nueva Edad Media, 1938;
Metafisica y Ciencia, 1941;
La Honda Inquietud, 1915;
Conocimiento y Creencia, 1916.

Three fundamental questions and a tenacious effort to answer them run throughout the entire thought of Nuñez Regüeiro, namely the three questions of Kant: What can I know? What must I do? What can I expect? Science as auch does not write finis to anything. We experience in science the same realm of contradictions and inconsistencies which we experience elsewhere. Fundamentally, this chaos is of the nature of dysteleology. At the root of the conflict lies a crisis of values. The problem of doing is above all a problem of valuing. From a point of view of values, life ennobles itself, man lifts himself above the trammels of matter, and the world becomes meaning-full. Is there a possibility for the realization of this ideal? Has this plan ever been tried out? History offers us a living example: The Fact of Jesus. He is the only possible expectation. In him and through him we come to fruition and fulfilment. Nuñez Regüeiro's philosophy is fundamentally religious. -- J.A.F.

Nyaya: (Skr.) One of the great systems of Indian philosophy (q.v.) going back to the Nyaya-sutras of Gotama (q.v.) and dealing with the logical approach to reality in a science of reasoning and epistemology designed to accomplish the practical aims of all Indian speculation. Having established perception (pratyaksa), inference (anumana), comparison (upamana), and testimony (sabdaq as sources of valid knowledge or truth, a doctrine of logical realism is arrived at in which the objective world is conceived independent of thought and mind. -- K.F.L.