Peirce’s Contributions to Baldwin’s Dictionary

Charles Sanders Peirce contributed several dozen definitions to the Dictionary of Philosophy and Psychology, which was edited by James Mark Baldwin and published by the Macmillan Company in 1901. The following list of definitions by Peirce, many of which are short essays, has been extracted from an online version, which was developed by Christopher D. Green at York University, Toronto.

These definitions may be of interest to philosophers, logicians, and historians for several reasons:  they present short summaries of themes that Peirce treated at length in publications; they indicate Peirce's thoughts on other topics that he seldom addressed in his writings; and they show the breadth and depth of Peirce's scholarship. The number of references give some indication:  Aristotle is first with 62 references, and Kant is second with 40; others include Schröder 14, Hamilton 13, Hegel 12, Aquinas 11, Leibniz 11, De Morgan 10, Plato 9, Duns Scotus 8, Ockham 5; those with 4 references each include Petrus Hispanus, Boole, Cantor, and Dedekind; many others have one, two, or three references.

The initials C.S.P. mark contributions by Peirce. Some of the definitions were coauthored with other contributors, two of whom were former students of Peirce's at Johns Hopkins University:  Christine Ladd-Franklin (C.L.F.) and John Dewey (J.D.). Others include R. Adamson (R.A.), James Mark Baldwin (J.M.B.), H. B. Fine (H.B.F.), E. Morselli (E.M.), and Andrew Seth Peingle-Pattison (A.S.P.P.). This list of definitions ends with the letter O because that is the last letter that has been finished in the online version of the dictionary. The list will be extended when the full dictionary becomes available.

Dualism (in philosophy). (1) A general tendency to divide any genus of objects of philosophical thought into two widely separate categories, as saints and sinners, truth and falsehood, &c.; opposed to the tendency to look for gradations intermediate between contraries. Especially (2) any theory which explains the facts of the universe by referring them to the action of two independent and eternally coexistent principles. Cf. PLURALISM. (C.S.P.- A.S.P.P.)

Economy (logical principle of). A principle maintained by E. Mach that general concepts are merely an adaptation for the economy of mental process. That they have that effect was noticed by Locke. (C.S.P.)

Empirical Logic: Ger. empirische Logik; Fr. logique empirique; Ital. logica empirica. The treatment of logic on the basis or from the point of view of a sensationalist or other markedly empiricist theory of knowledge. (R.A.- C.S.P.)

Equipollence or -cy [Lat. aequus, equal, + pollere, to be able]: Ger. Aequipollenz; Fr. équipollence; Ital. equipollenza. The relation between two propositional forms which represent the same fact. It translates the Gr. isodunamwn. (C.S.P.)

Genus (in logic) [Lat. genus, birth]: Ger. Genus; Fr. genre; Ital. genere. A class which contains within its extension, or is divisible into, smaller classes, called relatively species.

The significance of the term has always shared the ambiguity which is discernible in classification. Genera have been distinguished partly by reason of the obvious differences in the larger types of natural forms, partly by reference to the relatively arbitrary process of arranging in accordance with selected marks. The first or empirical factor is predominant in the popular sense of the term, and in much of the Aristotelian and Scholastic logic; the second has been insisted on in the more strictly formal logic. The divergence of the two views makes itself manifest at the limits of classification, at the conception of a summum genus and an infima species, which tend on the one view to be regarded as having a place in rerum natura, while on the other they are but ideal boundaries to an arbitrary process. (R.A.)

One of the Aristotelian rules of DIVISION (q.v.) in logic is that the differences of different genera are different, that is to say, cross-divisions are not to be made. This rule is signally violated in the modern classifications of chemistry, mathematics, and logic itself; but in biology, owing to the common origin of species, the classification is hierarchical, as Aristotle required. Cf. PREDICABLES (C.S.P.)

Given [AS. gifan]: Ger. (das) Gegebene; Fr. (la) donnée; Ital. (il) dato. One of the hypotheses of a problem; used also in the Latin form datum (of which it is a translation). In Greek mathematics, the corresponding word was also extended to whatever is determined in certain specified ways by a given hypothesis. The plural data is loosely applied to any unquestioned knowledge upon which a judgment is based, and in particular to our percepts, in the phrase 'data of experience.'

The English adjective, given, has an exceedingly convenient use to indicate that that which its noun denotes must be understood as specified (in the verification of what is said) previously to the specification of something mentioned before. Thus, 'Some woman is adored by any given man,' is said to avoid all possibility of understanding the statement as 'Some woman is adored by every man.' (C.S.P.)

Imaging (in logic): Ger. Abbildung; Fr. (in mathematics) représentation; Ital. rappresentazione. A term proposed to translate Abbildung in its logical use. In order to apprehend this meaning, it is indispensable to be acquainted with the history of the meanings of Abbildung. This word was used in 1845 by Gauss for what is called in English a map-projection, which is an incorrect term, since many such modes of representation are not geometrical rectilinear projections at all; and of those which Gauss had in view, but a single one is so. In mathematics Abbildung is translated representation; but this word is preempted in logic. Since Bild is always translated image, imaging will answer very well for Abbildung. If a map of the entire globe were made on a sufficiently large scale, and out of doors, the map itself would be shown upon the map; and upon that image would be seen the map of the map; and so on, indefinitely. If the map were to cover the entire globe, it would be an image of nothing but itself, where each point would be imaged by some other point, itself imaged by a third, &c. But a map of the heavens does not show the map itself at all. A Mercator's projection shows the entire globe (except the poles) over and over again in endlessly recurring strips. Many maps, if they were completed, would show two or more different places on the earth at each point of the map (or at any rate on a part of it), like one map drawn upon another. Such is obviously the case with any rectilinear projection of the entire sphere, excepting only the stereographic. These two peculiarities may coexist in the same map.

Any mathematical function of one variable may be regarded as an image of its variable according to some mode of imaging. For the real and imaginary quantities correspond, one to one and continuously, to the assignable points on a sphere. Although mathematics is by far the swiftest of the sciences in its generalizations, it was not until 1879 that Dedekind (in the 3rd edition of his recension of Lejeune-Dirichlet's Zahlentheorie, § 163, p. 470; but the writer has not examined the second edition) extended the conception to discrete systems in these words: 'It very often happens in other sciences, as well as in mathematics, that there is a replacement of every element w of a system of elements or things by a corresponding element w' [of a system W']. Such an act should be called a substitution. . . . But a still more convenient expression is found by regarding W' as the image of W, and w' of w, according to a certain mode of imaging.' And he adds, in a footnote: 'This power of the mind of comparing a thing w with a thing w', or of relating w to w', or of considering w' to correspond to w, is one without which no thought would be possible.' [We do not translate the main clause.] This is an early and significant acknowledgment that the so-called 'logic of relatives' — then deemed beneath the notice of logicians — is an integral part of logic. This remark remained unnoticed until, in 1895, Schröder devoted the crowning chapter of his great work (Exakte Logik, iii. 553-649) to its development. Schröder says that, in the broadest sense, any relative whatever may be considered as an imaging — 'nämlich als eine eventuell bald "undeutige," bald "eindeutige," bald "mehrdeutige" Zuordnung.' He presumably means that the logical universe is thus imaged in itself. However, in a narrower sense, he says, a mode of imaging is restricted to a relative which fulfils one or other of the two conditions of being never undeutig, or being never mehrdeutig. That is, the relation must belong to one or other of the two classes, the one embracing such that every object has an image, and the other such that no object has more than one image. Schröder's definitions (however interesting his developments) break all analogy with the important property of the imaging of continua noticed above. If this is to be regarded as essential, an imaging must be defined as a generic relation between an object-class and an image-class, which generic relation consists of specific relations, in each of which one individual, and no more, of the image-class stands to each individual of the object-class, and in each of which every individual of the image-class stands to one individual, and to no more, of the object-class. This is substantially a return to Dedekind's definition, which makes an imaging a synonym for a substitution. (C.S.P., H.B.F.)

Implicit (in logic). Said of an element or character of a representation, whether verbal or mental, which is not contained in the representation itself, but which appears in the strictly logical (not merely in the psychological) analysis of that representation.

Thus, when we ordinarily think of something, say the Antarctic continent, as real, we do not stop to reflect that every intelligible question about it admits of a true answer; but when we logically analyse the meaning of reality, this result appears in the analysis. Consequently, only concepts, not percepts, can contain any implicit elements, since they alone are capable of logical analysis. An implicit contradiction, or contradiction in adieto, is one which appears as soon as the terms are defined, irrespective of the properties of their objects. Thus there is, strictly speaking, no implicit contradiction in the notion of a quadrilateral triangle, although it is impossible. But, owing to exaggeration, this would currently be said to involve not merely an implicit, but an explicit contradiction, or contradiction in terms.

Any proposition which neither requires the exclusion from nor the inclusion in the universe of any state of facts or kind of object except such as a given second proposition so excludes or requires to be included, is implied in that second proposition in the logical sense of implication, no matter how different it may be in its point of view, or otherwise. It is a part of the meaning of the copula 'is' employed in logical forms of proposition, that it expresses a transitive relation, so that whatever inference from the proposition would be justified by the dictum de omni is implied in the meaning of the proposition. Nor could any rule be admitted as universally valid in formal logic, unless it were a part of the definition of one of the symbols used in formal logic. Accordingly, whatever can be logically deduced from any proposition is implied in it; and conversely. Whether what is implied will, or will not, be suggested by the contemplation of the proposition is a question of psychology. All that concerns logic is, whether all the facts excluded and required by the one proposition are among those so excluded or required by the other. (C.S.P.)

Inconsistency [Lat. in + con + sistere, to stand]: Ger. Unvereinbarkeit; Fr. inconsistance; Ital. incompatibilità. The relation between two assertions which cannot be true at once, though it may not be a direct contradiction; as between a statement of items and a statement of their total. Cf. CONSISTENCY.

A logical discrepancy, on the other hand, is a difference between two statements either difficult or impossible to reconcile with the credibility of both. It is said to be negative if one assertion omits an inseparable part of the fact stated in another; as when one witness testifies that A pointed a pistol at B, and another that A shot at B. It is positive if one asserts what the other denies. But even then it may often be conciliable (verträglich); that is, may not prove that either statement is in other respects untrustworthy. See Bachmann, Logik, §§ 214 ff.

'Inconsistent' is applied to an assertion, or hypothesis, which either in itself, or in copulation with another proposition with which it is said to be inconsistent, might be known to be false by a man devoid of all information except the meanings of the words used and their syntax.

Inconsistent differs from contradictory (see CONTRADICTION) in being restricted usually to propositions, expressed or implied, and also in not implying that the falsity arises from a relation of negation. 'That is John' and 'It is Paul' are inconsistent, but hardly contradictory. Moreover, contradictory is also used in a peculiar sense in formal logic. Cf. OPPOSITION. (C.S.P.)

Independence [Lat. in + de + pendere, to hang]: Ger. Unabhängigkeit; Fr. indépendance; Ital. indipendenza. (1) Two subjects are independent in so far as the possession of any character by the one does not require nor prevent the possession of any character by the other, unless these characters are directly or indirectly relative to the other individual.

(2) Two events are independent if either is equally probable whether the other takes place or not. (C.S.P.)

Index (in exact logic). A sign, or representation, which refers to its object not so much because of any similarity or analogy with it, nor because it is associated with general characters which that object happens to possess, as because it is in dynamical (including spatial) connection both with the individual object, on the one hand, and with the senses or memory of the person for whom it serves as a sign, on the other hand.

No matter of fact can be stated without the use of some sign serving as an index. If A says to B, 'There is a fire,' B will ask, 'Where?' Thereupon A is forced to resort to an index, even if he only means somewhere in the real universe, past and future. Otherwise, he has only said that there is such an idea as fire, which would give no information, since unless it were known already, the word 'fire' would be unintelligible. If A points his finger to the fire, his finger is dynamically connected with the fire, as much as if a self-acting fire-alarm had directly turned it in that direction; while it also forces the eyes of B to turn that way, his attention to be riveted upon it, and his understanding to recognize that his question is answered. If A's reply is, 'Within a thousand yards of here,' the word 'here' is an index; for it has precisely the same force as if he had pointed energetically to the ground between him and B. Moreover, the word 'yard,' though it stands for an object of a general class, is indirectly indexical, since the yard-sticks themselves are signs of the Parliamentary Standard, and that, not because they have similar qualities, for all the pertinent properties of a small bar are, as far as we can perceive, the same as those of a large one, but because each of them has been, actually or virtually, carried to the prototype and subjected to certain dynamical operations, while the associational compulsion calls up in our minds, when we see one of them, various experiences, and brings us to regard them as related to something fixed in length, though we may not have reflected that that standard is a material bar. The above considerations might lead the reader to suppose that indices have exclusive reference to objects of experience, and that there would be no use for them in pure mathematics, dealing, as it does, with ideal creations, without regard to whether they are anywhere realized or not. But the imaginary constructions of the mathematician, and even dreams, so far approximate to reality as to have a certain degree of fixity, in consequence of which they can be recognized and identified as individuals. In short, there is a degenerate form of observation which is directed to the creations of our own minds — using the word observation in its full sense as implying some degree of fixity and quasi-reality in the object to which it endeavours to conform. Accordingly, we find that indices are absolutely indispensable in mathematics; and until the truth was comprehended, all efforts to reduce to rule the logic of triadic and higher relations failed; while as soon as it was once grasped the problem was solved. The ordinary letters of algebra that present no peculiarities are indices. So also are the letters A, B, C, &c., attached to a geometrical figure. Lawyers and others who have to state a complicated affair with precision have recourse to letters to distinguish individuals. Letters so used are merely improved relative pronouns. Thus, while demonstrative and personal pronouns are, as ordinarily used, 'genuine indices,' relative pronouns are 'degenerate indices'; for though they may, accidentally and indirectly, refer to existing things, they directly refer, and need only refer, to the images in the mind which previous words have created.

Indices may be distinguished from other signs, or representations, by three characteristic marks: first, that they have no significant resemblance to their objects; second, that they refer to individuals, single units, single collections of units, or single continua; third, that they direct the attention to their objects by blind compulsion. But it would be difficult, if not impossible, to instance an absolutely pure index, or to find any sign absolutely devoid of the indexical quality. Psychologically, the action of indices depends upon association by contiguity, and not upon association by resemblance or upon intellectual operations. See Peirce, in Proc. Amer. Acad. Arts and Sci., vii. 294 (May 14, 1867). (C.S.P.)

Individual (in logic) [as a technical term of logic, individuum first appears in Boethius, in a translation from Victorinus, no doubt of atomon, a word used by Plato (Sophistes, 229 D) for an indivisible species, and by Aristotle, often in the same sense, but occasionally for an individual. Of course the physical and mathematical sense of the word were earlier. Aristotle's usual term for individuals is ta kaq ekasta, Lat. singularia, Eng. singulars.] Used in logic in two closely connected senses. (1) According to the more formal of these an individual is an object (or term) not only actually determinate in respect to having or wanting each general character and not both having and wanting any, but is necessitated by its mode of being to be so determinate. See PARTICULAR (in logic).

This definition does not prevent two distinct individuals from being precisely similar, since they may be distinguished by their heceeities (or determinations not of a generalizable nature); so that Leibnitz' principle of indiscernibles is not involved in this definition. Although the principles of contradiction and excluded middle may be regarded as together constituting the definition of the relation expressed by 'not,' yet they also imply that whatever exists consists of individuals. This, however, does not seem to be an identical proposition or necessity of thought; for Kant's Law of Specification (Krit. d. reinen Vernunft, 1st ed., 656; 2nd ed., 684; but it is requisite to read the whole section to understand his meaning), which has been widely accepted, treats logical quantity as a continuum in Kant's sense, i.e. that every part of which is composed of parts. Though this law is only regulative, it is supposed to be demanded by reason, and its wide acceptance as so demanded is a strong argument in favour of the conceivability of a world without individuals in the sense of the definition now considered. Besides, since it is not in the nature of concepts adequately to define individuals, it would seem that a world from which they were eliminated would only be the more intelligible. A new discussion of the matter, on a level with modern mathematical thought and with exact logic, is a desideratum. A highly important contribution is contained in Schröder's Logik, iii, Vorles. 10. What Scotus says (Quaest. in Met., VII. 9, xiii and xv) is worth consideration.

(2) Another definition which avoids the above difficulties is that an individual is something which reacts. That is to say, it does react against some things, and is of such a nature that it might react, or have reacted, against my will.

This is the stoical definition of a reality; but since the Stoics were individualistic nominalists, this rather favours the satisfactoriness of the definition than otherwise. It may be objected that it is unintelligible; but in the sense in which this is true, it is a merit, since an individual is unintelligible in that sense. It is a brute fact that the moon exists, and all explanations suppose the existence of that same matter. That existence is unintelligible in the sense in which the definition is so. That is to say, a reaction may be experienced, but it cannot be conceived in its character of a reaction; for that element evaporates from every general idea. According to this definition, that which alone immediately presents itself as an individual is a reaction against the will. But everything whose identity consists in a continuity of reactions will be a single logical individual. Thus any portion of space, so far as it can be regarded as reacting, is for logic a single individual; its spatial extension is no objection. With this definition there is no difficulty about the truth that whatever exists is individual, since existence (not reality) and individuality are essentially the same thing; and whatever fulfils the present definition equally fulfils the former definition by virtue of the principles of contradiction and excluded middle, regarded as mere definitions of the relation expressed by 'not.' As for the principle of indiscernibles, if two individual things are exactly alike in all other respects, they must, according to this definition, differ in their spatial relations, since space is nothing but the intuitional presentation of the conditions of reaction, or of some of them. But there will be no logical hindrance to two things being exactly alike in all other respects; and if they are never so, that is a physical law, not a necessity of logic. This second definition, therefore, seems to be the preferable one. Cf. PARTICULAR (in logic). (C.S.P.)

Inference [Lat. in + ferre, to bear]: Ger. Schliessen, Schluss; Fr. inférence; Ital. illazione (conclusione). (1) In logic: (a) the act of consciously determining the content of a cognition by a previous cognition or cognitions, in a way which seems generally calculated to advance knowledge.

In this sense the word differs from REASONING (q.v.) only in referring strictly to a single step of the process, or to what seems a single step. Unless the act is consciously performed, no logical control can be exercised; and this is sufficient reason for separating such acts from any operations otherwise analogous which may take place in the formation of percepts. To be conscious of determining a cognition by another, and not merely of making the one follow after the other, involves some more or less obscure judgment that the pair of representations, the determining and determined, belong to a class of analogous pairs, so that a general maxim is virtually obeyed in the act. There is, besides, a purpose of learning more of the truth. The representations concerned in inference are, it appears, always judgments (or propositions). Probably, if a pair of percepts were, in the very act of determining the one to accord with the other, looked upon as special cases of a class of pairs of percepts so related to one another that if one were true the other ought to be accepted, they would, ipso facto, become judgments.

(b) A pair (or larger set) of judgments, of which one (or all of them together but one) determines the remaining one, as in (a) above, the whole set being regarded as constituting together a cognition more complete than a judgment.

In this sense, inference is synonymous with argument. The latter word, it is true, only implies that the set of propositions might be thought, being perhaps written down and no longer even accepted by the author, while the former word implies that the movement of thought takes place. Moreover, an inference creates belief in the mind that makes it, while an argument may be a system of propositions put together with a view of creating belief in another mind, or perhaps merely to exhibit the logical relation between different beliefs. But these distinctions often vanish or lose all importance. When the determining judgment is a copulative proposition, its members may either be called the premises, or their compound may be called the PREMISE (q.v.). But when different beliefs are brought together in thought for the first time to form a copulative judgment, the premises must be taken as plural.

Several other logical meanings are in general use as more or less permissible inaccuracies of language. Thus, the determined judgment, or conclusion, may sometimes be conveniently called an 'inference.' The popular use of the word for a dubious illation, as in such a sentence as 'This is a proof positive, while that is only an inference,' is quite inadmissible. (C.S.P.)

Insolubilia [Lat. in + solvere, to loose; trans. of Aristotle's aporia; used mainly in plural]. A class of sophisms in which a question is put of such a nature that, whether it be answered affirmatively or negatively, an argument unimpeachable in form will prove the answer to be false.

The type is this: Given the following proposition:

This assertion is not true: is that assertion, which proclaims its own falsity, and nothing else, true or false? Suppose it true. Then,
Whatever is asserted in it is true, But that it is not true is asserted in it; Therefore, by Barbara, That it is not true is true; Therefore, It is not true. Besides, if it is true, that it is true is true. Hence, That it is not true is not true, But that it is not true is asserted in the proposition; Therefore, By Darapti, Something asserted in the proposition is not true; Therefore, the proposition is not true. On the other hand, suppose it is not true. In that case, That it is not true is true, But all that the proposition asserts is that it is not true; Therefore, By Barbara, All that the proposition asserts is true; Therefore, The proposition is true.

Besides, in this case, Something the proposition asserts is not true, But all that the proposition asserts is that it is not true; Therefore, By Bokardo, That it is not true is not altogether true; Therefore, That it is true is true; Therefore, it is true.

Thus, whether it be true or not, it is both true and not. Now, it must be either true or not, hence it is both true and not, which is absurd.

Only two essentially distinct methods of solution have been proposed. One, which is supported by Ockham (Summa totius logices, 3rd div. of 3rd part, cap. 38 and 45), admits the validity of the argumentation and its consequence, which is that there can be no such proposition, and attempts to show by other arguments that no proposition can assert anything of itself. Many logical writers follow Ockham in the first part of his solution, but fails to see the need of the second part. The other method of solution, supported by Paulus Venetus (Sophismata Aurea, sophisma 50), diametrically denies the principle of the former solution, and undertakes to show that every proposition virtually asserts its own truth. This method, therefore, denies the premise of the antithesis that 'all that the proposition asserts is that it is not true,' since, like every other proposition, it also asserts its own truth, and is therefore contradictory and false, not in what it expressly asserts, but in what it implicitly asserts. Some writers (as Fries) hold that because every proposition asserts its own truth, therefore nothing is a proposition which asserts its own falsity. See Aristotle, Sophisticae Elenchi, cap. 25. Other proposed solutions of little importance are given by Paulus Ventus, loc. cit. (C.S.P.)

Intention (in logic) [Lat. intentio, with the same meaning in Aquinas (Summa Theol., I. 9. 53, is the principal passage); in classical writers an act of attention (and so Aquinas, ibid., I. ii. 9. 38, art. 2, and elsewhere); from in + tendere, to stretch. Aquinas seems sometimes to use the term for a mode of being (ibid., I. ii. 9. 22) and sometimes for a relation (ibid., I. 9. 29 art. 1; 9. 76, art. 3, and esp art. 4)]. A concept, as the result of attention.

First intentions are those concepts which are derived by comparing percepts, such as ordinary concepts of classes, relations, &c. Second intentions are those which are formed by observing and comparing first intentions. Thus the concept 'class' is formed by observing and comparing class-concepts and other objects. The special class-concept, ens, or what is, in the sense of including figments as well as realities, can only have originated in that way. Of relative second intentions, four are prominent — identity, otherness, co-existence, and imcompossibility. Aquinas defined logic as the science of second intentions applied to first. (C.S.P.)

Involution [Lat. in + volvere, to roll]: Ger. Involution; Fr. involution; Ital. involuzione. A term of SYMBOLIC LOGIC (q.v. borrowed from algebra, where it means the raising of a base to a power. In logic it has two different senses. (1) Relative involution: let lwm denote any lover of a well-wisher of a man. That is, any individual A is denoted by lwm, provided there are in existence individuals B and C (who may be identical with each other or with A), such that A loves B, while B wishes well to C, and C is a man. Further, let lwm denote any individual A, if, and only if, there is in existence an individual C, who is a man, and who is such that taking any individual B whatever, if B is a well-wisher of C, then A is a lover B. The operation of combining l and w in this statement is termed 'progressive involution.' Again, let lwm denote any individual A, if, and only if, there is in existence an individual B, who is loved by A, and who is such that taking any individual C whatever, if C is wished well by B, then C is a man. The operation of combining w and m in this statement is termed 'regressive involution.' These designations were adopted because of the analogy of the general formulae to those of involution in the algebra of quantity.

These kinds of involution are not, at present, in use in symbolical logic; but they are, nevertheless, useful, especially in developing the conception of continuity. These two kinds of involution together constitute relative involution.

(2) Non-relative involution: consisting in the repeated introduction of the same premise into a reasoning; as, for example, the half-dozen simple premises upon which the Theory of Numbers is based are introduced over and over again in the reasoning by which its myriad theorems are deduced. In exact logic the regular process of deduction begins by non-relatively multiplying together all the premises to make one conjunctive premise, from which whatever can be deduced by using those premises as often as they are introduced as factors, can be deduced by processes of 'immediate inference' from that single conjunctive premise. But the general character of the conclusion is found to depend greatly upon the number of times the same factor is multiplied in. From this circumstance the importance and the name of non-relative involution arise. (C.S.P.)

Kind [AS. cynd, nature, from cynde, natural; same root as Gr. genoV, Lat. genus]: Ger. Art (the word 'kind' is also used to translate Ger. Gattung, for which see HEGEL'S TERMINOLOGY); Fr. genre; Ital. genere, specie. Before 'class' acquired its logical signification in Queen Anne's reign, kind was sometimes used for any collection of objects having a common and peculiar general character, simple or complex.

Thus, in Blundevile's Arte of Logicke, we read: 'Genus is a general kind which may be spoken of many things differing in special kind.' At other times, and more accurately, it was restricted to the species, or narrowest recognized class, or that which was supposed to be derived from one stock. Thus Wilson's Rule of Reason (1551) has: 'Genus is a general woorde, vnder the whiche diuerse kindes or sortes of thinges are comprehended.'

But before persons who picked their words had become ready to use 'class' as a mere logical extension, they had begun to avoid 'kind,' except when the emphasis of attention was placed upon the logical depth rather than the breadth. Watts's Logick (1724) illustrates this. This last is the ordinary popular sense of the word to-day; so that 'of this kind,' 'of this nature,' 'of this character' are interchangeable phrases. J. S. Mill, however, in his System of Logic, Bk. I. chap. vii. § 4, erected the word into a technical term of logic, at the same time introducing the term 'real kind.' His meaning, so far as it was determinate, was that classes are of two orders, the first comprising those which, over and above the characters which are involved in their definitions and which serve to delimit their extension, have, at most, but a limited number of others, and those following as 'consequences, under laws of nature,' of the defining characters; and the second, the real kinds, comprising those each of which has innumerable common properties independent of one another. As instances of real kinds, he mentions the class of animals and the class of sulphur; as an instance of a kind not real, the class of white things. It is important for the understanding of Mill's thought here, as throughout his work, to note that when he talks of 'properties,' he has in mind, mainly, characters interesting to us. Otherwise, it would not be true that all white things have few properties in common. By a 'law of nature' he means any absolute uniformity; so that it is hardly enough to assert that if all white things had any property P, this would be a 'consequence, under a law of nature,' of their whiteness; for it would be itself an absolute and ultimate uniformity. Mill says that if the common properties of a class thus follow from a small number of primary characters 'which, as the phrase is, account for all the rest,' it is not a real kind. He does not remark that the man of science is bent upon ultimately thus accounting for each and every property that he studies. The following definition might be proposed: Any class which, in addition to its defining character, has another that is of permanent interest and is common and peculiar to its members, is destined to be conserved in that ultimate conception of the universe at which we aim, and is accordingly to be called 'real.' (C.S.P.)

Knowledge (in logic). This word is used in logic in two senses: (1) as a synonym for COGNITION (q.v.), and (2), and more usefully, to signify a perfect cognition, that is, a cognition fulfilling three conditions: first, that it holds for true a proposition that really is true; second, that it is perfectly self-satisfied and free from the uneasiness of doubt; third, that some character of this satisfaction is such that it would be logically impossible that this character should ever belong to satisfaction in a proposition not true.

Knowledge is divided, firstly, according to whatever classification of the sciences is adopted. Thus, Kantians distinguish formal and material knowledge. See SCIENCE. Secondly, knowledge is divided according to the different ways in which it is attained, as into immediate and mediate knowledge. See IMMEDIACY AND MEDIACY (logical). Immediate knowledge is a cognition, or objective modification of consciousness, which is borne in upon a man with such resistless force as to constitute a guarantee that it (or a representation of it) will remain permanent in the development of human cognition. Such knowledge is, if its existence be granted, either borne in through an avenue of sense, external or internal, as a percept of an individual, or springs up within the mind as a first principle of reason or as a mystical revelation. Mediate knowledge is that for which there is some guarantee behind itself, although, no matter how far criticism be carried, simple evidency, or direct insistency, of something has to be relied upon. The external guarantee rests ultimately either upon authority, i.e. testimony, or upon observation. In either case mediate knowledge is attained by REASONING, which see for further divisions. It is only necessary to mention here that the Aristotelians distinguished knowledge oti, or of the facts themselves, and knowledge dioti, or of the rational connection of facts, the knowledge of the how and why (cf. the preceding topic). They did not distinguish between the how and the why, because they held that knowledge dioti is solely produced by SYLLOGISM (q.v.) in its greatest perfection, as demonstration. The term empirical knowledge is applied to knowledge, mediate or immediate, which rests upon percepts; while the terms philosophical and rational knowledge are applied to knowledge, mediate or immediate, which rests chiefly or wholly upon conclusions or revelations of reason. Thirdly, knowledge is divided, according to the character of the immediate object, into apprehensive and judicative knowledge, the former being of a percept, image, or Vorstellung, the latter of the existence or non-existence of a fact. Fourthly, knowledge is divided, according to the manner in which it is in the mind, into actual, virtual, and habitual knowledge. See Scotus, Opus Oxoniense, lib. I. dist. iii. quest. 2, paragraph beginning 'Loquendo igitur.' Fifthly, knowledge is divided, according to its end, into speculative and practical. (C.S.P., C.L.F.)

Laws of Thought: Ger. Denkgesetze; Fr. lois de la pensée; Ital. leggi del pensiero. The three formulas of identity, contradiction, and excluded middle have been widely so known, though the doctrine that they are three co-ordinate and sufficient laws of all thought or of all reasoning has been held by a comparatively small party which hardly survives; and it is not too much to say that the doctrine is untenable. But the designation is so familiar and convenient that those formulas may very well be referred to as 'the so-called three laws of thought.' The formulas have usually been stated by those who upheld the doctrine as follows: — 

I. The Principle of Identity: A is A.
II. The Principle of Contradiction: A is not not-A.
III. The Principle of Excluded Middle or Excluded Third: everything is either A or not-A.

It is noticeable that two of these propositions are categorical and the third disjunctive, a circumstance demanding explanation for those who hold the distinction of categorical, conditional, and disjunctive propositions to be fundamental.

The meaning of the formula of identity presents only one small difficulty. If the copula 'is' be taken in the sense of 'is, if it exists,' then the meaning of the formula is that no universal affirmative proposition having the same term as subject and predicate is false. If, however, the copula be understood to imply existence, the meaning is that no universal affirmative proposition is false in which the same term is subject and predicate, provided that term denotes any existing object. Or, the meaning may be that the same thing is true when the subject and predicate are the same proper name of an individual. In any case, it may properly be required that the precise meaning attached to the copula should be explained; and this explanation must in substance involve one or other of the above three statements; so that in any case the principle of identity is merely a part of the definition of the copula.

In like manner, if the word 'not is to be used in logical forms, its force should be explained with the utmost precision. Such an explanation will consist in showing that the relation it expresses belongs at once to certain classes of relations, probably not more than two, in view of the simplicity of the idea. Each of these two statements may be embodied in a formula similar, in a general way, to the formulas of contradiction and excluded middle. It has, therefore, seemed to Mill and to the 'exact' logicians that these two formulas ought together to constitute a definition of the force of 'not.'

Other writers have regarded all three laws as 'practical maxims.' But practically nobody needs a maxim to remind him that a contradiction, for example, is an absurdity. It might be a useful injunction to tell him to beware of latent contradictions; but as soon as he clearly sees that a proposition is self-contradictory, he will have abandoned it before any maxim can be adduced. Seeing, then, that such formulas are required to define the relation expressed by not, but are not required as maxims, it is in the former aspect that their true meanings are to be sought.

If it is admitted that they constitute a definition, they must conform to the rules of definition. Considered as part of a definition, one of the commonest statements of the principle of contradiction, 'A non est non-A,' offends against the rule that the definitum must not be introduced into the definition. This is easily avoided by using the form 'A est non non-A,' 'A is not not-A,' or every term may be subsumed under the double negation of itself. If this form is adopted for the principle of contradiction, the principle of excluded middle ought to be 'What is not not-A is A.' If, however, we prefer to state the principle of excluded middle as 'Everything is either A or not-A,' then we should state the principle of contradiction as 'What is, at once, A and not-A is nothing.' There is no vicious circle here, since the term 'nothing,' or 'non ens,' may be formally defined without employing the particle 'not' or any equivalent. Thus, we may express the principle of contradictions as follows:

Whatever there may be which is both A and not-A is X, no matter what term X may be.
In either formula, A may be understood to be restricted to being an individual, or it may be allowed to be any term, individual or general. In the former case, in order to avoid conflict with the fundamental law that no true definition asserts existence, a special clause should be added, such as 'if not-A there be.' In the latter case, it should be stated that by 'not-A' is not meant 'not some A,' but 'not any A,' or 'other than whatever A there may be.'

Bearing these points in mind, the formula 'A is not-not-A,' or 'A is other than whatever is other than whatever is A,' is seen to be a way of saying that the relation expressed by 'not' is one of those which is its own converse, and is analogous to the following:

Every rose is similar to whatever is similar to whatever is a rose; which again is similar to the following:
Every man is loved by whatever loves whatever is a man.
But if we turn to the corresponding formula of excluded middle, 'Not-not-A is A,' or 'Whatever is not anything that is not any A is A,' we find that its meaning cannot be so simply expressed. Supposing that the relation r is such that it is true that
Whatever is r to whatever is r to whatever is A is A,
it can readily be proved that, whether the multitude of individuals in the universe be finite or infinite, each individual is either non-r to itself and to nothing else, or is one of a pair of individuals that are non-r to each other and to nothing else; and conversely, if the universe is so constituted, the above formula necessarily holds. But it is evident that if the universe is so constituted, the relation r is converse to itself; so that the formula corresponding to that of contradiction also holds. But this constitution of the universe does not determine r to be the relation expressed by 'not.' Hence, the pair of formulas,
A is not not-A,
Not not-A is A,
are inadequate to defining 'not,' and the former of them is mere surplusage. In fact, in a universe of monogamously married people taking any class, the A's,
Every A is a non-spouse to whatever is non-spouse to every A,
Whatever is non-spouse to whatever is a non-spouse to every A is an A.
No such objection exists to the other pair of formulas:
Whatever is both A and not-A is nothing,
Everything is either A or not-A.
Their meaning is perfectly clear. Dividing all ordered pairs of individuals into those of the form A: B and those of the form A: A,
The principle of contradiction excludes from the relation 'not' all of the form A:A,
The principle of excluded middle makes the relation of 'not' to include all pairs of the form A:B.
>From this point of view, we see at once that there are three other similar pairs of formulas defining the relations of identity, coexistence, and incompossibility, as follows:
Whatever is A is identical with A; i.e. Identity includes all pairs A:A.
Whatever is identical with A is A; i.e. Identity excludes all pairs A:B.
Whatever is A is coexistent with A; i.e. Coexistence includes all pairs A:A.
Everything is either A or coexistent with A; i.e. Coexistence includes all pairs A:B.
Whatever is both A and incompossible with A is nothing; i.e. Incompossibility excludes all pairs A:A.
Whatever there may be incompossible with A is A; i.e. Incompossibility excludes all pairs A:B.
Much has been written concerning the relations of the three principles to forms of syllogism. They have even been called Die Principien des Schliessens, and have often been so regarded. Some points in reference to the meanings they have borne in such discussions require mention. Many writers have failed to distinguish sufficiently between reasoning and the logical forms of inference. The distinction may be brought out by comparing the moods Camestres and Cesare (see MOOD, in logic). Formally, these are essentially different. The form of Camestres is as follows:
    Every P is an M,
    Every S is other than every M;
∴ Every S is other than every P.
This form does not depend upon either clause of the definition of 'not' or 'other than.' For if any other relative term, such as 'lover of,' be substituted for 'other than,' the inference will be equally valid. The form of Cesare is as follows:
    Every P is other than every M,
    Every S is an M;
∴ Every S is other than every P.
This depends upon the equiparance of 'other than.' For if we substitute an ordinary relative, such as loves, for 'other than' in the premise, the conclusion will be
    Every S is loved by every P.
(See De Morgan's fourth memoir on the syllogism, Cambridge Philos. Trans., x. (1860) 354.) The two forms are thus widely distinct in logic; and yet when a man actually performs an inference, it would be impossible to determine that he 'reasons in' one of these moods rather than in the other. Either statement is incorrect. He does not, in strict accuracy, reason in any form of syllogism. For his reasoning moves in first intentions, while the forms of logic are constructions of second intentions. They are diagrammatic representations of the intellectual relation between the facts from which he reasons and the fact which he infers, this diagram necessarily making use of a particular system of symbols — a perfectly regular and very limited kind of language. It may be a part of a logician's duty to show how ordinary ways of speaking and of thinking are to be translated into that symbolism of formal logic; but it is no part of syllogistic itself. Logical principles of inference are merely rules for the illative transformation of the symbols of the particular system employed. If the system is essentially changed, they will be quite different. As the Boolians represent Cesare and Camestres, they appear, after literally translating the algebraic signs of those logicians into words, as follows:
    A that is B is nothing,
    C that is not B is nothing,
A that is C is nothing.
The two moods are here absolutely indistinguishable.

From the time of Scotus down to Kant more and more was made of a principle agreeing in enunciation, often exactly, in other places approximately, with our principle of contradiction, and in the later of those ages usually called by that name, although earlier more often principium primum, primum cognitum, pricipium identitatis, dignitas dignitatum, &c. It would best be called the Principle of Consistency. Attention was called to it in the fourth book of Aristotle's Metaphysics. The meaning of this, which was altogether different, at least in post-scholastic times, from our principle of contradiction, is stated in the so-called Monadoligie of Leibnitz (§ 31) to be that principle by virtue of which we judge that to be false which involves a contradiction, and the denial of the contradiction to be true. The latter clause involves an appeal to the principle of excluded middle as much as the former clause does to the formal principle of contradiction. And so the 'principle of contradiction' was formerly frequently stated. But, in fact, neither is appealed to; for Leibnitz does not say that the contradiction is to be made explicit, but only that it is to be recognized as an inconsistency. Interpreted too strictly, the passage would seem to mean that all demonstrative reasoning is by the reductio ad absurdum; but this cannot be intended. All that is meant is that we draw that conclusion the denial of which would involve an absurdity — in short, that which consistency requires. This is a description, however imperfect, of the procedure of demonstrative REASONING (q.v.), and does not relate to logical forms. It deals with first, not second, intentions. (C.S.P.)

It is unfortunate that 'contradictory' and 'principle of contradiction' are terms used with incongruent significations. If a and b are statements, they are mutually contradictory, provided that one or the other of them must be true and that both cannot be true; these are the two marks (essential and sufficient) of contradiction, or precise denial, as it might better be called. If a and b are terms, b is the precise negative of a (or the contradictory term to a), provided it takes in all of that which is other than a — that is, if everything must be one or the other (a or b) and if nothing can be both. These two properties constitute the definition of a pair of contradictories (whether terms or propositions), namely, they are mutually exclusive, and they are together exhaustive; expressed in the language of 'exact logic,' these properties are (writing  for the negative of x and + for or):

Together these properties constitute the requirements of contradiction or of exact negation; it is a very inelegant piece of nomenclature (besides that it leads to actual confusion) to refer to (1) alone as the 'principle of contradiction.' Better names for them are (1) exclusion and (2) exhaustion (in place of excluded middle). In the common phraseology we are obliged to commit the absurdity of saying that two terms or propositions may satisfy the 'principle of contradiction' and still not be contradictory (since they may lack the quality of being exhaustive). The mere fact that (1) has been called the principle of contradiction has given it a pretended superiority over the other which it by no means deserves; they are of equal importance in the conducting of reasoning processes. In fact, for every formal argument which rests upon (1) there is a corresponding argument which rests upon (2): thus in the case of the fundamental law of TRANSPOSITION (q.v.), which affirms the identity of these two propositions, (m) the student who is not a citizen is not a voter; (n) every student is either a citizen or not a voter; that (m) follows from (n) depends upon one of these principles, and that (n) follows from (m) depends upon the other. These two names, exhaustion and exclusion, have the great advantage that they permit the formation of adjectives; thus we may say that the test for the contradictoriness of two terms or propositions which are not on their face the negatives one of another is that they should be (1) mutually exclusive and (2) together exhaustive.

It may be noticed that if two terms are exhaustive but not exclusive, their negatives are exclusive but not exhaustive. Thus within the field of number, 'prime' and 'even' are exclusive (no number can be both) but not exhaustive (except in the limiting case of two, some numbers can be neither), while 'not even' and 'not prime' are exhaustive and not exclusive.

In the case of propositions, 'contrary' and 'subcontrary' are badly chosen names for the OPPOSITION (q.v.) of A and E, O and I, respectively, of the traditional logical scheme; they do not carry their meaning on their face, and hence are unnecessarily difficult for the learner to bear in mind. A and E should be said to be mutually exclusive (but not exhaustive), O and I to be conjointly exhaustive (but not exclusive). This relation of qualities is then seen to be a particular case merely of the above-stated general rule.

Again, 'no a is b' and 'all a is b' are exclusive but not exhaustive, while 'some a is b' and 'some a is not b' are exhaustive but not exclusive (provided in both cases that a exists).

Laws of thought is not a good name for these two characteristics; they should rather be called the laws (if laws at all) of negation. Properly speaking, the laws of thought are all the rules of logic; of these laws there is one which is of far more fundamental importance than those usually referred to under the name, namely, the law that if a is b and b is c, it can be concluded that a is c. This is the great law of thought, and everything else is of minor importance in comparison with it. It is singular that it is not usually enumerated under the name. Another law of thought of equal consequence with those usually so called is, according to Sigwart, the law that the double negative is equivalent to an affirmative,

But these are not fundamental, for from the principles of

it follows


Literature: for the history of these principles see UEBERWEG, Syst. d. Logik, §§ 75-80; PRANTL, Gesch. d. Logik (see 'principium' in the indices to the four volumes). There are additional notes in an appendix to HAMILTON, Lects. on Logic. (C.S.P.)

Leading of Proof: no concise foreign equivalents. The operation bringing up to attention, among propositions admitted to be true, certain relations between them which logically compel the acceptance of a conclusion. (C.S.P.)

Leading Principle: Ger. leitendes Prinzip; Fr. principe directeur; Ital. principio fondamentale. It is of the essence of reasoning that the reasoner should proceed, and should be conscious of proceeding, according to a general habit, or method, which he holds would either (according to the kind of reasoning) always lead to the truth, provided the premises were true; or, consistently adhered to, would eventually approximate indefinitely to the truth; or would be generally conducive to the ascertainment of truth, supposing there be any ascertainable truth. The effect of this habit or method could be stated in a proposition of which the antecedent should describe all possible premises upon which it could operate, while the consequent should describe how the conclusion to which it would lead would be determinately related to those premises. Such a proposition is called the 'leading principle' of the reasoning.

Two different reasoners might infer the same conclusion from the same premises; and yet their proceeding might be governed by habits which would be formulated in different, or even conflicting, leading principles. Only that man's reasoning would be good whose leading principle was true for all possible cases. It is not essential that the reasoner should have a distinct apprehension of the leading principle of the habit which governs his reasoning; it is sufficient that he should be conscious of proceeding according to a general method, and that he should hold that that method is generally apt to lead to the truth. He may even conceive himself to be following one leading principle when, in reality, he is following another, and may consequently blunder in his conclusion. From the effective leading principle, together with the premises, the propriety of accepting the conclusion in such sense as it is accepted follows necessarily in every case. Suppose that the leading principle involves two propositions, L and L', and suppose that there are three premises, P, P', P''; and let C signify the acceptance of the conclusion, as it is accepted, either as true, or as a legitimate approximation to the truth, or as an assumption conducive to the ascertainment of the truth. Then, from the five premises L, L', P, P', P'', the inference to C would be necessary; but it would not be so from L, L', P', P'' alone, for, if it were, P would not really act as a premise at all. >From P' and P'' as the sole premises, C would follow, if the leading principle consisted of L, L', and P. Or from the four premises L', P, P', P'', the same conclusion would follow if L alone were the leading principle. What, then, could be the leading principle of the inference of C from all five propositions L, L', P, P', P'', taken as premises? It would be something already implied in those premises; and it might be almost any general proposition so implied. Leading principles are, therefore, of two classes; and any leading principle whose truth is implied in the premises of every inference which it governs is called a 'logical' (or, less appropriately, a formal) leading principle; while a leading principle whose truth is not implied in the premises is called a 'factual' (or material) leading principle. (C.S.P.)

Lemma [Gr. lhmma, gain, an assumption, premise]: Ger. Hilfssatz, Lehnsatz; Fr. lemme; Ital. lemma. A theorem which interrupts the course of development of a mathematical theory, but which is inserted to supply a premise for one of the theorems.

This use of the word seems to go back to Euclid, at least; and even Aristotle uses the word — not a common one with him — in connection with geometry, in the first chapter of the Topics. With Aristotle, however, it means a premise; and with the Stoics, more particularly, the major premise of a syllogism. (C.S.P.)

Light of Nature [trans. of Lat. lumen naturae or naturale, a term used by Aquinas, Summa Theologiae, Pt. I, qu. 12, art. 13, and elsewhere. It is not necessary to suppose that he borrowed the term from the passage of Aristotle's De Anima, 430 a, 14, where the creative intellect is compared with light]: Ger. natürliches Licht; Fr. lumière naturelle (Pascal); Ital. lume naturale (Galileo). A natural power, or instinct, by which men are led to the truth about matters which concern them, in anticipation of experience or revelation. See LUMEN (also for literature).

The phrase is used in contradistinction to supernatural light. Tucker's Light of Nature pursued is a book written as a mild reaction against Locke and the Associationalists in the direction of the philosophy of common sense. (C.S.P.)

Limitative [Lat. limitare, to enclose]: Ger. limitativ (Urtheil); Fr. limitatif; Ital. limitativo. (1) Applied to a third quality of judgments, additional to affirmative and negative. The idea of such a third quality originated among the Romans from the difference between 'homo non est bonus' and 'homo est non bonus,' the latter being the limitative.

(2) Setting limits in the sense (2) given under LIMITING NOTION (q.v.).

It is one of the numerous cases in which accidents of language have affected accepted logical forms without any good reason. Boethius and others applied the infinitation to the subject also, which De Morgan has shown makes a valuable addition to logic. Wolff, however, limited the modification to the predicate, without showing any serious reason for such application. Kant adopted it because it rounded out his triad of categories of quality. His defence, as reported by Jäsche, is that the negative excludes the subject from the sphere of the predicate, while the unendliche, limitative, or infinite judgment puts it into the infinite sphere outside the predicate. It is to be remarked that Kant regards a positive mark as differing per se from a negative one, and, in particular, as having a far narrower extension. Like most of the old logicians, he virtually limited the universe of marks to such as arrest our attention. If that had been explicitly and consistently done, it would have constituted an interesting particular logic, in which there would be a material and not merely formal difference between affirmative and negative facts. It is probable that Kant also understood the affirmative proposition to assert the existence of its subject, while the negative did not do so; so that 'Some phoenixes do not rise from their ashes' would be true, and 'All phoenixes do rise from their ashes' would be false. The limitative judgment would agree with the affirmative in this respect. This was probably his meaning, and he did not observe that his limitative judgment, 'The human soul is immortal (nichtsterblich),' may be construed as equivalent to the conjunctive judgment, 'The human soul is not mortal, and it is the human soul.' No doubt Kant would have seen a world of difference between these two assertions. In that case he should have adopted a fourth quality, 'The human soul is not immortal.' (C.S.P.)

Limiting Notion: Ger. Grenzbegriff; Fr. notion-limite; Ital. concetto limite. (1) A term used by Kant in a single passage of the Krit. d. reinen Vernunft (1st ed., 255) to signify that a NOUMENON (q.v., ad fin.), which is a thing in itself regarded as an object of reason, is something to which experience cannot attain, but is the inconceivable something behind the phenomena. The passage reads: 'Der Begriff eines Noumenon ist also bloss ein Grenzbegriff, um die Anmassung der Sinnlichkeit einzuschränken, und also nur von negativem Gebrauche.' (J.M.B.- C.S.P.)

Logic [Gr. logikh]: Ger. Logik; Fr. logique; Ital. logica. Logic is a science which has not yet completed the stage of disputes concerning its first principles, although it is probably about to do so. Nearly a hundred definitions of it have been given. It will, however, generally be conceded that its central problem is the classification of arguments, so that all those that are bad are thrown into one division, and those which are good into another, these divisions being defined by marks recognizable even if it be not known whether the arguments are good or bad. Furthermore, logic has to divide good arguments by recognizable marks into those which have different orders of validity, and has to afford means for measuring the strength of arguments.

It is generally admitted that there is a doctrine which properly antecedes what we have called critic. It considers, for example, in what sense and how there can be any true proposition and false proposition, and what are the general conditions to which thought or signs of any kind must conform in order to assert anything. Kant, who first raised these questions to prominence, called this doctrine transcendentale Elementarlehre, and made it a large part of his Critic of the Pure Reason. But the Grammatica Speculativa of Scotus is an earlier and interesting attempt. The common German word is Erkenntnisstheorie, sometimes translated EPISTEMOLOGY (q.v.).

It is further generally recognized that another doctrine follows after critic, and which belongs to, or is closely connected with, logic. Precisely what this should contain is not agreed; but it must contain the general conditions requisite for the attainment of truth. Since it may be held to contain more, one hesitates to call it heuristic. It is often called Method; but as this word is also used in the concrete, methodic or methodeutic would be better.

For deciding what is good logic and what is bad, appeal is made by different writers to one or more, generally several, of these eight sources: to direct dicta of consciousness, to psychology, to the usages of language, to metaphysical philosophy, to history, to everyday observation, to mathematics, and to some process of dialectic. In the middle ages appeal was frequently made to authority.

The appeal to direct consciousness consists in pronouncing certain reasoning to be good or bad because it is felt to be so. This is a very common method. Sigwart, for example, bases all logic upon our invincible mental repulsion against contradiction, or, as he calls it, 'the immediate feeling of necessity' (Logic § 3, 2). Those who think it worth while to make any defence at all of this proceeding urge, in effect, that, however far the logician may push his criticisms of reasoning, still in doing so, he must reason, and so must ultimately rely upon his instinctive recognition of good and bad reasoning. Whence it follows that, in Sigwart's words, 'every system of logic must rest upon this principle.' It is, however, to be noted that among the dicta of direct consciousness, many pronounce certain reasonings to be bad. If, therefore, such dicta are to be relied upon, man not only usually has a tendency to reason right, but also sometimes has a tendency to reason wrong; and if that be so, the validity of a reasoning cannot consist in a man's having a tendency to reason in that way. Some say that the validity of reasoning consists in the 'definitive dictum' of consciousness; but it has been replied that certain propositions in Euclid were studied for two thousand years by countless keen minds, all of whom had an immediate feeling of evidence concerning their proofs, until at last flaws were detected in those proofs, and are now admitted by all competent persons; and it is claimed that this illustrates how far from possible it is to make direct appeal to a definitive pronouncement. Besides, say those who object to this method, all reasoning and inquiry expects that there is such a thing as the truth concerning whatever question may be under examination. Now, it is of the very essence of this 'truth,' the meaning of the expectation, that the 'truth' in no wise depends upon what any man to whom direct appeal can be made may opine about that question. A fortiori it does not depend upon whether I am satisfied with it or not. It is further insisted that there can be no genuine criticism of a reasoning until that reasoning is actually doubted; and no sooner is it actually doubted than we find that consciousness has revoked her dictum in its favour, if she ever made any. It is, indeed, maintained that so far from true is it that every system of logic must be based upon any instinctive recognition of good and bad reasoning, that it is quite impossible for any reasoning to be based upon such recognition in respect to that same reasoning. In reasoning, a man may feel sure he is right; but to 'rest' that confidence on nothing but itself is to rest it on nothing at all. If the fact that we must use our reasoning instinct in criticizing reasoning proves that we must appeal to nothing else in such criticism, it equally proves that we ought to follow the lead of that instinct without any logical control at all, which would be as much as to say that we ought not to reason at all. A man cannot criticize every part of his reasoning, since he cannot criticize the act of reasoning he is performing in the criticism, it is true. But he can criticize steps whose validity he doubts; and in doing so, ought to consider in what characters the validity of reasoning consists, and whether the reasoning in question possesses those characters.

Under an appeal to psychology is not meant every appeal to any fact relating to the mind. For it is, for logical purposes, important to discriminate between facts of that description which are supposed to be ascertained by the systematic study of the mind, and facts the knowledge of which altogether antecedes such study, and is not in the least affected by it; such as the fact that there is such a state of mind as doubt, and the fact that the mind struggles to escape from doubt. Even facts like these require to be carefully examined by the logician before he uses them as the basis of his doctrine. But many logicians have gone much further, and have avowedly based their systems upon one or another theory of psychology. Another class of logicians have professed to base logic upon a psychological theory of cognition. Of course, if this is done, such psychological doctrine is placed above logical criticism, or, at any rate, above logical support. For if the truth of a conclusion is known only from certain premises, it cannot be used to support those premises. Now, it may be doubted whether psychology is not, of all the special sciences, the one which stands most in need of appeal to a scientific logic.

Appeals to the usages of language are extremely common. They are made even by those who use algebraical notation in logic 'in order to free the mind from the trammels of speech' (Schröder, Logik, i. p. iii). It is difficult to see what can be hoped for from such a proceeding, unless it be to establish a psychological proposition valid for all minds. But to do this, it would be necessary to look beyond the small and very peculiar class of Aryan languages, to which the linguistic knowledge of most of those writers is confined. The Semitic languages, with which some of them are acquainted, are too similar to the Aryan greatly to enlarge their horizon. Moreover, even if other languages are examined, the value of any logical inferences from them is much diminished by the custom of our grammarians of violently fitting them to the Procrustean bed of Aryan grammar.

The objection which has been suggested to appeals to psychological results applies with far greater force to appeals to metaphysical philosophy, which, it will generally be conceded, can hardly take a step with security unless it rests upon the science of logic. Nevertheless, a great many logical treatises of various colours make it their boast that they are built upon philosophical principles.

Logicians occasionally appeal to the history of science. Such and such a mode of reasoning, it is said, for example, was characteristic of mediaevalism or of ancient science; such another produced the successes of modern science. If logic is to be based upon probable reasonings, as some logicians maintain that it must be, such arguments, if critically examined, must be admitted to have great weight. They will naturally be out of place in a system of logic which professes to demonstrate from certain initial assumptions that the kinds of reasoning it recommends must be accepted.

There is probably room for dispute as to whether logic need assert anything at all as an absolute matter of fact. If it does not, any appeal to experience would seem to be irrelevant. If it does, still the opinion may be that such assertions of logic are of so exceedingly broad and slight a nature that the universal experience of every man's every day and hour puts them beyond all doubt — such experiences as that the world presents appearances of variety, of law, and of the real action of one thing upon another. As appearances, these things do not seem likely ever to be doubted. If logic has need of any facts, and if such facts will suffice, no objection can well be made to an appeal to them.

The boundary between some parts of logic and pure mathematics in its modern treatment is almost evanescent, as may be seen in Dedekind's Was sind und was sollen die Zahlen (1888, Eng. trans. 1901). There are, however, departments of logic, such as the logic of probable inference (if that be regarded a part of logic), in which appeal is sometimes made to mathematical results, such as Bernoulli's law of high numbers. It seems to be the general opinion that nothing so difficult as mathematics can be admitted into, or be appealed to by, the science of logic, which has the peculiarity of consisting chiefly of truisms.

In mathematical reasoning there is a sort of observation. For a geometrical diagram or array of algebraical symbols is constructed according to an abstractly stated precept, and between the parts of such diagram or array certain relations are observed to obtain, other than those which were expressed in the precept. These being abstractly stated, and being generalized, so as to apply to every diagram constructed according to the same precept, give the conclusion. Some logicians hold that an equally satisfactory method depends upon a kind of inward observation, which is not mathematical, since it is not diagrammatic, the development of a conception and its inevitable transformation being observed and generalized somewhat as in mathematics; and those logicians base their science upon such a method, which may conveniently be termed, and is sometimes termed, a Dialectic. Other logicians regard such a method as either extremely insecure or as altogether illusory.

The generally received opinion among professors of logic is that all the above methods may properly be used on occasion, the appeal to mathematics, however, being less generally recognized.

Literature: the history of logic in Western Europe, down to the revival of learning, is given by PRANTL, Gesch. d. Logik im Abendlande. Upon the points upon which this author touches, he always affords valuable information, though his judgments are peremptory and slashing. Unfortunately, he omits much which was regarded by the authors of whom he treats as most important, because he does not himself so regard it. He also omits much which would be interesting to a reader taking a broader conception of logic. It is hardly necessary to say that upon some large subjects his views are controverted. Of the modern development of logic there is no satisfactory history; but there are notices good as far as they go in UEBERWEG, Syst. d. Logik (Eng. trans.); in the much earlier work of BACHMANN, Syst. d. Logik (1828); in HAMILTON, Lects. on Logic; and for later work in B. ERDMANN, Logik. CH. SIGWART, Logic (Eng. trans.), and WUNDT, Logik, may also be profitably consulted. See under the logical topics generally (e.g. EMPIRICAL LOGIC, FORMAL LOGIC, JUDGMENT, and PROPOSITION); and also BIBLIOG. C. (C.S.P., C.L.F.)

Logic (exact): Ger. exakte Logik; Fr. logique exacte; Ital. logica esatta. The doctrine that the theory of validity and strength of reasoning ought to be made one of the 'exact sciences,' that is, that generalizations from ordinary experience ought, at an early point in its exposition, to be stated in a form from which by mathematical, or expository, REASONING (q.v.), the rest of the theory can be strictly deduced; together with the attempt to carry this doctrine into practice.

This method was pursued, in the past, by Pascal (1623-62), Nicolas Bernoulli (1687-1759), Euler (1708-83), Ploucquet (1716-90), Lambert (1728-77), La Place (1749-1827), De Morgan (1806-71), Boole (1815-64), and many others; and a few men in different countries continue the study of the problems opened by the last two named logicians, as well as those of the proper foundations of the doctrine and of its application to inductive reasoning. The results of this method, thus far, have comprised the development of the theory of probabilities, the logic of relatives, advances in the theory of inductive reasoning (as it is claimed), the syllogism of transposed quantity, the theory of the Fermatian inference, considerable steps towards an analysis of the logic of continuity and towards a method of reasoning in topical geometry, contributions towards several branches of mathematics by applications of 'exact' logic, the logical graphs called after Euler and other systems for representing in intuitional form the relations of premises to conclusions, and other things of the same general nature.

There are those, not merely outside the ranks of exact logic, but even within it, who seem to suppose that the aim is to produce a calculus, or semi-mechanical method, for performing all reasoning, or all deductive inquiry; but there is no reason to suppose that such a project, which is much more consonant with the ideas of the opponents of exact logic than with those of its serious students, can ever be realized. The real aim is to find an indisputable theory of reasoning by the aid of mathematics. The first step in the order of logic towards this end (though not necessarily the first in the order of inquiry) is to formulate with mathematical precision, definiteness, and simplicity, the general facts of experience which logic has to take into account.

The employment of algebra in the investigation of logic is open to the danger of degenerating into idle trifling of too rudimentary a character to be of mathematical interest, and too superficial to be of logical interest. It is further open to the danger that the rules of the symbols employed may be mistaken for first principles of logic. An algebra which brings along with it hundreds of purely formal theorems of no logical import whatever must be admitted, even by the inventor of it, to be extremely defective in that respect, however convenient it may be for certain purposes. On the other hand, it is indisputable that algebra has an advantage over speech in forcing us to reason explicitly and definitely, if at all. In that way it may afford very considerable aid to analysis. It has been employed with great advantage in the analysis of mathematical reasonings.

Algebraic reasoning involves intuition just as much as, though more insidiously than, does geometrical reasoning; and for the investigation of logic it is questionable whether the method of graphs is not superior. Graphs cannot, it is true, readily be applied to cases of great complexity; but for that very reason they are less liable to serve the purposes of the logical trifler. In the opinion of some exact logicians, they lead more directly to the ultimate analysis of logical problems than any algebra yet devised. See LOGICAL DIAGRAM (or GRAPH).

It is logical algebra, however, which has chiefly been pursued. De Morgan invented a system of symbols, which had the signal advantage of being entirely new and free from all associations, misleading or otherwise. Although he employed them for synthetical purposes almost exclusively, yet the great generality of some of the conceptions to which they led him is sufficient to show that they might have been applied with great advantage in analysis. Boole was led, no doubt from the consideration of the principles of the calculus of probabilities, to a wonderful application of ordinary algebra to the treatment of all deductive reasoning not turning upon any relations other than the logical relations between non-relative terms. By means of this simple calculus, he took some great steps towards the elucidation of probable reasoning; and had it not been that, in his pre-Darwinian day, the notion that certain subjects were profoundly mysterious, so that it was hopeless, if not impious, to seek to penetrate them, was still prevalent in Great Britain, his instrument and his intellectual force were adequate to carrying him further than he actually went. Most of the exact logicians of to-day are, from the nature of the case, followers of Boole. They have modified his algebra by disusing his addition, subtraction, and division, and by introducing a sign of logical aggregation. This was first done by Jevons; and he proposed .|. , a sign of division turned up, to signify this operation. Inasmuch as this might easily be read as three signs, it would, perhaps, be better to join the two dots by a light curve, thus Y. Some use the sign + for logical aggregation. The algebra of Boole has also been amplified so as to fit it for the logic of relatives. The system is, however, far from being perfect. See RELATIVES (logic of).

Certain terms of exact logic may be defined as follows: — 

Aggregation. The operation of uniting two or more terms or propositions, called aggregants, to produce an aggregate term or proposition which is true of everything of which any aggregant is true, and false of everything of which all the aggregants are false. It is opposed to composition, which is the operation of producing from two or more terms or propositions, called the components, a new term or proposition, called their compound, which is true of all of which all the components are true, and false of all of which any are false.

Absorption, law of (Ger. Absorptionsgesetz). The proposition that if of two aggregants one contains the other as a component, the aggregate is identical with the latter.

Alternative proposition. A term preferred by some logicians to 'disjunctive,' because the latter term is often, as by Cicero and Aulus Gellius, understood to imply that one, and one only, of the alternatives is true. At the same time, the standard traditional example of a disjunctive was 'Socrates currit vel Plato disputat,' and the rule was 'Ad veritatem disiunctivae sufficit alteram partem esse veram.' Nevertheless, the narrower sense was also recognized, and the term alternative is perhaps preferable.

Associative. An operation combining two elements is associative if, and only if, in combining the result with a third element, it makes no difference whether the middle element be first combined with the last and the result with the first, or the other way, so long as the order of sequence is preserved. Addition and multiplication are associative, while involution is not so; for ten to the three-square power is a milliard, while ten cube squared is only a million. An associative algebra is an algebra in which multiplication is associative.

Commutative. An operation by which two elements are united is said to be commutative if, and only if, it makes no difference which is taken first. Thus, because twice three is thrice two, numerical multiplication is commutative.

Composition: see Aggregation, above.

Compound: see Aggregation, above.

Copula is often defined as that which expresses the relation between the subject-term and the predicate-term of a proposition. But this is not sufficiently accurate for the purposes of exact logic. Passing over the objection that it applies only to categorical propositions, as if conditional and copulative propositions had no copula, contrary to logical tradition, it may be admitted that a copula often does fulfil the function mentioned; but it is only an accidental one, and its essential function is quite different. Thus, the proposition 'Some favoured patriarch is translated' is essentially the same as 'A translated favoured patriarch is'; and 'Every mother is a lover of that of which she is a mother' is the same as 'A mother of something not loved by her is not.' In the second and fourth forms, the copula connects no terms; but if it is dropped, we have a mere term instead of a proposition. Thus the essential office of the copula is to express a relation of a general term or terms to the universe. The universe must be well known and mutually known to be known and agreed to exist, in some sense, between speaker and hearer, between the mind as appealing to its own further consideration and the mind as so appealed to, or there can be no communication, or 'common ground,' at all. The universe is, thus, not a mere concept, but is the most real of experiences. Hence, to put a concept into relation to it, and into the relation of describing it, is to use a most peculiar sort of sign or thought; for such a relation must, if it subsist, exist quite otherwise than a relation between mere concepts. This, then, is what the copula essentially does. This it may do in three ways: first, by a vague reference to the universe collectively; second, by a reference to all the individuals existent in the universe distributively; third, by a vague reference to an individual of the universe selectively. 'It is broad daylight,' I exclaim, as I awake. My universe is the momentary experience as a whole. It is that which I connect as object of the composite photograph of daylight produced in my mind by all my similar experiences. Secondly, 'Every woman loves something' is a description of every existing individual in the universe. Every such individual is said to be coexistent only with what, so far as it is a woman at all, is sure to be a lover of some existing individual. Thirdly, 'Some favoured patriarch is translated' means that a certain description applies to a select individual. A hypothetical proposition, whether it be conditional (of which the alternative, or disjunctive, proposition is a mere species, or vice versa, as we choose to take it) or copulative, is either general or ut nunc. A general conditional is precisely equivalent to a universal categorical. 'If you really want to be good, you can be,' means 'Whatever determinate state of things may be admissibly supposed in which you want to be good is a state of things in which you can be good.' The universe is that of determinate states of things that are admissable hypothetically. It is true that some logicians appear to dispute this; but it is manifestly indisputable. Those logicians belong to two classes: those who think that logic ought to take account of the difference between one kind of universe and another (in which case, several other substantiae of propositions must be admitted); and those who hold that logic should distinguish between propositions which are necessarily true or false together, but which regard the fact from different aspects. The exact logician holds it to be, in itself, a defect in a logical system of expression, to afford different ways of expressing the same state of facts; although this defect may be less important than a definite advantage gained by it. The copulative proposition is in a similar way equivalent to a particular categorical. Thus, to say 'The man might not be able voluntarily to act otherwise than physical causes make him act, whether he try or not,' is the same as to say that there is a state of things hypothetically admissible in which a man tries to act one way and voluntarily acts another way in consequence of physical causes. As to hypotheticals ut nunc, they refer to no range of possibility, but simply to what is true, vaguely taken collectively.

Although it is thus plain that the action of the copula in relating the subject-term to the predicate-term is a secondary one, it is nevertheless necessary to distinguish between copulas which establish different relations between these terms. Whatever the relation is, it must remain the same in all propositional forms, because its nature is not expressed in the proposition, but is a matter of established convention. With that proviso, the copula may imply any relation whatsoever. So understood, it is the abstract copula of De Morgan (Camb. Philos. Trans., x. 339). A transitive copula is one for which the mood Barbara is valid. Schröder has demonstrated the remarkable theorem that if we use IS in small capitals to represent any one such copula, of which 'greater than' is an example, then there is some relative term r, such that the proposition 'S IS P' is precisely equivalent to 'S is r to P and is r to whatever P is r to.' A copula of correlative inclusion of one for which both Barbara and the formula of identity hold good. Representing any one such copula by is in italics, there is a relative term r, such that the proposition 'S is P' is precisely equivalent to 'S is r to whatever P is r to.' If the last proposition follows from the last but one, no matter what relative r may be, the copula is called the copula of inclusion, used by C. S. Peirce, Schröder, and others. De Morgan uses a copula defined as standing for any relation both transitive and convertible. The latter character consists in this, that whatever terms I and J may be, if we represent this copula by is in black = letter, then from 'I is J' it follows that 'J is I.' From these two propositions, we conclude, by Barbara, that 'I is I.' Such copulas are, for example, 'equal to,' and 'of the same colour as.' For any such copula there will be some relative term r, such that the proposition 'S is P' will be precisely equivalent to 'S is r to everything, and only to everything, to which P is r.' Such a copula may be called a copula of correlative identity. If the last proposition follows from the last but one, no matter what relative r may be, the copula is the copula of identity used by Thomson, Hamilton, Baynes, Jevons, and many others.

It has been demonstrated by Peirce that the copula of inclusion is logically simpler than that of identity.


Dialogism. A form of reasoning in which from a single premise a disjunctive, or alternative, proposition is concluded introducing an additional term; opposed to a syllogism, in which from a copulative proposition a proposition is inferred from which a term is eliminated.

Syllogism: All men are animals, and all animals are mortal; ∴ All men are mortal.

Dialogism: Some men are not mortal; ∴ Either some men are not animals, or some animals are not mortal.

Dimension. An element or respect of extension of a logical universe of such a nature that the same term which is individual in one such element of extension is not so in another. Thus, we may consider different persons as individual in one respect, while they may be divisible in respect to time, and in respect to different admissible hypothetical states of things, &c. This is to be widely distinguished from different universes, as, for example, of things and of characters, where any given individual belonging to one cannot belong to another. The conception of a multidimensional logical universe is one of the fecund conceptions which exact logic owes to O. H. Mitchell. Schröder, in his then second volume, where he is far below himself in many respects, pronounces this conception 'untenable.' But a doctrine which has, as a matter of fact, been held by Mitchell, Peirce, and others, on apparently cogent grounds, without meeting any attempt at refutation in about twenty years, may be regarded as being, for the present, at any rate, tenable enough to be held.

Dyadic relation. A fact relating to two individuals. Thus, the fact that A is similar to B, and the fact that A is a lover of B, and the fact that A and B are both men, are dyadic relations; while the fact that A gives B to C is a triadic relation. Every relation of one order of relativity may be regarded as a relative of another order of relativity if desired. Thus, man may be regarded as man coexistent with, and so as a relative expressing a dyadic relation, although for most purposes it will be regarded as a monad or non-relative term.

Index (in exact logic): see sub verbo.

Many other technical terms are to be found in the literature of exact logic.

Literature: for the study of exact logic in its more recent development, excluding probability, the one quite indispensable book is SCHRÖDER, Algebra d. Logik; and the bibliography therein contained is so exhaustive that it is unnecessary to mention here any publications previous to 1890. Schröder's pains to give credit in full measure, pressed down and running over, to every other student is hardly less remarkable than the system, completeness, and mathematical power of his work, which has been reviewed by C. S. PEIRCE in the Monist, vii. 19-40, 171-217. See also C. S. PEIRCE, Studies in Logic; Pop. Sci. Mo., xii. 1; and Proc. Amer. Acad. Arts and Sci., vii. 287. Cf. SCIENTIFIC METHOD. (C.S.P.)

Logical [Lat. logicalis, from logica, logic]: Ger. logisch; Fr. logique; Ital. logico. Irrespective of any facts except those of which logic needs to take cognizance, such as the facts of doubt, truth, falsity, &c.

Logical possibility is, according to usage, freedom from all contradiction, explicit or implicit; and any attempt to reform the inaccuracy would only bring confusion.

Logical necessity is the necessity of that whose contrary is not logically possible.

Logical induction is an induction based on examination of every individual of the class to which the examination relates. Thus, conclusions from a census are logical inductions. While this mode of inference is a degenerate form of induction, it also comes into the class of dilemmatic reasoning.

Logical truth is a phrase used in three senses, rendering it almost useless.

1. The harmony of a thought with itself. Most usually so defined, but seldom so employed. So far as this definition is distinct, it makes logical truth a synonym for logical possibility; but, no doubt, more is intended (Hamilton, Lects. on Logic, xxvii).

2. The conformity of a thought to the laws of logic; in particular, in a concept, consistency; in an inference, validity; in a proposition, agreement with assumptions. This would better be called mathematical truth, since mathematics is the only science which aims at nothing more (Kant, Krit. d. reinen Vernunft, 1st ed., 294).

3. More properly, the conformity of a proposition with the reality, so far as the proposition asserts anything about the reality. Opposed, on the one hand, to metaphysical truth, which is an affection of the ens, and on the other hand to ethical truth, which is telling what a witness believes to be true (Burgersdicius, Inst. Met., chap. xviii).

Logical parts and whole. Parts and whole of logical extension.

Logical reasoning. Reasoning in accordance with a LEADING PRINCIPLE (q.v.) which thorough analysis, discussion, and experience have shown must lead to the truth, in so far as it is relied upon. But what Aristotle understood by a logical demonstration may be seen in his De generatione animalium, Lib. II. cap. viii.

Logical presumption. A Wolffian term for synthetic reasoning, that is, induction and analogy; for hypothetic reasoning was not recognized as reasoning at all. The uniformity of nature is called the principle of logical presumption.

Logical division. Division into logical parts.

Logical distinctness. That distinctness which results from logical analysis.

Logical actuality. Kant, in the Logik by Jäsche (Einleitung, vii), defines logical actuality as conformity to the principle of sufficient reason, consisting of the cognition having reasons and having no false consequences; and he makes this, along with logical possibility, to constitute logical truth, which is thus used in its second sense. But in the Critic of the Pure Reason, in discussing the functions of judgments (1st ed., 75), he says that an assertoric proposition asserts logical actuality (Wirklichkeit, which Max Müller wrongly translates 'reality'), and makes this phrase synonymous with logical truth (which is thus used in its third, and proper, sense).

Logical definition. A strict definition by genus and specific difference. Ockham and his followers objected to the designation on the ground that the logician, as such, had no occasion to define any ordinary term, such as man (Tractatus logices, Pt. I. chap. xxvi). (C.S.P.)

Logical Diagram (or Graph): Ger. logische Figur; Fr. diagramme logique; Ital. diagramma logico. A diagram composed of dots, lines, &c., in which logical relations are signified by such spatial relations that the necessary consequences of these logical relations are at the same time signified, or can, at least, be made evident by transforming the diagram in certain ways which conventional 'rules' permit.

In order to form a system of graphs which shall represent ordinary syllogisms, it is only necessary to find spatial relations analogous to the relations expressed by the copula of inclusion and its negative and to the relation of negation. Now all the formal properties of the copula of inclusion are involved in the principle of identity and the dictum de omni. That is, if r is the relation of the subject of a universal affirmative to its predicate, then, whatever terms X, Y Z may be, Every X is r to an X; and if every X is r to a Y, and every Y is r to a Z, every X is r to a Z. Now, it is easily proved by the logic of relatives, that to say that a relation r is subject to these two rules, implies neither more nor less than to say that there is a relation l, such that, whatever individuals A and B may be.

If nothing is in the relation l to A without being also in the same relation l to B, then A is in the relation r to B; and conversely, that, If A is r to B, there is nothing that is l to A except what is l to B.

Consequently, in order to construct such a system of graphs, we must find some spatial relation by which it shall appear plain to the eye whether or not there is anything that is in that relation to one thing without being in that relation to the other. The popular Euler's diagrams fulfil one-half of this condition well by representing A as an oval inside the oval B. Then, l is the relation of being included within; and it is plain that nothing can be inside of A without being inside B. The relation of the copula is thus represented by the spatial relation of 'enclosing only what is enclosed by.' In order to represent the negation of the copula of inclusion (which, unlike that copula, asserts the existence of its subject), a dot may be drawn to represent some existing individual. In this case the subject and predicate ovals must be drawn to intersect each other, in order to avoid asserting too much. If an oval already exists cutting the space in which the dot is to be placed, the latter should be put on the line of that oval, to show that it is doubtful on which side it belongs; or, if an oval is to be drawn through the space where a dot is, it should be drawn though the dot; and it should further be remembered that if two dots lie on the boundaries of one compartment, there is nothing to prevent their being identical. The relation of negation here appears as 'entirely outside of.' For a later practical improvement see Venn, Symbolic Logic, chap. xi. (C.S.P.)

Logomachy [Gr., taken from the First Epistle of Paul to Timothy, vi. 4 noswn peri xhthseiV kai logomaciaV, doting about questions and strifes of words]: Ger. Logomachie, Wortstreitigkeit; Fr. logomachie; Ital. contesa di parole. A contention (in which it is not essential that two parties should be active) not professedly relating to the use of words and phrases, but in which proper care exercised to make the ideas clear will show the critic, either that there is no important difference between the position attacked and that defended, or if there is, that the argumentation does not relate to such points.

Theology and subjects connected with it, such as the freedom of the will, have been the great theatre of such war. At present it is still kept up in logic; and other branches of philosophy are not entirely freed from it. Disputes about the propriety of modes of speech, however hot and silly they may be, are not logomachy. (C.S.P.)

Major and Minor (extreme, term, premise, satz, &c., in logic): Ger. Ober- and Unter- (Begriff, &c.); Fr. majeur and mineur; Ital. maggiore and minore. The subject and predicate of the conclusion of a syllogism are called the extremes (ta akra, by Aristotle), because they are only brought together by the agency of the third term, called, on that account, the middle term (o mesoV oroV, Aristotle). Of the two extremes, the one that is the predicate of the conclusion is called the major extreme (to meizon akron, Aristotle), because in a universal affirmative proposition (the typical formal proposition) its breadth is the greater, while the subject of the conclusion is the minor extreme (to elatton akron, Aristotle).

Whether the expressions major term and minor term, for the major and minor extremes, are grammatically accurate or not, they are consecrated by usage through the scholastic period. The major and minor premises are respectively those which contain the major and minor extremes. Aristotle (I. Anal. Pr., ix) calls the former h proV to meizon akron protasiV, 'the proposition about the major extreme.' (C.S.P.)

Mark [AS. mearc, a bound]: Ger. Merkmal; Fr. marque, attribut; Ital. segno (contrassegno), nota. To say that a term or thing has a mark is to say that of whatever it can be predicated something else (the mark) can be predicated; and to say that two terms or things have the same mark is simply to say that one term (the mark) can be predicated of whatever either of these terms or things can be predicated.

The word translates the Latin nota. It has many practical synonyms, such as quality, mode, attribute, predicate, character, property, determination, consequent, sign. Most of these words are sometimes used in special senses; and even when they are used in a general sense, they may suggest somewhat different points of view from mark. (C.S.P., C.L.F.)

A great oversight which had vitiated the entire discourse of logicians about marks, and had prevented them from fully understanding what marks are, was corrected by Augustus de Morgan when he observed that any collection whatever of individuals has some mark common and peculiar to them. That it is so will appear when we consider that nothing prevents a list of all the things in that collection from being drawn up. Now, the mere being upon that list, although it has not actually been drawn up, constitutes a common and peculiar mark of those individuals. Of course, if anybody tries to specify a number of individuals that have no common and peculiar mark, this very specification confers upon their common and peculiar mark a new degree of actuality.

On the other hand, if two marks are common and peculiar to precisely the same collection of things, they may, for the ordinary purposes of formal logic, be looked upon as the same mark. For it is indifferent to formal logic how objects are marked, whether in a simpler or more complex way. We may, therefore, regard the two marks as constituting together a single mark. Marks, after all, are not the object of logical study; they are only fictitious aids to thought. (C.S.P.)

Material Fallacy. This term originated with Whately (Encyc. Metropolitana, i. 218 b). Whately's material fallacies are those in which the conclusion does follow from the premises. Therefore, excluding the multiple interrogation, which is no syllogism, of the rest of Aristotle's thirteen, only the ignoratio elenchi and the petitio principii are material. Cf. FALLACY (also for foreign equivalents).

Aldrich had modified Aristotle's division into fallacies in dictione and fallacies extra dictionem; making a division into Sophismata in forma argumenti (sicubi conclusio non legitime consequatur ex praemissis), and Sophismata in materia argumenti (sicubi legitime non tamen vere concludere videatur syllogismus). Under the latter head he placed the ignoratio elenchi, the non causa pro causa, the non sequitur, and the petitio principii. Whately's distinction is — whether from a theoretical or a practical point of view — by far the most important that can be drawn among fallacies; so that besides the reason of priority, which ought itself to be final, the needs of the logician forbid us to depart from Whately's definition. Some logicians do not admit material fallacies among the number of fallacies, but consider them to be faults of method (Hamilton, Lects. on Logic, xxvi; Ueberweg, Syst. d. Logik, §§ 126, 137). E.E. Constance Jones (Elements of Logic as a Science of Propositions, § xxvi) reduces them to formal fallacies. Hyslop (Elements of Logic, chap. xvii) uses the term material fallacy, quite unjustifiably, to include all fallacies due to something in the matter of reasoning. (C.S.P.)

Material Logic: Ger. materielle Logik; Fr. logique matérielle; Ital. logica materiale. Formal logic classifies arguments by producing forms in which, the letters of the alphabet being replaced by any terms whatever, the result will be a valid, probable, or sophistic argument, as the case may be; material logic is a logic which does not produce such perfectly general forms, but considers a logical universe having peculiar properties.

Such, for example, would be a logic in which every class was assumed to consist of a finite number of individuals; so that the syllogism of transposed quantity would hold good. In most cases material logic is practically a synonym of applied logic. But a system like Hegel's may also properly be termed material logic. The term originated among the English Occamists of the 14th century, who declared Aristotle's logic to be material, in that it did not hold good of the doctrine of the Trinity. (C.S.P.)

Mathematical Logic: Ger. (1) Logik der Mathematik; Fr. (1) logique des mathématiques; Ital. (1) logica della matematica. (1) The logical analysis of mathematics. (C.S.P.)

(2) SYMBOLIC LOGIC (q.v.).

Literature (to 1): the logic of arithmetic is treated by DEDEKIND in his Was sind und was sollen die Zahlen? (Eng. trans. in Essays on Number, 1901). See also the ninth lecture of the third volume of SCHRÖDER, Logik; and FINE, Number System of Algebra. For the logic of the calculus, see the second edition of JORDAN, Cours d'Analyse; also CLIFFORD, Theory of Metrics, in his Mathematical Papers; WEBER, Algebra; and the papers of G. CANTOR, some of which are contained in the Acta Mathematica, ii, and subsequent ones in the Mathematische Annalen, 15, 17, 20, 21, 23, 46, 49. LISTING'S papers on topical geometry are two, one in the Gött. Abhand., the other in the Gött. Nachr. Several of RIEMANN'S papers are valuable in a logical point of view. See also CAUCHY, Théorie des Clefs. PETERSEN, Methods and Theories, shows how to solve problems in elementary geometry. Cf. MATHEMATICS. (C.S.P.)

Matter and Form: Ger. Materie (Stoff) und Form; Fr. la matière et la forme; Ital. materia e forma. The word matter (Lat. materia, which was used to translate the Gr. ulh) is often employed where the more appropriate Greek word would be swma, corpus, body; or to upokeimenon, subjectum, or even h npostasiV, translated person in theology. Form (Lat. forma, used to translate the Gr. morfh and eidoV, though the latter is more exactly represented by species) is often employed where schma, figure, or tupoV, shape, would be near equivalents. The Greek expressions morfh, paradeigma, eidoV, idea, to ti esti, to ti hn einai are pretty nearly synonymous.

The distinction of matter and form was first made, apparently, by Aristotle. It almost involves his metaphysical doctrine; and as long as his reign lasted, it was dominant. Afterwards it was in disfavour; but Kant applied the terms, as he did many others drawn from the same source, to an analogous but widely different distinction. In many special phrases the Aristotelian and Kantian senses almost coalesce, in others they are quite disconnected. It will, therefore, be convenient to consider: (1) the Aristotelian distinction; (2) the Kantian distinction; and (3) special applications.

The Aristotelian distinction. Not only was the distinction originated by Aristotle, but one of the two conceptions, that of matter, is largely due to him. Indeed, it is perhaps true that the Greek word for matter in the sense of material, ulh, was never understood in that general sense before Aristotle came to Athens. For the first unquestionable cases of that meaning occur in certain dialogues of Plato, concerning which — though there are no dates that are not open to dispute — it seems to the present writer that it is as certain as any such fact in the history of Greek philosophy that the earliest of them was written about the time of Aristotle's arrival. It is true that, as Aristotle himself says, matter was the earliest philosophical conception. For the first Ionian philosophers directed their thoughts to the question what the world was made of. But the extreme vagueness of the notion with them is shown by their calling it h arch, the beginning, by the nonsense of the question, and by many more special symptoms. If the philosophical conception of matter distinguished the metaphysics of Aristotle, that of Plato had been no less marked by its extraordinary development of the notion of form, to which the mixed morality and questioning spirit of Socrates had naturally led up; the morality, because the form is the complex of characters that a thing ought to have; the questioning, because it drew attention to the difference between those elements of truth which experience brutally forces upon us, and those of which reason persuades us, which latter make up the form. But Aristotle's distinction set form, as well as matter, in a new light.

It must not be forgotten that Aristotle was an Asclepiad, that is, that he belonged to a family which for generation after generation, from prehistoric times, had had their attention turned to vital phenomena; and he is almost as remarkable for his capacity as a naturalist as he is for his incapacity in physics and mathematics. He must have had prominently before his mind the fact that all eggs are very much alike, and all seeds are very much alike, while the animals that grow out of the one, the plants that grow out of the other, are as different as possible. Accordingly, his dunamis is germinal being, not amounting to existence; while his entelechy is the perfect thing that ought to grow out of that germ. Matter, which he associates with stuff, timber, metal, is that undifferentiated element of a thing which it must possess to have even germinal being. Since matter is, in itself, indeterminate, it is also in itself unknowable; but it is both determinable by form and knowable, even sensible, through form. The notion that the form can antecede matter is, to Aristotle, perfectly ridiculous. It is the result of the development of matter. He looks upon the problem from the point of view of a naturalist. In particular, the soul is an outgrowth of the body.

The scholastics, who regarded Aristotle as all but infallible, yet to whom the ideas of a naturalist were utterly foreign, who were thoroughly theological in their notions, admitted that the soul was a form. But then, they had great difficulty with those opinions of their master which depended upon his conceiving of matter as more primitive than form. Their notions of form were rather allied to those of Plato. The mode of being that, in some sense, anteceded individual existence, they would have held to be one in which there was form without matter, if awe of Aristotle had not caused them to modify the proposition in one way or another. A question, for example, which exercised them greatly was, how the form was restricted to individual existence? For Aristotle there could not be any such question, because he did not conceive of a form taking on individuality, but of an undifferentiated matter taking on, or rather developing, form, and individuality, perhaps, with it (412 a, 7).

The Kantian distinction. Aristotle refuses to consider any proposition as science which is not universal. He does not go so far as to say that all knowledge involves synthesis, but he often approaches doing so. In particular, he holds that matter is something in itself beyond our knowledge, but the existence of which has to be assumed in order to synthetize the opposites that are involved in all change. He expressly defines that as the function of the conception of matter. With Kant, the view that all knowledge involves synthesis — various acts of synthesis one over another -- is vastly more developed; and he, too, employs the terms matter and form as called for by such synthesis. But it is curious that while with Aristotle it is matter that is the quasi-hypothesis imported into the facts that the mind may synthetize, with Kant, on the other hand, it is form which performs this function. The matter of cognition consists of those elements which are brutally and severally forced upon us by experience. By the form he means the rational or intelligible elements of cognition, which he wishes, as far as possible, to regard as independent contributions of the mind itself, which we have no right to suppose are duplicated by anything corresponding to them in the thing. For the Aristotelian, all pure matter is exactly alike, equally devoid of all predicates, while the forms make all the variety of the universe. For the Kantian, on the other hand, matter is the manifold, while the pure forms are the few different modes of unity. Nevertheless, the Kantians — indeed, Kant himself (see the Critic of the Pure Reason, 1st ed., 266) — argued that they were using the terms in their old and accepted sense. What enabled them to give some speciousness to their contention was the circumstance that during the full century and more of neglect of the Aristotelian doctrine that had intervened, certain secondary senses of the term matter, especially that of corporeal matter, and that of a species of corporeal matter, had become relatively prominent.

Special senses. Although there is only one first or primary matter, absolutely indeterminate, yet Aristotle often uses the term in a modified sense as that which is relatively indeterminate; so that the last or second matter is the same as the form. But these phrases are also used in quite other senses, which need not here be specially noticed. Matter being taken relatively, the same thing can have this or that as its matter in different respects; and so matter is distinguished into materia ex qua, in qua, and circa quam. Materia ex qua is the material; silver is the materia ex qua of a dime. Materia in qua is the subject in which the form inheres; materia circa quam is the object. Aquinas illustrates the distinction by virtue, which is a form, and, as such, has no materia ex qua; but it has a subject in which it inheres and an object upon which it is exercised. Aquinas introduced the term signate matter. Matter of composition, or proximate matter, is that of which a thing consists; matter of generation, or remote matter, that from which it is developed, as a seed or egg.

The varieties of form are so numerous that they may best be taken in alphabetical order.

Absolute form: form abstracted from matter.

Accidental form: an accident, or that the presence of which constitutes an accident; as music is the accidental form of the musician.

Advenient form: a form subsequent to the final form.

Apprehended form = apprehended SPECIES (q.v.).

Artificial form: a form superinduced by art.

Assistant form: an agent aiding in the realization of a form, especially of that whose essential character is to move; as the angel who turns the heavens round once every twenty-four hours, or the captain of a ship.

Astral form. According to Gilbert (De Magnete), phenomena of electricity are produced by a material effluvium, while the action of a magnet takes place directly at a distance. Whatever it may be then which constitutes the magnetic field, not being matter, must be called form. Gilbert names it forma prima radicalis et astralis.

Common form: a form belonging to a species.

Completive form: used by Aquinas in the sense of the last of the series of forms which gradually bring a thing to fully developed existence. By Aristotle called last form.

Composite form: the form of a collective whole, so far as it is different from its parts.

Corporeal form: a form of a corporeal nature. This is used by Aquinas, Summa Theol., pars I. qu. 1xv. art. 4. See Material form.

Disponent form: a form rendering matter apt to receive another, principal, form. Thus, dryness in wood disposes it to receive combustibility.

Elementary form: one of the four combinations of hot and cold with moist and dry which were supposed to characterize the four elements.

Exemplar form: an idea.

Final form: see Completive form.

General form: the form of a genus; as we should now say a generic form.

Immaterial form: a form which neither depends upon matter while it is being made nor after it is made; a term employed in the theological doctrine of creation.

Incorruptible form: a form not subject to corruption.

Individual form: in one of the theories of individuation, was a form which by existing in matter acquired the power of individuating another form.

Informant form: a form which is a part of the thing of which it is the form.

Inherent form: a form which can only exist in a state of inherence in matter.

Intellective form: the mind as form.

Intelligible form: see Sensible form.

Intermediate form: a form having a middle position between an elementary and a completive form.

Material form: a term of Scotus, who defines it as follows: 'Formam materialem dico esse omnem illam, quae ex natura sua necessario inclinatur naturaliter, ut sit actus materiae, sive sit substantialis, sive accidentalis' (Op. Oxon., IV. i. 1); 'Ideo dici potest tertio modo.' But elsewhere (ibid., 1 Post. qu. ii.) he distinguishes two senses of the term: 'Forma materialis potest intelligi dupliciter. Uno modo dicitur, quae educitur de potentia materiae, vel quia utitur organo corporeo in operando: et isto modo forma intellectiva non est forma materialis. Alio modo dicitur forma materialis, quia perfectio materiae, et isto modo anima intellectiva est forma materialis, ideo aliquam variationem potest accipere a materia, quam perficit, quia ex materia et forma fit vere unum.' Perhaps the most accessible book from which to gain a hint of the nature of the difficulty which gives rise to this distinction is Bridges' edition of what is called The Opus Majus of Roger Bacon, ii. 507-11, cap. ii.

Mathematical form: an object of mathematical contemplation, and the result of mathematical abstraction.

Metaphysical form: form in the philosophical sense.

Native or natural form, forma in natura exsistens, forma naturae, form of a nature, is a term going back to John of Salisbury (Opera, ed. Giles, v. 92), and closely connected, if not synonymous, with material form. Certain questions started by Aristotle in Book V of the Metaphysics (of which there is an admirable periphrastic translation by Grote, Aristotle, 2nd ed., 619 ff.) gave rise to discussions in which the doctrine was compared with Christian beliefs; and the natural form plays a considerable part in such discussions.

Bacon adopted the term forma naturae. He did not grossly depart from the received meaning of the term, but owing to his occupying himself with inquiries quite antipodal to those of the scholastics, the two parties did not understand one another. Bacon means the physical explanation of a phenomenon, its occult modus operandi. Among the followers of Bacon we, at first, hear a great deal about forms. Boyle wrote whole books about them. But the distinction of matter and form was not calculated to further such inquiries as theirs. It is adapted to expressing phenomena of life. It might be twisted to such a purpose as Gilbert put it to (see Astral form), but it was not suited to the mechanical philosophy of Boyle, and only led to wordy and fruitless discussions.

Participate form: a form considered as it is united with matter.

Preparatory form: a term used by Boyle where disponent form would be more technical. He says, 'The preparatory form is but (if I may so speak) a harbinger that disposes the matter to receive a more perfect form, which, if it be not to be succeeded by any other more noble, is entitled the specific form of that body; as in the embryo, the vegetative and the sensitive soul is but preparatory to the rational, which alone is said to be the specific form of man' (Free Considerations about Subordinate Forms).

Physical form: such forms as may form the object of physical inquiries. Of course, the term was very differently understood during scholastic times and in the 17th century. But the above definition covers both uses.

Primary form. There is no such well-recognized term of metaphysics; but a remark of William Gilbert leads us to suppose that medical men attached some meaning to it.

Principal form is that which per se constitutes a species. Called also specific form.

Radical form: see Astral form.

Sensible form. Though it chances that Aristotle nowhere distinguishes morfh into aisqhth and nohth, yet his followers did. Sensible forms are those which the outward senses distinguish; intelligible are those which the intellect alone can distinguish.

Significate form: a Thomistic term, a form distinguished by a name.

Simple form: form without matter. 'Forma simplex, quae est purus actus, est solus deus,' says St. Thomas.

Specific form: see Principal form.

Subsistent form: a form capable of existing separate from matter, as Aquinas holds that the angels and departed spirits are.

Substantial form: a form which constitutes a nature, i.e. a species or genus. Thus, the accidental form of a musician is music; but his substantial form is the rational soul which makes him a man. When men's thoughts became turned from theology to the investigation of physics, those who were animated by the new spirit found themselves confronted with objections based upon allegations of substantial forms. That these substantial forms, so used, were merely a hindrance to the progress of science, was quite plain to them. But the objections were urged with a logical accuracy, born of centuries of study, with which the new men were utterly incapable of coping. Their proper course would have been quietly to pursue their own inquiries, and leave the theologians to square their results with philosophy as best they could. But circumstances did not permit this. The theologians had the popular intelligence and the arm of power on their side; and when an apparent opposition arose, they naturally exerted themselves to put it down. Thus, the innovators were led to protest against these senseless and harmful substantial forms; and they had to formulate their objections to them — a business for which they were entirely unfitted. But since the discoveries of the physicists were plainly adding to man's knowledge and power, while their antagonists were simply obstructive, the former soon carried the day in the general opinion of mankind. The history proves that there was something vicious about the theological application of substantial forms; but it in no degree goes to show that the physicists accurately defined the objection to that application. In reviewing the arguments at the present day, when the position of the mechanical philosophers is becoming almost as obsolete as that of the scholastic doctors, we first note that when the new men denied that the substantial forms were 'entities,' what they really had in mind was, that those forms had not such a mode of being as would confer upon them the power dynamical to react upon things. The Scotists, for it was they upon whom, as being in possession of the universities, the brunt of the battle fell, had in fact never called the substantial forms 'entities, 'a word sounding like a Scotistic term, but in fact the mere caricature of such a term. But had they used the word, nothing more innocent than the only meaning it could bear for them could be imagined. To call a form an 'entity' could hardly mean more than to call it an abstraction. If the distinction of matter and form could have any value at all, it was the substantial forms that were, properly speaking, forms. If the Scotists could really specify any natural class, say man — and physics was at that time in no condition to raise any just doubt upon that score — then they were perfectly justified in giving a name to the intelligible characteristic of that class, and that was all the substantial form made any pretension to being. But the Scotists were guilty of two faults. The first — great enough, certainly, but relatively inconsiderable — was often referred to, though not distinctly analysed and brought home to them. It was that they were utterly uncritical in accepting classes as natural, and seemed to think that ordinary language was a sufficient guarantee in the matter. Their other and principal fault, which may with justice be called a sin, since it involved a certain moral delinquency, was that they set up their idle logical distinctions as precluding all physical inquiry. The physicists and Scotists, being intent upon widely discrepant purposes, could not understand one another. There was a tolerably good excuse for the physicist, since the intention of the Scotist was of an abstract and technical kind, not easily understood. But there was no other excuse for the Scotist than that he was so drugged with his metaphysics that ordinary human needs had lost all appeal to him. All through the 18th century and a large part of the 19th, exclamations against the monstrousness of the scholastic dogma that substantial forms were entities continued to be part of the stock-in-trade of metaphysicians, and it accorded with the prevalent nominalism. But nowadays, when it is clearly seen that physical science gives its assent much more to scholastic realism (limited closely to its formal statement) than it does to nominalism, a view of the history more like that here put forward is beginning to prevail.

In the following terms, mostly Kantian, prepositional phrases express the qualifications.

Form of corporeity: a very common term of scholasticism, originating with Avicenna, and used by Aquinas (Summa Theol., pars i. cap. 1xvi. art. 2), but more particularly by Scotus (in his great discussion Opus Oxon., IV. dist. xi. 9. 3, beginning 'De secundo articulo dico') and by all his followers. The point is, that the rational soul, being purely spiritual, cannot confer corporeity upon the human body, but a special form, the form of corporeity, is requisite. Suarez and others, generally Thomists, as well as Henry of Ghent, denied this on the ground that a species has but one form. Thus a great metaphysical dispute arose. It sprung from the study of the doctrine of transubstantiation. See Cavellus, Suppl. ad quaest. Scoti in De Anima, disp. i, which is in the Lyons ed. of Scotus, tom. ii.

Form of cognition, in Kant's doctrine, is that element of knowledge which the matter of experience must assume in order to be apprehended by the mind. Kant seems to have been thinking of legal forms which must be complied with in order to give standing before a court. So an English sovereign, in order to be crowned, must, as a 'matter of form,' swear to an intensity of loathing for Romish dogmas which he probably regards with great coolness. Kant's definitions are chiefly the following: — 

'In the phenomenon, that which corresponds to the impression of sense, I call the matter of it; while that which constitutes the fact that manifoldness of the phenomenon is intuited as ordered in certain relations, I call the form of the phenomenon' (Krit. d. reinen Vernunft, 1st ed., 20).

'All cognition requires a concept, be it as imperfect and dark as you will; and this, in respect to its form, is always a universal which serves as a rule' (ibid., 106).

'The transcendental unity of the synthesis of the imagination is the pure form of all possible cognition, through which, consequently, all objects of possible experience must a priori be represented' (ibid., 118).

'There are two factors in cognition; first, the concept by which any object is thought — that is, the category; and secondly, the intuition by which that object is given. For if the concept had had no corresponding intuition, it would be a thought, no doubt, as far as its form goes; but having no object, no cognition whatsoever [he means, whether true or false] of anything would be possible by it; since, so far as I should know, there would be nothing, and perhaps could be nothing, to which such a concept would be applicable' (2nd ed. of the Deduction of the Categories, § 22).

'It is not more surprising that the laws of phenomena in nature must agree with the understanding and its a priori form, i.e. with its power of combining any manifold, than that the phenomena themselves must agree with the a priori form of sensuous intuition. For just as phenomena have no existence in themselves, but are merely relative to the mind, as having senses, so laws do not exist in the phenomena, but are merely relative to the mind in which the phenomena inhere, that mind exercising understanding' (and see the rest of this passage, ibid., § 26).

Form of forms. Francis Bacon says 'the soul may be called the form of forms,' which would be a pretty conceit, were it not plagiarized from the serious doctrine of Aristotle: o nouV eidoV eidwn (432 a, 2).

The terms matter and form are used in certain peculiar ways in logic. Speaking materialiter, the matter of a proposition is said to be its subject and predicate, while the copula is its form. But speaking formaliter, the matter of a proposition is, as we familiarly say, the 'matter of fact' to which the proposition relates; or as defined by the scholastics, 'habitudo extremorum adinvicem.' The second tractate of the Summulae of Petrus Hispanus begins with the words: 'Propositionum triplex est materia; scilicet, naturalis, contingens, et remota. Naturalis est illa in qua praedicatum essentia subiecti vel proprium eius; ut, homo est animal; vel, homo est risibilis. Contingens est illa in qua praedicatum potest adesse et abesse subiecto praeter subiecti corruptionem; ut, homo est albus, homo non est albus. Remota est illa in qua praedicatum non potest convenire cum subiecto; ut, homo est asinus.'

Of a syllogism, the proximate matter is the three propositions; the remote, the three terms. The form, which ought to be the ergo, by the same right by which the copula is recognized as the form of the proposition, is said to be 'apta trium propositionum dispositio ad conclusionem ex praemissis necessario colligendam.' But Kant, in the Logik by Jäsche, § 59, makes the premises the matter, and the conclusion the form. (C.S.P.)

Maxim (in logic). A widely received general assertion or rule.

The earliest writers, so far as has been shown, to use maxima as a substantive were Albertus Magnus and Petrus Hispanus. The former (Post. Anal., lib. I. cap. ii) makes maximae constitute the seventh of thirteen classes of propositions which may be accepted, though they are uncertain, so that they differ widely from dignitates, or axioms. He says, 'Maximae propositiones opinantur esse quae non recipiuntur nisi in quantum sunt manifestae. Et putat vulgus commune et alii simplices et non periti quod sint primae ex sui veritate communicantes omnem intellectum; sicut est ista propositio, Mendacium est turpe,' &c. Hamilton quotes, but gives an unverifiable reference to, a sentence in which Albertus makes maxima another name for a dignitas. Petrus Hispanus (Summulae, v) says, 'Maxima est propositio qua non est altera prior neque notior'; and he divides commonplace into two kinds, called Maxim and Difference of Maxim. This phraseology was so generally followed that it is surprising that Prantl's attribution of it to Albert of Saxony (who simply copies the Summulae here, almost verbatim) should have found any acceptance. Blundevile and other early writers of logic in English take the word from the Summulae. It was also adopted into English law. The meaning now tends to return to that used by Albertus. Kant (Krit. d. reinen Vernunft, 1st ed., 666) defines a maxim of reason as a subjective principle derived not from the character of the object, but from the interest of reason in such perfection of cognition as may be possible; and in the Critic of the Practical Reason he endeavours to make out something analogous in that sphere. In the Logik by Jäsche (Einleitung III) he defines a maxim as an inward principle of choice between different ends. (C.S.P.)

Method and Methodology, or Methodeutic: Ger. Methodenlehre; Fr. méthodologie, théorie de la méthode; Ital. teoria dei metodi, metodologia. A branch of logic which teaches the general principles which ought to guide an inquiry.

Owing to general causes, logic always must be far behind the practice of leading minds. Moreover, for the last three centuries thought has been conducted in laboratories, in the field, or otherwise in the face of the facts, while chairs of logic have been filled by men who breathe the atmosphere of the seminary. The consequence is that we can appeal to few works as showing what methodology ought to be. The first book of Bacon's Novum Organum is well enough, as far as it goes, and was no doubt useful in its day. Senebier's L'Art d'observer is instructive. Comte's Philosophie positive accomplished something. Whewell's History of the Inductive Sciences and other works have the advantage of being written by a man of great power of investigation himself, who drew his doctrine from the facts of scientific history. Mill's System of Logic is, no doubt, of considerable value, although the author knew too little of science. There is hardly one of the illustrations of fine method adduced in his first edition which has not been refuted. Beneke's Logik in praktischer Absicht was not altogether without value. Of great value, also, is Jevon's Principles of Science. Pearson's Grammar of Science is a work of great force, but unfortunately too much influenced by certain philosophical ideas. Wundt devotes two of the three volumes of his Logik to Methodenlehre.

The traditional doctrine of method is confined chiefly to rules of definition and division, which teach an exactness of thought much needed, but are marked by the total absence of modern ideas. Cf. SCIENTIFIC METHOD, and EVIDENCE. (C.S.P.)

Middle Term (and Middle) [trans. of terminus medius, medium, used by Boethius to translate Aristotle's o mesoV oroV, to meson]: Ger. Mittelbegriff; Fr. terme moyen; Ital. mezzo termine, termine medio. The adjective mesoV is applied in Greek to a third object additional to two others, when the idea of intervening can hardly be detected. It is, therefore, perhaps needless to seek further for Aristotle's intention in calling that term, by the consideration of which two others are illatively brought into one proposition as its subject and predicate, the middle term, or middle. It is the most important factor of Aristotle's theory of reasoning.

The same word means little more than third in the phrase 'principle of excluded middle,' which is, indeed, often called principium exclusi tertii. See LAWS OF THOUGHT. On the other hand, something which partakes of each of two disparate natures, and renders them capable of influencing one another, is called a tertium quid (Aristotle's h trith ousia). (C.S.P., C.L.F.)

Mixed [Lat. mixtum, from miscere, to mix]: Ger. vermischt; Fr. composé; Ital. misto. (1) Mixed proof: a proof which is partly analytic, partly synthetic.

(2) Mixed mode: a mode compounded of simple ideas of several kinds, put together to make one complex one (Locke, Essay concerning Human Understanding, Bk. II. chap. xii. § 5). See MODE.

(3) Mixed power: a power at once active and passive, because the principle of change is in itself. (C.S.P.)

Mnemonic Verses and Words (in logic). Aids to memory in logic, of the sort described under MNEMONICS (q.v.). (J.M.B.)

1. Instrumenta novem sunt, guttur, lingua, palatum. Quattuor et dentes, et duo labra simul.

The following mnemonic verses are contained in the Summulae Logicales of Petrus Hispanus, but were older, perhaps very much older.

2. 'Quae?' ca. vel hyp., 'Qualis?' ne. vel aff., u. 'Quanta?' univ. par. in. velsing. [What is the substance of a proposition? categorical or hypothetical. What is its quality? negative or affirmative. What is its quantity? universal, particular, indefinite, or singular.]

3. Simpliciter Feci, convertitur Eva, per acci, A sto per contra: sic fit conversio tota. Asserit A, negat E, sed universaliter ambae; Asserit I, negat O, sed particulariter ambo.

[E and I are converted simply; E and A, per accidens; A and O, per contrapositionem.]

4. Prae, contradic.; post, contra.; prae postque, subalter.
Non omnis, quidam non; omnis non, quasi nullus;
Non nullus, quidam; sed 'nullus non' valet 'omnis';
Non aliquis, nullus; 'non quidam non' valet 'omnis';
Non alter, neuter; 'neuter non' praestat 'uterque.'

[Non placed before omnis or nullus gives the contradictory proposition; placed after, the contrary; both before and after, the subalternate.]

5. Primus, Amabimus; Edentuli que, secundus;
Tertius, Illiace; Purpurea, reliquus.
Destruit u totum, sed a confirmat utrumque;
Destruit e dictum, destruit i que modum.
Omne necessariat; impossible, quasi nullus;
Possible, quidam; quidam non, possibile non.
E dictum negat, i que modum, nihil a, sed u totum.

[The first syllable of each of the four vocables Amabimus, Edentuli, Illiace, Purpurea, is for the possible mode; the second for the contingent; the third for the impossible; the fourth for the necessary. The vowel a signifies that both mode And 'dictum' are to be taken assertorically; e, that the dictum is to be denied; i, that the mode is to be denied; u, that both mode and dictum are to be denied. Each word refers to a line or order of equipollent modal forms.]

6. Tertus est quarto semper contrarius ordo.
Sit tibi linea subcontraria prima secundae.
Tertius est primo contradictorius ordo.
Pugnat cum quarto contradicendo secundus.
Prima subest quartae vice particularis habens se.
Hanc habet ad seriem se lege secunda sequentem.

[The relation of 'Sortem impossible est currere' and 'Sortem necesse est currere' is that of contraries; they cannot be true at once. The relation 'Sortem possibile est currere' and 'Sortem possible est non currere' is that of subcontraries; they cannot be false at once. The relation of 'Sortem possibile est currere' and 'Sortem impossibile est currere' is that of contradictories. The relation of 'Sortem possibile est non currere' and 'Sortem necesse est currere,' is likewise that of contradictories. 'Sortem possibile est currere' follows from 'Sortem necesse est currere,' as does 'Sortem possibile est non currere' from 'Sortem impossibile est currere.']

7. Sub. prae. prima, secunda prae. bis, tertia sub. bis.

[The first figure contains the middle term as subject and predicate; the second, the middle as predicated twice; the third, the middle twice as subject.]

8. Barbara, Celarent, Darii, Ferio, Baralipton,
Celantes, Dabitis, Fapesmo, Frisesomorum.
Cesare, Camestres, Festino, Baroko, Darapti,
Felapton, Disamis, Datisi, Bokardo, Ferison.

[These are original names of the syllogistic moods, which there is no sufficient reason for abandoning. The direct moods of the first figure are recognizable by their containing no sign of conversion, s, p, or k; the indirect moods (or moods of the fourth figure) by their having those signs attached either to the third vowel or to the first two. In the second figure, one of the signs s, p is attached to the first vowel, or to the second and third, or k is attached to the second. In the names of the moods of the third figure, s or p is attached to the second vowel, or to the first and third, or k to the first. There are also names for syllogism with weakened conclusions or strengthened premises, as well as for indirect moods of the first figure considered as belonging to a fourth. But the above rules will enable a reader to identify them. Thus, Bramantip can be nothing but Baralipton; while Barbari is Barbara with a weakened conclusion. Camenes can be nothing but Celantes; Dimaris nothing but Dabitis; Fesapo nothing but Fapesmo; Fresison nothing but Frisesomorum. A writer who introduces an m into the name of a mood containing an s or p only after its third vowel, or who omits m from the name of a mood having s or p after the first and second vowels, uses the fourth figure.]

9. Simpliciter vult s, verti p vero per acci. M vult transponi, k per impossibile duci.

Servat maiorem variatque secunda minorem;

Tertia maiorem variat servatque minorem.

[s, in the name of a mood, shows that the proposition denoted by the preceding vowel is, in a preferred mode of reduction, to be converted simply; p, that it is to be converted per accidens; m shows that the premises are to be transposed; k, that the preferred reduction is by reduction of the contradictory of the conclusion to an absurdity, this contradictory of the conclusion being, in the second figure, put in place of the minor premise (the major being retained), and in the third figure in the place of the major (the minor being retained).]

A great number of other memorial words and verses have been proposed by logicians. (C.S.P.)

Modality [Lat. modus; see MODE]: Ger. Modalität; Fr. modalité; Ital. modalità. There is no agreement among logicians as to what modality consists in; but it is the logical qualification of a proposition or its copula, or the corresponding qualification of a fact or its form, in the ways expressed by the modes possibile, impossibile, contingens, necessarium.

Any qualification of a predication is a mode; and Hamilton says (Lects. on Logic, xiv) that 'all logicians' call any proposition affected by a mode a modal proposition. This, however, is going much too far; for not only has the term usually been restricted in practice, from the age of Abelard, when it first appeared, until now, to propositions qualified by the four modes 'possible,' 'impossible,' 'necessary,' and 'contingent,' with only occasional extension to any others, but positive testimonies to that effect might be cited in abundance.

The simplest account of modality is the scholastic, according to which the necessary (or impossible) proposition is a sort of universal proposition; the possible (or contingent, in the sense of not necessary) proposition, a sort of particular proposition. That is, to assert 'A must be true' is to assert not only that A is true, but that all propositions analogous to A are true; and to assert 'A may be true' is to assert only that some proposition analogous to A is true. If it be asked what is here meant by analogous propositions, the answer is — all those of a certain class which the conveniences of reasoning establish. Or we may say the propositions which in some conceivable state of ignorance would be indistinguishable from A. Error is to be put out of the question; only ignorance is to be considered. This ignorance will consist in its subject being unable to reject certain potentially hypothetical states of the universe, each absolutely determinate in every respect, but all of which are, in fact, false. The aggregate of these unrejected falsities constitute the 'range of possibility,' or better, 'of ignorance.' Were there no ignorance, this aggregate would be reduced to zero. The state of knowledge supposed is, in necessary propositions, usually fictitious, in possible propositions more often the actual state of the speaker. The necessary proposition asserts that, in the assumed state of knowledge, there is no case in the whole range of ignorance in which the proposition is false. In this sense it may be said that an impossibility underlies every necessity. The possible proposition asserts that there is a case in which it is true.

Various subtleties are encountered in the study of modality. Thus, when the thinker's own state of knowledge is the one whose range of ignorance is in question, the judgments 'A is true' and 'A must be true' are not logically equivalent, the latter asserting a fact which the former does not assert, although the fact of its assertion affords direct and conclusive evidence of its truth. The two are analogous to 'A is true' and 'A is true, and I say so'; which are readily shown not to be logically equivalent by denying each, when we get 'A is false' and 'If A is true, I do not say so.'

In the necessary particular proposition and the possible universal proposition there is sometimes a distinction between the 'composite' and 'divided' senses. 'Some S must be P,' taken in the composite sense, means that there is no case, in the whole range of ignorance, where some S or other is not P; but taken in the divided sense, it means that there is some S which same S remains P throughout the whole range of ignorance. So 'Whatever S there may be may be P,' taken in the composite sense, means that there is, in the range of ignorance, some hypothetic state of things (or it may be the unidentifiable true state, though this can hardly be the only such case) in which there either is no S, or every S there is is P; while in the divided sense, it means that there is no S at all in any hypothetic state but what is some hypothetic state or other is P. When there is any such distinction, the divided sense asserts more than the composite in necessary particular propositions, and less in possible universal. But in most cases the individuals do not remain identifiable throughout the range of possibility, when the distinction falls to the ground. It never applies to necessary universal propositions or to possible particular propositions.

Some logicians say that 'S may be P' is not a proposition at all, for it asserts nothing. But if it asserted nothing, no state of facts could falsify it, and consequently the denial of it would be absurd. Now let S be 'some self-contradictory proposition,' and let P be 'true.' Then the possible proposition is 'Some self-contradictory proposition may be true,' and its denial is 'No self-contradictory proposition can be true,' which can hardly be pronounced absurd. It is true that those logicians usually take the form 'S may be P' in the copulative sense 'S may be P, and S may not be P,' but this only makes it assert more, not less. The possible proposition, then, is a proposition. It not only must be admitted among logical forms, if they are to be adequate to represent all the facts of logic, but it plays a particularly important part in the theory of science. See SCIENTIFIC METHOD. At the same time, according to the view of modality now under consideration, necessary and possible propositions are equipollent with certain assertory propositions; so that they do not differ from assertory propositions as universal and particular propositions differ from one another, but rather somewhat as hypothetical (i.e. conditional, copulative, and disjunctive), categorical, and relative propositions differ from one another — perhaps not quite so much.

According to this view, logically necessary and possible propositions relate to what might be known, without any knowledge whatever of the universe of discourse, but only with a perfectly distinct understanding of the meanings of words; geometrically necessary and possible propositions, to what a knowledge of the properties of space does or does not exclude; physical necessity, to what a knowledge of certain principles of physics does or does not exclude, &c. But when we say that of two collections one most be correspondentially greater than the other, but each cannot be correspondentially greater than the other, it has not been shown how this kind of necessity can be explained on the above principles.

The earliest theory of modality is Aristotle's, whose philosophy, indeed, consists mainly in a theory of modality. The student of Aristotle usually begins with the Categories; and the first thing that strikes him is the author's unconsciousness of any distinction between grammar and metaphysics, between modes of signifying and modes of being. When he comes to the metaphysical books, he finds that this is not so much an oversight as an assumed axiom; and that the whole philosophy regards the existing universe as a performance which has taken its rise from an antecedent ability. It is only in special cases that Aristotle distinguishes between a possibility and an ability, between a necessity and a constraint. In this, he is perhaps nearer the truth than the system of equipollencies set forth above.

Kant seems to have been the first to throw any light upon the subject. To the old distinction between logical and real possibility and necessity, he applied two new pairs of terms, analytic and synthetic, and subjective and objective. The following definitions (where every word is studied) certainly advanced the subject greatly: — 

'1. Was mit den formalen Bedingungen der Erfahrung (der Anschauung und den Begriffen nach) übereinkommt, ist möglich.

'2. Was mit den materialen Bedingungen der Erfahrung (der Empfindung) zusammenhängt, ist wirklich.

'3. Dessen Zusammenhang mit dem Wirklichen nach allgemeinen Bedingungen der Erfahrung bestimmt ist, ist (existirt) nothwendig' (Krit. d. reinen Vernunft, 1st ed., 219).

Kant holds that all the general metaphysical conceptions applicable to experience are capable of being represented as in a diagram, by means of the image of time. Such diagrams he calls 'schemata.' The schema of the possible he makes to be the figure of anything at any instant. The schema of necessity is the figure of anything lasting through all time (ibid., 144, 145). He further states (ibid., 74, footnote; Jäsche's Logik, Einl. ix, and elsewhere) that the possible proposition is merely conceived but not judged, and is a work of the apprehension (Verstand); that the assertory proposition is judged, and is, so far, a work of the judgment; and that the necessary proposition is represented as determined by law, and is thus the work of the reason (Vernunft). He maintains that his deduction of the categories shows that, and how, the conceptions originally applicable to propositions can be extended to modes of being — constitutively, to being having reference to possible experience; regulatively, to being beyond the possibility of experience.

Hegel considers the syllogism to be the fundamental form of real being. He does not, however, undertake to work over, in the light of this idea, in any fundamental way, what is ordinarily called logic, but which, from his point of view, becomes merely subjective logic. He simply accepts Kant's table of functions of judgment, which is one of the most ill-considered performances in the whole history of philosophy. Consequently, what Hegel says upon this subject must not be considered as necessarily representing the legitimate outcome of his general position. His followers have been incompetent to do more. Rosenkranz (Wissenschaft d. logischen Idee) makes modality to represent the superseding of the form of the judgment and to be the preparation for that of the syllogism. In the Encyclopädie, Hegel's last statement, §§ 178-80, we are given to understand that the judgment of the Begriff has for its contents the totality (or, say, conformity to an ideal). In the first instance, the subject is singular, and the predicate is the reflection of the particular object upon the universal. That is, this or that object forced upon us by experience is judged to conform to something in the realm of ideas. But when this is doubted, since the subject does not, in itself, involve any such reference to the ideal world, we have the 'possible' judgment, or judgment of doubt. But when the subject is referred to its genus, we get the apodictic judgment. But Hegel had already developed the ideas of possibility and necessity in the objective logic as categories of Wesen. In the Encyclopädie the development is somewhat as follows: Wirklichkeit is that whose mode of being consists in self-manifestation. As identity in general (the identity of Sein and Existenz) it is, in the first instance, possibility. That is to say, apparently, bare possibility, any fancy projected and regarded in the aspect of a fact. It is possible, for example, that the present Sultan may become the next Pope. But in the second movement arise the conceptions of the Zufällig, Aeusserlichkeit, and 'condition.' The Zufällig is that which is recognized as merely possible: 'A may be, but A may not be'; but it is also described by Hegel as that which has the Grund, or antecedent of its being, in something other than itself. The Aeusserlichkeit seems to be the having a being outside the ground of its being — an idea assimilated to caprice. That which such Aeusserlichkeit supposes outside of itself, as the antecedent of its being, is the presupposed condition. The third movement gives, in the first instance, 'real possibility.' In this we find the conceptions of 'fact' (Sache), 'activity' (Thätigkeit), and 'necessity.'

Lotze and Trendelenburg represent the first struggles of German thought to rise from Hegelianism. The most remarkable characteristic of Lotze's thought is, that he not only sees no urgency for unity of conception in philosophy, but holds that such unity would inevitably involve a falsity. He represents a judgment as a means of apprehending becoming, in opposition to the concept, which apprehends being; but he says that the business of the judgment is to supply the cement for building up concepts. Accordingly, he has no doctrine of modality as a whole, but merely considers three cases, between which he traces no relation. Necessity may arise either out of the universal analytic judgment, the conditional judgment, or the disjunctive judgment. By the 'judgment' is meant the meaning of a proposition. Lotze finds that the meaning of the analytical judgment is illogical, since it identifies contraries. However, the meaning of this meaning is justified by its not meaning to mean that the terms are identical, but only that the objects denoted by those terms are identical. The analytic proposition is, therefore, admissible, because it is practically meant to mean a particular proposition, that is, one in which the predicate is asserted of all the particulars. And the justification of the proposition, whose use was to be to connect elements of terms, is that, meant not as it is meant, but as it is meant to be meant, these elements are identical and do not need to be connected. In this way Lotze vindicates the necessity of the analytical categorical proposition. Coming next to conditionals, by thought of the same order, he finds that, assuming that the universe of real, intelligible objects is 'coherent,' we may be justified in asserting that the introduction of a condition X into a subject S gives rise to a predicate P as an analytical necessity; and for this purpose, when it is once accomplished, it does not matter whether the ladder of the assumption of coherence remains or is taken away. Lotze treats the disjunctive proposition last, as if it were of a higher order, following Hegel in this respect. But what was excusable for Hegel is less so for Lotze, since he himself had signalized the significance of impersonal propositions, such as 'it rains,' 'it thunders,' 'it lightens,' whose only subject is the universe. Now, if there is any difference between 'If it lightens, it thunders,' and 'Either it does not lighten or it thunders,' it is that the latter considers the actual state of things alone, and the former a whole range of other possibilities. However, Lotze considers last the propositional form 'S is P1 or P2 or P3.' Properly, this is not a disjunctive proposition, but only a proposition with a disjunctive predicate. Lotze considers it a peculiar form, because it cannot be represented by an Euler's diagram, which is simply a blunder. The necessity to which it gives rise must, therefore, either be the same as the conditional necessity, or else differ from it merely by greater simplicity. For other sound objections to Lotze's theory see Lange, Logische Studien, ii.

Trendelenburg (Logische Untersuch., xiii) maintains that possibility and necessity can only be defined in terms of the antecedent (Grund), though me might, perhaps, object to the translation of Grund by so purely formal a word as 'antecedent,' notwithstanding its harmony with Aristotle. If all conditions are recognized, and the fact is understood from its entire Grund, so that thought quite permeates being — a sort of phrase which Trendelenburg always seeks — there is 'necessity.' If, on the other hand, only some conditions are recognized, but what is wanting in Grund is made up in thought, there is 'possibility.' In itself, an egg is nothing but an egg, but for thought it may become a bird. Trendelenburg will, therefore, neither admit, with Kant, that modality is originally a mere question of the attitude of the mind, nor with Hegel, whom he criticizes acutely, that it is originally objective.

Sigwart, who holds that logical questions must ultimately be decided by immediate feeling, and that the usages of the German language are the best evidence of what that feeling is, denies that the possible proposition is a proposition at all, because it asserts nothing. He forgets that if a proposition asserts nothing, the denial of it must be absurd, since it must exclude every possibility. Now, the denial of 'I do not know but that A may be true' is 'I know A is not true,' which is hardly absurd. Sigwart, it is true, in accordance with usages of speech, takes 'A may be true' in what the old logicians called the sensus usualis, that is, for the copulative proposition 'A may be true, and further A may be not true.' But this does not make it assert less, but more, than the technical form. In regard to the necessary proposition, Sigwart, following his guide, the usages of speech, finds that 'A must be true,' asserts less than 'A is true,' so that from the latter the former follows, but not at all the latter from the former. This may be true for the usages of German speech, just as such phrases as 'beyond every shadow of doubt,' 'out of all question,' and the like, in our vernacular commonly betray the fact that there is somebody who not only doubts and questions, but flatly denies, the proposition to which they are attached. Bradley accepts the sensational discovery of Sigwart.

Lange (loc. cit.) thinks the matter is put in the clearest light by the logical diagrams usually attributed to Euler, but really going back to Vives. 'We, therefore, here again see,' he says, 'how spatial intuition, just as in geometry, verifies (begründet) a priority and necessity.' (C.S.P.)

Modification and Variation (mental). The same distinction between these terms is recommended as that given under MODIFICATION (in biology). Cf. VARIATION. (J.M.B.)

Modulus [Lat. modus, a mode]. (1) Proposed by Schröder (Ger. Modul; Fr. not in use; Ital. modulo, suggested  — E.M.) for the four relative terms upon which the logic of dual RELATIVES (q.v.) hinges; namely, 'Not,' 'Same as,' Excluded from a universe containing,' and 'With, or within a universe containing.'

These terms were first called by Peirce the 'definite dual relatives of second intention'; he now think it might be well to term these the four 'cardinals,' or four cardinal dual relatives.

Literature: PEIRCE, in Studies in Logic by Members of the Johns Hopkins University, 191, and Amer. J. Math., iii. 47; SCHRÖDER, Algebra d. Logik, iii. 117.


Modus ponens and Modus tollens [Lat.]. Two ways of reasoning from a conditional proposition or consequence. The modus ponens from the consequence and the antecedent infers the consequent; the modus tollens from the consequence and the falsity of the consequent infers the falsity of the antecedent, thus:

Modus Ponens. If A is true, C is true; A is true; ∴ C is true.
Modus Tollens. If A is true, C is true; C is false; ∴ A is false.

A third way of reasoning, namely, from the truth of the antecedent and falsity of the consequent to the falsity of the consequence, is generally overlooked. See HYPOTHETICAL (syllogism). (C.S.P.)

Monad (Monadism, Monadology) [Gr. monaV, unit]: Ger. Monade; Fr. monade; Ital. monade. In ancient philosophy, the unit in arithmetic, or unity as opposed to duality; it figures in this sense in the numerical speculations of the Platonic school and the later Pythagoreans. The special case of the number two, considered as unit or constituent of being, was known to the Pythagoreans as the Dyad (for Zenocrates' doctrine of the Dyad see ONE). (A.S.P.P.- J.M.B.)

(1) With the Pythagoreans, the monad was the number one considered, as well as we can make out, as the first creative deity (Zeller).

(2) In other Greek schools a monad is simply an individual. With the Atomists, an atom.

(3) In the philosophy of Leibnitz a monad is a being pursuing its development according to an inward law, in pre-established harmony with other beings. The idea may be illustrated by two pendulums, each moving according to a formula of its own. This illustration is used by Leibnitz himself. This theory has been resuscitated by Renouvier (La Nouvelle Monadologie, Paris, 1898).

(4) In the logic of RELATIVES (q.v.), a proposition with one term left blank, to be filled in if the proposition is to be completed. In chemistry: a radicle with one free bond. (C.S.P.)

Multitude (in mathematics) [Lat. multitudo]: Ger. Mächtigkeit, Cardinalzahl; Fr. puissance; Ital. moltitudine. That relative character of a collection which makes it greater than some collections and less than others. A collection, say that of the A's, is greater than another, say that of the B's, if, and only if, it is impossible that there should be any relation r, such that every A stands in the relation r to a B to which no other A is in the relation r.

The precise analysis of the notion is due to G. Cantor, whose definition is, however, a little different in its mode of expression, since it is more abstract. He defines the character in these words: 'By Mächtigkeit or cardinal number of a collection (Menge) M, we mean the universal concept, which by the help of our active faculty of thought results from the collection M by abstraction from the characters of the different members (Elemente) of that collection and from the order in which they are given (Gegebensein).

A cardinal number, though confounded with multitude by Cantor, is in fact one of a series of vocables the prime purpose of which, quite unlike any other words, is to serve as an instrument in the performance of the experiment of counting; these numbers being pronounced in their order from the beginning, one as each member of the collection is disposed of in the operation of counting. If the operation comes to an end by the exhaustion of the collection, the last cardinal number pronounced is applied adjectivally to the collection, and expresses its multitude, by virtue of the theorem that a collection the counting of which comes to an end, always comes to an end with the pronunciation of the same cardinal number.

If the cardinal numbers are considered abstractedly from their use in counting, simply in themselves, as objects of mathematical reasoning, stripped of all accidents not pertinent to such study, they become  indistinguishable from the similarly treated ordinal numbers, and are then usually called ordinal numbers by the mathematico-logicians. There is small objection to this; yet it is to be remarked that they are ordinal in different senses in grammar and in the logic of mathematics. For in grammar they are called ordinal as being adapted to express the ordinal places of other things in the series to which those things belong; while in the logic of mathematics the only relevant sense in which they are ordinal is as being defined by a serial order within their own system. The definition of this order is not difficult; but the syntax of ordinary language does not lend itself to the clear expression of such relations in the manner in which they ought to be expressed in order to bring out their logical character. It must, therefore, be here passed by. In fact, none of the doctrines of logic can be satisfactorily expressed under the limitations here imposed, however simple they may be. The doctrine of ordinal numbers is by Dedekind (Was sind und was sollen die Zahlen?) made to precede that of the cardinal numbers; and this is logically preferable, if hardly so imperative as Schröder considers it.

The doctrine of the so-called ordinal numbers is a doctrine of pure mathematics; the doctrine of cardinal numbers, or, rather, of multitude, is a doctrine of mathematics applied to logic. The smallest multitude is most conveniently considered to be zero; but this is a question of definition. A finite collection is one of which the syllogism of transposed quantity holds good. Of finite collections, it is true that the whole is greater than any part. It is singular that this is often taken as the type of an axiom, although it has from early times been a matter of familiar knowledge that it is not true of infinite collections. Every addition of one increases a finite multitude. An infinite collection cannot be separated into a lesser collection of parts all smaller than itself.

The multitude of all the different finite multitudes is the smallest infinite multitude. It is called the denumeral multitude. (Cantor uses a word equivalent to denumerable; but the other form has the advantage of being differentiated from words like enumerable, abnumerable, which denote classes of multitudes, not, like denumeral, a single multitude.) Following upon this is a denumeral series of multitudes called by C. S. Peirce the first, second, &c. abnumerable multitudes. Each is the multitude of possible collections formed from the members of a collection of the next preceding multitude. They seem to be the same multitudes that are denoted by Cantor as Alephs. The first of them is the multitude of different limits of possible convergent series of rational fractions, and therefore of all the quantities with which mathematical analysis can deal under the limitations of the doctrine of limits. (The imaginaries do not increase the multitude.) What comes after these is still a matter of dispute, and is perhaps of inferior interest. The transition to continuity is, however, a matter of supreme importance for the theory of scientific method; nor is it a very complicated matter; but it cannot be stated under the limitations of expression here imposed upon us. (C.S.P., H.B.F.)

Literature: see NUMBER.

Name (in logic). Two quite different sorts of terms are called in logic names. A proper name serves to call to mind an individual object of experience well known both to the speaker and hearer (for if the object is not known to the hearer it is only just beginning to fulfil for him the function of a proper name), and to show that it is that object concerning which information is furnished or desired. Many proper names are names of collective individuals; and a few are grammatically plural, as the Gracchi. A common name, usually now called a class-name in logic, though common name is better, has a signification as well as a denotation. That is to say, it conveys the idea that whatever it may be that is spoken of it is of a certain indicated general description, which may be in some sense negative.

Abstract names are common names of fictitious objects which correspond to predicates. At first sight they appear to be mere convenient superfluities; for to say that opium has a soporific virtue, is precisely the same thing as to say that opium puts people to sleep. But closer examination shows that abstract words enable us to express relations which could not otherwise be expressed. A relation is something true of a set of objects. But abstractions enable us to express a fact true of a set of sets of objects. Every collective name is an abstract name; and it would be a want of discrimination to say that numbers are superfluities. Moreover, when we see what the true nature of abstract names is, we must confess that their objects may be just as real as the objects of concrete names. They are fictitious only in the sense of having been made up out of concrete names. An abstract name may be regarded as the name of a fictitious individual; and when this individual is perfectly indescribable, like the quality of a simple sensation, the abstract name may perhaps be more like a proper name than like a common name.

Names are divided into names of first imposition and names of second imposition, which latter are names applicable to words, as pronoun, conjunction, &c. The precise definition is given by Ockham, Logica, Pars I, cap. xi. Names are also divided into names of first and of second intention. See TERM. (C.S.P.)

Necessary (in logic): Ger. nothwendig; Fr. nécessaire; Ital. necessario. That is necessary which not only is true, but would be true under all circumstances.

Something more than brute compulsion is, therefore, involved in the conception; there is a general law under which the thing takes place. Thus necessity, in the philosophical sense, is quite opposed to any 'Noth' that 'kennt kein Gebot.' Springing from law, and thus being essentially rational, it would perhaps be more accurately described as persuasive than as compulsive.

The Stoics defined the necessary as 'that which, being true, is not susceptible of becoming false, or, if it be so, is prevented from ever becoming false' (Diog. Laer., vii. 75). Kant defines the necessary as that which is a priori certain (Krit. d. reinen Vernunft, 1st ed., 125).

Necessary adjunct: a phrase which a very improper usage makes to signify a property, that is, an inessential predicate, not only belonging at all times to every individual of the species of which it is a necessary adjunct, but further, belonging to nothing else.

Necessary cause: one which acts by a necessity of its nature and is not free.

Necessary object, says Kant, is one which is determined according to concepts by the connection of perceptions (Krit. d. reinen Vernunft, 1st ed., 234).

Necessary sign: a sure indication. (C.S.P.)

Necessity [Lat. necessitas]: Ger. Nothwendigkeit; Fr. nécessité; Ital. necessità. (1) The state or condition that cannot be otherwise than it is; that must be just as it is.

(2) The principle in virtue of which the condition of the universe as a whole, or any particular part of it, is rendered, both as to its existence and quality, inevitable. Opposed to both freedom and chance, but especially, in its strictly philosophical use, to CHANCE (q.v.) or contingency. That which has the property of necessity is said to be necessary.

It is frequently used to designate the chief principle of those philosophies which admit only the principle of cause and effect, and which deny purposiveness to the universe. Technically, various forms of it have been recognized. (1) Logical (also metaphysical) necessity: the necessity of thought in virtue of which a truth, either immediate or inferential, must be conceived in such and such a manner; thus freedom itself would be a logical necessity if it followed, in accordance with the principles of identity and non-contradiction, from conceded premises. (2) Mathematical necessity: the similar logical relationship of parts of a demonstration or construction in mathematical reasoning. (3) Physical (also natural) necessity: that which arises from laws of nature or which arises in the course of nature from the principle of causation: mechanism, the 'reign of law'; invariable sequence, according to modern writers, e.g. J.S. Mill. (4) Moral necessity: that required by moral law, by the moral order of the universe; that which follows from the nature of God as a moral governor; also used in a narrower sense, as equivalent to 'practical' necessity, which is neither logical nor physical, but the result of a certain need or demand regarded as of fundamental importance (see POSTULATE).

These distinctions we owe directly to Leibnitz, and they are most fully developed in his Théodicée. According to him there are three main types. (a) Metaphysical, logical, geometrical: that which cannot be otherwise than as it is without self-contradiction; absolute necessity. (b) Physical necessity: that of the order of nature, which might conceivably be otherwise, but which follows from the will of God, who has chosen the best world; hypothetical necessity. (c) Moral necessity: that which animates a moral being, even God himself, in the choice of good. Since a perfectly moral being would have a perfectly adequate conception of the good, it would by moral necessity choose it. In this sense, physical necessity depends upon moral necessity. The term is also used in a strictly logical sense, equivalent to APODICTIC (q.v.), and also to designate the opposite of those theories which assert free will (necessitarianism: see DETERMINISM, and WILL).

In the Pre-Socratics, necessity was a quasi-mythical expression for the law or order of the cosmos, as in the teaching of Parmenides that the goddess at the centre of the world is Necessity — an (apparently) Pythagorean conception which finds expression in the myth of Er (Plato, Rep., Bk. X), where the entire universe is made to revolve upon an axis of necessity. Heraclitus used the idea (in the form of destiny) to account for the fact that a certain balance and system is observed in all change. With the Atomists (Leucippus) it becomes (anagkh) a definite philosophical concept; the atoms, darting about at random, impinge upon one another; from the aggregations thus formed, there is, of necessity, a whirling motion set up. With Plato (aside from incidental and non-technical use of it as equivalent to the force of proof and demonstration) necessity is the co-author, with nouV, of the sensible world; as irrational it is blind, indifferent to good, since nouV alone is the principle of ends, or of the good, and hence that which keeps the world in a state of partial non-being and which prevents its arriving at completion (Timaeus, 48, 56, 68). Aristotle repeats the same idea (De An. part., IV. ii. 677). Matter resists form, and thus hinders NATURE (q.v.) from arriving at its actualization. (The idea seems to be that in part matter lends itself to the realization of purposes, but in part has an impetus of its own which is quite indifferent to ends.) In this indifference matter is thus contingent — it may or may not present certain traits. As such it is tuch, chance; so that necessity in the physical sense, and chance in the teleological, are practically one and the same thing. Hence, in his logical writings necessity has quite another meaning. Of future events, we cannot make a necessary assertion; the general tendency of nature may be thwarted by chance. Hence our judgment is not of determinate truth. On the other hand, of universals, of past events, &c., any judgment is either necessarily true or false. Here the tendency comes out to identify necessity with the immanent logical rationale or any subject, that from which perfectly definite consequences follow. The Stoics fuse the various senses of necessity —  that of (a) the source of physical world-order, (b) the universal of reason form which determined conclusions result, and (c) the natural (or temporal) causal antecedent (Zeller, Stoics, Epicureans, and Sceptics, 170-82, and Windelband, History, 181). Since the Atomists did not work out their own idea systematically, and even presupposed a more or less random movement upon which necessity supervened, we may fairly regard the Stoics as the authors of the conviction that everything, everywhere, is controlled by necessity admitting of no exception — in other words, of the idea of the universality of natural causation, which is fate. This conception is common to what is called fatalism, also, in oriental philosophies: the hypothesis of a fixed and immutable world decree.

Spinoza carries the fusion still further by expressly identifying the whole causal relationship with the logical or mathematical — the world follows from the nature of God by the same necessity that various truths follow from a geometrical definition. (It was partly in reaction from Spinoza that Leibnitz made the distinctions referred to above.) It was characteristic of the whole rationalistic school (see RATIONALISM) to identify reality with the requirements of logical necessity, as manifested in the principles of identity and non-contradiction; and if, like Leibnitz, they made a distinction between truths of reason and truths of matter of fact (which are empirical), and thus avoided the Spinozistic identification of logical relationship with natural sequence, it was a concession to common sense rather than a philosophic implication of their system. Kant introduces a new motive. On the one hand, growing natural science had given to the conception of necessity (causal relationship) in nature a solidity and concreteness which it could not have had in earlier writers; on the other hand, he rejects the dogmatic identification of the laws being with those of logical thought. Hence his theory makes causality and thus necessity absolutely true of all nature, or the world of phenomena, by regarding causation as a category involved in the presentation of the world of sense to an experiencing subject. The source of necessity is thus found in the understanding as applied to sense; so that it may fairly be said that Kant restores in a critical and constructive way that which he had rejected in a dogmatic and formal way, namely, the origin of necessity in reason. At least, this path was followed by his idealistic successors, finding its outcome in the expression of Hegel (Logic, § 158), that 'freedom is the truth of necessity,' that is to say, that the determination of one phase of the objective world by another is at bottom but the self-determination of conscious mind, so that the necessary object, when experienced completely, appears as a co-operating factor in the development of free spiritual life. (J.D.)

Literature: Works on metaphysics and logic; G. TAROZZI, La dottrina della necessità (2 vols., 1895-7). (J.M.B., E.M.)

The following distinctions are usual:

Internal necessity springs from the nature of the subject of the necessity; external necessity comes from the outside.

Internal necessity is either absolute or secundum quid. Absolute necessity belongs to that whose being otherwise would involve contradiction. Necessity secundum quid is that which depends upon some matter of fact. Thus the Aristotelians held that a body falls to the ground by a necessity of its own nature, without external force or agency; yet it is easily prevented from falling.

External necessity, also called necessity ex hypothesi, because depending on an external condition, is distinguished in whatever ways the necessary is distinguished in the doctrine of the MODAL (q.v.), and, in particular, in reference to the sensus compositus and sensus divisus. In addition, external necessity is divided according as the realization of the condition precedes, is contemporaneous with, or follows after, the necessary result. Necessity from a previous condition is either that due to God's fore-knowledge or it is causal. Causal necessity (used also in modern logic) is either necessity of compulsion or necessity of determination.

Necessity determined by a subsequent condition is either ex hypothesi finis or ex hypothesi eventus (as the apostle says, 'it is necessary that offences should come'). Necessity ex hypothesi finis is either ad esse or ad bene esse.

Another common distinction is between necessity in causando, in essendo, and in praedicando, phrases which explain themselves.

Still another threefold distinction, due to Aristotle (I Anal. post., iv), is between necessity de omni (to kata pantoV), per se (kaq auto), and universaliter primum (kaqolou prwton). The last of these, however, is unintelligible, and we may pass it by, merely remarking that the exaggerated application of the term has given us a phrase we hear daily in the streets, 'articles of prime necessity.' Necessity de omni is that of a predicate which belongs to its whole subject at all times. Necessity per se is one belonging to the essence of the species, and is subdivided according to the senses of per se, especially into the first and second modes of per se.

Among modern distinctions we may mention that of Benno Erdmann between predicative and deductive necessity. The former seems to be necessity for a judgment being as it is in order to express what is in its immediate object.

Logical necessity is determined by the laws of the understanding, according to Kant (Krit. d. reinen Vernunft, 1. Aufl., 76).

Metaphysical necessity is that of God's existence.

Simple = absolute necessity. See above.

The adjectives by which different kinds of necessity are usually distinguished include absolute, antecedent, causal, comitant, composite, consequent, deductive, disjunct, disjunctive, external, formal, hypothetical, immediate, internal, logical, material, mediate, metaphysical, modal, moral, physical, practical, predicative, prime, simple, teleological, unconditional. (C.S.P.)

Negation [Lat. negatio, which translates Gr. apofasiV]: Ger. Verneinung; Fr. négation; Ital. negazione. Negation is used (1) logically, (2) metaphysically. In the logical sense it may be used (a) relatively, and (b) absolutely. Used relatively, when applied to a proposition, it may be understood (a) as denying the proposition, or (b) as denying the predicate.

(1) In its logical sense, negation is opposed to affirmation, although, when it is used relatively, this is perhaps not a convenient contrary term; in its metaphysical sense, negative is opposed to positive (fact, &c.).

The conception of negation, objectively considered, is one of the most important of logical relations; but subjectively considered, it is not a term of logic at all, but is pre-logical. That is to say, it is one of those ideas which must have been fully developed and mastered before the idea of investigating the legitimacy of reasonings could have been carried to any extent.

The treatment of the doctrine of negation affords a good illustration of the effects of applying the principle of PRAGMATISM (q.v.) in logic. The pragmatist has in view a definite purpose in investigating logical questions. He wishes to ascertain the general conditions of truth. Now, without of course undertaking to present here the whole development of thought, let it be said that it is found that the first step must be to define how two propositions can be so related that under all circumstances whatsoever,

The truth of the one entails the truth of the other,
The truth of the one entails the falsity of the other,
The falsity of the one entails the truth of the other,
The falsity of the one entails the falsity of the other.
This must be the first part of logic. It is deductive logic, or (to name it by its principal result) syllogistic. At all times this part of logic has been recognized as a necessary preliminary to further investigation. Deductive and inductive or methodological logic have always been distinguished; and the former has generally been called by that name.

In order to trace these relations between propositions, it is necessary to dissect the propositions to a certain extent. There are different ways in which propositions can be dissected. Some of them conduce in no measure to the solution of the present problem, and will be eschewed by the pragmatist at this stage of the investigation. Such, for example, is that which makes the copula a distinct part of the proposition. It may be that there are different ways of useful dissection; but the common one, which alone has been sufficiently studied, may be described as follows:

Taking any proposition whatever, as 'Every priest marries some woman to some man,'

we notice that certain parts may be struck out so as to leave a blank form, in which, if the blanks are filled by proper names (of individual objects known to exist), there will be a complete proposition (however silly and false). Such blank forms are, for example:

Every priest marries some woman to ______,
_____ marries _____ to some man,
_____ marries _____ to _____.

It may be that there is some language in which the blanks in such forms cannot be filled with proper names so as to make perfect propositions, because the syntax may be different for sentences involving proper names. But it does not matter what the rules of grammar may be.

The last of the above blank forms is distinguished by containing no selective word such as some, every, any, or any expression equivalent in force to such a word. It may be called a PREDICATE (q.v., sense 2) or rhma. Corresponding to every such predicate there is another, such that if all the blanks in the two be filled with the same set of proper names (of individuals known to exist), one of the two resulting propositions will be true, which the other is false; as

Chrysostom marries Helena to Constantine;
Chrysostom non-marries Helena to Constantine.

It is true that the latter is not good grammar; but that is not of the smallest consequence. Two such propositions are said to be contradictories, and two such predicates to be negatives of one another, or each to result from the negation of the other. Two propositions involving selective expressions may be contradictories; but in order to be so, each selective has to be changed from indicating a suitable selection to indicating any selection that may be made, or vice versa. Thus the two following propositions are contradictories:

Every priest marries some woman to every man;
Some priest non-marries every woman to some man.

It is very convenient to express the negative of a predicate by simply attaching a non to it. If we adopt that plan, non-non-marries must be considered as equivalent to marries. It so happens that both in Latin and in English this convention agrees with the usage of the language. There is probably but a small minority of languages of the globe in which this very artificial rule prevails. Of two contradictory propositions each is said to result from the negation of the other.

The relation of negation may be regarded as defined by the principles of contradiction and excluded middle. See LAWS OF THOUGHT. That is an admissible, but not a necessary, point of view. Out of the conceptions of non-relative deductive logic, such as consequence, coexistence or composition, aggregation, incompossibility, negation, &c., it is only necessary to select two, and almost any two at that, to have the material needed for defining the others. What ones are to be selected is a question the decision of which transcends the function of this branch of logic. Hence the indisputable merit of Mrs. Frankin's eight copula-signs, which are exhibited as of co-ordinate formal rank. But, so regarded, they are not properly copulas or assertions of the relation between the several individual subjects and the predicate, but mere signs of the logical relations between different components of the predicate. The logical doctrine connected with those signs is of considerable importance to the theory of pragmatism.

For the negation of modals see MODAL.

Conversion by negation = CONTRAPOSITION (q.v.).

Negant or negative negation is the negation effected by attaching the negative particle to the copula in the usual Latin idiom, 'Socrates non est stultus,' in contradistinction to infinite (aoristh), or infinitant, negation, which is effected by attaching the negative particle to the predicate, 'Socrates est non stultus.'

Kant revived this distinction in order to get a triad to make out the symmetry of his table of categories; and it has ever since been one of the deepest and dearest studies of German logicians. No idea is more essentially dualistic, and distinctly not triadic, than negation. Not-A = other than A = a second thing to A. Language preserves many traces of this. Dubius is between two alternatives, yea and nay.

(2) In the metaphysical sense, negation is the mere absence of a character or relation that is regarded as positive. It is distinguished from privation in not implying anything further.

Spinoza's celebrated saying, of which the Schellings have made so much, 'omnis determinatio est negatio,' has at least this foundation, that determinatio to one alternative excludes us from another. The same great truth is impressed upon youth in the utterance: 'You cannot eat your cake and have it too'. (C.S.P., C.L.F.)

Negative [Lat. negativa; a term appearing first in logic in Boethius, in place of the previous abdicativa, although negatio was much earlier. It translates Aristotle's apofatikh. Cognate words were used by Plato, and even earlier]: Ger. verneinend; Fr. négatif; Ital. negativo. Involving NEGATION (q.v.), either in the second application of the logical sense, or in the metaphysical sense given under that term.

Negative abstraction is an act of abstraction derived from considering something which does not possess the character considered.

Negative (or necessary) condition: see NECESSARY AND SUFFICIENT CONDITION.

Negative criterion: a criterion which is a negative condition; a test. Most criteria are of this sort.

Negative discrepancy: see DISCREPANCY.

Negative distinction: a mutual real distinction separating anything from its negation; as the distinctions of heat and cold (no heat), light and darkness (no light), sound and silence (no sound).

Negative idea: see Negative name.

Negative mark: a mark which consists in the non-occurrence of a positive phenomenon under certain conditions.

Negative name: a common name which characterizes an object by its want of some character. 'I appeal,' says Locke, 'to every one's own experience, whether the shadow of a man, though it consists of nothing but the absence of light (and the more the absence of light is, the more discernible is the shadow), does not, when a man looks on it, cause as clear and positive an idea in his mind as a man himself, though covered over with clear sunshine? And the picture of a shadow is a positive thing. Indeed, we have negative names, which stand not directly for positive ideas, but for their absence, such as insipid, silence, nihil, &c., which words denote the positive ideas, taste, sound, being, with a signification of their absence' (Essay concerning Human Understanding, II. viii. 5).

Negative negation: see NEGATION.

Negative syllogism: any syllogism of the second figure, or the modus tollens, where the reasoning turns upon the change of quality. The canon of syllogism, that nothing can be concluded from two negatives, is inaccurate. What is requisite, in non-relative syllogism, is that the middle term should be once distributed and once undistributed. Darapti and Felapton, which appear to violate this rule, only do so because one of the premises, so far as it is efficient, is virtually a particular. What is requisite is, that one of the interlocutors should select the individual denoted by the middle term in one premise and the other in the other.

Negative whole is one which has no parts; as God, the soul, &c. (C.S.P.)

Nominal [Lat. nominalis, pertaining to a name]: Ger. nominal; Fr. nominal; Ital. nominale. Relating to a logical term, whether expressed in language or merely a concept of the mind, and not to anything real. Cf. NOMINALISM, and REALISM.

Nominal definition (definitio nominis): the declaration of the essence of a word or expression, that is, the necessary and sufficient conditions of its applicability, or the enumeration of marks which suffice, but do not more than suffice, to give the meaning of the term, understanding by the 'meaning,' not the whole idea it may convey, but so much as it would require to be intended to convey in order to be a suitable word. Leibnitz says, 'Habemus quoque discrimen inter definitiones nominales, quae notas tantum rei ab aliis discernendae continent, et reales, ex quibus constat rem esse possibilem, et hac ratione satisfit Hobbio, qui veritates volebat esse arbitrarias, quia ex definitionibus nominalibus penderent, non considerans realitatem definitionis in arbitrio non esse, nec quaslibet notiones inter se posse coninungi. Nec definitiones nominales sufficiunt ad perfectam scientiam, nisi quando aliunde constat rem definitam esse possibilem.' This mode of making the distinction has been approved by many nominalists, as J. S. Mill. It cannot satisfy the realists, who demand of the real definition that it should express the real generating nature of the real species which it defines. As for the possibility of the thing, if by that is meant logical possibility, the nominal definition suffices. If more than that is meant, it is out of the province of definition to prove or declare a thing to be possible; a 'problem' proves such possibility.

Nominal mode, in the doctrine of modals: a mode of a proposition expressed by an adjective, as 'Sortem currere est contingens.' (C.S.P.)

Nomology [Gr. nomoV, law, + logoV, doctrine]: for equivalents see the next topic. The science which investigates laws, as general psychology and general physics; contradistinguished from classificatory and explanatory science. Hamilton says, 'We have a science which we may call the nomology of mind — nomological psychology' (Lects. on Met., vii). (C.S.P.)

Non-A (in logic): same in the other languages. An expression occurring in the usual forms of statement of the principles of contradiction and excluded middle. It is a term which denotes whatever is supposed not to be denoted by A, and denotes nothing more. (C.S.P.)

Non-contradiction. The 'law' of non-contradiction' is another name for the principle of CONTRADICTION (q.v.). See also LAWS OF THOUGHT. (C.S.P.)

Nonsequitur [Lat. for 'it does not follow]'. A name which belongs to the slang of the universities for the fallacia consequentis (called by Aristotle o para to epomenon elegcoV, De Sophist. Elen., 167 b 1), which is, strictly speaking, a fallacy which arises from a simple conversion of a universal affirmative, or transposing a protasis and apodsis, or condition and consequent.

Thus Aristotle tells us that the Eleatic Melissus argued that the universe is ungenerated, since nothing can be generated by what does not previously exist. The universe, then, not being generated, had no beginning; and having no beginning, it is infinite. But, as Aristotle remarks, although everything generated has a beginning, it does not follow (non sequitur, ouk anagkh de touto sumbainein) that everything that has a beginning is generated. A fever, for example, is not generated. Such fallacies are extremely common. De Morgan (Formal Logic, 268) gives this example: 'Knowledge gives power, power is desirable, therefore knowledge is desirable.' But though whatever is desirable has some desirable effect, it does not follow that whatever has any desirable effect is desirable. An attack of yellow fever has the desirable effect of rendering it unlikely the patient will for a long time have another; still, it is not itself desirable.

But the majority of logicians not only confound this fallacy with the post hoc, ergo propter hoc, which Aristotle considers immediately after, but even define it as 'failure in the formal inadequacy of the reason' (Sidgwick, Fallacies, II. ii. 4), or as 'the introduction of new matter into the conclusion, which is not contained in the premises' (Hyslop, Logic, xviii. 2), or as 'the simple affirmation of a conclusion which does not follow from the premises' (De Morgan, loc. cit.), or as 'any argument which is of so loose and inconsequent a character that no one can discover any cogency in it' (Jevons, Lessons in Logic, xxi), or 'to assume without warrant that a certain conclusion follows from premises which have been stated' (Creighton, Introductory Logic, § 46). Very many logicians omit it altogether, which is better.

Aristotle, however, could not express himself more precisely: 'O para to epomenon elegcoV dia to oiesqai antistrefein thn akolouqhoin. That is, 'from thinking that the consequentia can be converted.' That is to say, thinking that because 'If A, then C,' therefore 'If C, then A.' Owing to the neglect of fallacies by the more scientific logicians, it is not easy to cite many who define the fallacy correctly. The Conimbricenses (than whom no authority is higher) do so (Commentarii in Univ. Dialecticam Arist. Stagir., In lib. Elench., q. i. art. 4); also Eustachius (Summa Philos., Tom. I, pars. III, tract. iii, disput. iii. 9. 3); also Cope, an admirable student of Aristotle, in his note on the Rhetorics, B. cap. xxiv. See also the Cent. Dict., under 'Fallacy.' (C.S.P.)

Noology [Gr. nouV, reason, + logoV, theory]: Ger. Noologie; Fr. noologie; Ital. noologia (the equivalents are suggested). That part of philosophy which deals with intuitive truths of reason; as distinct from Dianoiology, which deals with truths discursively or demonstratively established.

A term suggested by Sir William Hamilton, Reid's Works, note A, § v, but having no currency. Hamilton probably derived it from Kant (Krit. d. reinen Vernunft, 643). It is used by Crusius for psychology. (J.D.)

Norm (and Normality) [Lat. norma, a carpenter's square, a rule]: Ger. Norm, Normalität; Fr. norme, normalité; Ital. norma, normalità. (1) A standard type or pattern from which continuous departures are possible in opposite directions. (C.S.P.)

Nota notae [Lat.]: The logical principle Nota notae est nota rei ipsius, that is, the predicate of the predicate is the predicate of the subject, which is laid down in several places by Aristotle as the general principle of syllogism. The principle passages are as follows: — 

'When one thing is predicated of another as its subject, whatever is said of the predicate can also be said of the subject' (Categ., iii. 1 b 10).

'Whatever is said of the predicate will hold also of the subject' (Categ., v. 3 b 4).

'We say that something is predicated universally when nothing can be admitted as coming under the subject of which the predicate will not hold; and the same thing holds of negation' (1 Anal. pr., i. 24 b 28). The term nota notae is from the first words of the original of this passage.

'Of whatever the species is predicated, the genus is predicable' (Topics, D. i. 121 a 25).

Some writers (as Hamilton, Lects. on Logic, App. VI. ii) imagine a distinction between the nota notae and the dictum de omni. Some have been so extravagant as to attribute the former to Kant, in whose Falsche Spitzfindigkeit (1762, ii) it is very likely that the precise phrase 'nota notae est nota rei ipsius' first occurs, though similar phrases, such as 'cui convenient notae eidem quoque convenit nomen,' are common in Wolf's and other logics of the 18th century. But it is clear that in Aristotle's mind it was one principle, essentially that which De Morgan well called the principle of the 'transitiveness of the copula.'

Aristotle, in the last but one of the above passages, seems to regard the nota notae as following from the definition of universal predication. To say that 'Any S is P' is to say that of whatever S is true, P is true. This amounts to deriving the transitiveness of the copula from the transitiveness of illation. If from A follows B and from B follows C, then from A follows C. This, again, is equivalent to the principle that to say that from the truth of X follows the truth of the consequence that from Y follows Z, is the same as to say that from the joint truth of X and Y follows Z. (C.S.P.)

Numerical [Lat. numerus, number]: Ger. Zahl- (in compounds, as Zahldifferenz); Fr. numérique; Ital. numerico, numerale. If two bodies move in the same orbit and differ in no respect but that of being at any one instant in different places, they are said to be numerically different. Whether or not it is quite accurate to say that they differ only in this, that there are two of them, it is sufficiently so to account for the origin of the phrase. Numerical difference is individual difference, apart from all qualitative unlikeness. Numerical identity is being strictly the same individual. Cf. the different topics IDENTITY, DIFFERENCE, and INDIVIDUAL.

This adjective in logical phrases usually translates the Greek ariqmw. Some writers have doubted whether the Greek word is here to be understood in an arithmetical sense, and have seemed to suspect that it was a relic of some original and different signification of the word. But this is hardly called for.

A numerically definite syllogism is one the force of which depends upon the relations of numbers; as 'Most of the men at a certain gathering wore dress-coats, and most of them had white neckties. Hence, some of those who wore dress-coats had white neckties.' (C.S.P.)

Observation [Lat. observatio, from observare, to look up]: Ger. Beobachtung; Fr. observation; Ital. osservazione. Attentive experience; especially, an act of voluntarily attentive experience, usually with some, often with great, effort. Cf. the following topics.

More or less fixity in the object is requisite. Indeed, experience supposes that its object reacts upon us with some strength, much or little, so that it has a certain grade of reality or independence of our cognitive exertion. All reasoning whatever has observation as its most essential part. Whatever else there is in the act of reasoning is only preparatory to observation, like the manipulation of a physical experiment.

Much stress has been laid upon the distinction between 'sciences of observation' and 'sciences of experiment'; and undoubtedly there is a great contrast between the proceedings, let us say, of the anatomist and of the physiologist. Although the anatomist has to make many experiments (with stains, for example), yet the stress of his labour comes upon the act of observation; while the preparations for observation of the physiologist are far more elaborate, and the mere act of observation itself often very easy and coarse. The difference is, however, chiefly one of degree, and from a philosophical point of view is of quite secondary importance. (C.S.P., J.M.B.)

Obversion [Lat. obversio, a turning]. Hamilton (Lects. on Logic, xiv, and especially Appendix V. iii) states that CONVERSION (q.v., also for foreign equivalents) in logic is sometimes called obversion.

This is a surprising statement, which neither he nor his editors are able to support by citations. It is, therefore, no unlikely that Hamilton took it at second hand.

Bain (Logik, Pt. I. Bk. I. chap. iii. § 27) says: In affirming one thing, we must be prepared to deny the opposite: "the road is level," "it is not inclined," are not two facts, but the same fact from its other side. This process is called obversion.' Bain gives no reference. The regular scholastic name for the process he describes — a name given by Abelard (Dialectica, 225) — is infinitatio. This word is very common (see, for example, Albertus Magnus in II. Peri hermeneias, iii; Ockham, Logica, II. xii, xiii; and the index to Prantl, Logik, iv). But somebody may have got the notion that it was 'barbarous,' and have preferred to use a more classical-sounding designation. (C.S.P.)

Opposition (in logic). One of Aristotle's POSTPREDICAMENTS (q.v.). There are said, in the book of Categories (cap. x), to be four kinds of opposites. Relative opposites are relate and correlate of a disquiparent relation. Contrary opposites are the most unlike species of the same genus, as black and white, sickness and health. The third kind of opposition is between a habit and its privation, as sight and blindness. The fourth kind is between affirmation and negation. This passage has prevented the word opposite from taking any definite meaning in philosophy. (C.S.P.)

Organon [Gr.]: the same in the other languages. Since neither the Aristotelian definition of a speculative science, nor of a practical science, nor of an art, seemed to suit logic very well, the early peripatetics and commentators denied that it was either a science or an art, and called it an instrument, organon; but they did not precisely define their meaning. It was negative chiefly. The collection of Aristotle's logical treatises, when it was made, thus came to be called the Organon.

Francis Bacon, disapproving of Aristotle's methods, wished all that to be laid aside; and he consequently called his work, which was designed to be a guide for establishing a systematic inductive procedure, Novum Organum. The name was afterwards imitated by sundry authors, as Lambert in his Neues Organon, and Whewell in his Novum Organum Renovatum. (C.S.P.)

Vague. A proposition is vague when there are possible states of things concerning which it is intrinsically uncertain whether, had they been contemplated by the speaker, he would have regarded them as excluded or allowed by the proposition. By intrinsically uncertain we mean not uncertain in consequence of any ignorance of the interpreter, but because the speaker's habits of language were indeterminate. (C.S.P.)