Page:Encyclopædia Britannica, Ninth Edition, v. 14.djvu/825

LOGIC 801

tration of elementary logical relations by numerical or algebraic signs or by diagrammatic schemata. The expression has the signification which it bears in mathematical analysis, and implies that the general relations of dependence among objects of thought, of whatsoever kind, in correspondence with which operations of perfectly general character are carried out, shall be represented by symbols, the laws of which are determined by the nature of these relations or by the laws of the corresponding operations. The mere use of abbreviations for the objects of treatment is not the application of a symbolic method^[1]; but so soon as the general relations of, or general operations with, these objects are represented by symbols, and the laws of such symbols stated as deductions therefrom, there arises the possibility of a symbolic development or method of treatment, which may lead to more or less expanded results according as the significance of the symbolic laws is more or less general. Thus quantity, whether discrete or continuous, presents, as an aspect of phenomena, relations of a highly general kind, offers itself as object of operations of a highly general kind, and is therefore peculiarly the subject of symbolic treatment. Currently, indeed, the treatment of quantity is assumed to have the monopoly of symbolism, but such an assumption is not self-evidently true, and it is permissible to inquire whether matters non-quantitative do not present relations of such generality that they, too, can be symbolically dealt with. It is, however, a further question whether the generality of the relations and therefore the significance of the symbols in such cases, although subject to some special conditions not necessarily involved in the nature of quantity, do not spring from the fact that we treat the matters as quantities of a special kind, and so insensibly find ourselves applying quantitative methods. In other words, it remains to be investigated, after the preliminary definitions and axioms of any symbolic method have been laid down, whether the conception of thought with which we start, or a special feature distinctly quantitative in character, has been the truly fruitful element in after-development of the system.^[2]

The first step in any symbolic logic must evidently be the determination of the nature and laws of the symbols, and, as these follow from the nature of the operations of thought, the first step is like wise a statement of the essential characteristic of thinking. As above noted, there have been adopted various modes of expressing this characteristic, and in some cases the mode adopted is not one from which any generally applicable symbolic rules of procedure could have followed.^[3] Two only require here to be noted, as representing special views: first, that which proceeds from the idea of thought as essentially the process of grouping, classing, determining a definite set of objects by a mark or notion; and second, that which proceeds more generally from the conception of thought as consisting of a series of self-identical units, to be variously combined in obedience to the law of self-identity.^[4] Adopting the first view, we find that processes capable of symbolic representation, by the customary algebraic signs of addition, subtraction, equivalence, multiplication, and division, have a perfectly general significance in reference to the combination, separation, equalization of classes, to the imposition and removal of restriction on a class; that to the symbols there can therefore be assigned a set of general laws; and that any peculiarity of these symbolic laws which differentiates them from the laws of like symbols in mathematical analysis is deducible from the notion of thought with which we started, and is consequently to be carried along with them in all the after development. Symbolic representation of relations of classes follows with equal directness from the general notion that by any such relation a new group is determined in reference to the original groups, or rather that the position or negation of a new group (or series of groups) is given, definitely or indefinitely, as the result of such a relation.

With the aid of the symbolic laws so reached, the logical problem as such may then be approached. Given any number of logical terms (i.e., classes, or, as it may be better put, positions and negations) connected together by any relations, to determine completely any one in reference to the others, or to express any one in terms of the others. The symbolic procedure, expounded with marvellous ingenuity and success by Boole, may take various forms, and may be simplified by many analytical devices, but consists essentially in determining systematically how given positions and negations, definite or indefinite, combine with or neutralize one another. A more detailed account of these formal processes is beyond our limits.^[5]

The first question which suggests itself in connexion with Boole's symbolic logic is the necessity or advisability of retaining the reference to classes, or the description of thought as classification. Do the symbolic laws really depend to any extent on the logical peculiarities of class arrangement? Mr Venn, who emphasizes this feature in Boole's scheme, has, however, done good service in leading up to a different explanation. The general reference to objects, which is also noted as implied in all Boole's formulae, has nothing to do with the possible difference of conceptualist or materialist doctrines of the proposition, and, in fact, as all distinctions of thing and quality, resemblance and difference, higher and lower, subject and predicate vanish, or are absorbed in the more general principle underlying the symbolic method, phrases such as classification, extension, intension, and the like should be banished as not pertinent. Nay, the usual distinctions of quantity and even of quality either disappear or acquire a new significance when they are brought under the scope of the new principle. "What symbolic logic works upon by preference is a system of dichotomy, of x and not x, y and not y, and so forth."^[6] In other words, quantitative differences require to find expression through some combination of the positions and negations of the elements making up the objects dealt with.^[7] while the usual qualitative distinctions are merged in the position or negation of various combinations.

The whole phraseology then of classification and its allied processes seems needless when used to denote the simple determination of objects thought. The literal signs express, not "classes," but units, determined in and for thought as self-identical. For this reason then it appears that the view of the foundations of the symbolic methods of logic taken in Grassmann's Begriffslehre is more thoroughgoing, and more closely represents the underlying principles, than that involved in Boole's formulæ and expounded in detail by Mr Venn.

Grassmann, as above stated, deduces logical relations as a particular class of the determinations necessarily attaching to all quantities (i. e. , determined contents of thought). Abstraction being made of all peculiarities which may be due to their special constitution, quantities exhibit certain formal relations when they are combined (added, subtracted, &c. ). Each quantity is a unit of thought, a definite positum, and of such units there are but two classes, elements and complexes. Units of thought, which are self-identical, and therefore subject to the specific law that addition of each to itself or multiplication of it by itself yields as result only the original unit, are notions. The theory of notions, therefore, is the development of the general formal relations of units under the special restrictions imposed by their nature.^[8]

There appears very clearly in Grassmann's treatment the essence of the principle on which symbolic logic proceeds. Thought is viewed as simply the process of positing and negating definite contents or units, and the operations of logic become methods for rendering explicit that which is in each case posited or negated. To apply symbolic methods, we require units as definite as those of quantitative science, and the only laws we can employ are those which spring from the nature of units as definite. Now it seems a profound error to reduce the whole complex process of thinking to this reiterated position of self-identical units. Undoubtedly if we start from any given fact of thought, as, e.g., a judgment, and inquire what can be exhibited as involved in it, we have before us a problem of analysis, the solution of which must take form in a series of positions and negations, but our thinking is not therefore as a whole mere analysis. The synthetic process by which connexions of thought among the objects of our conscious experience are established is not the mechanical aggregation of elementary parts. The relations which give intelligible significance to our experience are not simply those of identity and non-identity. It is an altogether abstract and external view of thought, resting in all probability on an obscure metaphysical principle,^[9] that would treat it as in essence the composition and decomposition of elementary atoms, of πρῶτα, as Antisthenes would have called them. It has, indeed, been imagined that a symbolic logic might be developed which should be independent in all its fundamental axioms of any metaphysical or psychological assumptions, but this is an illusion. No logical method can be developed save from a most definite conception of the essential nature and modus operandi of thinking, and any system of symbolic logic finds it necessary, if it is to be complete and consistent, to adopt some such view as that above criticized, to regard thought as purely analytic, as dealing with compounds or

1. ↑ Thus one would not describe Aristotle's use of letters for the forms of his syllogisms, nor the current logical abbreviations of S, P, and M in like case as being, in any true sense of the word, symbolic. On the subject generally the instructive work of Mr Venn (Symbolic Logic, 1881) should be consulted. Mr Venn has not only in this work expounded the foundations and main theorems of Boole's logic with a care and skill that leave nothing to be desired but he has independently of many real contributions to logical analysis, put in its true light the nature of symbolic method in logic. He has rendered it impossible, even for the outsider, to complain that symbolic logic is an arbitrary application of mathematical method to logical material.
2. ↑ An excellent note on symbolic logic will be found in Lotze, Logik (2d ed , 1880) pp. 256-59.
3. ↑ Some of these, as, e.g., Lambert's and Plouequot's, are noted and discussed by Mr Venn, Symbolic Logic, xxxii.-xxxvi. and passim).
4. ↑ The first is the view taken by Boole (and expounded with great fulness in Venn, as above); the second is that of the brothers Grassmann (in the Formenlehre, 1872, especially bk. ii. Die Begriffslehre oder Logik).
5. ↑ Mr Venn's work is here again invaluable. Jevons's Principles of Science and Studies in Deductive Logic should be consulted. Schröder's Operationskreis des Logikkalculs contains some very elegant and simple methods.
6. ↑ Venn, as above, p. 162.
7. ↑ Where this is impossible, as in the case of the truly particular or indeterminate judgment, symbolic methods encounter almost insurmountable difficulties.
8. ↑ See Die Begriffslehre oder Logik (1872). p. 43. Schröder (op. cit.) follows Grassmann, though with the use of class phraseology.
9. ↑ As above noted, p. 800.