conveyed to students except in the critico-historical fashion. Even if a teacher, in these critical rather than constructive days, seeks to expound his ultimate view of things to a class of students, it is to them but one other added to the tale of historical systems, and the chances, in any particular case, are against the supposition of its being of equal value with the greater philosophical constructions that have weathered the storms of time. As the crown of a philosophical education, students are to be taught to think for themselves; and to this end there seems no other way but that of bringing before them a representation of the thinking of the best minds of the race. On this vital point there is no difference between Mr. Stewart and me. I object only to the arbitrary way in which he seems to shut up the student to converse with this single thinker or that, whereas I would give the student, after due preparation, the free choice of all. And as a last word I repeat after due preparationscientific and other.

EDITOR.

VI.-CRITICAL NOTICES.

Der Operationskreis des Logikkalkuls. Von Dr. ERNST SCHRÖDER, ordentlichem Professor der Mathematik an der Polytechnischen Schule in Karlsruhe. Leipzig: Teubner, 1877.

This tractate, of only 37 pages, contains the clearest and most elegant exposition yet given of the mathematical or algebraic doctrine of logical reasoning. In essentials the author agrees with Boole, and his work may be regarded as in many points a simplification, in some points a rectification, of the elaborate processes first fully stated in the Laws of Thought. To Boole's method Schröder objects that the several steps in the symbolical processes are not in themselves interpretable or intelligible, and that certain elements are introduced and employed which cannot but be regarded as altogether foreign to the nature of logical inference. In place of Boole's algebraic method, he would therefore substitute forms capable of symbolic statement and subject to definite symbolic laws, but deduced carefully from the nature of the quantities symbolised, and at each stage intuitively interpretable.

Schröder, like Boole and all who have adopted the quasi-mathematical view of logical processes, starts from the consideration of classes as the elements of reasoning. Classes of things are the only logical quantities and the laws of symbolic operation are immediate

expressions for the various relations of classes to one another. In this mode of restricting attention to the quantitative relations of classes, Schröder agrees in the main with R. Grassmann, to whom he refers, and to whom some of the theorems in the work are due. Grassman's Begriffslehre oder Logik, the second part of a more comprehensive treatise on quantitative reasoning in general (Die Formenlehre oder Mathematik), deserves attention. He appears to have written in ignorance of any previous attempts at symbolical representation of reasoning, and it can hardly be said that he has worked out his principles to their full extent. Most of the theorems stated in the Logik with considerable display of mathematical proof, are merely translations into symbols of the ordinary logical laws of relation between notions in extent. When

the same quantitative method is applied to content, the results are not generally of much value. Grassmann expounds no general

theorem of elimination which can be made of service in the solution of complicated problems, and though he has handled syllogism, the results are not of the first importance.

Taking as the foundation for his logical calculus the view of symbols as representing classes, the symbolic laws and processes are with Schröder dependent on the nature of class relations. In section first, the specifically logical processes are stated as four in number: two direct-Multiplication or Determination, and Addition or Collection; two indirect or inverse-Division or Abstraction, and Subtraction or Exception. The inverse processes however may be superseded through the operation of a fifth process, Opposition or Negation. In section second, the longest of the four, the principles of the calculus, so far as the direct processes are concerned, are stated with the needful definitions, postulates and axioms. The explanations of symbolic Addition (a+b) and Multiplication (ab), together with the proofs of the cumulative (ab ba; a + b ba) and associative [a(be) = ab(c); a + (b + c) = (a + b) + c] laws of these two processes do not essentially differ from those of Boole. On page 12, however, is given a theorem which is not directly used by Boole; the omission, indeed, is one of the peculiarities of Boole's system. It is an evident deduction from the relation of super- and sub-ordinate notions, and may be stated symbolically thus: a = a + ab.

=

=

The introduction of the negative of any term leads to a statement of the useful principle that of one and the same class-symbol or of equivalent class-symbols the complements are equivalent, complement being that class-symbol which added to any other gives the result 1 or the universe of thinkable things, or which multiplied into the other gives the result 0 or the non-existent. It follows that of each term there is only one negative; thus a + a1 = 1; aa1 = 0.

Theorem 14 (p. 14) is substantially Boole's formula for the development of any logical function, but it receives a somewhat different statement, thus: "Any class b can be expressed in a linear homogeneous manner with regard to any other class a in the form b = = xa+ya,” x and y being indeterminate class-symbols which may

have the values 0 or 1. The proof given is elegant; and a useful form of equation, b= (ab + ua ̧)a + (a,b+va)a1, is deduced from it.

The

Theorem 15 gives a very simple explanation of the rule for multiplying developments according to the same arguments. result of the multiplication is found by multiplying the coefficients of the similar terms.

Theorems 17 to 20 are the most original in Schröder's work. In 17 it is shown that any logical equation a=b is capable of being resolved into the form

ab1+ab=0; ab + a,b1 = 1; (a + b1) (a + b) = 1.

By this theorem he is able to dispense with the process of transposition, which can only be employed under definite conditions.

In resolving these equations the negatives of complex terms are constantly involved. Theorems 18 and 19 contain methods for finding these negatives. "The negative of a product is the sum of the negatives of the factors," (ab),=a+b; "the negative of a sum is the product of the negatives of the members,” (a+b), = a,b.... Similarly the negative of a developed term is found by substituting for all the coefficients their negatives; thus (ab1 + a1b), = ab+a1b1; for the first member may be regarded as completely developed with regard to one of the quantities, though it is not developed with regard to both.

These propositions lead to the fundamental theorem of Elimination and by simplifying this process they render superfluous much of the algebraic machinery introduced by Boole. Theorem 20 is thus stated: "The equation xa + ya1 = 0 is equivalent to the two equations xy=0 and aux1+y, u being an arbitrary class." Since uyu (Y1 + y) + y = uy1 + uy + y = uy1 +y, and since, if xy=0, x,y=y, the second equation may be written in the forms a = (u+y)x1, or a = u(y1 + y)×1, or a=ux1Y1 + Y.

=

The inspection of this theorem shows us that by its means we can eliminate any given term from any equation (xy=0 being the result of eliminating a) of the given form, i.e., since by theorem 17 any equation can be thrown into this form, we can eliminate any term from any logical equation. In the same manner we can state any logical quantity in terms of all the others involved in the original equation. The close relation between this method of elimination and that stated by Boole does not require to be pointed out.

The third section of the work applies the method to one of the more complicated examples solved by Boole. The superiority in logical intelligibility of Schröder's solution must be admitted; its superiority in brevity is not so clear.

Section fourth takes up the inverse processes of Subtraction and Division, shows how these are capable of being brought under the same forms of solution as have been expounded for Addition and Multiplication, and points out the peculiar condition, that of disjunctive relation between the terms, necessary for applying them.

As has been said the peculiar merit of Schröder's method is the closeness with which it keeps to the logical realities expressed in

mathematical symbols. It is thus in every sense of the word a logical calculus; no law or process is admitted which has not a logical significance, and there is no step taken which is not susceptible of interpretation in logical language. Thus it approaches more closely to Prof. Jevons's method of indirect inference than to Boole's algebraic forms, and it enables us to perceive with more clearness than was possible in the case of Boole's logic, the worth of the symbolic representation of reasoning. Apart from any opinion as to the nature of the judgment, and therefore without pronouncing upon the philosophic validity of the doctrine that all logical quantities are classes, we must admit that after the preliminary process of throwing the premisses into quantitative form has been gone through, the symbolic method allows us to deal easily and compendiously with highly complex and involved reasonings. If we represent notions by symbols and their relations by algebraic signs, and if by introducing contradictory terms we can state exhaustively possible alternatives, then we can avoid the confusion incident to carrying the whole signification of our notions through the train of reasoning. But there is no more generality in the symbolic laws and processes than in the logical laws and processes which they express. We have in no sense brought logic under a more general quantitative science, as at first sight appeared to be the case with Boole's method. Even the process of elimination, which in Boole was effected by devices only dimly recognisable as logical, is in Schröder's system nothing but a complex application of the ordinary formal rules of logical inference. It is a convenient, mechanical contrivance, founded on logical forms, and capable of translation into them.

A competent review of these various attempts to simplify logical processes by the use of algebraic symbols is a desideratum in logical literature.

De la Conscience en Psychologie et en Morale.

BOUILLIER. Paris: Germer Baillière.

R. ADAMSON,

Par FRANCISQUE 1872.

The word Consciousness, according to M. Bouillier, has many significations, but is to be used as expressing "simple, spontaneous consciousness, embracing all internal phenomena and all mental states as the primitive fact of the intellectual and moral life, the condition, the essence even, of every idea and of every feeling. It is not definable; its omnipresence renders circumscription impossible.

The beginnings of consciousness are slow and gradual. The first sensation is a vague impression of easiness or uneasiness, followed immediately by, if not contemporaneous with, the faintest perception of resistance on the part of the bodily organs or of a foreign body. The beginning of consciousness coincides with the beginning of existence; the moment of conception is also the moment of the first consciousness. This is the boldest hypothesis but also the best and

most philosophical. Maine de Biran is right when he says: "To live is to feel".

What is the place of consciousness in a theory of the human mind? Is it, or is it not, a special faculty? If consciousness is a special faculty it ought to be conceivable at least apart from any other faculty. But it is not so. No psychological analysis, however subtle, can make it appear that to think and to know that one thinks, to will or to feel and to know that one wills or feels, are not one and the same thing. Leibnitz and Kant, when they speak of "imperceptible perceptions" and "unconscious representations," are not, indeed, speaking with rigorous exactness, but they do not mean to identify themselves with those who hold that consciousness is a special faculty of the mind. They only mean, by these phrases, facts on the threshold of consciousness, but not outside it. Others, however, Schelling, Schopenhauer, Herbart, Hartmann, &c., hold that consciousness is an ordinary but not necessary accompaniment of mental operations. But this is not the case. All possible diminutions in the consciousness of such and such an idea or sensation are conceivable, provided these diminutions do not reach the extreme limit of zero-in which ease nothing would remain to which the name of sensation or idea could be given without the most singular abuse of reasoning and language. There is no ground whatever for any distinction between consciousness and the phenomena of consciousness.

Again, if consciousness is a special faculty it ought to have its own object, its distinct domain. But there is no such object, and no such domain. All facts, known or felt, are facts of consciousness, but there are no facts which can be called peculiarly its own. Consciousness is not a new element added to other psychological elements to enlighten and complete them, to make them facts of consciousness; it is the generative and essential element of all the powers of the soul, of sensation and volition not less than of intelligence itself.

Consciousness is not, then, a special faculty. Far from being shut up, so to speak, in any one part of the soul, it is that which envelops it, that which contains all its phenomena and all its faculties. Far from representing only one class of phenomena and being only one special faculty, all phenomena and all faculties are but its transformations and modifications. Consciousness is not a part of the Ego, it is the Ego in its entirety-the stuff of which it is made. It is not only the connecting link, but the very essence, of the powers of the soul— the reality of realities, the fact of facts.

English psychology generally, looking at the soul from the outside, if we may so say, sees nothing but phenomena, relations, laws of association; it finds no being, no faculties, nothing but apparitions and trains of apparitions. For it is only within that the reason of these phenomena can be found, and the force which produces and governs them. But an appeal to consciousness itself brings out the fact that we feel this force within us, perceive it clearly, in its permanence and identity, through and over all the phenomena which pass and vanish incessantly.