of these propositions) is interesting and deserves notice. The third of these primitive propositions is of course exceedingly complicated and does not commend itself as true so readily as the primitive propositions of Principia Mathematica, but it is very interesting to have found it and to have shown how the more familiar propositions can be deduced. In the same issue of the Proc. Camb. Phil. Soc. C. E. van Horn uses the same indefinable calling it "p deltas q." Then from a certain group of three primitive propositions he claims to be able to deduce the propositions required for logic. These propositions consist of the principle of inference as given in Principia Mathematica, a proposition analogous to prop. 1.71 in Principia Mathematica, by means of which it is possible to deduce that p deltas q is an elementary proposition, when p and q are elementary propositions; and a third proposition stating that if p and q are of the same truth value, p and p deltas q are of oppositive truth values, and that if p and q are of opposite truth values, p deltas q is true. However, in the development of this system, it is clear that some axiom is required to connect "deltas" with "of the same truth value." It is not difficult to find vicious circles in some of the demonstrations (cf. the criticism of Nicod at the end of his paper, ibid., p. 40).

C. I. Lewis (Journal of Phil., Psy., and Scientific Methods, 1917, XIV, 350-355) gives a further statement of his views on the nature of material implication (cf. Mind, 1912, number 84; 1914, number 90) and continues his criticism of Russell's use of the notion.

A translation of part of Frege's Grundgesetze is given in The Monist (1917, XXVII, 114-127). A. E. Heath (ibid., pp. 1-56) contributes an interesting account of Grassmann's work. D. M. Wrinch (ibid., pp. 83-104) gives a sketch of Bernard Bolzano's life and his pioneer work in mathematical logic. Philip E. B. Jourdain (ibid., pp. 142-151) discusses existence and distinguishes the "entity" of a number from its "existence."

N. Wiener (Trans. Amer. Math. Soc., 1917, XVIII, 65-72) attacks a very general problem in Boolean algebras, finding the necessary and sufficient conditions that a relation between any number of elements will remain invariant with reference to all transformations of the algebra into itself which may be expressed in the symbolism and which leave it a Boolean algebra.

An extract from Couturat's unpublished Manuel de Logistique is given in the Rev. de Métaphys. (1917, XXIV, 15-58). The

nature of propositional functions, formal and material implication, and real and apparent variables is examined. The article concludes with a sketch of the more elementary parts of the calculus of classes and the definition of 0 and 1. Part of an unfinished treatise written by Couturat sometime before 1902 is given in the same periodical (ibid., pp. 291-313), which is interesting as an exposition of the frequency view of probability originally put forward by Venn and the mathematical writers, as opposed to the sufficiency view held by Bosanquet, Lotze, and Sigwart. Probable is said to be significantly predicable only of indeterminate judgments (by which is meant, presumably, propositional functions) and not of events or determinate judgments. The probability of a judgment is then defined to be the ratio of the number of cases when it is true to the number of cases when it is true or false. The article ends with a short account of the elementary part of the calculus of probabilities. Tenney L. Davis (Journal of Phil., Psy., and Scientific Methods, 1917, XIV, 421-440) gives an interesting study of the theory of probabilities, discussing some of the philosophical problems as well as the calculus.

F. Enriques (Rev. de Métaphys., 1917, XXIV, 149-164), in an article on the mathematical infinite draws a distinction between a potential infinite and an actual infinite. As an illustration he gives the set

·3, 33, 333,....

Now, this ordered set can be given in two ways. (1) The general term consists of so many threes and the terms are ordered by the relation "less than." (2) Each term is obtained from the term before, the first term being 3. In the first case we are said to have an actual and in the second case a potential infinite. The real antithesis, however, seems to be that in the second case we have the terms given as the field of a relation which is one-one, relating only consecutive terms, whereas in the first case the terms are given as the field of a relation which is transitive and symmetrical, and thus relates any early term to any later one. In his brilliant work on the ancestral relation (see Begriffsschrift, 1879, Part III, pp. 55-87, Grundgesetze der Arithmetik, Vol. I, 1893, §§ 45-46), Frege has shown how to manufacture a relation having the formal characteristics of a series from a one-one-relation: "less than" is the ancestral relation obtained from the relation between terms in (2). There seems to be no fundamental logical distinction with respect

to the nature of the infinite as Enriques suggests. The question of the validity of the assumption of the existence of classes is discussed, but no mention is made of the work of Whitehead and Russell (cf. Principia Mathematica, Vol. I) which suggests a method of logical construction by means of which this assumption can be avoided.

A. Padoa (Rev. de Métaphys., 1917, XXIV, 315-325) discusses the general problem of changing the primitive ideas in a deductive system. In an interesting article on the function of symbolism in mathematical logic (Scientia, 1917, XXI, 1-12) Philip E. B. Jourdain answers certain charges brought by Rignano against mathematical logic (ibid., 1916, XX). He points out that, though the aim of symbolism in mathematics and logic has been until recently to make the process of reasoning mechanical by pointing out algebraic analogies, the aim in modern logic has been to increase the subtlety of our thought by emphasizing differences. An account of the work of Peano, Frege, and Russell is given.

In an article, entitled "The Organisation of Thought" in a book of the same title (London, 1917), Prof. A. N. Whitehead investigates the relation between science and logic, and describes some of the features of modern logic.

Prof. L. P. Saunders (Mind, 1917, number 101) in a criticism of Mr. Russell's Lowell Lectures discusses the nature of logical constructions. The subject is interesting to the mathematical logician, in that, on the validity of this method rests the validity of all but the most elementary parts of mathematical logic, for classes, relations, and numbers are all logical constructions. D. M. Wrinch (ibid., 1917, number 104) attempts to answer some of the points brought up by Professor Saunders in his attack.

Raphael Demos (Mind, 1917, number 102) contributes an interesting article on particular negative propositions. He assumes that there are no negative facts and deduces that negative propositions must have reference to the world of positive facts. Particular negative propositions are said to be ambiguous descriptions of positive propositions and they are treated as Russell treats descriptive phrases (see Principia Mathematica, Vol. I). It seems, however, to the reviewer a little unwise to base a theory on such a disputable point as the non-existence of negative facts.

DOROTHY MAUD WRINCH.

CAMBRIDGE, ENGLAND.

MONISM AND DUALISM.

Are monism and dualism incompatible? or are they complementary aspects of existence? Is human experience such as to be satisfied by either one of these attitudes? or does it demand both?

Our point of view is determined, not immediately by being-initself, but by our experience of being. Therefore if man may experience being in two distinct ways he also may see being from two different points of view.

According to Plato's theory of knowledge man may know being either through sense-perception produced by the material manifestations of being, or through immediate intuition of unmanifested being. This thought reconciles monism and dualism. And the fact that it does so would seem to be good reason for accepting it.

Through sense-perception we know the material world. This world is so constituted that it generally leads us to a dualistic point of view. Only those who confine their attention to the physical side of nature come to look on the world as the manifestation of a single principle. Supersensible experience, on the other hand, may contain no duality, and therefore may lead naturally to a monistic point of view. If thus sense-perception and supersensible experience constitute two distinct spheres of experience which never combine in human consciousness, and which are separated by a gulf impassable by thought, it is quite proper that one of these spheres of experience be interpreted dualistically while the other demands a monistic point of view. The student of nature may be right in seeing the world as a manifestation of two principles. The monist, for whom perhaps the world does not exist while he sees being from the supersensible point of view, may also be right in seeing all as one.

The universe is made of forces. All these forces are probably alike. As far as we know the differences between things lie entirely in the difference of arrangement of the forces of which the things are made. The arrangement of the forces results from their local motion. When the local motion of the forces in a piece of wood is increased by throwing it into the fire, these forces cease to make a piece of wood and instead form themselves into smoke and flame and ashes. This local motion, in conjunction with the local arrange

ment which results directly from the motion, is called energy. Each thing in the universe is a distinct thing by virtue of the energy which it contains.

It is the nature of energy not to stay where it is put, but to dissipate itself. This peculiarity of energy has led to the formulation of the Law of Dissipation of Energy. According to this law energy is never transmitted without being at the same time dissipated. We also know that our sensations result from transmissions of energy, either from things to our bodies, or from our bodies to things. No concentration of energy, however great, could in the least affect our senses so long as the energy is locked up in the thing. Energy produces sensation only when it is transmitted from one thing to another. According to the Law of Dissipation of Energy this transmission results in dissipation. Hence it follows that we are able to perceive the universe only in so far as it is going to pieces. Some things, as the sun for instance, are perceived because they are going to pieces. Other things, as for instance a stone seen by the light of the sun, are perceived because another thing is going to pieces. However, the stone also is going to pieces, although at such a slow rate that that part of its own energy which it liberates produces no sensation. Every part of the universe is unceasingly going to pieces. This is the peculiarity of the universe which enables us to perceive it and to know it, and which, therefore, makes it our universe. This is the reason why we are sometimes tempted to think that the universe never does anything but go to pieces.

But when we reflect we find that the conception of a universe which is just going to pieces throughout infinite time is repugnant to the intellect. The intellect revolts against this impressionistic attitude and boldly infers that there is also creation going on in the universe. Creation consists in concentration of energy. According to the Law of Dissipation of Energy concentration of energy cannot result from transmission of energy. Therefore we are compelled to assume that energy is created at certain times, or possibly at all times.

All mechanical process requires that the elemental forces in it remain unchanged. So long as the forces involved in a process remain unchanged no energy can be created or destroyed in this process. This is the Law of Conservation of Energy. Hence it follows that creation of energy cannot result from the mechanical