Russell's Theory Research Articles

On the theory of types

In this paper the theory of logical types will be examined, and certain departures from it will be suggested. Though the purpose of the paper is not primarily expository, an approach has been possible which presupposes no familiarity with special literature. Matters at variance with such an approach have been confined to appendices and footnotes.In the early pages the logical paradoxes will be considered—an infinite series of them, of which Russell's paradox is the first. Then Russell's simple theory of types will be formulated, in adaptation to a minimal set of logical primitives: inclusion and abstraction. Two aspects of the theory will be distinguished: an ontological doctrine and a formal restriction. It will be found that by repudiating the former we can avoid certain unnatural effects of the type theory—notably the reduplication of logical constants from type to type, and the apparent dependence of finite arithmetic upon an axiom of infinity. But the formal restriction itself has unnatural effects, which survive, even in an aggravated form, after the type ontology has been dropped. A liberalization of the formal restriction will be proposed which removes the more irksome of these anomalies.

Prof. H. N. Russell

PROF. H. N. RUSSELL, professor of astronomy at Princeton University, U.S.A., is one of the best-known American astronomers. He first became prominent through his theory of stellar evolution, according to which a star, contracting continuously throughout its history, passed successively through a ‘giant5 stage of rising, and a ‘dwarf state of falling, surface temperature. This theory, regarded somewhat sceptically at first, gained general acceptance through the discovery, by means of Adams and Kohlschutter's spectroscopic method for determining absolute magnitudes, of the reality of the distinction between giant and dwarf stars. The development of the theory of stellar constitution called for some amendment of the original postulates of Russell's theory, and it is characteristic of him that he made no attempt to preserve ideas based on insufficient data, but became a leader in the reformulation which was seen to be necessary. He was among the earliest to realize the importance of Saha's theory of ionization in stellar atmospheres, and took a leading part in working out its implications, always with a keen appreciation of the limitations placed on theoretical possibilities by difficulties inherent in methods of observation. The development of laboratory spectroscopy attracted him strongly, and with the collaboration of Prof. F. A. Saunders he obtained the first evidence of the co-operation of extra-nuclear electrons in producing line spectra: ‘Russell-Saunders coupling’ is now a well-established, and the most frequently occurring, type of such co-operation. More recently he has made important contributions to the analysis of complex spectra corresponding to various stages of ionization—particularly those of metals prominent in celestial spectra.

On the minimum stress theory of cable breakdowns

For the rational and economical design of electric cables, it is important to know the relation between the dimensions of the cable and its breakdown strength. Many different theories have been proposed, in the past, such as the maximum stress theory, the average stress theory, Russell's theory, and Osborne's theory, all of them conflicting. Recently a new theory has been proposed by Fernie that the minimum stress, namely that at the sheath of a cable is the limit. It is the purpose of this paper to discuss Fernie's theory and data, inasmuch as it is so diametrically opposed to some of the earlier theories. It seems quite plausible that insulating materials have a specific breakdown stress. Fernie having discovered, as he states, that the minimum stresses were constant in his tests, feels forced to abandon this idea and attempts to explain his results in terms of a limiting value of stress at the sheath, namely the minimum value. An analysis of his test results, however, does not seem to justify him inasmuch as, although his minimum stresses were much more constant than the maximum stresses, they were by no means constant, and in fact, it could be claimed with almost equal justice that his test results vindicated the average stress theory. Since, however, Fernie's experimental minimum stresses present a certain degree of constancy, this phenomenon (which remains to be proved) is investigated further. It is shown that if it be assumed (1) that the inner layers of insulation may be overstressed without complete rupture of the cable due to the stable equilibrium of the remainder of the insulation, and (2) that insulating materials have a critical breakdown gradient, a direct result of these two hypotheses is that the minimum stress at breakdown is a constant, though it is not in itself the criterion. It may be concluded therefore, that Fernie's experimental data are not sufficient to justify his claim that the minimum stress is a constant, and that if later tests should prove the constancy of minimum stress, this phenomenon could be explained otherwise than by assuming that the minimum stress itself is the limit.

On the Minimum Stress Theory of Cable Breakdowns

For the rational and economical design of electric cables, it is important to know the relation be-between the dimensions of the cable and its breakdown strength. Many different theories have been proposed in the past, such as the maximum stress theory, the average stress theory, Russell's theory, and Osborne's theory, all of them conflicting. Recently a new theory has been proposed by Fernie that the minimum stress, namely that at the sheath of a cable is the limit. It is the purpose of this paper to discuss Fernie's theory and data, inasmuch as it is so diametrically opposed to some of the earlier theories. It seems quite plausible that insulating materials have a specific breakdown stress. Fernie having discovered, as he states, that the minimum stresses were constant in his tests, feels forced to abandon this idea and attempts to explain his results in terms of a limiting value of stress at the sheath, namely the minimum value. An analysis of his test results, however, does not seem to justify him inasmuch as, although his minimum stresses were much more constant than the maximum stresses, they were by no means constant, and in fact, it could be claimed with almost equal justice that his test results vindicated the average stress theory. Since, however, Fernie's experimental minimumn stresses present a certain degree of constancy, this phenomenon (which remains to be proved) is investigated further.

THE SECULAR VARIATIONS OF CLIMATE

After a brief discussion of the secular variations of the climate of the earth, the following theories, proposed for the explanation of these variations are described. None of these theories is sufficient to account for the known variations of climate, and some require dynamical conditions which are not in accord with celestial mechanics. These theories are: The effect of the temperature of the interior of the earth. Emanations of radioactive substances in the earth's crust. Variations of solar energy according to Russell's theory of stellar evolution. Variations of the inclination of the earth's axis (precessional motion and variations of latitude). The “pendulation” theory. Variations in the carbon dioxide content of the air. Volcanic action; dislocation of parts of the earth's crust: distribution of sea and land. Cosmic causes. It is concluded that no single theory is likely to be the right one. but that the factors represented by several combine to give the known secular variations of climate.

The Radiation of the Stars

SINCE the publication of Homer Lane's paper “On the Theoretical Temperature of the Sun” in 1870, many writers have discussed the internal state of a star, considered as a globe of gas in equilibrium under its own gravitation. Recent observational work gives encouragement to these investigations, for it is now known that numerous stars are in a truly gaseous condition with mean densities similar to that of our atmosphere. To such stars the results for a perfect gas may fairly be applied, whereas stars, such as the sun, with densities greater than water must necessarily deviate widely from the theoretical conditions. The stars which are in a perfectly gaseous state correspond to the “giants” on H. N. Russell's theory,1 or to the stars of rising temperature on Lockyer's principle of classification; the denser “dwarfs” are outside the scope of this discussion. The two series coalesce for spectral type B, which marks the highest temperature attained.

Russell's Theory of Types

Russell's Theory of Types