An Addition to the Logical Square of Opposition.

By THE REV. J. CLARK MURRAY, LL.D.

(Read May 23, 1883.)

I do not think it necessary in this paper to enter into an explanation of the general logical doctrine of the opposition of propositions, as my intention is merely to indicate a useful addition to the diagram commonly used to illustrate that doctrine. That diagram is a square, at the four angles of which are placed the symbols of the four kinds of propositions, A, E, I, O, while the four sides of the square, along with its two diagonals, are supposed to illustrate the relation between the two opposite propositions of the several kinds of opposition. The relation between two opposites, however, is but very imperfectly exhibited by the simple square; and it occurred to me, that a very slight addition to this diagram would render it a great deal more serviceable for its purpose. This addition, accordingly, I have been accustomed to use among my students for many years.

The addition is based on the nature of the relation between opposites, which the square is designed to symbolize. Now, the relation of logical opposition is one that is expressed by the inferences which may be drawn from one opposite to another. If, for example, two propositions are so diametrically opposed, that one of them must be true and the other must be false, then that opposition is expressed by saying, that you can infer the falsity of the one from the truth of the other, and the truth of the one from the falsity of the other. In other cases, however, as is well known, the relation of two propositions is such, that you can infer only the falsity of one from the truth of the other, or only the truth of one from the falsity of the other, and so on. Now, the object of the proposed improvement on the logical diagram is to exhibit these varions relations to the eye. But here a difficulty arises, which has been very generally overlooked by logicians. The word some, which is the common expression of particularity in a proposition, is beset by an important ambiguity. It may mean either "some at least,” that is, "possibly all," or "some at most,” that is, "some only, not all." Even such an eminent expositor of logical doctrine as Whately confounds these two meanings; and his account of the opposition of propositions is consequently in part unintelligible.

Now, the only way to avoid this ambiguity in an illustrative diagram is to adopt two squares, each intended to exhibit the relations of opposites under one interpretation of the word some. Then the relations are fully displayed by means of arrows between the symbols of the two propositions forming each pair of opposites. The arrows point in the direction in which inferences may be drawn. A transverse bar on the stem of an arrow symbolizes

falsity or negation, while a ring or small circle is a sign of truth or affirmation. With these explanations the diagrams are subjoined.

ROYAL SOCIETY OF CANADA.

TRANSACTIONS

SECTION III.

MATHEMATICAL, PHYSICAL AND CHEMICAL SCIENCES

PAPERS FOR 1882 and 1883.

I.-The Relations of the Natural Sciences.

By THOMAS STERRY HUNT, M.A., LL.D. (Cantab.), F.R.S.

(Read May 25, 1882.)

The occasion which brings us together is one which should mark a new departure in the intellectual history of Canada. Science and letters find but few votaries in a country like this, where the best energies of its thinkers are necessarily directed to devising means of subduing the wilderness, opening the ways of communication, improving agriculture, building-up industries, and establishing upon a proper basis schools in which the youth of the country may be instructed in those arts and professions which are among the first needs of civilized society. The teachers under such conditions can do little more than interpret to their pupils so much of the wisdom of the past, and of contemporary science, as may suffice for the immediate wants of the country, and will have but scanty leisure for original investigation in the field of knowledge. There are, however, never wanting earnest and curious minds who feel an almost irresistible impulse to labor in this field, to enlarge the bounds of thought, and to grapple with the great problems of man and nature. To foster this spirit, to encourage its beginnings, and to extend the influence of its example, should be the aim of wise statesmen and legislators who seek to elevate their kind and ennoble their nation knowing that the brightest glories and the most enduring honors of a country are those which come from its thinkers and its scholars.

The world's intellectual workers are, from the very nature of their lives of thought and study, separated in some degree from the mass of mankind. They feel, however, not less than others, the need of human sympathy and co-operation, and out of this need have grown academies and learned societies devoted to the cultivation of letters and of science. The records of these bodies in Florence, in Rome, in Paris, in London, and elsewhere, are the records of scientific progress for the last three centuries. Such bodies do not create thinkers and workers, but they give to them a scientific home, a centre of influence, and the means of making known to the world the results of their labors.

It was with a wise forethought that more than a century since Franklin and his friends founded at Philadelphia the American Philosophical Society. Its planting then seemed premature, but its vigorous growth during a century has served to show that the seed was not too early sown. That, however, unlike many of the academies of the old world, to which we have adverted, had no formal recognition from the State, and there came a period in the growth of the American Union when the need of an official scientific body was felt. Thus it was that nineteen years ago, in the midst of the great civil war, the American Congress authorized the erection of a National Academy of Sciences, to which, as an American citizen, I have the honor to belong. The aim proposed in founding that Academy was to gather together what was best and highest in the scientific life of the nation, and moreover to organize a body of councillors to which the executive authority could always look for advice and direction in scientific matters relating to the interests of the State. In that Sec. III., 1882. 1