Studies and Exercises in Formal Logic

by John Neville Keynes

EXERCISES.

132. Illustrate by means of the Eulerian diagrams (1) the relation between A and E, (2) the relation between I and O, (3) the conversion of I, (4) the contraposition of O, (5) the inversion of E. [K.]

133. A denies that none but X are Y ; B denies that none but Y are X. Which of the five class relations between X and Y (1) must they agree in rejecting, (2) may they agree in accepting? [C.]

134. Take all the ordinary propositions connecting any two terms, combine them in pairs so far as is possible without contradiction, and represent each combination diagrammatically. [J.]

CHAPTER VI.

PROPOSITIONS IN EXTENSION AND IN INTENSION.

135. Fourfold Implication of Propositions in Connotation and Denotation.—In dealing with the question whether propositions assert a relation between objects or between attributes or between objects and attributes, logicians have been apt to commit the fallacy of exclusiveness, selecting some one of the given alternatives, and treating the others as necessarily excluded thereby. It follows, however, from the double aspect of names—in extension and intension—that the different relations really involve one another, so that all of them are implied in any categorical proposition whose subject and predicate are both general names.[177] If any one of the relations is selected as constituting the meaning of the proposition, the other relations are at any rate involved as implications.

[¹⁷⁷] In the discussion that follows we limit ourselves to the traditional scheme of propositions.

The problem will be made more definite if we confine ourselves to a consideration of connotation and denotation in the strict sense, as distinguished from comprehension and exemplification, our terms being supposed to be defined intensively.[178] Both subject and predicate will then have a denotation determined by their connotation, and hence our 178 proposition may be considered from four different points of view, which are not indeed really independent of one another, but which serve to bring different aspects of the proposition into prominence. (1) The subject may be read in denotation and the predicate in connotation; (2) both terms may be read in denotation; (3) both terms may be read in connotation; (4) the subject may be read in connotation and the predicate in denotation.