The Modern-Traditional Logic of Keynes and Johnson by A.N. Prior When I first studied logic, in the 1930s, my text-books were Stebbings Modern Introduction to Logic and Maces Principles of Logic; & those of us who
wished to do further reading in the subject were encouraged to tackle two works to which both Mace & Stebbing were heavily indebted, Johnsons Logic & the Formal
Logic of J.N. Keynes. Behind that lies a chapter in the history of logic which has yet
to be written, & ought to be written. Keynes & Johnson were & are good books, even great books, and there is much in them which, either directly or through Stebbing & Mace & others, has become part of the almost unconscious background of 20thcentury logical thought. The two books grew together, & the picture of their growth, which it is not difficult to build by studying the four successive editions of Keynes, & then Johnson, is a fascinating one. [In one way the relation between Johnson and Keynes was very like that between Mill and Whately. Johnsons Logic, like Mills, dealt extensively with
induction, & on the deductive side was designed to be read in close connection with Dr. Keyness classical work {2} Mills System was designed to be read in
connection with Whatelys Elements. But there was another relation between them. In
his second] The first edition of Keynes appeared in 1884 under the title Studies &
Exercises in Formal Logic, including a Generalization of Logical Processes in their
Application to Complex Inferences. Its final three parts dealt with Terms,
Propositions & Syllogisms in a substantially traditional manner, while Part IV was devoted to a study of conjunctive & disjunctive terms, being a little indebted to Boole & Schroeder & much more to de Morgan, Jevons & Venn. This arrangement was reproduced in the two following editions (1887 &1894), but the modern material of Part IV came increasingly to affect, & to some extent to be absorbed in, the treatment of the traditional logic in Parts I to III, which in the 4th & final edition (1906) Part IV became simply a large appendix. Part I of Johnsons {3} Logic appeared 1921, Part II
in ; Part III in . In one way the relation of Johnson to Keynes was like that of Mill to Whately at an earlier period Johnsons Logic dealt extensively with induction, &
on the deductive side was designed to be read in connection with Dr Keyness classical work, as Mills System was designed to be read in connection with
Whatelys Elements. But there was another side to their relationship. In all editions
but the final, Keynes expressed his debt to the informed assistance of Johnson, & numerous footnotes indicate this indebtedness in detail; & apart from this extensive informal collaboration, the third (1894) edition owes much to Johnsons articles on the Logical Calculus in Mind for 1892.
Edited by Max Cresswell and Adriane Rini. The original is kept in the Prior collection at Bodleian Library, Oxford, Box 6. The page numbers in the original are put in curly braces.
§1. The Seven Relations between Propositions & between Classes As a final
illustration of the story of growth which I have mentioned, we must[?] take something which is now a commonplace matter the theory of the seven logical relations. In Jevonss Principles of Science, Chapter VI we are told that any one
proposition or group of propositions may be classed with respect to another proposition or group of propositions, as 1. Equivalent, 2. Impossible, 3. Consistent, 4. Contradictory, 5. By way of illustration {4} all immortals are not men is given as equivalent to all men are mortals, some mortals are men as inferrible from it, but not equivalent, all not-men are not mortals as consistent with it but not inferrible, & all men are immortals as of course contradictory. Keynes in the second edition of his Formal Logic has at the end of his treatment of immediate inferences a new
section (§88) on Mutual Relations of Propositions, in which he refers to this passage from Jevons, & enlarges Jevonss four relations to five, by dividing what Jevons calls contradictions into (a) contradictions in the ordinary sense, & (b) inconsistency that is not contradiction. In the third edition this material is transferred to the chapter on opposition, as Section 57, though it is observed in a footnote that in this extension of the theory of logical relations between propositions some knowledge of immediate inferences is assumed; & the number of relations is enlarged to six a pair of propositions may be (1) equivalent (as All S is P & All not-P is not-S); (2) one formally inferable from the
other but not vice versa (as Some P is S & All S is P; (3) such that one must be true &
both may be {5} (as Some S is P & Some not-S is P); (4) independent (as All S is P &
All P is S); (5) contrary; & (6) contradictory. The new item here is (3), which
corresponds to subcontrariety in the traditional square. It is clearly the transfer of the material from the discussion of immediate inference to that of opposition which have suggested the addition. In the 4th edition (§84) the same possibilities are given, but with a re-numbering. Possibility (2) is now labeled (2) or (3), & the remaining numbers go up one; the new distinction is clearly that between the case in which the given proposition is implied by the other & the case in which it implies it. And this is connected with another development. In the third edition, towards the end of the chapter on the Diagrammatic Representation of Propositions there is a new section (§93) in which Eulers five diagrams are enlarged to seven, his last to cases being each subdivided into the case in which the two classes exhaust the universe between them & the case in which they do not. To represent the universe, Eulers pairs of circles are here surrounded by a single large circle. In the corresponding section (§130) of the fourth edition, we are invited {6} at the end to compare this sevenfold scheme of class relations with the sevenfold scheme of relations between propositions given in §84. This parallel naturally did not become obvious to Keynes until he had enlarged the six propositional relations of the third edition to seven. In Johnson I.iii.9, we are told a proposition p may be either (1) co-implicant to
another proposition q, (2) super-implicant to it, (3) sub-implicant to it, (4)
independent of it, (5) sub-opponent to it, (6) super-opponent to it, or (7) co-opponent to it. In traditional logic, he goes on the seven forms of relation are known respectively by the names equipollent, superaltern, subaltern, independent, sub-contrary, contrary, contradictory. This is inaccurate; so far as I know, the term
superaltern is innovation of Johnsons: in traditional logic (& in Keynes) both A and I, & both E & O, are called subalterns, the universal being subalternant to its particulars & the particulars subalternate to its universal. And in I.ix.7, Johnson reproduces Keyness enlargement of Eulers diagrams (saying simply that they are due to Euler), but with Keyness outer circle replaced by a square. {7} [With regard to terms he says that P may be co-incident to it, sub-incident, super-incident, inter-sectant, super-remainder[?], sub-remaninder[?] or co-remainder to Q.] Neither Keynes nor Johnson refers explicitly to de Morgan, who had listed the same seven possibilities in his Formal Logic ch IV, but without diagrams; & as Keynes worked
them and in connection with the diagrams it is possible that he had by that time forgotten the passage in de Morgan.