Johnson, William Ernest

(1858-1931), logician

by R. B. Braithwaite, rev. Raffaella Simili

© Oxford University Press 2004 All rights reserved

Johnson, William Ernest (1858-1931), logician, was born at 1 Millington Road, Cambridge, on 23 June 1858, the fifth child and second son of William Henry Farthing Johnson, proprietor and headmaster of Llandaff House School, Cambridge, and his wife, Harriet, daughter of Augustine Gutteridge Brimley, of Cambridge, and half-sister of the essayist George Brimley. He was the brother of the writers George William Johnson and Reginald Brimley Johnson. Educated at his father's school, at the Perse School, Cambridge, and at the Liverpool Royal Institution School, he entered King's College, Cambridge, as a mathematical scholar in 1879, and was eleventh wrangler in the mathematical tripos of 1883. For some years he gained his living as a mathematical coach in Cambridge, until openings were found for him in teaching for the moral sciences tripos and as a lecturer in psychology and in the theory of education to the Cambridge Women's Training College and for the university teachers' training syndicate. He married in 1895 Barbara Keymer, daughter of Charles William Heaton, lecturer in chemistry at Charing Cross Hospital, London. After his wife's death in 1904 his sister Fanny made a home for him and his two sons. Johnson had no permanent position at Cambridge until 1902, when he was appointed to the newly created Sidgwick lectureship in moral sciences in the university and was elected to a fellowship at King's College. He held these positions for the rest of his life.

Johnson suffered all his life from ill health: bronchial troubles, together with a natural shyness, kept him to his house or college rooms. But he was none the less recalled by John Maynard Keynes as a sociable man who loved conversation. He would sit by a fire at Llandaff House, wrapped in the red shawl which became so characteristic of him, conversing with undergraduates. His pupils (who numbered a large proportion of those reading moral sciences in the university) and people who attended his lectures, which were delivered conversationally and with frequent digressions, were aware of a lovable personality and were infected with his exacting subordination of originality to clarity and truth. He was an accomplished pianist, and often entertained students and guests by playing a large grand piano. He was also interested in football and mountain climbing. He was nonconformist in religion, and remained a free-trader all his life.

From 1884 Johnson was an associate and a member of the Society for Psychical Research (SPR), which was founded in 1882 and whose first president was Henry Sidgwick. He was also mathematical assistant to the first research officer of the SPR, his sister Alice Johnson; he initiated experiments on cross-correspondence in automatic writing. Another sister of his, Fanny, was also a member of the SPR.

As well as lecturing on logic, Johnson lectured on philosophical and mathematical economics (he published a substantial article on 'The pure theory of utility curves' in the Economic Journal of Science, December 1913); but it was as a logician that he became known to the learned world when he published 'The logical calculus' (Mind, 1892), 'Analysis of thinking' (Mind, 1918), and above all when he developed his ideas in three volumes of a treatise on Logic (part 1, 1921; part 2, 1922; part 3, 1924), assisted by a pupil, Naomi Bentwick. A fourth volume, dealing with probability, was projected but never completed; its first three chapters were published posthumously in Mind in 1932. The Logic brought Johnson fame outside Cambridge: the universities of Manchester (1922) and Aberdeen (1926) conferred honorary degrees upon him, and in 1923 he was elected a fellow of the British Academy.

Johnson's 'Logical calculus' starts from the proposition as the unity of thought and introduces the logic of terms as subsidiary to the logic of propositions, making a significant reversal of the customary logical order developed subsequently by W. MacColl and B. Russell. Moreover, using primitive propositions rather than algebraical symbols, his work on logic is in the tradition of philosophical logicians such as Jevons, Venn, and J. M. Keynes rather than that of mathematical logicians, such as George Boole and Russell. Johnson was not in full sympathy with these last developments, and devoted time and energy to providing a critique of Russell; but his definition of logic as 'the analysis and criticism of thought' did much to break down previous restrictions upon its scope.

Johnson's logic presents two crucial aspects: formal calculus and philosophical analysis. His emphasis on the concepts of meaning and significance induced him to distinguish between inference and implication, usually both expressed by 'if ... then'. It also caused him to attach great importance to the relation of the thinker to the proposition which is the object of thought, thus showing that this 'epistemic' aspect could not be ignored in logic. In his discussion of deduction he emphasized, at a time when the symbolic logicians had not realized its importance, the difference between the premisses of a deduction and the logical principles in accordance with which the conclusion is drawn.

Johnson introduced an illuminating distinction between properties of different degrees of generality now best known as the theory of determinables. A 'determinable' is one of the broad bases of distinction which may be found in objects, such as colour, shape, size. The theory introduces a new form of names (categorical) by which propositions such as 'red is a colour' can be expressed, where a determinate is linked to a determinable in a very different way from that which links a member to a class, for example, 'Plato is a man'. His treatment of probability as a field of rational but not certain belief was similar to that of J. M. Keynes, who in his Treatise on Probability (1921) acknowledged his debt to Johnson. Probability expresses a certain degree of rational belief, between two propositions which Johnson called 'supposal' (an evidence given as certain) and 'proposal' (a hypothesis).

From a technical point of view Johnson made some other interesting contributions to the theory of probability: for instance, that known later as R. Carnap's principle, and one subsequently referred to as 'exchangeability' or 'symmetry', named by Johnson 'Postulate of permutation' (Postulate 3). He remained an active philosopher to the end of his life, and continued to lecture until a few months before his death, which took place at St Andrew's Hospital, Northampton, on 14 January 1931. He was buried at Granchester, Cambridgeshire.


R. B. Braithwaite, 'W. E. Johnson', Cambridge Review (30 Jan 1931)
C. D. Broad, 'William Ernest Johnson', PBA, 17 (1931), 491-514
C. D. Broad, review, Mind, new ser., 31 (1922), 496-510
C. D. Broad, 'Mr Johnson on the logical foundations of science', Mind, new ser., 33 (1924), 242-61; new ser., 33 (1924), 369-84
R. Simili, 'W. E. Johnson e il concetto di proposizione', Atti convegno storia della logica (1983)
G. H. von Wright, 'Broad on induction and probability', The philosophy of C. D. Broad, ed. P. A. Schlipp (c.1959)
A. N. Prior, 'Determinables, determinates and determinants', Mind, new ser., 58 (1949), 1-20, 178-94
C. A. Mace, ed., British philosophy in the mid-century (1957), 11-61
R. B. Braithwaite, 'Philosophy', University studies, Cambridge, 1933, ed. H. Wright, 1-32
R. Simili, ed., L'epistemologia di Cambridge (1987)
Proceedings of the Society for Psychical Research, 2 (1884)

CUL, notes on his 'Advanced logic' lectures taken by G. E. Moore

D. Banner, oils, 1927, King's Cam.
W. Stoneman, photograph, 1930, NPG
D. H. Banner, portrait; known to be in family possession in 1949

Wealth at death  
£2086 4s. 10d.: resworn probate, 12 March 1931, CGPLA Eng. & Wales

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