FOOTNOTES:
1 Since the above was written Mr. Harley has read an account of Stanhope’s logical remains at the Dublin Meeting (1878) of the British Association. The paper will be printed in Mind. (Note added November, 1878.)
2 Leibnitii Opera Philosophica quæ extant. Erdmann, Pars I. Berolini, 1840, p. 94.
3 Erdmann, p. 102.
4 Ibid. p. 98.
5 Erdmann, p. 100.
6 Fifth Edition, 1860, p. 158.
7 Section 120.
8 See his “Remarks on Boole’s Mathematical Analysis of Logic.” Report of the 36th Meeting of the British Association, Transactions of the Sections, pp. 3–6.
9 Hamilton’s Lectures, vol. iv. p. 319.
10 Ibid. p. 326.
11 Pure Logic, or the Logic of Quality apart from Quantity; with Remarks on Boole’s System, and on the Relation of Logic and Mathematics. London, 1864, p. 3.
12 La Philosophie Positive, Mai-Juin, 1877, tom. xviii. p. 456.
13 Inventum Novum Quadrati Logici, &c., Gissæ Hassorum, 1714, 8vo.
14 See Ueberweg’s System of Logic, &c., translated by Lindsay, p. 302.
15 Since the above was written M. Liard has republished this exposition as one chapter of an interesting and admirably lucid account of the progress of logical science in England. After a brief but clear introduction, treating of the views of Herschel, Mill, and others concerning Inductive Logic, M. Liard describes in succession the logical systems of George Bentham, Hamilton, De Morgan, Boole, and that contained in the present work. The title of the book is as follows:—Les Logiciens Anglais Contemporains. Par Louis Liard, Professeur de Philosophie à la Faculté des Lettres de Bordeaux. Paris: Librairie Germer Baillière. 1878. (Note added November, 1878.)
16 Spectator, September 19, 1874, p. 1178. A second portion of the review appeared in the same journal for September 26, 1874, p. 1204.
17 Mind: a Quarterly Review of Psychology and Philosophy. No. II. April 1876. Vol. I. p. 206.
18 Portions of this work have already been published in my articles, entitled “John Stuart Mill’s Philosophy Tested,” printed in the Contemporary Review for December, 1877, vol. xxxi. p. 167, and for January and April, 1878, vol. xxxi. p. 256, and vol. xxxii. p. 88. (Note added in November, 1878.)
19 Mind, vol. i. p. 222.
20 Fortnightly Review, New Series, April 1875, p. 480. Lecture reprinted by the Sunday Lecture Society, p. 24.
21 Sir W. Thomson’s words are as follows (Cambridge Mathematical Journal, Nov. 1842, vol. iii. p. 174). “When x is negative, the state represented cannot be the result of any possible distribution of temperature which has previously existed.” There is no limitation in the sentence to the laws of conduction, but, as the whole paper treats of the results of conduction in a solid, it may no doubt be understood that there is a tacit limitation. See also a second paper on the subject in the same journal for February, 1844, vol. iv. p. 67, where again there is no expressed limitation.
22 Pp. 25–26. The parentheses are in the original, and show Professor Tait’s corrections in the verbatim reports of his lectures. The subject is treated again on pp. 168–9.
23 Theory of Heat 1871, p. 245.
24 The Senses and the Intellect, Second Ed., pp. 5, 325, &c.
25 Max Müller, Lectures on the Science of Language, Second Series, vol. ii. p. 63; or Sixth Edition, vol. ii. p. 67. The view of the etymological meaning of “intellect” is given above on the authority of Professor Max Müller. It seems to be opposed to the ordinary opinion, according to which the Latin intelligere means to choose between, to see a difference between, to discriminate, instead of to unite.
26 Hartley on Man, vol. i. p. 359.
27 Principles of Psychology, Second Ed., vol. ii. p. 86.
28 Pure Logic, or the Logic of Quality apart from Quantity, 1864, pp. 10, 16, 22, 29, 36, &c.
29 Brewster, Treatise on New Philosophical Instruments, p. 273. Concerning this method see also Whewell, Philosophy of the Inductive Sciences, vol. ii. p. 355; Tomlinson, Philosophical Magazine, Fourth Series, vol. xl. p. 328; Tyndall, in Youmans’ Modern Culture, p. 16.
30 Formal Logic, p. 38.
31 Hallam’s Literature of Europe, First Ed., vol. ii. p. 444.
32 Outline of a New System of Logic, London, 1827, pp. 133, &c.
33 An Investigation of the Laws of Thought, pp. 27, &c.
34 Formal Logic, pp. 82, 106. In his later work, The Syllabus of a New System of Logic, he discontinued the use of the sign.
35 Principles of Psychology, Second Ed., vol. ii. pp. 54, 55.
36 Pure Logic, or the Logic of Quality, p. 14.
37 Pure Logic, pp. 18, 19.
38 Ueberweg’s System of Logic, transl. by Lindsay, pp. 442–446, 571, 572. The anticipations of the principle of substitution to be found in the works of Leibnitz, Reusch, and perhaps other German logicians, will be noticed in the preface to this second edition.
39 Substitution of Similars (1869), p. 9.
40 Port-Royal Logic, transl. by Spencer Baynes, pp. 212–219. Part III. chap. x. and xi.
41 Description of a Notation for the Logic of Relatives, resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic. By C. S. Peirce. Memoirs of the American Academy, vol. ix. Cambridge, U.S., 1870.
42 On the Syllogism No IV., and on the Logic of Relations. By Augustus De Morgan. Transactions of the Cambridge Philosophical Society, vol. x. part ii., 1860.
43 Observations on Boole’s Laws of Thought. By the late R. Leslie Ellis; communicated by the Rev. Robert Harley, F.R.S. Report of the British Association, 1870. Report of Sections, p. 12. Also, On Boole’s Laws of Thought. By the Rev. Robert Harley, F.R.S., ibid. p. 14.
44 Jevons’ Elementary Lessons in Logic, pp. 41–43; Pure Logic, p. 6. See also J. S. Mill, System of Logic, Book I. chap. ii. section 5, and Shedden’s Elements of Logic, London, 1864, pp. 14, &c. Professor Robertson objects (Mind, vol. i. p. 210) that I confuse singular and proper names; if so, it is because I hold that the same remarks apply to proper names, which do not seem to me to differ logically from singular names.
45 Professor Robertson has criticised my introduction of “Substantial Terms” (Mind, vol. i. p. 210), and objects, perhaps correctly, that the distinction if valid is extra-logical. I am inclined to think, however, that the doctrine of terms is, strictly speaking, for the most part extra-logical.
46 Mathematical Analysis of Logic, Cambridge, 1847, p. 17. An Investigation of the Laws of Thought, London, 1854, p. 31.
47 Pure Logic, p. 15.
48 “Velut si dicam, Sol, Sol, Sol, non tres soles effecerim, sed uno toties prædicaverim.”
49 Book i., Part iv., Section 5.
50 Laws of Thought, p. 29. It is pointed out in the preface to this Second Edition that Leibnitz was acquainted with the Laws of Simplicity and of Commutativeness.
51 Prior Analytics, i. cap. xxvii. 3.
52 Encyclopædia Britannica, Eighth Ed. art. Logic, sect. 37, note. 8vo. reprint, p. 79.
53 De Morgan, On the Root of any Function. Cambridge Philosophical Transactions, 1867, vol. xi. p. 25.
54 Syllabus of a proposed System of Logic, §§ 122, 123.
55 Elementary Lessons in Logic, p. 86.
56 Outline of the Laws of Thought, § 87.
57 Treatise on Natural Philosophy, vol. i. p. 161.
58 Treatise on Natural Philosophy, vol. i. p. 6.
59 Todhunter’s Plane Co-ordinate Geometry, chap. ii. pp. 11–14.
60 An explanation of this and other technical terms of the old logic will be found in my Elementary Lessons in Logic, Sixth Edition, 1876; Macmillan.
61 Elementary Lessons in Logic, pp. 67, 79.
62 Pure Logic, p. 19.
63 An Outline of the Necessary Laws of Thought, Fifth Ed. p. 161.
64 Mansel’s Aldrich, p. 103, and Prolegomena Logica, p. 221.
65 Elements of Logic, Book II. chap. iv. sect. 4.
66 Aldrich, Artis Logicæ Rudimenta, p. 104.
67 Examination of Sir W. Hamilton’s Philosophy, pp. 452–454.
68 Pure Logic, pp 76, 77.
69 Pure Logic, p. 65. See also the criticism of this point by De Morgan in the Athenæum, No. 1892, 30th January, 1864; p. 155.
70 Boole’s Laws of Thought, p. 106. Jevons’ Pure Logic, p. 69.
71 On the Syllogism, No. iii. p. 12. Camb. Phil. Trans. vol. x, part i.
72 See Horsley, Philosophical Transactions, 1772; vol. lxii. p. 327. Montucla, Histoire des Mathematiques, vol. i. p. 239. Penny Cyclopædia, article “Eratosthenes.”
73 Euclid, Book x. Prop. 117.
74 Philosophical Magazine, December 1852; Fourth Series, vol. iv. p. 435, “On Indirect Demonstration.”
75 Philosophical Magazine, Dec. 1852; p. 437.
76 Mind; a Quarterly Review of Psychology and Philosophy; October, 1876, vol. i. p. 487.
77 Whewell, History of the Inductive Sciences, vol. i. p. 222.
78 Formal Logic, p. 124. As Professor Croom Robertson has pointed out to me, the second and third premises may be thrown into a single proposition, D = DeBC ꖌ DEbc.
79 Pp. 55–59, 81–86.
80 See his work called The Process of Thought adapted to Words and Language, together with a Description of the Relational and Differential Machines. Also Philosophical Transactions, [1870] vol. 160, p. 518.
81 Philosophical Transactions [1870], vol. 160, p. 497. Proceedings of the Royal Society, vol. xviii. p. 166, Jan. 20, 1870. Nature, vol, i. p. 343.
82 Syllabus of a proposed system of Logic, §§ 57, 121, &c. Formal Logic, p. 66.
83 Lectures on Metaphysics, vol. iv. p. 369.
84 Bowen, Treatise on Logic, Cambridge, U.S., 1866; p. 362.
85 The contents of this and the following section nearly correspond with those of a paper read before the Manchester Literary and Philosophical Society on December 26th, 1871. See Proceedings of the Society, vol. xi. pp. 65–68, and Memoirs, Third Series, vol. v. pp. 119–130.
86 Proceedings of the Manchester Literary and Philosophical Society, 6th February, 1877, vol. xvi., p. 113.
87 Montucla. Histoire des Mathématiques, vol. iii. p. 373.
88 British Quarterly Review, No. lxxxvii, July 1866.
89 Mind, October 1876, vol. i. p. 484.
90 Pure Logic, Appendix, p. 82, § 192.
91 Elementary Lessons in Logic (Macmillan), p. 123. It is pointed out in the preface to this Second Edition, that the views here given were partially stated by Leibnitz.
92 Syllabus of a Proposed System of Logic, p. 29.
93 It has been pointed out to me by Mr. C. J. Monroe, that section 14 (p. 339) of this paper is erroneous, and ought to be cancelled. The problem concerning the number of paupers illustrates the answer which should have been obtained. Mr. A. J. Ellis, F.R.S., had previously observed that my solution in the paper of De Morgan’s problem about “men in the house” did not answer the conditions intended by De Morgan, and I therefore give in the text a more satisfactory solution.
94 Montucla, Histoire, &c., vol. iii. p. 388.
95 Wallis, Of Combinations, &c., p. 119.
96 James Bernoulli, De Arte Conjectandi, translated by Baron Maseres. London, 1795, pp. 35, 36.
97 Arithmeticæ Theoria. Ed. Amsterd. 1704. p. 517.
98 Rees’s Cyclopædia, art. Cipher.
99 Œuvres Complètes de Pascal (1865), vol. iii. p. 302. Montucla states the name as De Gruières, Histoire des Mathématiques, vol. iii. p. 389.
100 Histoire des Mathématiques, vol. iii. p. 378.
101 Bernoulli, De Arte Conjectandi, translated by Francis Maseres. London, 1795, p. 75.
102 Wallis’s Algebra, Discourse of Combinations, &c., p. 109.
103 Œuvres Complètes, vol. iii. p. 251.
104 See also Galton’s Lecture at the Royal Institution, 27th February, 1874; Catalogue of the Special Loan Collection of Scientific Instruments, South Kensington, Nos. 48, 49; and Galton, Philosophical Magazine, January 1875.
105 Wallis, Of Combinations, p. 116, quoting Vossius.
106 Philosophical Transactions (1803), vol. xciii. p. 193.
107 Hofmann’s Introduction to Chemistry, p. 36.
108 Works, edited by Shaw, vol. i. pp. 141–145, quoted in Rees’s Encyclopædia, art. Cipher.
109 Nature, vol. i. p. 553.
110 Formal Logic, p. 172.
111 Philosophical Magazine, 4th Series, vol. i. p. 355.
112 Transactions of the Royal Society of Edinburgh, vol. xxi. part 4.
113 Philosophical Magazine, 4th Series, vol. vii. p. 465; vol. viii. p. 91.
114 Memoirs of the Manchester Literary and Philosophical Society, 3rd Series, vol. iv. p. 347.
115 Letters on the Theory of Probabilities, translated by Downes, 1849, pp. 36, 37.
116 Encyclopædia Metropolitana, art. Probabilities, p. 396.
117 Elements of Logic, Book III. sections 11 and 18.
118 Encyclopædia Metropolitana, art. Probabilities, p. 400.
119 Philosophical Transactions (1767). Abridg. vol. xii. p. 435.
120 Transactions of the Edinburgh Philosophical Society, vol. xxi. p. 375.
121 Montucla, Histoire des Mathématiques, vol. iii. p. 386.
122 Leibnitz Opera, Dutens’ Edition, vol. vi. part i. p. 217. Todhunter’s History of the Theory of Probability, p. 48. To the latter work I am indebted for many of the statements in the text.
123 Positive Philosophy, translated by Martineau, vol. ii. p. 120.
124 System of Logic, bk. iii. chap. 18, 5th Ed. vol. ii. p. 61.
125 Montucla, Histoire, vol. iii. p. 405; Todhunter, p. 263.
126 Essay concerning Human Understanding, bk. iv. ch. 14. § 1.
127 Philosophical Magazine, 4th Series, vol. i. p. 354.
128 Essay concerning Human Understanding, bk. ii. chap. xxi.
129 De Rerum Natura, bk. ii. ll. 216–293.
130 Cambridge Philosophical Transactions (1830), vol. iii. pp. 369–372.
131 Observations on the Nature and Tendency of the Doctrine of Mr. Hume, concerning the Relation of Cause and Effect. Second ed. p. 44.
132 Ibid. p. 97.
133 System of Logic, bk. II. chap, iii.
134 Inductive Logic, pp. 13, 14.
135 Bain, Deductive Logic, pp. 208, 209.
136 System of Logic. Introduction, § 4. Fifth ed. pp. 8, 9.
137 Ibid. bk. II. chap. iii. § 5, pp. 225, &c.
138 These are the figurate numbers considered in pages 183, 187, &c.
139 Commercium Epistolicum. Epistola ad Oldenburgum, Oct. 24, 1676. Horsley’s Works of Newton, vol. iv. p. 541. See De Morgan in Penny Cyclopædia, art. “Binomial Theorem,” p. 412.
140 Bk. ii. chap. iv.
141 Philosophical Transactions (1866), vol. 146, p. 334.
142 Budget of Paradoxes, p. 257.
143 Proceedings of the Royal Society (1872–3), vol. xxi. p. 319.
144 Life of Galileo, Society for the Diffusion of Useful Knowledge, p. 102.
145 Professor Bowen has excellently stated this view. Treatise on Logic. Cambridge, U.S.A., 1866, p. 354.
146 Roscoe’s Spectrum Analysis, 1st edit., p. 98.
147 Euler’s Letters to a German Princess, translated by Hunter. 2nd ed., vol. ii. pp. 17, 18.
148 Lavoisier’s Chemistry, translated by Kerr. 3rd ed., pp. 114, 121, 123.
149 Euler’s Letters, vol. ii. p. 21.
150 Lardner, Edinburgh Review, July 1834, p. 277.
151 Mémoires par divers Savans, tom. vi.; quoted by Todhunter in his History of the Theory of Probability, p. 458.
152 Poisson, Recherches sur la Probabilité des Jugements, Paris, 1837, pp. 82, 83.
153 Kirchhoff’s Researches on the Solar Spectrum. First part, translated by Roscoe, pp. 18, 19.
154 Edinburgh Review, No. 185, vol. xcii. July 1850, p. 32; Herschel’s Essays, p. 421; Transactions of the Cambridge Philosophical Society, vol. i. p. 43.
155 Evans’ Ancient Stone Implements of Great Britain. London, 1872 (Longmans).
156 Herschel, Outlines of Astronomy, 1849, p. 565; but Todhunter, in his History of the Theory of Probability, p. 335, states that the calculations do not agree with those published by Struve.
157 Philosophical Transactions, 1767, vol. lvii. p. 431.
158 Philosophical Magazine, 3rd Series, vol. xxxvii. p. 401, December 1850; also August 1849.
159 History, &c., p. 334.