- Boole , G. 1847 . The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning Cambridge and London repr. 1958, New York), 1. Reprinted in G. Boole, Studies in logic and probability (ed. R. Rhees: 1952, London); see p. 49. This work is hereafter cited as ‘MAL’. An acquaintance with the main traits of the methods used by Boole in this treatise, though not necessary for the understanding of the present paper, would help to a better comprehension of the claims which are made in it.
- I have dealt with another of the influences which contributed to the birth of Boole's logic in The influence of Boole's search for a universal method in analysis on the creation of his logic Annals of science 1977 34 163 176
- A short list of works about the life and the philosophy of Sir William Hamilton includes Stirling J.H. Sir William Hamilton London 1865 J. Veitch, Hamilton (1882, Philadelphia and Edinburgh); W. H. S. Monk, Sir William Hamilton (1881, New York); and S. A. Grave, The Scottish philosophy of the common sense (1960, Oxford). Hamilton's logical ideas are found in Lectures on metaphysics and logic (4 vols., ed. H. L. Mansel and J. Veitch: 1859–1861, Edinburgh and London), vols. 3 and 4.
- Hamilton , W. 1829 . On the philosophy of the unconditioned, in reference to Cousin's infinitoabsolute . Edinburgh review , 50 : 194 – 221 . According to Hamilton's opinions as expressed in this paper, human thought imposes conditions upon its subjects of knowledge in the sense that when thinking of something, our mind subsumes it under a concept. In this way only ‘the conditioned’ becomes an object of knowledge, whereas the absolute or ‘unconditioned’ cannot be so. But the conditioned lies in the middle of two opposite unconditioneds. These two unconditioneds are not conceivable, but at least we know that one of them is true and the other false, this due to the principles of contradiction and of excluded middle. It thus seems that for Hamilton, truth is attained by taking a middle way between opposite absolute alternatives, which per se are not objects of thought.
- Hamilton , W. 1852 . “ On the study of mathematics, as an exercise of the mind ” . In Discussions on philosophy and literature 257 – 313 . London repr. from Edinburgh review, 62 (1836), 409–455. This work is hereafter cited as Hamilton, ‘Study of mathematics’.
- Whewell , W. 1835 . Thoughts on the study of mathematics as part of a liberal education Cambridge
- Hamilton , W. Study of mathematics 258 – 258 .
- Hamilton , W. Study of mathematics 274 – 274 .
- Hamilton , W. Study of mathematics 275 – 275 .
- Hamilton , W. Study of mathematics 281 – 281 .
- Hamilton , W. Study of mathematics 281 – 282 .
- Hamilton , W. Study of mathematics 297 – 297 .
- Hamilton , W. Study of mathematics 299 – 299 .
- Hamilton , W. Study of mathematics 309 – 309 .
- MacFarlane , A. 1916 . Lectures on ten British mathematicians of the XIX century Vol. 23 , London and New York In 1849 (=432), De Morgan was 43 years old.
- Graves , R.P. 1889 . Life of Sir William Rowan Hamilton Vol. 3 , 370 – 370 . London and Dublin It is essential not to confuse the Irish mathematician William Rowan Hamilton with the Scottish philosopher William Hamilton, De Morgan's opponent; both men attained deserved fame by their own merits in two completely different contexts.
- Jevons , S. Encylopaedia Britannica , 9th ed. Vol. 7 , 65 – 65 .
- De Morgan , S. 1882 . Memoir of Augustus De Morgan London The outline of De Morgan's life given in the text is based on this memoir.
- MAL , 1 – 1 .
- MAL , 11 – 11 .
- de Morgan , A. 1846 . On the structure of the syllogism . Trans. Cambridge Phil. Soc. , 8 : 379 – 408 .
- Hamilton , W. 1865 . Lectures Vol. 4 , 251 – 323 . London in
- de Morgan , A. 1846 . On the structure of the syllogism . Trans. Cambridge Phil. Soc. , 8 : 379 – 408 . reprinted in On the syllogism (edited with an introduction by Peter Heath: 1966, London), 1–17. This work is hereafter cited as De Morgan, ‘Syllogism’ and is cited from Heath's edition. Heath's ‘Introduction’ to De Morgan's ‘Syllogism’ is the best account of the controversy we are dealing with. This introduction, together with De Morgan's own version of the issue in his Formal logic (see footnote 35 below), and the papers which Hamilton and De Morgan published during the controversy are the sources of the account given in this paper.
- De Morgan , A. Syllogism , 1 – 1 .
- De Morgan , A. Syllogism , 2 – 2 .
- De Morgan , A. Syllogism , 8 – 8 .
- De Morgan , A. Syllogism , 8 – 8 .
- De Morgan , A. Syllogism , 8 – 8 . Supposing that m and n are the number of Ys in the first and second premises, respectively (Y being the middle term), De Morgan noticed that Aristotle considered only those cases in which one of the two, m or n, is unity and the other is zero. In that case m + n = 0 + 1 or 1 + 0. But for De Morgan m + n can be larger than 1, and both m and n may have values different from 1 and 0. Thus in De Morgan's theory, syllogisms such as: are acceptable if one gives to the words ‘most of’ the quantity ½ + a (which means a number larger than one-half of the number of elements constituting the universe to which the propositions refer).
- De Morgan , A. Addition to ‘Syllogism , 17 – 17 . De Morgan states that Hamilton might have preceded him in the construction of a theory of the quantification of the predicate. Nevertheless, De Morgan makes clear that he did not know the actual content of Hamilton's theory.
- De Morgan , S. 1882 . Memoir of Augustus De Morgan 160 – 160 . London
- De Morgan , A. 1847 . Formal logic, or the calculus of inference necessary and probable 303 – 303 . London De Morgans italics.
- De Morgan , A. 1847 . Formal logic, or the calculus of inference necessary and probable 313 – 313 . London
- Hamilton , W. 1850 . “ Requirements for a prize essay on the new analytic of logical forms ” . In An essay on the new analytic of logical forms Edited by: Baynes , T.S. New York in repr. 1971 see p. xi. Baynes's work is cited in text and footnotes as Essay.
- Hamilton , W. 1865 . Lectures Vol. 4 , 251 – 323 . London in This work must be a revised version of what Hamilton sent to De Morgan.
- Hamilton , W. 1850 . “ Requirements for a prize essay on the new analytic of logical forms ” . In An essay on the new analytic of logical forms Edited by: Baynes , T.S. xi – xi . New York repr. 1971
- Baynes , T.S. Essay ,
- Hamilton's symbolical propositions are not actually proper mathematical equations, but rather linear formulas in which the ordinary mathematical symbols do not appear Baynes's Essay 76 77
- Hamilton , W. 1865 . Lectures Vol. 4 , 254 – 254 . London
- Hamilton , W. 1865 . Lectures Vol. 4 , 260 – 260 . London MAL, 71–72.
- Baynes , T.S. Essay , 4 – 4 .
- Baynes , T.S. Essay , 7 – 7 .
- A , De Morgan . 1847 . Formal logic, or the calculus of inference necessary and probable 303 – 303 . London The two systems to which the quotation refers are those presented in ‘Syllogism’ and its ‘Addition’.
- De Morgan , A . 1847 . Formal logic, or the calculus of inference necessary and probable 303 – 303 . London
- De Morgan , A . 1847 . Statement in answer to an assertion made by Sir William Hamilton, Bart. London
- Hamilton , W. 1847 . A letter to Augustus De Morgan, Esq., on his claim to an independent rediscovery of a new principle in the theory of the syllogism London and Edinburgh
- De Morgan , A. 1847 . Statement in answer to an assertion made by Sir William Hamilton, Bart. 9 – 9 . London
- A. , De Morgan . 1847 . Statement in answer to an assertion made by Sir William Hamilton, Bart 14 – 14 . London
- Hamilton , W. 1847 . A letter to Augustus De Morgan, Esq., on his claim to an independent rediscovery of a new principle in the theory of the syllogism 4 – 4 . London and Edinburgh
- Hamilton , W. 1847 . A letter to Augustus De Morgan, Esq., on his claim to an independent rediscovery of a new principle in the theory of the syllogism 31 – 31 . London and Edinburgh
- Hamilton , W. 1847 . A letter to Augustus De Morgan, Esq., on his claim to an independent rediscovery of a new principle in the theory of the syllogism 32 – 32 . London and Edinburgh
- MacFarlane , A. 1916 . Lectures on ten British mathematicians of the XIX century Vol. 23 , 23 – 23 . London and New York
- Brody , B. 1967 . The rise of the algebra of logic , 59 – 68 . Ann Arbor : University Mierofilms .
- P. Heath, in his Introduction to De Morgan's ‘Syllogism’ xvi – xvi .
- MAL , 11–14 81 – 81 .
- MAL , 12 – 12 .
- MAL , 13 – 13 .
- MAL , 13 – 13 .
- MAL , 13 – 13 .
- MAL , Boole's italics.
- MAL , 9 – 9 .
- MAL , 7 – 8 .
- MAL , chapter 4.
- MAL , 40 – 40 .
- MAL , Boole's italics.
- MAL , 41 – 41 .
- MAL , 82 – 82 .
- De Morgan , A. Syllogism , 20 – 20 .
- MAL , 40 – 40 . 49–50, 52, 64, 65, 77
- Boole , M. Everest . 1878 . Home side of a scientific mind . The university magazine , 1 : 105 – 114 . 173–183, 327–336, 456–460 (p. 107).
- MAL , 4 – 5 . A testimony concerning Boole's interest and ability for languages is found in ‘George Boole, F.R.S.’, an 1866 biography of Boole by Robert Harley which is reprinted in G. Boole, Studies (footnote 1), 425–472 (p. 429). It is also interesting to notice that Boole, in a footnote to p. 5 of his introduction to The mathematical analysis of logic, indicates that he had studied German in, among other books, a German grammar by Becker, in which the similarity between the processes of reasoning and of language formation is stated.
- Laita . 1977 . The influence of Boole's search for a universal method in analysis on the creation of his logic . Annals of science , 34 : 163 – 176 .
- A defence of the claim that Boole held a philosophical conception involving a theory of mind with psychological and pedagogical implications appears in Laita L.M. A study of the genesis of Boolean logic University Microfilms Ann Arbor 1976 a doctoral dissertation submitted to the Graduate School of the University of Notre Dame. This dissertation was directed by Professor Michael J. Crowe.
- Hamilton , W. 1865 . Lectures Vol. 4 , London in
- T.S. , Baynes . Essay , 4 – 4 .
- Hamilton , W. 1829 . On the philosophy of the unconditioned, in reference to Cousin's infinitoabsolute . Edinburgh review , 50 : 194 – 221 .
- Boole's operator v appears seemingly arbitrarily introduced in chapter 2 of MAL. Nevertheless, it is a symbol which resulted coherently as a result of a mathematical process in chapter 7. In more detail, it appeared as the interpretation of the expansion of an elective function (as Boole called his logical formulas) of the form x(1-y). The coefficient 0/0 appeared in one of the terms of the sum. Boole associated the ‘indefiniteness’ of 0/0 with the indefinite operator v. See MAL 72 77
Annals of Science
Volume 36, 1979 - Issue 1