Abstract
It is my aim in this paper to take an especially broad look at the logic of Boole. This does not mean that I think details are not important, but the more you go into them the more you get caught at a particular spot in space and time—say Lincoln in the year 1847—and Boole seems correspondingly further away from us. If the details of Boole’s logic are left aside, what remains is a science. The question therefore arises of how much Boole’s concept of a science determined his logic. This is the vital question of the following exposition.
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References
Macmillan, Barclay and Macmillan, Cambridge 1847 (referred to hereafter as MAL). Reprinted in Collected Logical Works by George Boole. Vol. I. Studies in Logic and Probability, Rush Rhees (ed.), The Open Court, La Salle Ill. 1952 (referred to hereafter as CLW), pp. 45–124. MAL was recently reprinted in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, W.B. Ewald (ed.), Clarendon Press, Oxford 1996, vol. I, pp. 451–509, and also as a monograph (with an introduction by J. Slater) by Thoemmes Press, Bristol 1998. References are to the original 1847 pagination.
The idea still haunted Boole when he wrote An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities, Walton and Maberley, London 1854 (referred to hereafter as LT). There, in a footnote (LT 406), he recognizes the possibility of sciences erected on fundamental ideas which ‘might be wholly different from those with which we are at present acquainted’.
‘On the Theory of Probabilities, and in particular on Mitchell’s Problem of the Distribution of Fixed Stars’, The Philosophical Magazine, Series 4, vol. i, 1851, pp. 521–30. Reprinted in CLW, pp. 247–59. The quotation is on p. 252 of the reprint.
At LT 50 Boole takes up the theme of the possibility of other logics and stresses the hypothetical character of this train of thought. It does not imply, he points out, that our logic ‘is the product either of chance or of arbitrary will. This remark is also true, by the way, if logic is considered as part of our culture.
Taylor and Walton, London 1851 (referred to hereafter as CoS). Reprinted in CLW, pp. 187–210; references are to the pagination of the reprint.
From this circumstance one might expect an uplifting speech lacking depth, but Boole ‘took considerable pains over the matter of the lecture’ because he intended to publish it and was not willing ‘to labour for a merely temporary object’ as he wrote in a letter to Augustus De Morgan. See Desmond MacHale, George Boole—His Life and Work, Boole Press Dublin 1985 (referred to hereafter as MacHale), pp. 97f. The letter from which the foregoing quotations are taken is also reprinted on pp. 54f. of G.C. Smith, The Boole-De Morgan Correspondence 1842–1864, Clarendon Press, Oxford 1982.
His System of Logic, Ratiocinative and Inductive appeared in 1843 and is referred to by Boole more than once, beginning with the preface to MAL.
His History of the Inductive Sciences and The Philosophy of the Inductive Sciences appeared in 1837 and 1840 respectively. Boole refers to the latter work at LT 406.
I borrow these terms from Ferdinand de Saussure, who uses them to distinguish two types of linguistics (i.e. ‘linguistique synchronique’ and ‘linguistique diachronique’).
The following quotation from her writings is taken from Luis Maria Laita: ‘Boolean Algebra and Its Extra-logical Sources: The Testimony of Mary Everest Boole’, History and Philosophy of Logic 1, 1980, pp. 37–60. The quotation is on p. 58; additions are mine. The anecdote is treated exhaustively by Volker Peckhaus in his Logik, Mathesis universalis und allgemeine Wissenschaft—Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert, Akademie Verlag, 1997, pp. 222ff. In the Nachlass of Boole there is a fragment in which he deals with ‘a recently published tract of Leibnitz’, drawing parallels to his own laws of logic. This fragment is printed in George Boole, Selected Manuscripts on Logic and Its Philosophy, Ivor Grattan-Guinness and Gérard Bornet (eds), Birkhäuser Verlag, Basel 1997 (referred to hereafter as SMLP), pp. 185ff.
Book 1, Chapter xi. Boole cites Aristotle in Greek; the above translation is by A.E. Wardman.
It is not my intention here to give an interpretation of Aristotle but to elucidate Boole’s concept of ‘unity’. It is true that Aristotle criticized the Platonist conception of a science ‘which embraces everything’ on the grounds that ‘being is said in many ways’ (Metaphysics Book 1, Ch. 9). But whether this is enough to make him unsuitable as an exponent of the doctrine of the ‘unity of science’, as Prof. Elisabeth Schwartz argued in the discussion of this paper, depends exactly on the concept of unity which one applies. Boole was probably thinking of the principle of contradiction as an example of an ‘axiom’ which is common to all sciences. He discusses it in light of his reading of Aristotle (Metaphysics Book 4, Ch. 3) at LT 49f. and also at SMLP I64ff. (N.B. Metaphysics Book 4 was formerly referred to as Book 3; this explains why Boole refers to Book 3, Ch. 3 in his footnote at LT 49.)
It depends of course on how (correct) thinking is defined. Boole did nothing other than what we would do today: he consulted the leading scientists of his time. This is, in my opinion, the proper role to assign for instance to Richard Whately (1787–1863) and even to Immanuel Kant (1724–1804). But this has become obscured because the role has long since been passed on to others. The result of this confusion is that Boole is blamed for having formalized Aristotelian logic. But his greatness lies not in what he formalized but in how he did so.
SMLP 126. The quotation is also found in an excerpt at CLW 14.
The Social Aspects of Intellectual Culture, George Purcell, Cork 1855, pp. 8f. MacHale, pp. 122–5, contains an extensive extract from this small brochure; the passage quoted is on p. 123.
Portrayed at some length at CoS 204f.
Weltkreis, Berlin 1928 (referred to hereafter as Aufbau).
George Allen & Unwin, London 1919 (referred to hereafter as Introduction).
Introduction 61.
SMLP 120. Boole speaks there (pp. 119–20) of the analogy ‘between the operations of thought in Logic and its operation within a particular sphere of the science of number’.
Carnap uses this example to introduce the concept of a structural description in his Aufbau, p. 17.
LT 320 contains a different but cognate determination of the ‘final object of science’.
Compare also LT 404: ‘...if the process of reasoning be carefully analyzed, it will appear that abstraction is made of all peculiarities of the individual to which the conclusion refers...’.
For instance in the last chapter of LT at 415f. Boole says in a survey of philosophical systems of all ages: ‘The attempts of speculative minds [...] have differed less in the forms of theory which they have produced, than through the nature of the interpretations which have been assigned to those forms’.
I suppose that Boole would have accepted a proof of the existence of God which would conclude from the existence of an orderly creation to the existence of a being which caused this order. Thus he says (CoS 191): ‘Science... may... be compared to some stately temple, whose materials have been brought together from many distant regions... in whose fair proportions and goodly order, we read the traces of the designing mind’. Boole is careful not to make the intellect of men the cause of order in nature according to the strategy of Kantian transcendental idealism. ‘The native powers of the mind, cast abroad amid a world of mere chance and disorder, could never have realised the conception of law’ (CoS 190).
This passage is also quoted by Mary Hesse, ‘Boole’s Philosophy of Logic’, Annals of Science 8, 61–81 on p. 66. The phrase ‘Natural History of Thought’ is also used at SMLP 53.
‘We may by restricting the canon of interpretation confine our expressed results within the limits of the scholastic logic; but this would only be to restrict ourselves to the use of a part of the conclusions to which our analysis entitles us’ (MAL 34).
Epistemologically the clue for the incomprehensibility may lie for Boole in the capability of the human mind to grasp general conceptions. This capability is restricted because humans need signs to have something perceptible—a ‘sensuous element, without which the conception could not have been thought’ (CLW 226; see also SMLP 68)—and their mental power is finite. Compare also what Boole says about the intelligibility of God at SMLP 200–202. The reason why the existence of a single all-uniting ‘superstructure’—the subject of this ‘higher logic’—is plausible for Boole could lie in the fact that for him nature and mind were designed by the same supreme being. One should be careful to note that for Boole this does not mean that the laws of the mind could be reduced to the laws of nature. They are still normative in the sense that they are subject to an authority which does not reign by blind necessity as in the realm of nature but ‘with reference to an ideal standard and a final purpose’ (LT 421). A last remark: it is very tempting to draw the border between class logic and the ‘higher logic’ with the help of the difference between an object-language and a metalanguage. But this idea is alien to Boole and his time. Boole knew, however, the difference in logic between operations which depend upon ‘laws of combination’ and those which depend upon ‘interpretation’, as he makes clear in a letter to Arthur Cayley (SMLP 192). This allows for the possibility of modestly identifying the ‘higher logic’ as that logic which comprises ‘laws of combination’ only.
See SMLP 125, where Boole claims that no one before him has given ‘a complete statement of the elementary operations involved in Conception...[and]... an analysis of the laws of that faculty’.
The Cambridge & Dublin Mathematical Journal 3, 1848, pp. 183–98. Reprinted in CLW, pp. 125–40. The passage specifically referred to is on p. 136 of the reprint.
In the discussion of this paper Theodore Hailperin pointed out that in spite of Boole’s well-known adherence to the exclusive ‘or’ in logic, his calculations nevertheless contain steps which are not allowed on an exclusive understanding of ‘or’.
In the world of things, Boole says (SMLP 146), the operation of subtraction ‘does presuppose the condition that when a number x is to [be] diminished by another number y, x should be greater than y’. But, Boole goes on, in algebra this restriction does not hold. At SMLP 43 he proposes a thought experiment in which it is imagined that we could conceive of particular numbers only, such as 10, 20, 30 etc. and of no others. If we were to investigate the laws of these numbers, we could make calculations as usual, as long as the results were interpretable, although it would mean ‘the intermediate steps being uninterpretable for these numbers’ (Boole’s emphasis).
Boole’s Logic and Probability, North-Holland, Amsterdam 1986 (2nd ed., revised and enlarged), pp. 89f. (emphasis mine, slightly paraphrased). I want to point out that Hailperin does not find the talk of embeddability quite appropriate for Boole and that I have chosen a broader and somewhat different concept of structure here than he does.
It is significant that Boole places the symbolical methods at the end of his Treatise on Differential Equations (Macmillan, Cambridge 1859); see there pp. vi-viii. Likewise he places his Mathematical Analysis of Logic at the end of the historical development of logic and emphasises that without the ‘aids of symbolical language’ we would have ‘to abandon the hope of further conquest’ in the field of logic (MAL 10).
The collection of cases is not useless though; it can help to find the one single instance which is necessary to find absolute truth.
Cf. his The Construction of Social Reality, The Free Press, New York 1995.
This is the text of the talk as distributed at the conference entitled ‘Boole 1997: One Hundred Fifty Years of Mathematical Analysis of Logic’, held at Lausanne in September 1997, with some additions in the footnotes which reflect the discussion. I thank Elaine Lerf for proofreading, James Gasser for careful editing and Andreas Bächli for advice in connection with the objection of Prof. Elisabeth Schwartz dealt with in footnote 12. This paper has benefited considerably from the results of research carried out by the author with support from the Swiss National Science Foundation during the period 1991–1995.
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Bornet, G. (2000). George Boole and the Science of Logic. In: Gasser, J. (eds) A Boole Anthology. Synthese Library, vol 291. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9385-4_14
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