Abstract
By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s Begriffsschrift, on the other hand, is a visual tool to scrutinize concepts and inferences, and is a calculus only in the thin sense that every possible transition between sentences is fully and unambiguously specified in advance. While Frege’s outlook has dominated much of philosophical thinking about logical symbolism, we believe there is value—particularly in light of recent interest in the role of notations in mathematics and logic—in reviving Boole’s idea of an intrinsic link between, as he put it, a ‘calculus’ and a ‘directive method’ to solve problems.
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Notes
Wilson (2006, p. 523).
These expressions should be approached with caution, as they have been used for various purposes and interpreted in numerous different ways; for further discussion, see Sect. 6 below, in particular note 50. Note that Frege generally writes lingua characterica, but that other authors typically use lingua characteristica instead. Nothing much hinges on this, but for reference, here is the source of the discrepancy: Leibniz, who coined the phrase, actually used the latter spelling; the former (mistaken) spelling comes from a spurious title for one of Leibniz’s manuscripts introduced by a much-circulated eighteenth-century edition (Raspe 1765), which was then used by Trendelenburg (1856), Frege’s main source on the topic. We thank an anonymous reviewer for this remark.
See below, especially footnote 14, p. 7.
We are here glossing on Boole’s own account (see in particular (Boole 1854, ch. XV, pp. 226–242), and not aiming for a historically accurate rendition of the goals of syllogistic.
Boole (1854, p. 59).
In fact, Boole uses indefinite class symbols ambiguously, a difficulty that we shall point out here but ignore in the sequel, as it does not bear on our main points. In most settings, he takes such symbols to be absolutely indefinite, that is, to denote a class that can be empty, equal to the full universe of discourse, or anything in between. But Boole also translates the Aristotelian form ‘Some \(A\) is \(B\)’ as \(va=vb\), in which—if the traditional interpretation of such forms is to be preserved—\(v\) has to be interpreted as an indefinite non-empty class. For a careful discussion written from a Boolean perspective, see Venn (1881, ch. VI–VII).
Boole (1854, pp. 134–137). We follow Boole’s rendition of Aristotle; the passage in question is Nicomachean Ethics II.5 (1105b20–1106a15).
In traditional tables of the canonical syllogistic forms, there are other restrictions, which Boole neglects (it is expected that the major term will come first, for instance). The discrepancy arises because Boole does not fully do justice to traditional logic: he takes syllogistic’s classification of inference forms as a full-fledged theory of reasoning, whereas traditional textbooks would also contain a broader theory exploring how non-canonical pieces of reasoning are to be brought into one of the standard forms. This need not detain us further, as we are only concerned with Boole’s own portrayal of syllogistic.
There is yet another form that can be requested of the conclusion: one may want the list of those combinations of the selected terms that are excluded by the premises, i.e., that correspond to classes that the premises force to be empty (for more on this, see Sect. 4 below). In Boole’s system, this amounts to seeking an equation of the form \(V=0\), in which \(V\) is a sum of combinations of class symbols or their negations; such an equation is equivalent to having each of the members of the sum be separately equal to 0. This form is the most exhaustive, being equivalent to the premises (or to the premises once elimination has been performed, if some terms have been eliminated), whereas equations of the form \(x=\ldots \) will usually be weaker.
Boole (1854, p. 10). Note that Boole uses the word ‘premise’ to refer to the data, or given relations, of the kind of logical problems he considers; the word thus takes on a broader meaning than may be usual (i.e., it does not refer merely to the starting points of some individual inference).
Boole (1854, p. 165).
Boole (1854, pp. 168–169)
Boole (1854, pp. 178–179).
Schröder (1877, pp. 25–28), Lotze (1884, pp. 219–221) = Lotze (1912, pp. 265–267), Wundt (1880, vol. 1, pp. 356–357), Frege (1979, pp. 39–45) = Frege (1969, pp. 45–51), Venn (1881, pp. 280–281), and McColl (1878, pp. 23–25). That this single problem has been solved by logicians of various outlooks whose solutions would repay further comparison has already been noted by Gabriel (1989, p XXIII).
See Boole (1854), pp. 146–147.
As Boole puts it, ‘It will be observed, that in each of the three data, the information conveyed respecting the properties \(A\), \(B\), \(C\), and \(D\), is complicated with another element, \(E\), about which we desire to say nothing in our conclusion. It will hence be requisite to eliminate the symbol representing the property \(E\) [\(\ldots \)]’ (Boole 1854, p. 146).
Note that \(\Delta \) is our own notation to denote some arbitrary algebraic expression.
In our case, we indeed get \(U = T = {\bar{c}}d+c{\bar{d}}+{\bar{b}}{\bar{c}}{\bar{d}}\) and \(W = ST = 0\) (it is easy to see that the product ST vanishes, because each combination of a summand from S and a summand from T multiplies out to a term containing both a symbol and its negation, e.g. c and \({\bar{c}}\), and thus to zero), but on the face of it, we obtain a nonzero term for \(V = {\bar{S}}\). However, one can also see that \({\bar{S}}{\bar{T}} = 0\), which means that \(V{\bar{U}} = 0\): in terms of classes, V is included in U, so that in the solution \(a = U + uV\), where u is an indefinite class term, V is redundant and can be discarded—thereby correctly falling back on Boole’s solution (Schröder 1877, p. 27).
For instance, as discussed in the previous section, one might ask for the expression of \(ab\) in terms of \(c\) and \(d\), that is, seek a relation of the form \(ab=\ldots \) instead of the simpler \(a=\ldots \) or \(b=\ldots \). Boole would then introduce an auxiliary term \(t\) with an additional premise \(t=ab\), then eliminate \(a\) and \(b\) as well as other unwanted terms and proceed as above, seeking a solution of the form \(t=\ldots \), where \(t\) can ultimately be replaced by \(ab\) again (Boole 1854, pp. 140–142).
See note 6 above.
See Boole (1854, p. 84): ‘Every primary proposition can thus be resolved into a series of denials of the existence of certain defined classes of things, and may, from that system of denials, be itself reconstructed.’
Jevons (1887, p. 113); this sentence is missing from the 1874 edition.
Venn (1881, p. 301).
For a fuller discussion of Lotze’s outlook on this point as well as of parallels with Frege, see Heis (2013) as well as Gabriel (1989). Note that Frege, although he knew Lotze’s book, may not have been aware that the second edition contained an addendum treating the very problem he chose to discuss in his manuscript on Boole; see Gabriel, op. cit, p. XX and XXIII.
Frege (1993b, p. 97).
Frege (1979, p. 46).
This is assuming his solution is corrected as per footnote 42 below; his own version has only twelve premises.
Note that this is only true because of the specific form of the nested conditionals Frege uses here; if there were, say, negation signs anywhere else on the top line than just before the letter, this would cease to hold.
This is assuming that Frege’s mistakes are corrected. For the record, here is how this should be done. The editors (see Frege 1979, note 1 p. 41) suggest adding a premise (10)\(^\prime \) and replacing Frege’s premise (12) by their (12)\(^\prime \) (shown below). Additionally, the following amendments are required. First, one should introduce equivalent variants of (10)\(^\prime \) and (12)\(^\prime \), namely (10)\(^{\prime \prime }\) and (12)\(^{\prime \prime }\):
Second, in Frege’s solution of the first question, one should check that combining either of (10)\(^{\prime \prime }\) or (12)\(^{\prime \prime }\) with either of (8) or (9) only produces tautologies. Third, in the list at the top of p. 44, all occurences of ‘(17)’ should be replaced by ‘(19)’.
In fact, Frege notices that a further simplification is possible: \(B\) can be eliminated by cut from (7) and (16), yielding
so that (6) and (17) together contain the full solution to Boole’s first question.
In Aristotelian logic, ‘converting’ means inverting subject and predicate, e.g., going from ‘Some \(A\) is \(B\)’ to ‘Some \(B\) is \(A\)’ or from ‘All \(A\) are \(B\)’ to ‘Some \(B\) are \(A\)’.
Boole (1854, pp. 10–11). (Emphasis in the original.)
The idea to use these phrases to describe distinct traditions in logic goes back, as far as we know, to Philip Jourdain in his preface to Couturat (1914); it was sharpened and popularized by Van Heijenoort (1967), then taken up by various authors to shed light on other aspects of the history of logic (for further references, see for instance Kusch 1989, notes 5 and 6 p. 289). A further complication is that already in Frege’s time, the phrases (which ultimately come from Leibniz through the reconstruction by Trendelenburg 1856) were used in senses quite different from his, most notably by Schröder in his review of the Begriffsschrift. For discussion, see Sluga (1987) and Peckhaus (2004).
Grattan-Guinness (1997, p. xliv), his emphasis.
Grattan-Guinness (1992, p. 44).
A good reference point here is John Stuart Mill, whom Boole quotes at the beginning of his first logical tract (see below). Mill argues that the language of algebra is prototypically mechanical: ‘The complete or extreme case of the mechanical use of language, is when it is used without any consciousness of a meaning [\(\ldots \)]. This extreme case is nowhere realized except in the figures of arithmetic, and still more, the symbols of algebra [\(\ldots \)]. Its perfection consists in the completeness of its adaptation to a purely mechanical use. The symbols are mere counters [\(\ldots \)]. There is nothing, therefore, to distract the mind from the set of mechanical operations which are to be performed upon the symbols [\(\ldots \)].’ (Mill 1974, pp. 707–708.) But he goes on to argue that such a mechanical language is in fact radically inappropriate outside of arithmetic: ‘On all other subjects, instead of contrivances to prevent our attention from being distracted by thinking of the meaning of our signs, we ought to wish for contrivances to make it impossible that we should ever lose sight of that meaning even for an instant.’ (Ibid, p. 710.)
Boole (1847, p. 2).
As Grattan-Guinness points out, Boole credits the mind with a capacity to grasp ‘general laws from particular cases’, for instance, which appears to be outside the scope of his system (Grattan-Guinness 1997, p. xliv).
See Machale (2018, pp. 182–184).
For further discussion on this, see Rohr (2020, section 1.5).
Boole (1854, p. 11).
See for instance Landy et al. (2014), which also surveys some of the relevant experimental literature.
See for instance Vold and Schlimm (2020).
See Schröder (1895).
See Brady (2000), in particular ch. 8 and 9.
On Peirce’s view of analysis as the main task of logic, see for instance Pietarinen (2016).
Incidentally, Peirce also developed technical criticisms of Schröeder’s approach of the topic, on which see Brady (2000, pp. 153–155).
Peirce (1897, pp. 193–194).
Peirce (1902, p. 24).
Ibid.
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Acknowledgements
Many thanks to Tabea Rohr, Nicolas Michel, David Makinson, and an anonymous reviewer, as well as to (virtual) audiences at the ‘Orange County and Inland Empire Seminar in the History and Philosophy of Mathematics and Logic’ and at the ‘Mathématiques XIXe-XXIe siècles, histoire et philosophie’ seminar of the SPHere research team (CNRS, Paris). This work was supported by the Social Sciences and Humanities Research Council, Canada.
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Waszek, D., Schlimm, D. Calculus as method or calculus as rules? Boole and Frege on the aims of a logical calculus. Synthese 199, 11913–11943 (2021). https://doi.org/10.1007/s11229-021-03318-x
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DOI: https://doi.org/10.1007/s11229-021-03318-x