Abstract
This paper studies what can reasonably be regarded as truth-table devices in one of Boole’s late manuscripts as a way of addressing Boole’s relation to modern propositional logic. A careful investigation of the divergences between those table devices and our current conception of truth tables offers an opportunity to reassess the singularity of Boole’s logical system, especially concerning the relation between its linguistic and mathematical aspects. The paper explores Boole’s conception of the compositional structure of symbolic expressions, the genesis of table devices from his method of development into normal forms, and the non-logical origin of the constants 0 and 1 as dual terms. Boole’s system of logic is in this way shown to be chiefly concerned with the problem of the formal interpretability conditions of symbolic expressions, rather than with the truth conditions of logical propositions.
This paper was supported by the Formal Epistemology—the Future Synthesis project, in the framework of the Praemium Academicum program of the Czech Academy of Sciences, to which I am very much indebted.
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Notes
- 1.
See, for instance, Gödel’s recently published 1939 logic course at Notre Dame [1], which constitutes a flawless account of the historical, conceptual, and technical significance of truth tables for the understanding and development of propositional logic.
- 2.
See [11] for Grattan-Guinness’s argument in favor of Shosky’s position.
- 3.
Unlike the rest of the names in this list, Russell’s relation to the Boolean tradition was not direct, but mediated in a decisive way by Peano’s original elaboration, which, in turn, explicitly relies on the formulations of Boole, Schröder and Peirce (see Russell’s [14, §2] and Peano’s [15, §§VII, XII, XVI, XVII, XXI]).
- 4.
For Boole’s temporal interpretation of secondary propositions, see [18].
- 5.
- 6.
See the introduction to Sect. 3 below.
- 7.
Extracts from this manuscript were published in 1952 by Rhees in [21], but the editor, manifestly interested above all in Boole’s perspective on the problem of reasoning, skipped the section containing the “tables” in question, mentioning only that “[t]he logical symbolism is the same as in the Laws of Thought” [21, p. 238]. The publication of the entire document had then to await Grattan-Guinness and Bornet’s edition of Boole’s manuscripts in 1997 [22], the same year as Shosky’s paper.
- 8.
Which borrows the form from tables such as (1), as we can infer from the first column of the first of the three tables in the manuscript.
- 9.
For a standard presentation of Boolean algebras and rings, see, for example, [23].
- 10.
Reference to a larger number of works following this approach can be found in [27].
- 11.
- 12.
- 13.
See, for instance, [2, p. 167].
- 14.
See, for instance, [3, § 5.101].
- 15.
For Gödel’s use of truth tables for the decidability of propositional calculus, see [1, p. 23].
- 16.
See [1, § 1.1.8].
- 17.
For an interesting study putting forward the notion of expression in Lagrange, see [30].
- 18.
- 19.
Boole is clear about this point: “We might justly assign it as the definitive character of a true Calculus, that it is a method resting upon the employment of Symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation.” [16, p. 4].
- 20.
See [17, XI, § 5].
- 21.
For the reduction of hypotheticals to categoricals, see [33].
- 22.
As in this passage, in which truth and falsehood are said to concern only “a branch” of logic: “A denial must be a denial of the truth of a proposition and there is a branch of Logic […] which relates to propositions in their special attributes of truth and falsehood and in the relations flowing from those attributes” [22, p. 59].
- 23.
Boole’s modifications in the treatment of secondary propositions in LT do not significantly change this decisive feature of his approach. See [17, p. 169 sqq.].
- 24.
Indeed, the meaning of uninterpretable algebraic expressions, such as \(\sqrt {-1}\) or non-convergent series expansions, was a common concern for the English algebraists and occasioned the originality of their approach. For an extensive treatment of this problem in the most general terms, see Babbage’s [34].
- 25.
- 26.
- 27.
- 28.
For example: “Let ϕ(xy) = x(1 − y), then ϕ(10) = 1, ϕ(11) = 0, ϕ(01) = 0, ϕ(00) = 0” [16, p. 74].
- 29.
This definition is, as we now know, full of subtlety and danger. However, Boole adroitly avoids defining 1 as the class of all classes and prevents the confusion between both by attributing to them different semiological status: in Boole’s own terms, while 1 is a “symbol” (as much as x or y), classes are referred to by “letters” such as X or Y. See [16, p. 15].
- 30.
Boole will further associate the fact that the fundamental law of logic is of second degree to the dual nature of logical symbols (since from x 2 = x we can have x(1 − x) = 0, which he interprets as the law of contradiction). Incidentally, this offers him occasion to consider the possibility of the fundamental law being of higher degree, thus explicitly envisaging the existence of logics whose terms would accept more than two possibilities [17, p. 49–50]. However, Boole never explicitly relates the degree and factorization of that fundamental equation to the duality of 0 and 1, which could have been made through their status as roots of that equation.
- 31.
Interestingly, Boole remarks that the first of those two results can lead to 1 = 0, which he interprets as the “nonexistence of the logical Universe,” and indicate the attempt “to unite contradictory Propositions in a single equation” [16, p. 65].
- 32.
Boole explicitly associates the whole procedure with that of solving linear differential equations, “arbitrary elective symbols in the one, occupying the place of arbitrary constants in the other” [16, p.70].
- 33.
For instance, Boole tried to confer a logical signification on the uncomfortable symbols \(\frac {0}{0}\) and \(\frac {1}{0}\), much like in the case of 0 and 1, by associating them with indefiniteness and infinity [17, pp. 90–91]. In the manuscripts he will also refer—no less unconvincingly—to the four coefficients in terms of “logical categories,” now considering \(\frac {1}{0}\) to be the symbol of impossibility [22, pp. 99–100].
- 34.
The text, entitled “On the Foundations of the Mathematical Theory of Logic and on the Philosophical Interpretation of Its Methods and Processes” [22, pp. 63–104], provides a complete overview of his system, insisting on its foundational aspects. It was intended as a preliminary philosophical introduction to the system for the purpose of an application of the theory of probabilities.
- 35.
The presence of this condition can be traced back to MAL, recognizable under the form {ϕ(xy…)}n = ϕ(xy…), which Boole presents as a condition introducing “symmetry into our Calculus” [16, p. 66]. In LT, Boole acknowledges it, under the form V (1 − V ) = 0, as “the condition of interpretability of logical functions,” and devotes a whole chapter to exploring some of its properties [17, p. 93 and ch. X]. In those pages, such a condition inspired, however, a rather convoluted method for reducing every step of a calculation to an interpretable form.
- 36.
For the problem of division in Boole, see [19].
- 37.
In the paragraphs preceding his presentation of the tables, Boole explicitly provides the derivation of conditions for the additive expression [22, p. 92].
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Gastaldi, J.L. (2022). Boole’s Untruth Tables: The Formal Conditions of Meaning Before the Emergence of Propositional Logic. In: Béziau, JY., Desclés, JP., Moktefi, A., Pascu, A.C. (eds) Logic in Question. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-94452-0_7
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