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Leibniz's interpretation of his logical calculi

Published online by Cambridge University Press:  12 March 2014

Nicholas Rescher
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Extract

The historical researches of Louis Couturat saved the logical work of Leibniz from the oblivion of neglect and forgetfulness. They revealed that Leibniz developed in succession several versions of a “logical calculus” (calculus ratiocinator or calculus universalis). In consequence of Couturat's investigations it has become well known that Leibniz's development of these logical calculi adumbrated the notion of a logistic system; and for these foreshadowings of the logistic treatment of formal logic Leibniz is rightly regarded as the father of symbolic logic.

It is clear from what has been said that it is scarcely possible to overestimate the debt which the contemporary student of Leibniz's logic owes to Couturat. This gratitude must, however, be accompanied by the realization that Couturat's own theory of logic is gravely defective. Couturat was persuaded that the extensional point of view in logic is the only one which is correct, an opinion now quite antiquated, and shared by no one. This prejudice of Couturat's marred his exposition of Leibniz's logic. It led him to battle with windmills: he viewed the logic of Leibniz as rife with shortcomings stemming from an intensional approach.

The task of this paper is a re-examination of Leibniz's logic. It will consider without prejudgment how Leibniz conceived of the major formal systems he developed as logical calculi – that is, these systems will be studied with a view to the interpretation or interpretations which Leibniz himself intends for them. The aim is to undo some of the damage which Couturat's preconception has done to the just understanding of Leibniz's logic and to the proper evaluation of his contribution.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 19 , Issue 1 , March 1954 , pp. 1 - 13
Copyright
Copyright © Association for Symbolic Logic 1954

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References

1 Couturat's exposition of Leibniz's logical work is contained in his La Logique de Leibniz (Paris, 1901)Google Scholar, and the previously unpublished writings on which this is based are given in his Opuscules et fragments inédits de Leibniz (Paris, 1903)Google Scholar. Couturat discusses Leibniz's logical calculi in the eighth chapter, Le Calcul Logique, of Logique.

2 See Church's definition in The Dictionary of Philosophy, edited by Runes, D. (New York, n.d.)Google Scholar. With this compare Leibniz's discussion on pages 204–207 of volume seven of Die philosophischen Schriften von G. W. Leibniz edited by Gerhardt, C. I. (Berlin, 1890)Google Scholar.

3 The extensional interpretation of logic is, he claims, “la seule qui permette de soumettre la logique au traitement mathématique” (Logique, p. 32).

4 A limitation must be mentioned. We deal only with the mature portion of Leibniz's logical work, not with his earlier efforts, prior to 1679. Regarding these, reference should be made to Dürr's, Karl article, Leibniz' Forschungen im Gebiet der Syllogistik, in Leibniz zu seinem 300. Geburtstag (Berlin, 1949), to the exposition of Leibniz's arithmetic treatment of logic on pages 126–129Google Scholar of Łukasiewicz's, JanAristotle's syllogistic (Oxford, 1951)Google Scholar, and, of course, to Couturat's Logique.

5 Leibniz possessed to an insufficient extent the distinction – basic to the concept of a rule of inference – between a statement of the system and a statement about the system in a meta-language. However, he is often sufficiently close to an appreciation of the distinction in question to justify the explicit formulation of rules of inference. Thus, for example, in the marginalia given on pages 223–227 of vol. 7 of Phil. Schr. (Gerhardt), Leibniz distinguishes between verae propositiones, i.e., assertions of the system, and principiae calculi, i.e., rules for obtaining further assertions from given ones.

6 Leibniz's classic definition of equality, “Eadem sunt quorum unum in alterius locum substitui potest, salva veritate” (Phil. Sehr. (Gerhardt), vol. 7, p. 219)Google Scholar, is defective both as regards the distinction of use and mention, and that between object and meta-language. Our statement has repaired these defects.

7 Quicquid conclusum est in Uteris quibusquam indefinitis, idem intelligi debit conclusum in aliis quibuscunque easdem conditionibus habentibus, ut quia verum est ab est a, etiam verum erit bc est b, imo et bcd est bc (Phil. Schr. (Gerhardt), vol. 7, p. 224Google Scholar).

8 Leibniz cannot, in view of footnote 5, give a wholly adequate statement of this rule. However, he does come quite close. First some usages must be explained. A proposition vera is what we should term an asserted statement (of the system) (Phil. Schr. (Gerhardt), vol. 7, pp. 218 ff.Google Scholar). Statements play an axiomatic role if they are either self-evident truths (propositiones per se verae) or are arbitrarily assumed and asserted without proof (propositiones positae) (Loc. cit.). If a proposition is of the form Si …, ergo ——, i.e., if it is an implication, it is a consequentia. (Loc. cit. Compare the scholastic usage as discussed in the third chapter of Boehner's, P.Medieval logic (Manchester, 1952).Google Scholar) Now Leibniz states, “Proposition vera est, quae ex positis et per se veris per consequentias oritur,” i.e., “A statement is asserted which is obtained from posited statements and statements true-in-themselves (i.e., from the axioms) by utilizing implications” (Phil. Schr. (Gerhardt), vol. 7, p. 219)Google Scholar. No matter how this is taken, it would appear that the modus ponens rule is implied.

9 Besides the treatment in Couturat's Logique, there is an illuminating discussion of all three of these systems in Dürr's, Karl article, Die mathemalische Logik von Leibniz, Studio philosophica (Basel), vol. 7 (1947), pp. 87–102Google Scholar.

10 Cf. footnote 4.

11 Phil. Schr. (Gerhardt), vol. 7, pp. 219–221Google Scholar.

12 Phil. Schr. (Gerhardt), vol. 7, pp. 221–227Google Scholar. Couturat's discussion of this system is given on pages 336–343 of Logique.

13 In Leibniz's own notation, this would be written: “Eadem sunt a et non-non-a.” We will content ourselves with this one example.

14 On pages 219–221 of volume 7 of Phil. Schr. (Gerhardt), Leibniz offers an ingenious proof of 5, based on the fact that well-formed formulas of the calculus of this system must have one of a limited number of forms. This is, perhaps, the earliest example of what has come to be known as a syntactic metatheorem.

15 Si b est a et c est a, etiam bc erit a (Phil. Schr. (Gerhardt), vol. 7, p. 222Google Scholar). Couturat renders this as, “Si a est c, ou b est c, on peut affirmer que ab est c.” explaining in a footnote that Leibniz erroneously says “et” (Logique, p. 340). But Leibniz is not wrong, and 14 follows from 4, 10, and 13, all of which Couturat renders correctly.

16 A “term” a is proper if there is no “term” b such that a est bnon-b. The propriety condition is essential to the consistency of the system, as is shown by the following refutation of an unqualified 18:

(1) bnon-b est non-b by 17

(2) bnon-b est b by 16

(3) non-b est non-(bnon-b) by 3, (2)

(4) (bnon-b) est non-(bnon-b) by 4, (1), (3).

Leibniz does not always state the propriety condition explicitly, when required. However, in one of his writings he indicates that he understands the traditional categorical propositions always to contain the tacit assumption that the terms which enter are proper: “In omnibus tamen tacite assumitur terminum ingredientem esse Ens” (Phil. Schr. (Gerhardt), vol. 7, p. 214)Google Scholar. The formulation is in the terminology of Leibniz's second version of logical calculus.

17 In view of 2, this can be put in the form

19* If a is proper: If a est b, then a non est non-b

by substituting “non-b” for “b” in 19. This requires mention because, as we shall see, 19* plays an important role in the interpretations.

18 See footnote 8.

19 I will therefore confine myself here to referring each of the forty assertions we shall encounter to one occurrence in Leibniz's writings. I have selected in each case the occurrence which, to my knowledge, may be presumed to be the earliest chronologically. Regarding assertions 5, 14, 21, and 27, see the footnotes ad hoc. Reference may be made to vol. 7 of Phil. Schr. (Gerhardt) for assertions 1 (p. 218)Google Scholar, 4 (p. 218), 6 (p. 225), 7 (p. 218), 10–13 (p. 222), 15 (p. 223), 16–17 (p. 218), 18 (p. 224), 23 (p. 212), 25 (p. 230), 26 (p. 237), 28 (p. 232), 29 (p. 239), 30–31 (p. 232), 32–36 (p. 229), 37 (p. 234), 38 (p. 230), and 39–40 (p. 233). Finally, refer to Couturat's Opuscules et fragments for assertions 2–3 (p. 379), 8–9 (p. 365), 19–20 (p. 378), 22 (p. 233), and 24 (p. 261)Google Scholar.

20 See especially the Generates inquisitiones de analysi notionum et veritatum (Opuscules et fragments (Couturat), pp. 356–399), and the essays cited in footnote 49Google Scholar.

21 As objects (entia), Leibniz holds, one can take either all actually existing things, or else all which are (logically) possible. The dictum de omni et nullo must then be taken in the appropriate sense (Phil. Schr. (Gerhardt), vol. 7, p. 214)Google Scholar.

22 Substantivum [n.b.] est quod includit nomen Ens vel res; Adjectivum quod non includit. Ita animal est substantivum, seu idem quod ens animale. Rationale est adjectivum, fit enim demum substantivum, si adjicias Ens, dicendo Ens rationale, vel per compendium una voce (si jocari licet) Rational. Ut ex termino Ens animale: animal, (n.b. Hae definitiones usui scholae sunt accommodatae, sed in characteristibus [i.e., in the symbolism] necesse non est differentiam nominis substantivi atque adjectivi apparare, neque ilia vero usum habet ullum (Phil. Schr. (Gerhardt), vol. 7, p. 227Google Scholar).

23 Witness Couturat's comment on the passage cited in the previous footnote: “Cette influence scolastique se révèle par les définitions des termes de la logique traditionelle (grammaticale) dont Leibniz reconnaît lui-même l'inutilité” (Logique, p. 337, notes)Google Scholar.

24 Couturat holds that the extensional view of logic is “la seule qui permette de soumettre la Logique au traitement mathématique” (Logique, p. 32). This prejudice on his part leads Couturat to hold Leibniz's intensional point of view responsible for the shortcomings – generally rather imagined than actual – of his logical work (Logique, pp. 30–32, 353–54, 359–62, 373–77, and elsewhere).

25 One of the clearest expressions of this concern is the essay Difficultates logicae (Phil. Schr. (Gerhardt), vol. 7, pp. 211–217)Google Scholar.

26 Regarding Leibniz's solution see the writer's article. Contingence in the philosophy of Leibniz, Philosophical review, vol. 61 (1952), pp. 26–39CrossRefGoogle Scholar.

27 Opuscules et fragments (Couturat), pp. 356–399Google Scholar, and cf. also ibid., pp. 261–264.

28 Opuscules et fragments (Couturat), pp. 229–231Google Scholar.

29 Phil. Schr. (Gerhardt), vol. 7, pp. 211–217Google Scholar.

30 Opuscules et fragments (Couturat), pp. 232–237Google Scholar.

31 Opuscules et fragments (Couturat), pp. 421–423Google Scholar. Couturat's discussion of this system is given on pages 344–362 of Logique.

32 Opuscules et fragments (Couturat), p. 233Google Scholar. Cf. footnote 16.

33 Opuscules et fragments (Couturat), p. 233Google Scholar. By 3, an equivalent formulation of 21 is

21*. non-Ens est non-A, unless A est non-Ens.

Thus Leibniz states, “Non Ens est mere privativum, sive non-Y, id est non-A, non-B, non-C, etc., idque quod vulgo dicunt nihili nulla esse proprietates” (Ibid., p. 356). Again, another formulation, in virtue of 19, is

21**. non-Ens non est A, unless A est non-Ens.

Leibniz gives this also: “Esto N non est A, N non est B, item N non est C, et ita porro, tunc dici potest N est Nihil [i.e., non-Ens]. Huc pertinet quod vulgo dicunt,, non Entis nulla esse Attributa” (Couturat, , Logique, p. 349, notesGoogle Scholar).

Couturat is patently misguided when he remarks in discussing this last passage (Ibid., and cf. p. 353, notes) that, “cette définition, inspirée, comme on voit, de la tradition scholastique, n'a aucune valeur. Tout au contraire, on définit à présent le zéro logique comme le terme qui est contenu dans tous les autres (en extension), comme le sujet de tous les prédicats possibles.”

34 The method of proof by cases facilitates the check.

35 If non-A is proper, non-A est Ens, whence non-Ens est A. Thus non-Ens is of (virtually) universal intension, and so the intension of Ens is null.

36 Ens is the class of all things (entia). See footnote 21.

37 Such sets of symbolic versions of categorical propositions are given in many places, including pp. 211–217 of vol. 7 of Phil. Schr. (Gerhardt), and pp. 232-33Google Scholar of Opuscules et fragments (Couturat). Couturat's apparent denial (Logique, p. 30) notwithstanding, the intensional interpretation of this second system is adequate to classical syllogistic logic.

38 Opuscules et fragments (Couturat), pp. 300 (top), 384–385, et alGoogle Scholar.

39 See assertions 16 and 17.

40 Logique, pp. 30–32.

41 See footnote 24.

42 Cum dico A est B, et A et B sunt propositiones, intellego ex A sequi B.” “A est B” is held to be the symbolic version of “A infert B” or “B sequitur ex A” (Couturat, , Logique, p. 355, notesGoogle Scholar).

43 A is necessary iff A = Ens. A is impossible if non-A is necessary, and it readily follows that, “Quod continet Bnon-B, idem est quod impossibile” (Opuscules et fragments (Couturat), p. 368)Google Scholar.

44 Regarding Leibniz's conception of this interpretation see especially the Generales inquisitiones (cf. footnote 27).

45 See Couturat, , Logique, p. 355Google Scholar.

46 Couturat, . Logique, p. 355Google Scholar.

47 The concept of containment provided the central idea underlying Leibniz's interpretations of his first two systems. Already in the second system the notation “continet” was occasionally used in place of “est.”

48 Opuscules et fragments (Couturat), pp. 292–321Google Scholar, and cf. pp. 267–270. Couturat's discussion of this system is given on pages 362–385 of Logique.

49 Number XVI, pp. 208–210, number XIX, pp. 228–235, and number XX, pp. 236–247. The last two are available in an English translation in the appendix of Lewis's, C.I.Survey of symbolic logic (Berkeley, 1918)Google Scholar.

50 Leibniz nowhere explicitly states this associative law. However he uses it in proofs, and he writes sums without parentheses (Phil. Schr. (Gerhardt), pp. 228 ff.Google Scholar). [Note that Leibniz is elsewhere scrupulous in their use, eg., (Opuscules et fragments (Couturat) pp. 356 ff)Google Scholar]. K. Dürr also adds this associative law in his exposition of Leibniz's logic, rightly saying that, “Diese Ergänzung dient lediglich dazu, das Verständnis des Systems von Leibniz zu erleichtern; es wird dadurch an dem System nichts Wesentliches verändert” (Die mathematische Logik von Leibniz, p. 100)Google Scholar.

51 That is why this system is presented as a Non inelegans specimen demonstrandi in abstractis (Phil. Schr. (Gerhardt), vol. 7, pp. 228 ff.Google Scholar; cf. Lewis, , Survey of symbolic logic, pp. 373–379)Google Scholar. This essay – especially the third definition and the various scholia – shed much light on Leibniz's conception of this third system of logical calculus.

52 Lewis's evaluation of Leibniz's logic is of interest: “It is a frequent remark upon Leibniz' contributions to logic that he failed to accomplish this or that, or erred in some respect, because he chose the point of view of intension instead of that of extension. The facts are these: … He preferred the point of view of intension, or connotation, partly from habit and partly from rationalistic inclination. … This led him into some difficulties which he might have avoided by an opposite inclination or choice of example, but it also led him to make some distinctions the importance of which has since been overlooked and to avoid certain difficulties into which his commentators have fallen.” (Survey of symbolic logic, p. 14.)

53 This serves to explain why some modern logicians find Leibniz's third system the most satisfactory: it is the least exclusively logical, the most abstract.