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From Leibniz to Frege: Mathematical Logic Between 1679 and 1879
FROM LEIBNIZ TO FREGE: MATHEMATICAL LOGIC BETWEEN 1679 AND 1879 CHRISTIAN THIEL Aachen, F.R.G.
Celebrating the centenary of a book is not unusual at all, but it has problems all its own. If the book is rather well known and is a work of high standing, as we may fairly say of Frege’s Begrzjkschrijt, then it is as difficult to avoid platitudes as to escape from the temptation to snatch at novel and unthought-of aspects of the work, its background, or its after-effects. I plead guilty to yielding to a similar weakness the day I had to announce the title of my address, a title the pretentiousness of which had been clear to me all along but which grew into a torment in the course of my subsequent studies. But let me jump into the matter by pointing out to what extent it is yet a rewarding enterprise to review the two hundred years of formal logic from 1679 to 1879, even if mathematical logic in the sense of van Heijenoort’s Source Book (from which the title of this address obviously was purloined), did not spring into life with Leibniz, nor with Boole, as I have been assured by many of my British colleagues, but only with Frege just a hundred years ago.* Paul Lorenzen, a scholar who will not easily be suspected of exaggerated flattery concerning Platonist logicians, has called Frege’s Begrzrsschrift “a logical masterpiece comparable in originality and import only to Aristotle’s Analytics” and William and Martha Kneale in their exposition of Frege’s doctrines, have come to the opinion that “it is not unfair either to his predecessors or to his successors to say that 1879 is the most im-
* I gratefully acknowledge my debt to Dr. Mark Kulstad for valuable suggestions and criticism of an earlier draft. LORENZEN, P., 1960, Die Entstehung der exakren Wissenschafren(Berlin/Giittingen/Heidelberg), p. 156. 755
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portant date in the history of the subject.’ If I do not completely misjudge these and many similar statements, it is not only the singular magnitude of Frege’s logical achievements that they refer to; it is also the bewildering impression that Frege created his logic, as it were, ex nihilo. And indeed, as to one of his most remarkable achievements, quantificational logic, it would not be appropriate to say that Frege worked in ignorance of previous contributions to the subject, since there was actually nothing he could possibly have ignored; so that, with van Heijenoort, “one cannot but marvel at seeing quantification theory suddenly coming full-grown into the w ~ r l d ” .To ~ sum up, whereas on other logical and philosophical topics, Frege was in fact ignorant of earlier work-as were his contemporaries, and as are still many of ours, in spite of Angelelli’s inestimable Studies on Gottlob Frege and Traditional Philosophy,4 -regarding the main content of his Begriffschrift, Frege was a genuine pioneer and a finisher at the same time. Frege himself did not think that he was a pioneer only, but a continuator as well: in the preface to the BegriJkschrVt he explicitly refers to Leibniz’s ideas on a calculus of logic and a characteristica universalis. Except for an incidental reference to Aristotle this is the only reference of this kind in the whole book. It was only as a reaction to Schroder’s unfavourable review of the Begrifsschrift that he released some information about his previous study of logical systems, but there is no reason for suspecting that he read Boole, McColl, Robert Grassmann, and Schroder only after the latter’s Personally, I am a little puzzled by Frege’s remark in the same preface that reads: “Quite alien to my mind have been those endeavours to create an artificial simiIarity [i.e. between logic and arithmetic, Th.] by regarding a concept as the sum of its marks werkmale, Th.]” (Bs., p. IV). This formulation has been connected with Schroder’s calculus of domains by Jourdain and by Wilma Papst (for which there is some support from the posthumously published KNEALE,W. and M., 1962, The Development of Logic (Oxford), p. 51. VAN HEIJENOORT, J., 1967, Introduction to Begriffsschrif, in: From Frege to Giidel. A source book in mathematical logic, 1879-1931. ed. J. van Heijenoort (Cambridge, Mass.) p. 3b. ANGELELLI, I., 1967, Studies on Gottlob Frese and TraditionalPhilosophy (Dordrecht). SCHR~DER, E., 1881, Review of Begriffsschrifi, Zeitschrift f i i Mathematik und Physik, V O ~ .25. pp. 81-94. FREGE, G., 1883. Ueber den Zweck der Begriffsschrift,Jenaische Zeitschriftfiir Naturwissenschaft vol. 16 (1883), Supplement, pp. 1-10.
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early manuscripts), but taken literally it would refer to intensional interpretations of logical calculi, manifesting Frege’s familiarity with this subject of contemporary controversies (a reading supported by Frege’s later avowal in favour of extensional logic in opposition to a logic of content). More mysterious than this detail for devoted historians is the fact that, apart from the statement that Frege’s starting point for his train of thought leading to the Begrz&wchrift was arithmetic (Bs., p. VIII), and a tiny remark (Bs., p. 4) disclosing that he had once experimented with another system, there is absolutely no trace of any “prehistory”, neither as to previous occupation with logic, nor as to the development of that peculiar notational system which Frege remained the only one to use. Not a single one of the seventeen lectures and reports that Frege gave to the Mathematical Society of Jena between 1866 and 1881 seems to have anything to do with logic-and the seven of which summaries in Frege’s own hand have been preserved provably do not. Taking Frege’s pioneer work in logic and its impact on the present state of the art to be sufficiently investigated, let us ask what had been going on in logic since Leibniz-some of whose “germs of thought” Frege decided to foster-and let us ask why logic between Leibniz and Frege had practically no influence at all on the latter. For the regrettably short and incomplete survey that can be given here, it will suffice to outline the most important directions and to indicate particular achievements. Roughly, I will touch upon Leibniz, on the intensional logic of the socalled Leibniz School, on Saccheri, on the algebra of logic with its treatment of classes, propositions, and relations, and finally upon Frege again. I will, however, abstain from a chronological enumeration of works and achievements, most of which will be known to this audience, at least regarding the algebra of logic. Instead, I will use, or if you like, abuse this presentation for arguing a case that I believe deserves consideration and support at an occasion like this meeting to commemorate what van Heijenoort has called “perhaps the most important single work ever written in logic”’. What I wish to argue for is nothing less than a reconsideration or even re-assessment of the historiography of logic. And while I do not have complaints about the presentation of Frege in our texts on history of logic, I certainly do have complaints about many other cases, some of which I will mention later.
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VAN
HEUENOORT, J., loc. cit. (see note 3), p. 3.
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It seems that Leibniz was the first scholar in possession of a clear conception of a calculus as a set of rules for performing operations of a strictly determined kind on graphical patterns, whether on strings of letters taken from some alphabet, or on geometrical diagrams. It is well known that he made at least three differentiated attempts at establishing a calculus of logic. In his first attempts, he tries to turn elementary number theory, to wit, the theory of divisibility, to advantage in the field of Aristotelian categorical syllogistic. After having failed with the assignment of prime numbers to elementary concepts and composite numbers to complex concepts, this leaving out of account what has been called “negative concepts”, Leibniz associates pairs of numbers with the subject and the predicates of propositions:
s * (sly s2> 9 Pc-)(P,,P,>, in order to express the traditional standard forms of syllogistic in the following form:
s z p -PI Is1 A P 2 b z sip * ( P i , S z ) = I h ( S i , P z ) = 1 , SOP c-) 1SUP, Sep t, 1Sip 9
(where (u, b) is the greatest common divisor of a and b). Rewriting the premises of a syllogism arithmetically in the manner indicated, one can check whether the correlate of the conclusion may be inferred also in arithmetic. Until quite recently, it was almost generally agreed upon that this calculus fails, too, and that Leibniz did not follow it up just because of this insight. Lukasiewicz in his book on Aristotelian syllogistic is the only one I know of who explicitly states that Leibniz’s calculus no. 1 is a correct arithmetical interpretation of categorical syllogistic. Whether it is or not depends on the definition of validity. In a paper read at the Third International Leibniz Congress here in Hanover in 1977, I have tried to show that Leibniz first had an incorrect definition of validity and tended to give up his arithmetical interpretation, but that he himself found the correct definition immediately afterwards, and that with this definition the calculus is entirely correct.’ THIEL, Ch., Leibnizens Definition der Iogischen Allgemeingultigkeit und der “arithmetische Ka/kiil”. in: Theoria cum praxi. Proceedings of the Ill. International Leibniz Congress, Hanover 1977 (Wiesbaden 1980), vol. 3, pp. 14-22
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So we do not know why Leibniz abandoned the arithmetical calculus, but we can be sure the reason was not its alleged failure in the domain of syllogistic. The reason may have been very trivial: Leibniz was not occupied with logic all the time; between 1675 and 1684 he made decisive progress with his infinitesimal calculus and may have been taken up with problems in this field. On the other hand, it is quite clear that even an operative arithmetical calcuhs for syllogistic was far too narrow for Leibniz’s conception of logic, which tended to a general calculus of concepts, extensional and intensional, and was not restricted to the traditional standard forms like “Every A is B , and which moreover was conceived to take up and extend Joachim Jungius’ treatment of nonsyllogistic inferences in the Logica Hamburgensis (1638, a work highly esteemed by Leibniz), which means, to work in the direction of a logic of relations, and finally to include a logic of propositions by extending Leibniz’s genei-al calculus de continente et contento to relations of entailment between propositions. It is to both extensions that Leibniz’s statement in the Nouveaux Essais refers, saying that, “there exist sound asyllogistic inferences which it would be impossible to prove by any syllogi~rn”.~ The result, in Leibniz’s second and third attempt at a calculus of logic (1686 and 1690), are some fragmentary lattice-theoretical systems, admitting of various interpretations, extensional, intensional, geometrical, and others, “Every A is B being interpreted, e.g., intensionally as “ A = AB” in 1686, and as “ A = A+B” in 1690, when Leibniz had a calculus with +, -, and =, #, and <. These calculi have been expounded so often and in detail that I will abstain from another description of them, and just mention a so far uncorroborated conjecture of mine concerning Leibniz’s change to the socalled “algebraic calculi” that I have called “lattice-theoretical” in the preceding sentence. In more elaborate books on the history of logic, we find James, i.e. Jacob Bernoulli mentioned with his book, The Parallelism between Logical and Algebraic Arguments lo, and we are told, e.g. in Styazhkin’s History of Mathematical Logic from Leibniz to Peano ‘I, that Bernoulli, “following Leibniz ... only noted the analogy that exists between the laws of formal logic and the methods of elementary algebra” @
NE IV, XVII 5 4; see also 11, 5 9.
I , Parallefisms ratiocinii Iogici et algebraici, Basel 1685; also in: Opera, ed. G. Cramer, I, Geneva 1744, repr. Brussels 1968. STYAZHKM, N.I., 1969, History of Mathematical Logic from Leibniz to Peano (Cam-, bridge, MassJLondon) (Russian original 1964).
lo BERNOULLI, JAC.
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(p. 95). As I do not know of any algebraically oriented logical work done
by Leibniz previous to 1685-leaving out of account the combinatorial investigations into syllogistics in the Dissertatio 12-1 would suspect that Leibniz’s experiments with algebraic calculi were stimulated by Bernoulli’s work in spite of the fact that the latter was composed rather sketchily and lacked profundity, so that Leibniz certainly could not have profited from this work to the same extent as later from Jacob Bernoulli’s insights into pr0babi1ity.l~ And indeed, Leibniz’s conception had become so broad as to render the exploitation of algebraical methods more difficult: In his letter to Gabriel Wagner on the utility of the art of reasoning or logic (1696) he states: “I have come to the opinion that even algebra borrows its advantages from a much higher art, to wit, from the true logic’’.14 The question whether Leibniz deserves an eminent place in the history of mathematical logic is not sacrilegious. The answer depends not only on our definition of “mathematical logic” but also on our conceptions of history and historiography. If history is what actually happened and had a significant effect, then Leibniz has to be omitted from the history of logic in his lifetime, for the writings referred to as containing his great and fruitful ideas in logic were completely unknown to his contemporaries. Part of them did not appear in print until Raspe’s edition of 1765, and most of them appeared only at the beginning of our century when Couturat made them accessible in 1901 and 1903, such that in Leibniz’s lifetime it is mainly his remarks in letters to contemporary scholars that could have had any traceable influence. He who conceives of the history of logic as the development of logical ideas, however, would be entitled to assign a rather minor place to Leibniz when talking about the logic of the Neuzeit, and I may for similar reasons be forgiven my decision to omit the logical work of Bolzano and of Robert Grassmann from this paper as they did not really influence the development of logic within the period we are concerned with. Before passing to the so-called Leibniz School in logic, I should say a few words on Gerolamo Saccheri (1667-1733) who is mainly being remembered for his Euclides ab omni naevo vindicatus, i.e. Euclid Cleared of Every Flaw, published in Milan in 1733, and containing a good portion LEIBNIZ,G. W., 1666, Dissertatio de Arte Combinutoria (Leipzig). BERNOULLI, JAC., 1713, Ari Conjectandi, Basel, repr. Brussels 1968. LBIBMZ,G. W., 1696, Schreiben an Gabriel Wagner. Vom Nutzen der Vernunfrkumt oder Logik. 1696. Erdm. 418 ff., quotation on p. 424 b.
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of unstrived-for development of theorems of a non-Euclidean geometry. Less known is Saccheri’s Logicu Demonstrutivu, which saw three editions, an anonymous one in 1697, the others in 1701 and in 1735.” This work deserves a place in the history of logic not only for Saccheri’s interest in a logical analysis and description of mathematical proof techniques in geometry, but also because of two contributions presented in its eleventh chapter. Saccheri here develops a “via nobilior”, i.e. a “more noble way” than previously followed, in logic generally as well as earlier in his book. His first idea is that of proving a theorem by deriving it from its own contradictory.’6 According as the theorem is afEmative or negative, this means proceeding by (lu+u)+u
or
( u + la)-, l u ,
o r rather, l U + U
a+ l u
i.e., as rules of inference. Saccheri’s goal was a consistency proof for (or rather, a proof of the truth of) a postulate system, say for geometry, by applying this “principle of necessary truth”, i.e. deriving each of the postulates from its own contradictory. This method has been taken up in our century by Josiah Royce ”. Indeed, l u + u , l b + b ,... 11(u A b A ...) AUQUSTAE TAURINORUM (Torino) 1697;Ticini Regii (Pavia) 1701 (Heinrich Scholz’s correction of Vailati’s date “1701” to “1702” in his Abriss der Geschichte der Logik, FreiburglMunchen a1959,a1967,p.37,is itself mistaken); Augustae Ubiorum warn) 1735. l6 See p, 80 of the 1697 edition: “Sumam contradictorium propositionum dernonstrandarum, ex eoque ostensiuh, ac direct6 propositum eliciam” (p. 82 of the 1701 and p. 130 of the 1735 edition). As will be seen, my interpretation diverges from Angelelli’s and Hoorman’s who identify this principle with Clavius’ Law, and the via negativa with indirect proof without distinguishing reductio ad absurdum from a --t 1a < l a . Saccheri, in the Scholium to chapter 11, presents the via negativa as a deductlo ad impossibile as is further corroborated by his example, “v. g. quod, si modus AA concluderet in secunda figura, omnis, vel aliquis syllogismus AA esset syllogismus EA.” (p. 89 of the 1697edition). JOSIAHROYCE, Prinzipien der Logik. In :Encyclopddie der philosophischen Wissenschafren, in Verbindung mit Wilhelm Windelband herausgegeben von Arnold Ruge. Erster Band: Logik. Tubingen 1912, pp. 61-136. The original English version appeared in the first volume of the (unfinished) English edition: T h e Principles of Logic. In: Sir Henry Jones (ed.), Encyclopaedia of the Philosophical Sciences, vol. I: Logic (London 1913). pp. 67-135. A separate edition was published in New York in 1961. l6
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is even constructively valid. This method was put to use in Saccheri’s Euclides, and earlier in his Logica Demonstrativa in connection with the via nobilior for a justification of the syllogisms by a sort of self-application which I do not have the time here to expound but on which we have valuable commentaries by Angelelli 1975, Hamblin 1975, and Hoorman 1976.” Whereas Saccheri worked independently of Leibniz’s logical experiments, the writers of the so-called Leibniz School were influenced by Leibniz although part of their work precedes Raspe’s edition of 1765. For example, Johann Andreas von Segner’s Specimen Logicae appeared in Jena in 1740, Georg Johann von Holland’s correspondence with Lambert started previous to 1764, the year in which the first two volumes of Johann Heinrich Lambert’s Neues Organon appeared in print. All these writers, stimulating each other by correspondence and small treatises, worked along Leibnizian lines, but restricting themselves (with few exceptions) to the intensional interpretation of their calculi. Lambert makes use of a quantification of the predicate-a device concerning which Hamilton and De Morgan claimed the priority of invention some sixty years later-and distinguishes two cases of each standard form, e.g. in the universal affirmative proposition “All A is B”, Case I: A = B where the converse is also universal, and Case 11: A < B where the converse is particular. In other cases, a standard form had to be represented by two formulae, and the calculi became very complicated, too complicated for Ploucquet, Lambert, Holland and others to handle in a satisfactory manner. John Venn in his Symbolic Logic has attributed this failure to the intensional standpoint which he also blamed for the incompleteness and inutility of Leibniz’s endeavours. This has become the standard value judgment on the Leibniz School, including Castillon, whose calculus was considered to be the most consequential. Couturat has strongly emphasized this point of view with regard to Leibniz. C. I. Lewis stated it plainly: “This movement produced nothing directly which belongs to the history of symbolic logic. ...The record of symbolic logic on the continent is a record of failure, in England, ANGELELLI, I., 1975, On Saccheri’s use of the “ConsequentiaMirabilis”, Akten des II. Intern. Leibniz-Kongresses 1972, Bd. IV, (Wiesbaden 1975), pp. 19-26; C. L. IIAMBLIN, Saccherian arguments and the self-applicationof logic, Australasian Journal of Philosophy, vol. 53, pp. 157-160; CYRILF. A. HOORMAN, Jr., A further examination ofsaccheri’s use of the “ConsequentiaMirabilis”, Notre. Dame Journal of Formal Logic vol. 17 (1976). pp. 239-247.
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a record of success. The continental students habitually emphasized intension, the English, extension’’ Meanwhile, formal logic on the continent had suffered a serious set-back at the hands of Kantian transcendental logic which, though in no way inccmpatible with formal logic, detracted, at least in Germany, interest frcm formal logic which, moreover, was treated very contemptuously by Kant whose own lectures on logic might have been quite suitable to support this attitude, but who had ridiculed Leibniz’s idea of a logical calculus and a universal characteristic already in his Nova Dilucidatio.20 Hegel’s judgment, and Lotze’s in 1841, were in no way more favourable, and in spite of my respect for investigations into the very foundations of formal logic as we find them in most textbooks of “logic” during the last century, I cannot overlook the fact that the intellectual climate on the continent had indeed become suffocative for symbolic logic by the middle of the 19th century. Typical of this is the fate of Moritz Wilhelm Drobisch’s logic in its original form, entitled A New Exposition of Logic, According to Its Simplest Relations, with a logico-mathematical Appendix 21. The appendix must be mentioned here since in the first edition, there is a paragraph with the title, “algebraic construction of the simplest forms of judgment and a derivation of inferences founded thereupon” 2 2 , where Drobisch develops an extensional caIculus of classes and elementary judgments, improving the intensional systems of Ploucquet, Lambert, Jacob Bernoulli, and Gergonne, all of whcm he explicitly mentions. Unfortunately, Drobisch felt obliged to withdraw this valuable paragraph in the second and later editions because of a severe criticism put forward by Friedrich Adolf Trendelenburg, who in his Logical Investigations 23 criticized formal logic as taught by Leibniz, Twesten, Drobisch and others
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LEWIS,C. I., 1918, A survey of symbolic logic (Berkeley, abridged New York 1960), p. 36 f. KANT,I., Principiorum primorum cognitionis metaphysicae nova dilucidatio (& c.), 1755, Akad. I. (Berlin 1902/10), pp. 385-416 (esp. p. 390). DROBISCH, M. W., Neue Darstellung der Logik nach ihren einfachsten Verhaltnissen. Nebst einem logisch-mathematisrhen Anhang (Leipzig 1836); Neue Darstellung der Logik nach ihren einfachsten Verhaltnissen, mit Riicksicht auf Mathematik und Naturwissenschuft, Zweite, vollig umgearbeitete Auflage (Leipzig 1851); Dritte neu bearbeitete Auflage (Leipzig 1863); Vierte verbesserte Auflage (Leipzig 1875); Fiinfte Auflage (Leipzig 1887). op. cit. (see note 21), Algebraische Construction der einfachsten Urtheilsformen und darauf gegrihdete Ableitung der Schliisse, pp. 131-136 of the 1836 edition. I* TRENDELENBURG, F. A., Logische Untersuchungen (Berlin 1840, Leipzig ‘1862).
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for its allegedly unjustified treatment of concepts by arithmetical analogies. It is a curious fact that the same Trendelenburg, who claimed that Leibniz did not conceive logic as a formal discipline at all, published in 1856 an essay, On Leibniz’s Project of a Universal Characteristic 24, in which he also sketched the prehistory of this project since Raymundus Lullus, thereby drawing attention to the universal language movement, and contributing essentially to the revival of interest in Leibniz’s logic. Incidentally, this essay of Trendelenburg of 1856 is the earliest text in which I have been able to find the German expression Begrirsschrift, although used in a very general sense for any notation that brings the formation of a sign in touch with the content of the concept it denotes, as Trendelenburg finds it in numerals and other implements of science. The so-called algebra of logic, crowned with the names of De Morgan, Boole, Peirce, and S:hriider, has been so thoroughly investigated and expounded in the literature that it can be taken to be the best known period in the history of formal logic. Let me just remind you of De Morgan’s foundation of the theory of relations, surpassed only by Peirce’s work in the subject, whose calculus of relations could already profit from Boole’s calculus of classes. Boole had started with a successful attempt at an axiomatized Aristotelian syllogistic, in order to defend one of De Morgan’s claims in his quarrel with Hamilton. The result was The Mathematical Analysis of Logic, being an Essay towards a Calculus of Deductive Reasoning (Cambridge and London 1847), the exposition of the intended system being given in An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probability (London and Cambridge, 1854). Boole was either ignorant or intentionally negligent of the logical work of his immediate predecessors, and he did not devote much thought to the problems of extension and intension, of existential import, and of empty classes, problems that had hampered previous logicians and were to return later, but of which C. I. Lewis says rightly: “It is well that, with Boole, they are given a vacation long enough to get the subject started in terms of a simple and general procedure” (Survey, p. 51). A peculiarity TRENDELENBURG, F. A., Uber Leibnizens Entwurf einer allgemeinen Charakteristik (Vorgetragen zur Feier des Leibniztages 1856). Philosophische Abhandlungen der Koniglichen Akademie der Wissenschaftenzu Berlin. Aus dem Jahre 1856 (Berlin 1857), pp. 37-69 (also published separately); reprinted in F. A. T., Historische Beitruge zur Philosophie. Dritter Band. VermischteAbhandlungen(Berlin 1867), pp. 1-47. The expression “Begriffsschrift” on p. 39 of the 1857, p. 4 of the 1867 printing.
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of Boole’s, which he shares with Frege, is his conviction of the interrelation of logic, language, and mathematics, a thought that impressed itself on many of his logical conceptions, perhaps even on the parallelism claimed by Boole to hold between the algebra of classes and that of propositions or of 0 and 1, later adjusted by Peirce who pointed out the special status of the fact that if x # 0, then x = 1. Let me return to Frege by relating his Begrifsschrut to the algebra of logic, following his admirably clear statements in his paper, On the Purpose of the “Begriffsschrif” (1882), written as an address to the Jena Society for Medicine and Natural Sciences to defend himself against Schroder’s reproach that Frege was, firstly, ignorant of earlier work and that secondly, his BegriJ7khr$t marked a regress in comparison with the algebra of logic. Frege points out that his purpose was not the presentation of abstract logic by formulae, but the creation of an instrument capable of expressing a content in a more precise and more perspicuous way than could be done by natural language. He finds fault with Boole’s classification of propositions into primary and secondary ones, this leaving out of account existential propositions, and misses in Boole’s system a notation for individuals (as different from singletons), and for the falling of an individual under a concept. Moreover, granting the possibility of interpreting Boole’s calculus as dealing with classes on the one hand, and as dealing with propositions on the other hand, Frege does not find a bridge between the two, a bridge for passing from a proposition to the concepts contained in it, and vice versa. And as to the notation, this latter problem makes “0” and “1” ambiguous in arithmetical contexts where “0” and “1” already have their ordinary numerical meaning. To sum up, Frege thinks that Schroder was mistaken in trying to compare two systems of logic which depart from different starting points and aim at the fulfilment of different purposes. As manuscripts from the Nachlass show, it was quite clear to Frege that, in spite of the term “Begrifsschrijt”, which in this respect is rather ill-chosen, he did not start from concepts in order to combine them into propositions, but from propositions, decomposing them into function and arguments. Whereas in other respects he views himself as in the tradition of Leibniz, he is quite aware of the fact that here his system is opposed to that tradition. In my opinion, it will not be necessary to go into the system of the Begrifsschrijit here. Quantificational logic in Frege’s setting is close enough to its modern formulation to be considered well known itself, and I will mention only a point that is usually misunderstood. Introducing a notation
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for quantification, i.e. the universal quantifier and bound variables, Frege explicitly says that in “ @ ( A ) ” , “@” may be considered as the argument of a function, too, and may then be replaced by a German letter in a quantified proposition. It is, therefore, incorrect to say that Frege established only first-order logic in the Begriffsschrift, and that he illegitimately substituted bound functional letters for bound individual letters in some places of the third part of the Begri#kwhrvt where the concepts of sequence and successor are being defined. In the Begrtflsschrijt, the “a” of
“
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@(a) ’’ is typically ambiguous. 25
To many of you, what I have said in this paper will be proof enough for the delicacy of the problem of how to write or even to approach the history of logic. There are, first and almost trivially, two directions of approach. One may include everything that seems relevant from the view-point of the historiographer’s time, or one may, on the other hand, try to do justice to historical authors by taking seriously their contemporary problems, and even by trying to gain a better understanding of them by looking for their origins. Secondly, we have to make a decision whether we want to write a history of causes and effects, a history of stimuli and influences, or a history of aims and reasons, whether we want to investigate Wirkungsgeschichte in the traditional sense which will not be changed in principle by incorporating social and economical aspects and developments (desirable as this would be for the history of formal logic as it has been done for mathematics to some extent at least), or what Jiirgen Mittelstruss has called Griindegeschichte. My vote is for the latter, a history of means and ends, aims and reasons, of needs, goals, and purposeful actions. This does not strip Wirkungsgeschichte of its status as an indispensable basis of GriindeThis is to correct van Heijenoort’s remark on p. 3 of his introduction to Frege’s Begriffsschrqt quoted in note 3 above. The correctness of Frege’s inferences has also been pointed out by Terrell Ward Bynum (Onan Alleged Contradicrion Lurking in Frege’s Begriflsschrift, Notre Dame Journal of Formal Logic, vol. 14 (1973), (pp. 285-287) who thinks that while Frege “does not yet have all the machinery or the terminology to precisely spell out the distinction between what he would later call ‘first-level’ and ‘second-level’ functions, he never confuses the two” (p. 285). But closer inspection of Frege’s text shows (end of 5 10, 5 11. i. e. pp. 18-19, and pp. 60-62. p. 68 footnote) that the typical ambiguity is fully intentional. Indeed, in the Begriflsschrijit expression es
“
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@(a)
”
the letter ‘‘a” stands for “argument”, the letter “@” for “function”.
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geschichte which I would like to be understood as a history of (in our case) logic with particular emphasis on reasons for the various developments, but as well as a history of reasons themselves, their interconnections and changes. No doubt this is a beautiful challenge and programme, but I am afraid there is some troublesome work to be done before it can be taken up: there are scores of false data and erroneous judgements to be cleared away, and to be replaced by correct and more reliable ones. Let me just mention a small number of instances. Most historians of logic have not been able to see personally all the texts they quote and refer to; they rely on older historians of logic and carry over incorrect data as well as erroneous evaluations from these. How else could it be explained that Bardili, Victorin, Twesten are counted among the early representatives of mathematical logic in many histories of logic although there is practically nothing mathematical in them except variables A , B , ... and perhaps the plus, the minus and the equality sign? Why is Twesten referred to as a Leibnizian and as a member of the Leibniz School, while being strongly influenced by Kant, the structure of whose logic he took over? Why is Friedrich von Castillon (1747-1814), author of RPfiexions sur la logique (1802) and the MPmoires sur un nouvel algorithme logique (1803), mistaken for his father, Jean or Giovanni Francesco de Castillon (1708179I), practically everywhere, from John Venn via Shearman, C . I. Lewis, A. Church's Bibliography, Bochebski's Formal Logic, Styazhkin's History up to modern authors of 1970? 26 Going back to primary sources as far as they are accessible, and searching for items that have seemingly disappeared, seem to me the presently most important tasks to be taken care of before a reliable and informative history of mathematical logic can be written. It would be unfair and incompetent not to mention the work that has so far come closest to the demand stated: Wilhelm Risse's Die Logik der Neuzeit, a comprehensive enterprise that has so far grown into two volumes covering the period from 1500 to 1780. But it cannot be overlooked that Risse's is a history of logic in a broad sense, mathematical logic playing a subordinate part except in Leibniz, and even here there is a certain lack of apprehension of more recent research done by others with results that might have illuminated or made accessible obscure or difficult texts and passages. A true re-assessment of the history of the
'' See my paper Zur Beurteilung der intensionalen Logik bei Leibniz und Castillon, Akten des 11. Intern. Leibniz-Kongresses, Hannover 1972, Bd. IV (Wiesbaden 1975), pp. 27-37.
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subject will require much closer cooperation between historians and mathematical logicians; let us hope that it will come about. Let me come to an end with a short systematical remark that might be relevant for the historical evaluation of Frege’s logic. As is well known, Frege developed his Begriflsschrgt further and presented an intricate and profound system of logic and part of set theory in his Grundgesetze der Arithmetik, the first volume of which was published in 1893, the second in 1903. The destiny of this system is widely known: it proved inconsistent by admitting the derivation of the Zermelo-Russell antinomy. But after the wreckage of the Grundgesetze by this antinomy, Frege and other logicians came to believe that the system might be saved by modifying the principle of abstraction which had served to supply Frege’s coursesof-values. LeSniewski and later Quine have shown that Frege’s own modification was insufficient for his purposes, but there remains the simple question whether by seizing the principle of abstraction we got hold of the real trouble-maker. As to this question, it seems to be little known even among experts that the fuse leading to the Zermelo-Russell antinomy is already hidden in the Begrijhchrijl in spite of the absence of courses-of-values or extensions of concepts in this system. In what sense this claim is justified will become clear from an analysis of Frege’s Appendix to the second volume of the Grundgesetze. After presenting a meticulous derivation of the antinomy, and a shorter version using his elementhood symbol “n”, Frege blames his fundamental law V for the catastrophe. To be more precise, he blames one direction of it. Splitting (and slightly changing)
into
and
FROM LEIBNIZ TO FREGE
769
the latter part appears to be essential for Frege’s derivation of the antinomy. To raise this above all doubt, Frege decides to make his argumentation independent of courses-of-values altogether by deriving the falsehood of Vb without reference to them. The idea is to consider Vb as a special case of
f ( 4= g(a) MJ3j-m))
=Mg(-glP))
and to derive the negation of this formula in the system. Checking this derivation, one finds that Frege makes use of the fundamental laws IIa and IIb, of theorems Ig, IIIa, and IIIe, and that he uses substitution, detachment, contraposition, transitivity, and universal generalization as his rules. But substitution and detachment are precisely the rules of the Begrzfsschrijt system, the other rules appealed to are provably admissible, the fundamental law IIa is the Begr@sschrift axiom 58, theorems IIIa and IIIe are axioms 52 and 54, while Ig can easily be proved as a theorem of the Begrifsschrift, and IIb is understood by Frege to be contained in axiom 58, as indicated above and as may be seen from his 5 10 and his procedure in the third part of the text, especially on pp. 6 0 4 ~ 2 . ~ ~ In other words, the derivation in the Appendix can be carried out in the Begrzfsschrijt of 1879 as well, the result being a theorem stating that for any second-level function that takes an argument of the second kind (i.e., a one-place first-level function), there are two concepts yielding the same value when taken as arguments of the function although there are objects falling under one of them but not under the other. As this is valid for any second-level function whatsoever, it will also hold for Frege’s course-of-values function &#J ( E ) , contrary to its purpose : there are concepts O(() and Y ( { ) which yield the same value when taken as arguments, i.e. E@(E) = C;Y(a), although there is at least one object A such that @ ( A ) but not Y(A). Hence, @(() and U({) do not both extend to the same objects; their extensions are different. As E@(e) = drY(a), neither Z@(E) nor drY(a) can be the extension of the concepts @({) and Y ( { ) in the traditional sense. This, of course, obstructs Frege’s definition of number as an extension (or course-of-values)-unless some other step or expression in his derivation can be made responsible for the unpleasant
a’
Cf. note 25.
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result. The problem of identity of sets has returned through the back door as the problem of identity of functions. I will pursue this question further elsewhere, and restrict myself to the remark that the feature of Frege’s Begriffsschrijit just sketched refutes the wide-spread opinion that his system of 1879 is not only less precise than that of 1893-which is true-but also essentially weaker. And let us not forget, lastly, that it is, in spite of certain deficiencies, the beginning of mathematical logic in the narrower and modern sense, even though mathematical logic a century ago means mathematical logic before Peano’s axioms of 1889, before Hilbert’s Foundations of Geometry of 1899, before Zermelo’s axioms of set theory, before Brouwer’s intuitionistic critique of 1907, before Principia Mathematica, before Hilbert’s Programme, before recursive functions, and before Godel’s theorems. Was not Frege’s Begrirsschriff the memorable first step of a flight of stairs leading to a rapidly expanding world of logic?
Celebrating the centenary of a book is not unusual at all, but it has problems all its own. If the book is rather well known and is a work of high standing, as we may fairly say of Frege’s Begrzjkschrijt, then it is as difficult to avoid platitudes as to escape from the temptation to snatch at novel and unthought-of aspects of the work, its background, or its after-effects. I plead guilty to yielding to a similar weakness the day I had to announce the title of my address, a title the pretentiousness of which had been clear to me all along but which grew into a torment in the course of my subsequent studies. But let me jump into the matter by pointing out to what extent it is yet a rewarding enterprise to review the two hundred years of formal logic from 1679 to 1879, even if mathematical logic in the sense of van Heijenoort’s Source Book (from which the title of this address obviously was purloined), did not spring into life with Leibniz, nor with Boole, as I have been assured by many of my British colleagues, but only with Frege just a hundred years ago.* Paul Lorenzen, a scholar who will not easily be suspected of exaggerated flattery concerning Platonist logicians, has called Frege’s Begrzrsschrift “a logical masterpiece comparable in originality and import only to Aristotle’s Analytics” and William and Martha Kneale in their exposition of Frege’s doctrines, have come to the opinion that “it is not unfair either to his predecessors or to his successors to say that 1879 is the most im-
* I gratefully acknowledge my debt to Dr. Mark Kulstad for valuable suggestions and criticism of an earlier draft. LORENZEN, P., 1960, Die Entstehung der exakren Wissenschafren(Berlin/Giittingen/Heidelberg), p. 156. 755
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portant date in the history of the subject.’ If I do not completely misjudge these and many similar statements, it is not only the singular magnitude of Frege’s logical achievements that they refer to; it is also the bewildering impression that Frege created his logic, as it were, ex nihilo. And indeed, as to one of his most remarkable achievements, quantificational logic, it would not be appropriate to say that Frege worked in ignorance of previous contributions to the subject, since there was actually nothing he could possibly have ignored; so that, with van Heijenoort, “one cannot but marvel at seeing quantification theory suddenly coming full-grown into the w ~ r l d ” .To ~ sum up, whereas on other logical and philosophical topics, Frege was in fact ignorant of earlier work-as were his contemporaries, and as are still many of ours, in spite of Angelelli’s inestimable Studies on Gottlob Frege and Traditional Philosophy,4 -regarding the main content of his Begriffschrift, Frege was a genuine pioneer and a finisher at the same time. Frege himself did not think that he was a pioneer only, but a continuator as well: in the preface to the BegriJkschrVt he explicitly refers to Leibniz’s ideas on a calculus of logic and a characteristica universalis. Except for an incidental reference to Aristotle this is the only reference of this kind in the whole book. It was only as a reaction to Schroder’s unfavourable review of the Begrifsschrift that he released some information about his previous study of logical systems, but there is no reason for suspecting that he read Boole, McColl, Robert Grassmann, and Schroder only after the latter’s Personally, I am a little puzzled by Frege’s remark in the same preface that reads: “Quite alien to my mind have been those endeavours to create an artificial simiIarity [i.e. between logic and arithmetic, Th.] by regarding a concept as the sum of its marks werkmale, Th.]” (Bs., p. IV). This formulation has been connected with Schroder’s calculus of domains by Jourdain and by Wilma Papst (for which there is some support from the posthumously published KNEALE,W. and M., 1962, The Development of Logic (Oxford), p. 51. VAN HEIJENOORT, J., 1967, Introduction to Begriffsschrif, in: From Frege to Giidel. A source book in mathematical logic, 1879-1931. ed. J. van Heijenoort (Cambridge, Mass.) p. 3b. ANGELELLI, I., 1967, Studies on Gottlob Frese and TraditionalPhilosophy (Dordrecht). SCHR~DER, E., 1881, Review of Begriffsschrifi, Zeitschrift f i i Mathematik und Physik, V O ~ .25. pp. 81-94. FREGE, G., 1883. Ueber den Zweck der Begriffsschrift,Jenaische Zeitschriftfiir Naturwissenschaft vol. 16 (1883), Supplement, pp. 1-10.
a
*
FROM LEIBNIZ TO FREGB
757
early manuscripts), but taken literally it would refer to intensional interpretations of logical calculi, manifesting Frege’s familiarity with this subject of contemporary controversies (a reading supported by Frege’s later avowal in favour of extensional logic in opposition to a logic of content). More mysterious than this detail for devoted historians is the fact that, apart from the statement that Frege’s starting point for his train of thought leading to the Begrz&wchrift was arithmetic (Bs., p. VIII), and a tiny remark (Bs., p. 4) disclosing that he had once experimented with another system, there is absolutely no trace of any “prehistory”, neither as to previous occupation with logic, nor as to the development of that peculiar notational system which Frege remained the only one to use. Not a single one of the seventeen lectures and reports that Frege gave to the Mathematical Society of Jena between 1866 and 1881 seems to have anything to do with logic-and the seven of which summaries in Frege’s own hand have been preserved provably do not. Taking Frege’s pioneer work in logic and its impact on the present state of the art to be sufficiently investigated, let us ask what had been going on in logic since Leibniz-some of whose “germs of thought” Frege decided to foster-and let us ask why logic between Leibniz and Frege had practically no influence at all on the latter. For the regrettably short and incomplete survey that can be given here, it will suffice to outline the most important directions and to indicate particular achievements. Roughly, I will touch upon Leibniz, on the intensional logic of the socalled Leibniz School, on Saccheri, on the algebra of logic with its treatment of classes, propositions, and relations, and finally upon Frege again. I will, however, abstain from a chronological enumeration of works and achievements, most of which will be known to this audience, at least regarding the algebra of logic. Instead, I will use, or if you like, abuse this presentation for arguing a case that I believe deserves consideration and support at an occasion like this meeting to commemorate what van Heijenoort has called “perhaps the most important single work ever written in logic”’. What I wish to argue for is nothing less than a reconsideration or even re-assessment of the historiography of logic. And while I do not have complaints about the presentation of Frege in our texts on history of logic, I certainly do have complaints about many other cases, some of which I will mention later.
’
VAN
HEUENOORT, J., loc. cit. (see note 3), p. 3.
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CH.THIEL
It seems that Leibniz was the first scholar in possession of a clear conception of a calculus as a set of rules for performing operations of a strictly determined kind on graphical patterns, whether on strings of letters taken from some alphabet, or on geometrical diagrams. It is well known that he made at least three differentiated attempts at establishing a calculus of logic. In his first attempts, he tries to turn elementary number theory, to wit, the theory of divisibility, to advantage in the field of Aristotelian categorical syllogistic. After having failed with the assignment of prime numbers to elementary concepts and composite numbers to complex concepts, this leaving out of account what has been called “negative concepts”, Leibniz associates pairs of numbers with the subject and the predicates of propositions:
s * (sly s2> 9 Pc-)(P,,P,>, in order to express the traditional standard forms of syllogistic in the following form:
s z p -PI Is1 A P 2 b z sip * ( P i , S z ) = I h ( S i , P z ) = 1 , SOP c-) 1SUP, Sep t, 1Sip 9
(where (u, b) is the greatest common divisor of a and b). Rewriting the premises of a syllogism arithmetically in the manner indicated, one can check whether the correlate of the conclusion may be inferred also in arithmetic. Until quite recently, it was almost generally agreed upon that this calculus fails, too, and that Leibniz did not follow it up just because of this insight. Lukasiewicz in his book on Aristotelian syllogistic is the only one I know of who explicitly states that Leibniz’s calculus no. 1 is a correct arithmetical interpretation of categorical syllogistic. Whether it is or not depends on the definition of validity. In a paper read at the Third International Leibniz Congress here in Hanover in 1977, I have tried to show that Leibniz first had an incorrect definition of validity and tended to give up his arithmetical interpretation, but that he himself found the correct definition immediately afterwards, and that with this definition the calculus is entirely correct.’ THIEL, Ch., Leibnizens Definition der Iogischen Allgemeingultigkeit und der “arithmetische Ka/kiil”. in: Theoria cum praxi. Proceedings of the Ill. International Leibniz Congress, Hanover 1977 (Wiesbaden 1980), vol. 3, pp. 14-22
a
FROM LEIBNIZ TO FREOE
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So we do not know why Leibniz abandoned the arithmetical calculus, but we can be sure the reason was not its alleged failure in the domain of syllogistic. The reason may have been very trivial: Leibniz was not occupied with logic all the time; between 1675 and 1684 he made decisive progress with his infinitesimal calculus and may have been taken up with problems in this field. On the other hand, it is quite clear that even an operative arithmetical calcuhs for syllogistic was far too narrow for Leibniz’s conception of logic, which tended to a general calculus of concepts, extensional and intensional, and was not restricted to the traditional standard forms like “Every A is B , and which moreover was conceived to take up and extend Joachim Jungius’ treatment of nonsyllogistic inferences in the Logica Hamburgensis (1638, a work highly esteemed by Leibniz), which means, to work in the direction of a logic of relations, and finally to include a logic of propositions by extending Leibniz’s genei-al calculus de continente et contento to relations of entailment between propositions. It is to both extensions that Leibniz’s statement in the Nouveaux Essais refers, saying that, “there exist sound asyllogistic inferences which it would be impossible to prove by any syllogi~rn”.~ The result, in Leibniz’s second and third attempt at a calculus of logic (1686 and 1690), are some fragmentary lattice-theoretical systems, admitting of various interpretations, extensional, intensional, geometrical, and others, “Every A is B being interpreted, e.g., intensionally as “ A = AB” in 1686, and as “ A = A+B” in 1690, when Leibniz had a calculus with +, -, and =, #, and <. These calculi have been expounded so often and in detail that I will abstain from another description of them, and just mention a so far uncorroborated conjecture of mine concerning Leibniz’s change to the socalled “algebraic calculi” that I have called “lattice-theoretical” in the preceding sentence. In more elaborate books on the history of logic, we find James, i.e. Jacob Bernoulli mentioned with his book, The Parallelism between Logical and Algebraic Arguments lo, and we are told, e.g. in Styazhkin’s History of Mathematical Logic from Leibniz to Peano ‘I, that Bernoulli, “following Leibniz ... only noted the analogy that exists between the laws of formal logic and the methods of elementary algebra” @
NE IV, XVII 5 4; see also 11, 5 9.
I , Parallefisms ratiocinii Iogici et algebraici, Basel 1685; also in: Opera, ed. G. Cramer, I, Geneva 1744, repr. Brussels 1968. STYAZHKM, N.I., 1969, History of Mathematical Logic from Leibniz to Peano (Cam-, bridge, MassJLondon) (Russian original 1964).
lo BERNOULLI, JAC.
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(p. 95). As I do not know of any algebraically oriented logical work done
by Leibniz previous to 1685-leaving out of account the combinatorial investigations into syllogistics in the Dissertatio 12-1 would suspect that Leibniz’s experiments with algebraic calculi were stimulated by Bernoulli’s work in spite of the fact that the latter was composed rather sketchily and lacked profundity, so that Leibniz certainly could not have profited from this work to the same extent as later from Jacob Bernoulli’s insights into pr0babi1ity.l~ And indeed, Leibniz’s conception had become so broad as to render the exploitation of algebraical methods more difficult: In his letter to Gabriel Wagner on the utility of the art of reasoning or logic (1696) he states: “I have come to the opinion that even algebra borrows its advantages from a much higher art, to wit, from the true logic’’.14 The question whether Leibniz deserves an eminent place in the history of mathematical logic is not sacrilegious. The answer depends not only on our definition of “mathematical logic” but also on our conceptions of history and historiography. If history is what actually happened and had a significant effect, then Leibniz has to be omitted from the history of logic in his lifetime, for the writings referred to as containing his great and fruitful ideas in logic were completely unknown to his contemporaries. Part of them did not appear in print until Raspe’s edition of 1765, and most of them appeared only at the beginning of our century when Couturat made them accessible in 1901 and 1903, such that in Leibniz’s lifetime it is mainly his remarks in letters to contemporary scholars that could have had any traceable influence. He who conceives of the history of logic as the development of logical ideas, however, would be entitled to assign a rather minor place to Leibniz when talking about the logic of the Neuzeit, and I may for similar reasons be forgiven my decision to omit the logical work of Bolzano and of Robert Grassmann from this paper as they did not really influence the development of logic within the period we are concerned with. Before passing to the so-called Leibniz School in logic, I should say a few words on Gerolamo Saccheri (1667-1733) who is mainly being remembered for his Euclides ab omni naevo vindicatus, i.e. Euclid Cleared of Every Flaw, published in Milan in 1733, and containing a good portion LEIBNIZ,G. W., 1666, Dissertatio de Arte Combinutoria (Leipzig). BERNOULLI, JAC., 1713, Ari Conjectandi, Basel, repr. Brussels 1968. LBIBMZ,G. W., 1696, Schreiben an Gabriel Wagner. Vom Nutzen der Vernunfrkumt oder Logik. 1696. Erdm. 418 ff., quotation on p. 424 b.
la
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of unstrived-for development of theorems of a non-Euclidean geometry. Less known is Saccheri’s Logicu Demonstrutivu, which saw three editions, an anonymous one in 1697, the others in 1701 and in 1735.” This work deserves a place in the history of logic not only for Saccheri’s interest in a logical analysis and description of mathematical proof techniques in geometry, but also because of two contributions presented in its eleventh chapter. Saccheri here develops a “via nobilior”, i.e. a “more noble way” than previously followed, in logic generally as well as earlier in his book. His first idea is that of proving a theorem by deriving it from its own contradictory.’6 According as the theorem is afEmative or negative, this means proceeding by (lu+u)+u
or
( u + la)-, l u ,
o r rather, l U + U
a+ l u
i.e., as rules of inference. Saccheri’s goal was a consistency proof for (or rather, a proof of the truth of) a postulate system, say for geometry, by applying this “principle of necessary truth”, i.e. deriving each of the postulates from its own contradictory. This method has been taken up in our century by Josiah Royce ”. Indeed, l u + u , l b + b ,... 11(u A b A ...) AUQUSTAE TAURINORUM (Torino) 1697;Ticini Regii (Pavia) 1701 (Heinrich Scholz’s correction of Vailati’s date “1701” to “1702” in his Abriss der Geschichte der Logik, FreiburglMunchen a1959,a1967,p.37,is itself mistaken); Augustae Ubiorum warn) 1735. l6 See p, 80 of the 1697 edition: “Sumam contradictorium propositionum dernonstrandarum, ex eoque ostensiuh, ac direct6 propositum eliciam” (p. 82 of the 1701 and p. 130 of the 1735 edition). As will be seen, my interpretation diverges from Angelelli’s and Hoorman’s who identify this principle with Clavius’ Law, and the via negativa with indirect proof without distinguishing reductio ad absurdum from a --t 1a < l a . Saccheri, in the Scholium to chapter 11, presents the via negativa as a deductlo ad impossibile as is further corroborated by his example, “v. g. quod, si modus AA concluderet in secunda figura, omnis, vel aliquis syllogismus AA esset syllogismus EA.” (p. 89 of the 1697edition). JOSIAHROYCE, Prinzipien der Logik. In :Encyclopddie der philosophischen Wissenschafren, in Verbindung mit Wilhelm Windelband herausgegeben von Arnold Ruge. Erster Band: Logik. Tubingen 1912, pp. 61-136. The original English version appeared in the first volume of the (unfinished) English edition: T h e Principles of Logic. In: Sir Henry Jones (ed.), Encyclopaedia of the Philosophical Sciences, vol. I: Logic (London 1913). pp. 67-135. A separate edition was published in New York in 1961. l6
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is even constructively valid. This method was put to use in Saccheri’s Euclides, and earlier in his Logica Demonstrativa in connection with the via nobilior for a justification of the syllogisms by a sort of self-application which I do not have the time here to expound but on which we have valuable commentaries by Angelelli 1975, Hamblin 1975, and Hoorman 1976.” Whereas Saccheri worked independently of Leibniz’s logical experiments, the writers of the so-called Leibniz School were influenced by Leibniz although part of their work precedes Raspe’s edition of 1765. For example, Johann Andreas von Segner’s Specimen Logicae appeared in Jena in 1740, Georg Johann von Holland’s correspondence with Lambert started previous to 1764, the year in which the first two volumes of Johann Heinrich Lambert’s Neues Organon appeared in print. All these writers, stimulating each other by correspondence and small treatises, worked along Leibnizian lines, but restricting themselves (with few exceptions) to the intensional interpretation of their calculi. Lambert makes use of a quantification of the predicate-a device concerning which Hamilton and De Morgan claimed the priority of invention some sixty years later-and distinguishes two cases of each standard form, e.g. in the universal affirmative proposition “All A is B”, Case I: A = B where the converse is also universal, and Case 11: A < B where the converse is particular. In other cases, a standard form had to be represented by two formulae, and the calculi became very complicated, too complicated for Ploucquet, Lambert, Holland and others to handle in a satisfactory manner. John Venn in his Symbolic Logic has attributed this failure to the intensional standpoint which he also blamed for the incompleteness and inutility of Leibniz’s endeavours. This has become the standard value judgment on the Leibniz School, including Castillon, whose calculus was considered to be the most consequential. Couturat has strongly emphasized this point of view with regard to Leibniz. C. I. Lewis stated it plainly: “This movement produced nothing directly which belongs to the history of symbolic logic. ...The record of symbolic logic on the continent is a record of failure, in England, ANGELELLI, I., 1975, On Saccheri’s use of the “ConsequentiaMirabilis”, Akten des II. Intern. Leibniz-Kongresses 1972, Bd. IV, (Wiesbaden 1975), pp. 19-26; C. L. IIAMBLIN, Saccherian arguments and the self-applicationof logic, Australasian Journal of Philosophy, vol. 53, pp. 157-160; CYRILF. A. HOORMAN, Jr., A further examination ofsaccheri’s use of the “ConsequentiaMirabilis”, Notre. Dame Journal of Formal Logic vol. 17 (1976). pp. 239-247.
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a record of success. The continental students habitually emphasized intension, the English, extension’’ Meanwhile, formal logic on the continent had suffered a serious set-back at the hands of Kantian transcendental logic which, though in no way inccmpatible with formal logic, detracted, at least in Germany, interest frcm formal logic which, moreover, was treated very contemptuously by Kant whose own lectures on logic might have been quite suitable to support this attitude, but who had ridiculed Leibniz’s idea of a logical calculus and a universal characteristic already in his Nova Dilucidatio.20 Hegel’s judgment, and Lotze’s in 1841, were in no way more favourable, and in spite of my respect for investigations into the very foundations of formal logic as we find them in most textbooks of “logic” during the last century, I cannot overlook the fact that the intellectual climate on the continent had indeed become suffocative for symbolic logic by the middle of the 19th century. Typical of this is the fate of Moritz Wilhelm Drobisch’s logic in its original form, entitled A New Exposition of Logic, According to Its Simplest Relations, with a logico-mathematical Appendix 21. The appendix must be mentioned here since in the first edition, there is a paragraph with the title, “algebraic construction of the simplest forms of judgment and a derivation of inferences founded thereupon” 2 2 , where Drobisch develops an extensional caIculus of classes and elementary judgments, improving the intensional systems of Ploucquet, Lambert, Jacob Bernoulli, and Gergonne, all of whcm he explicitly mentions. Unfortunately, Drobisch felt obliged to withdraw this valuable paragraph in the second and later editions because of a severe criticism put forward by Friedrich Adolf Trendelenburg, who in his Logical Investigations 23 criticized formal logic as taught by Leibniz, Twesten, Drobisch and others
’’.
LEWIS,C. I., 1918, A survey of symbolic logic (Berkeley, abridged New York 1960), p. 36 f. KANT,I., Principiorum primorum cognitionis metaphysicae nova dilucidatio (& c.), 1755, Akad. I. (Berlin 1902/10), pp. 385-416 (esp. p. 390). DROBISCH, M. W., Neue Darstellung der Logik nach ihren einfachsten Verhaltnissen. Nebst einem logisch-mathematisrhen Anhang (Leipzig 1836); Neue Darstellung der Logik nach ihren einfachsten Verhaltnissen, mit Riicksicht auf Mathematik und Naturwissenschuft, Zweite, vollig umgearbeitete Auflage (Leipzig 1851); Dritte neu bearbeitete Auflage (Leipzig 1863); Vierte verbesserte Auflage (Leipzig 1875); Fiinfte Auflage (Leipzig 1887). op. cit. (see note 21), Algebraische Construction der einfachsten Urtheilsformen und darauf gegrihdete Ableitung der Schliisse, pp. 131-136 of the 1836 edition. I* TRENDELENBURG, F. A., Logische Untersuchungen (Berlin 1840, Leipzig ‘1862).
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for its allegedly unjustified treatment of concepts by arithmetical analogies. It is a curious fact that the same Trendelenburg, who claimed that Leibniz did not conceive logic as a formal discipline at all, published in 1856 an essay, On Leibniz’s Project of a Universal Characteristic 24, in which he also sketched the prehistory of this project since Raymundus Lullus, thereby drawing attention to the universal language movement, and contributing essentially to the revival of interest in Leibniz’s logic. Incidentally, this essay of Trendelenburg of 1856 is the earliest text in which I have been able to find the German expression Begrirsschrift, although used in a very general sense for any notation that brings the formation of a sign in touch with the content of the concept it denotes, as Trendelenburg finds it in numerals and other implements of science. The so-called algebra of logic, crowned with the names of De Morgan, Boole, Peirce, and S:hriider, has been so thoroughly investigated and expounded in the literature that it can be taken to be the best known period in the history of formal logic. Let me just remind you of De Morgan’s foundation of the theory of relations, surpassed only by Peirce’s work in the subject, whose calculus of relations could already profit from Boole’s calculus of classes. Boole had started with a successful attempt at an axiomatized Aristotelian syllogistic, in order to defend one of De Morgan’s claims in his quarrel with Hamilton. The result was The Mathematical Analysis of Logic, being an Essay towards a Calculus of Deductive Reasoning (Cambridge and London 1847), the exposition of the intended system being given in An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probability (London and Cambridge, 1854). Boole was either ignorant or intentionally negligent of the logical work of his immediate predecessors, and he did not devote much thought to the problems of extension and intension, of existential import, and of empty classes, problems that had hampered previous logicians and were to return later, but of which C. I. Lewis says rightly: “It is well that, with Boole, they are given a vacation long enough to get the subject started in terms of a simple and general procedure” (Survey, p. 51). A peculiarity TRENDELENBURG, F. A., Uber Leibnizens Entwurf einer allgemeinen Charakteristik (Vorgetragen zur Feier des Leibniztages 1856). Philosophische Abhandlungen der Koniglichen Akademie der Wissenschaftenzu Berlin. Aus dem Jahre 1856 (Berlin 1857), pp. 37-69 (also published separately); reprinted in F. A. T., Historische Beitruge zur Philosophie. Dritter Band. VermischteAbhandlungen(Berlin 1867), pp. 1-47. The expression “Begriffsschrift” on p. 39 of the 1857, p. 4 of the 1867 printing.
FROM LEIBNIZ TO FREGE
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of Boole’s, which he shares with Frege, is his conviction of the interrelation of logic, language, and mathematics, a thought that impressed itself on many of his logical conceptions, perhaps even on the parallelism claimed by Boole to hold between the algebra of classes and that of propositions or of 0 and 1, later adjusted by Peirce who pointed out the special status of the fact that if x # 0, then x = 1. Let me return to Frege by relating his Begrifsschrut to the algebra of logic, following his admirably clear statements in his paper, On the Purpose of the “Begriffsschrif” (1882), written as an address to the Jena Society for Medicine and Natural Sciences to defend himself against Schroder’s reproach that Frege was, firstly, ignorant of earlier work and that secondly, his BegriJ7khr$t marked a regress in comparison with the algebra of logic. Frege points out that his purpose was not the presentation of abstract logic by formulae, but the creation of an instrument capable of expressing a content in a more precise and more perspicuous way than could be done by natural language. He finds fault with Boole’s classification of propositions into primary and secondary ones, this leaving out of account existential propositions, and misses in Boole’s system a notation for individuals (as different from singletons), and for the falling of an individual under a concept. Moreover, granting the possibility of interpreting Boole’s calculus as dealing with classes on the one hand, and as dealing with propositions on the other hand, Frege does not find a bridge between the two, a bridge for passing from a proposition to the concepts contained in it, and vice versa. And as to the notation, this latter problem makes “0” and “1” ambiguous in arithmetical contexts where “0” and “1” already have their ordinary numerical meaning. To sum up, Frege thinks that Schroder was mistaken in trying to compare two systems of logic which depart from different starting points and aim at the fulfilment of different purposes. As manuscripts from the Nachlass show, it was quite clear to Frege that, in spite of the term “Begrifsschrijt”, which in this respect is rather ill-chosen, he did not start from concepts in order to combine them into propositions, but from propositions, decomposing them into function and arguments. Whereas in other respects he views himself as in the tradition of Leibniz, he is quite aware of the fact that here his system is opposed to that tradition. In my opinion, it will not be necessary to go into the system of the Begrifsschrijit here. Quantificational logic in Frege’s setting is close enough to its modern formulation to be considered well known itself, and I will mention only a point that is usually misunderstood. Introducing a notation
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for quantification, i.e. the universal quantifier and bound variables, Frege explicitly says that in “ @ ( A ) ” , “@” may be considered as the argument of a function, too, and may then be replaced by a German letter in a quantified proposition. It is, therefore, incorrect to say that Frege established only first-order logic in the Begriffsschrift, and that he illegitimately substituted bound functional letters for bound individual letters in some places of the third part of the Begri#kwhrvt where the concepts of sequence and successor are being defined. In the Begrtflsschrijt, the “a” of
“
-&-
@(a) ’’ is typically ambiguous. 25
To many of you, what I have said in this paper will be proof enough for the delicacy of the problem of how to write or even to approach the history of logic. There are, first and almost trivially, two directions of approach. One may include everything that seems relevant from the view-point of the historiographer’s time, or one may, on the other hand, try to do justice to historical authors by taking seriously their contemporary problems, and even by trying to gain a better understanding of them by looking for their origins. Secondly, we have to make a decision whether we want to write a history of causes and effects, a history of stimuli and influences, or a history of aims and reasons, whether we want to investigate Wirkungsgeschichte in the traditional sense which will not be changed in principle by incorporating social and economical aspects and developments (desirable as this would be for the history of formal logic as it has been done for mathematics to some extent at least), or what Jiirgen Mittelstruss has called Griindegeschichte. My vote is for the latter, a history of means and ends, aims and reasons, of needs, goals, and purposeful actions. This does not strip Wirkungsgeschichte of its status as an indispensable basis of GriindeThis is to correct van Heijenoort’s remark on p. 3 of his introduction to Frege’s Begriffsschrqt quoted in note 3 above. The correctness of Frege’s inferences has also been pointed out by Terrell Ward Bynum (Onan Alleged Contradicrion Lurking in Frege’s Begriflsschrift, Notre Dame Journal of Formal Logic, vol. 14 (1973), (pp. 285-287) who thinks that while Frege “does not yet have all the machinery or the terminology to precisely spell out the distinction between what he would later call ‘first-level’ and ‘second-level’ functions, he never confuses the two” (p. 285). But closer inspection of Frege’s text shows (end of 5 10, 5 11. i. e. pp. 18-19, and pp. 60-62. p. 68 footnote) that the typical ambiguity is fully intentional. Indeed, in the Begriflsschrijit expression es
“
-&-
@(a)
”
the letter ‘‘a” stands for “argument”, the letter “@” for “function”.
FROM LEIBNIZ TO FREGB
767
geschichte which I would like to be understood as a history of (in our case) logic with particular emphasis on reasons for the various developments, but as well as a history of reasons themselves, their interconnections and changes. No doubt this is a beautiful challenge and programme, but I am afraid there is some troublesome work to be done before it can be taken up: there are scores of false data and erroneous judgements to be cleared away, and to be replaced by correct and more reliable ones. Let me just mention a small number of instances. Most historians of logic have not been able to see personally all the texts they quote and refer to; they rely on older historians of logic and carry over incorrect data as well as erroneous evaluations from these. How else could it be explained that Bardili, Victorin, Twesten are counted among the early representatives of mathematical logic in many histories of logic although there is practically nothing mathematical in them except variables A , B , ... and perhaps the plus, the minus and the equality sign? Why is Twesten referred to as a Leibnizian and as a member of the Leibniz School, while being strongly influenced by Kant, the structure of whose logic he took over? Why is Friedrich von Castillon (1747-1814), author of RPfiexions sur la logique (1802) and the MPmoires sur un nouvel algorithme logique (1803), mistaken for his father, Jean or Giovanni Francesco de Castillon (1708179I), practically everywhere, from John Venn via Shearman, C . I. Lewis, A. Church's Bibliography, Bochebski's Formal Logic, Styazhkin's History up to modern authors of 1970? 26 Going back to primary sources as far as they are accessible, and searching for items that have seemingly disappeared, seem to me the presently most important tasks to be taken care of before a reliable and informative history of mathematical logic can be written. It would be unfair and incompetent not to mention the work that has so far come closest to the demand stated: Wilhelm Risse's Die Logik der Neuzeit, a comprehensive enterprise that has so far grown into two volumes covering the period from 1500 to 1780. But it cannot be overlooked that Risse's is a history of logic in a broad sense, mathematical logic playing a subordinate part except in Leibniz, and even here there is a certain lack of apprehension of more recent research done by others with results that might have illuminated or made accessible obscure or difficult texts and passages. A true re-assessment of the history of the
'' See my paper Zur Beurteilung der intensionalen Logik bei Leibniz und Castillon, Akten des 11. Intern. Leibniz-Kongresses, Hannover 1972, Bd. IV (Wiesbaden 1975), pp. 27-37.
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subject will require much closer cooperation between historians and mathematical logicians; let us hope that it will come about. Let me come to an end with a short systematical remark that might be relevant for the historical evaluation of Frege’s logic. As is well known, Frege developed his Begriflsschrgt further and presented an intricate and profound system of logic and part of set theory in his Grundgesetze der Arithmetik, the first volume of which was published in 1893, the second in 1903. The destiny of this system is widely known: it proved inconsistent by admitting the derivation of the Zermelo-Russell antinomy. But after the wreckage of the Grundgesetze by this antinomy, Frege and other logicians came to believe that the system might be saved by modifying the principle of abstraction which had served to supply Frege’s coursesof-values. LeSniewski and later Quine have shown that Frege’s own modification was insufficient for his purposes, but there remains the simple question whether by seizing the principle of abstraction we got hold of the real trouble-maker. As to this question, it seems to be little known even among experts that the fuse leading to the Zermelo-Russell antinomy is already hidden in the Begrijhchrijl in spite of the absence of courses-of-values or extensions of concepts in this system. In what sense this claim is justified will become clear from an analysis of Frege’s Appendix to the second volume of the Grundgesetze. After presenting a meticulous derivation of the antinomy, and a shorter version using his elementhood symbol “n”, Frege blames his fundamental law V for the catastrophe. To be more precise, he blames one direction of it. Splitting (and slightly changing)
into
and
FROM LEIBNIZ TO FREGE
769
the latter part appears to be essential for Frege’s derivation of the antinomy. To raise this above all doubt, Frege decides to make his argumentation independent of courses-of-values altogether by deriving the falsehood of Vb without reference to them. The idea is to consider Vb as a special case of
f ( 4= g(a) MJ3j-m))
=Mg(-glP))
and to derive the negation of this formula in the system. Checking this derivation, one finds that Frege makes use of the fundamental laws IIa and IIb, of theorems Ig, IIIa, and IIIe, and that he uses substitution, detachment, contraposition, transitivity, and universal generalization as his rules. But substitution and detachment are precisely the rules of the Begrzfsschrijt system, the other rules appealed to are provably admissible, the fundamental law IIa is the Begr@sschrift axiom 58, theorems IIIa and IIIe are axioms 52 and 54, while Ig can easily be proved as a theorem of the Begrifsschrift, and IIb is understood by Frege to be contained in axiom 58, as indicated above and as may be seen from his 5 10 and his procedure in the third part of the text, especially on pp. 6 0 4 ~ 2 . ~ ~ In other words, the derivation in the Appendix can be carried out in the Begrzfsschrijt of 1879 as well, the result being a theorem stating that for any second-level function that takes an argument of the second kind (i.e., a one-place first-level function), there are two concepts yielding the same value when taken as arguments of the function although there are objects falling under one of them but not under the other. As this is valid for any second-level function whatsoever, it will also hold for Frege’s course-of-values function &#J ( E ) , contrary to its purpose : there are concepts O(() and Y ( { ) which yield the same value when taken as arguments, i.e. E@(E) = C;Y(a), although there is at least one object A such that @ ( A ) but not Y(A). Hence, @(() and U({) do not both extend to the same objects; their extensions are different. As E@(e) = drY(a), neither Z@(E) nor drY(a) can be the extension of the concepts @({) and Y ( { ) in the traditional sense. This, of course, obstructs Frege’s definition of number as an extension (or course-of-values)-unless some other step or expression in his derivation can be made responsible for the unpleasant
a’
Cf. note 25.
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result. The problem of identity of sets has returned through the back door as the problem of identity of functions. I will pursue this question further elsewhere, and restrict myself to the remark that the feature of Frege’s Begriffsschrijit just sketched refutes the wide-spread opinion that his system of 1879 is not only less precise than that of 1893-which is true-but also essentially weaker. And let us not forget, lastly, that it is, in spite of certain deficiencies, the beginning of mathematical logic in the narrower and modern sense, even though mathematical logic a century ago means mathematical logic before Peano’s axioms of 1889, before Hilbert’s Foundations of Geometry of 1899, before Zermelo’s axioms of set theory, before Brouwer’s intuitionistic critique of 1907, before Principia Mathematica, before Hilbert’s Programme, before recursive functions, and before Godel’s theorems. Was not Frege’s Begrirsschriff the memorable first step of a flight of stairs leading to a rapidly expanding world of logic?