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T H E PROPOSITIONAL LOGIC O F BOETHIUS K A R L D U R R Professor of Philosophy University of Zurich 1 9 5 1 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM PRINTED IN THE NETHERLANDS DXUKKERIJ HOLLAND N.V.. AMSTERDAM PREFACE The text of the treatise “The Propositional Logic of Boethius” was finished in 1939. Prof. Jan Lukasiewicz wished at that time to issue it in the second volume of “Collectanea Logica”; as a result of political events, he was not able to carry out his plan. In 1938, I published an article in “Erkenntnis” entitled “AUS- sagenlogik im Mittelalter”; this article included the contents of a paper which I read to the International Congress for the Unity of Science in Cambridge, England, in 1938 (Cf. Erkenntnis, vol. 7, pp. 160-168). The subject matter of this paper touched upon that of the above-mentioned treatise. Recently an article of Mr. Rend van den Driessche, “Sur le ‘de syllogismo hypothetico’ de Boece”, was published in the journal “methodos” (vol. I, no. 3). Mr. van den Driessche referred in this article to the article on propositional logic in the middle ages, which had appeared in “Erkenntnis”. This reminded me of my yet-unpublished treatise on the propositional logic of Boethius. I wish to thank the editors of “Studies in Logic” and the North- Holland Publishing Company, for the fact that a treatise which owing to unfortunate circumstances remained unpublished, now will be made available to the learned world in English. University of Ziirieh, Switzerland July 31, 1950 K. DURR ABBREVIATIONS .. . Ab. C. Ouvrages inedits d'Ab6lard par M. Victor Cousin (1836). Ab. G. Peter Abaelards philosophische Schriften, hrg. von Bernhard Geyer (1919-1939). Ar. Arist. Graece ex rec. Imm. Bekkeri (1831). Be. Oskar Becker: Zur Logik der Modalitaten. Jahrbuch fur Philosophie and ph6nomenologische Forschung, Bd. 11 ( 1930). . . . Bm. Anitii Manlii Severini Boethi opera, quae extant, omnia. Basileae (1570). . Br Samuel Brandt : Entstehungszeit und zeitliche Folge der Werke des Boethius. Philologus, Bd. LXII (1903). Ca. Rudolf Carnap: Logische Syntax der Sprache (1934). CAG Commentaria in Aristotelem Graeca cdita consilio et auctoritate aoademiae litterarum regiae Borussicae. Ch. C. West Churchman : On linite and infinite modal systems. The Journal of symbolic logic, Vol. 3 (1938). Ci. M. Tullii Ciceronis Topica. Gr. Martin Grabmann : Bearbeitungen und Buslegungen der Aristotelischen Logik aus der Zeit von Peter Abaelard bis Petrus Hispanus. Abh. Preuss. Akad. Wiss., Phil. Hist. Kbsso Nr. 5 (1937). H. F. Hausdorff: Mengenlehre (1927). Le. C. I. Lewis: A Survey of Symbolic Logic (1918). L.a.L. Clarence Irving Lewis and Cooper Harald Langford : Sym- bolic Logic (1932). X ABBREVIATIONS Lu . Jan Lukasiewicz : Zur Geschichte der Aussagenlogik. Erkenntnis, Vo1. 5 (1935). Not. Die Schriften Notkers und seiner Schule, herausgegeben von Paul Piper, Vol. I (1895). PM Alfred Xorth Whitehead and Bertrand Russel1 : Principia Matheinatica, Vol. I (1925). Pr . Carl Prantl: Geschichte der Logik im Abendlande (1855- 1870). Sch. Heinrich Scholz : Geschichte dcr Logik (1935). Ta . Alfred Tarski : Der Wahrheitsbegriff in den forma,lisiertcn Sprachen. Studis Philosopliica, Vol. I (1935). INTRODUCTION $ 1. THE TWO BOOKS OF BOETHIUS ON THE THEORY OF THE PROPOSITION It is the unique property of propositional logic that the variables which are used are propositional variables, i.e. variables whose values are propositions. Among the logical writings of the man whom, for short, is called “Boethius’’ and whose full name is “Anicius Manlius Severinus Boethius”, we find two which can be characterized as presentations of propositional logic. The first of these is entitled “de syllogismo hypothetico” (on the hypo thetical syllogism). Incidentally, it should be noted that this title, as Samuel Brandt has shown, does not originate with Boethius, and it would be more correct to give the book the title “de hypotheticis syllogismis” (on hypothetical syllogisms) (Cf. Br. p. 238). Nevertheless, one does well to quote the work under its incorrect title “de syllogismo hypothetico” as long as the old editions are in use. The second book is a commentary on the Topics of Cicero. Here we do not consider the entire commentary, but only certain sections; we will indicate later which sections come into consideration (Cf. infra 9 38). 0 2. EDITIONS The two books are to be found in old editions of the complete works of Boethius. Among these we mention: (1) The edition which appeared in Venice in 1492 and 1499; we will refer to this as the “first edition”. (2) The edition wich appeared in Base1 in 1546 and 1570 which is called the Basel-edition. (3) The edition of Migne which forms volumes 63 and 64 of the collection “Patrologia Latina” and which appeared in 1847 in its first and in 1860 in its second edition. 2 INTRODUCTION According to expert judgement, the first edition is to be considered the best text. The edition of Migne appears to have no real scientific value, but is apparently only a reprint of the Basel edition (Cf. Br. p. 147). I have been able to verify by the comparison of im- portant selections that errors in the Basel edition are repeated in that of Migne, and I discovered only one place where such an error was corrected in the edition of Migne. Some works of Boethius have appeared recently in separate editions; however these do not include the two works that interest us here. As a result, only the first edition and the Basel edition can serve as sources. This is not to say that we are concerned with a text that can be regarded as secure from a philological standpoint. Nevertheless, it is to be assumed that the interpretation of Boethius’ logical theory is scarcely affected by this. We are primarily concerned with the logical formulae, and these formulae can be recognized with sufficient certainty on the basis of the texts which these editions give. We will quote primarily from the Basel edition and we can justify this by the fact that it is this edition which is usually referred to. In the notes of the well-known work of Carl Prantl “Geschichte der Logik im Abendlande” we find many selections from the logical writings of Boethius. Since this work is handier and more circulated than the old collected works of Boethius it seems more practical simply to refer where possible to the work of Prantl. In this connection, it is to be noted that Prantl himself quotes the Basel edition; a reference to the notes in Prantl’s work is, therefore, indirectly, a quotation of the Basel edition. Q 3. THE PERIOD OF ORIGIN OF THE TWO BOOKS The period of origin of the two books of Boethius, falls in the interval between 523 A.D. and 510 A.D. (Cf. Br. p. 268). The commentary on Cicero’s Topics was written after “de syllogismo hypothetico”. At the beginning of the fifth book of the commentary on Cicero’s Topics, Boethius points out that he has exhaustively MORE PRECISE CHARACTERIZATION 3 handled all hypothetical syllogisms in other books (Cf. Boe. p. 823). There can be no doubt that he has in mind the two books which form “de syllogism0 hypothetico”. This justifies our conclusion that the work referred to was already completed when Boethius composed the commentary on Cicero’s Topics. Q 4. MORE PRECISE CHARACTERIZATION OF BOETHIUS’ PROPOSIT- IONAL LOGIC At the beginning of this treatise, we declared that the logic which is represented in the two works of Boethius, may be characterized as propositional logic. We add the remark that all of the sentences that have an independent value (i.e. that do not occur only as auxillary sentences) in this logic were deductive rules, or, which comes to the same thing, inference schemes. In this connection we recall the explanation of Clarence Irving Lewis in the book “Symbolic Logic”: “Exact logic can be taken in two ways: (1) as a vehicle and canon of deductive interference, or (2) as that subject which comprises all principles the statement of which is tautological” (Cf. L.a.L., p. 235). We can now say that the logic of Boethius belongs to the first of these two forms of exact logic. Boethius’ aim is not to set up sentences which are tautological, but rather to present all of the deductive rules, I THE SOURCES OF “DE SYLLOGISM0 HYPOTHETICO” § 5. THE PROBLEM OF BOETHIUSSO’U RCES We should here like to pose the question, to what extent it can be established what the sources of “de syllogismo hypothetico” were. In this investigation we proceed from the testimony of Boethius. This is to be found at the beginning of “de syllogismo hypothetico”. We will cite this in English translation. Boethius speaks to a certain Symmachus who is named in the title of the work. “What I have found in a few Greek authors brief and confused and in no Latin authors, that I dedicate to your insight, after its having been completed by our long, but successful exertions. When you had acquired a comprehensive knowledge of categorical sylIogisms, you often desired information about hypothetical syllogisms, concerning which Aristotle wrote nothing. Theo- phrastus, a man versed in every science, works out only the main points. Although Eudemos touches on the subjects more broadly, he proceeds in such a fashion that he seems to have only thrown out a few seeds without having harvested any of the fruit (Cf. Boe. p. 606). We note here the expressions “Categorical syllogisms” (categorici syllogismi) and “hypothetical syllogisms” (hypothetici syllogismi). These two expressions each characterize a certain kind of logic. In his article “Zur Geschichte der Aussagenlogik”, Jan Lukasiewicz opposed propositional logic to the logic of names (Cf. Lu. p. 111). If we adopt this usage, we can say: The theory of categorical syllogisms is the logic of names and the theory of hypothetical syllogisnis is propositional logic. In the quotation from Boethius it is remarked that in the Aristotelean writings one finds only the theory of categorical syllogisms. If one remembers that the theory of categorical syllogisms is identical with the logic of names, we can deduce from this sentence THE LOGIC OF THE OLDER PERIPATETICS 5 that in the writings of Aristotle only the logic of names is represented. This statement agrees with Lukasiewicz’s remark that Aristotle’s syllogistics embodies a fragment of the logic of names (Cf. Lu. p. 121). Boethius relates the theory of hypothetical syllogisms with Theophrastus and Eudemos; it is unmistakable that he regards these two peripatetics as the first to develop this kind of logic. This last fact is significant, because according to the interpretation now represented by Jan Lukasiewicz and Heinrich Scholz, which we may call the modern interpretation, the Stoics either founded propositional logic or at least developed it to a higher level (Cf. Lu. p. 112 and Sch. p. 31). This raises the question of the relation between Boethius and the Stoic logic. Boethius explains that he had not .found a presentation of the theory of hypothetical syllogisms in any Latin authors. This point also demands closer investigation, since one could name Latin authors who give a presentation of the theory of hypothetical syllogisms and who must be brought in consideration when we discuss the sources of the work “de syllogismo hypothetico”. The historical problems that are posed in this connection will be handled in an order which can be taken from the following in- dication : (1) The extent to which Boethius was influenced by the older peripatetics is to be investigated. (2) The determination of the relation of the work to the Stoic logic is to be checked. (3) The Latin writings which are related in theme to the work of Boethius and which could have served as sources are to be named and closely examined. The possibility that the Greek work, originating from the (4) school of Ammonios Hermeiu which gives a presentation of the theory of hypothetical syllogisms, served as a source of Boethius’ work is to be checked. 0 6. THE LOGIC OF THE OLDER PERIPATETICS In order to establish what the theory of the older peripatetics