THE
PROPOSITIONAL LOGIC OF
BOETHIUS
KARL DURR Professor of Philosophy University of Zurich
1 9 5 1
N O R T H - H O L L A N D P U B L I S H I N G COMPANY AMSTERDAM
PRINTED I N THE NETHERLANDS DXUKKERIJ HOLLAND N.V.. AMSTERDAM
PREFACE The text of the treatise “The Propositional Logic of Boethius” was finished in 1939. Prof. J a n 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. I n 1938, I published an article in “Erkenntnis” entitled “AUSsagenlogik 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 This reminded me of my yet-unpublished treatise on the propositional logic of Boethius.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and the NorthHolland Publishing Company, for the fact that a treatise which owing to unfortunate circumstances remained unpublished, now will be made available to the learned world in English.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 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. Gr.
M. Tullii Ciceronis Topica. 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). F. Hausdorff: Mengenlehre (1927). C. I. Lewis: A Survey of Symbolic Logic (1918). Clarence Irving Lewis and Cooper Harald Langford : Symbolic Logic (1932).
H. Le. L.a.L.
opera, quae extant,
X
ABBREVIATIONS
Lu .
J a n 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 (18551870).
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 hypothetical 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 t o 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 important 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. I n 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. I n 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. THEPROBLEM
OF
BOETHIUS’ SOURCES
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. Theophrastus, 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. I n 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 O F 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 indication : (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. (4) The possibility that the Greek work, originating from the 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
I n order to establish what the theory of the older peripatetics
6
THE SOURCES OF ”DE SYLLOGISMO HYPOTHETICO”
was, we are directed to the information to be found in Greek commentaries on Aristotle’s Prior Analytics. Aristotle speaks, in several places, of hypothetical syllogisms. These remarks give the commentators occasion to point out what the doctrines of the older peripatetics on hypothetical syllogisms were. There are two commentaries to be taken into consideration, namely : (1) The commentary of Alexander of Aphrodisias. (2) The commentary of Ioannes Philoponos. If we compare the time of origin of the comnienaries with that of Boethius, we note that the first commentary precedes Boethius’ work by several centuries, while the second coincides temporally with him. We may establish this coincidence if we note that the “de syllogismo hypothetico” and the commentary of Ioannes Philoponos were composed in the first decades of the sixth century A.D. Alexander remarks that Theophrastus mentions hypothetical syllogisms in his own Analytics, and that Eudemos and several of his followers mentioned these syllogisms (CAG I1 1, p. 390, 1-2). The author points out of what types of syllogisms it may said that they are first mentioned, not by Aristotle, but by the older peripatetics. Those types of these syllogisms are described, namely : ( 1) hypothetical syllogisms, whose first premiss is a conditional and whose second premiss is termed “added”. ( 2 ) hypothetical syllogisms, whose first premiss is a disjunction. (3) hypothetical syllogisms, whose first premiss is the negation of a disjunction. (Cf. CAG I1 1, p. 390, 3-6). In the commentary of Ioannes Philoponos is found the remark that one should know that the pupils of Aristotle, including Theophrastus, Eudemus and others, and also the Stoics, wrote comprehensive books on the hypothetical syllogism (Cf. CAG XI11 2 , p. 242, 18-21). A comparison of these authorities leads to the following results. The later commentator expresses himself less clearly than the earlier. Where the earlier says that Theophrastus and Eudemos founded the study of hypothetical syllogisms themselves, the later says only that the pupils of Aristotle, like Theophrastus and
THE LOGIC OF THE OLDER PERIPATETICS
7
Eudemos developed this theory. On the other hand, we find in the later commentator one assertion that cannot be found in the earlier, namely that this branch of logic was presented in comprehensive books. The pure hypothetical syllogisms are a subclass of the hypothetical (Cf. CAG I1 1, p. 326, 8 and CAG XI11 2, p. 302, 9). I n both commentaries the name of Theophrastus occurs in connection with this class of syllo,’aisms. Alexander explains that Theophrastus designated pure hypothetical syllogisms “syllogisms by analogy’’ and points out that this designation is due to the fact that the premisses are analogous to each other and that the conclusion is analogous to the premisses (Cf. CAG I1 1, p. 326, 10-12). Indeed, one may say that in pure hypothetical syllogisms the premisses and conclusion are analogous, i.e. similar, in that they are all conditional sentences (i.e. implications). Ioannes Philoponos explains that Theophrastus designated those syllogisms as pure hypothetical in which both premisses and the conclusions are composed with the help of an hypothesis (Cf. CAG XI11 2, p. 302, 9-10). Both authorities agree that Theophrastus used a certain expression to designate the pure hypothetical syllogisms. But they disagree in that, according to one authority, Theophrastus used the expression “syllogisms by analogy” to designate these syllogisms, while, according to the other, he used the expression “pure hypothetical syllogisms” in this sense. We should prefer here the earlier commentator to the later. That Ioannes Philoponos on this point makes an assertion which is clearly incorrect we regard as evidence that the source writings that Alexander used were no longer available. I n Alexander’s commentary we find a report that is of significance in this connection. Alexander calls attention to the fact that Theophrastus in the first book of his Analytics explains that for pure hypothetical syllogisms the second figure is the one where the premisses agree in their antecedent clauses but not in the consequent, while the third figure is the one where the premisses
8
THE SOURCES OX “DE SYLLOGISMO HYPOTHETICO”
agree in their consequent clauses but not in the antecedent. The commentator points out, that he, in contrast to Theophrastus, regards those hypothetical syllogisms as of the second figure which agree in their consequent clauses and those which agree in their antecedent as of the first figure (Cf. CAG I1 1, p. 328, 2-5). The expression “first figure’’ is not used here; however, there is no doubt that if Theophrastus used the expressions “second figure” and “third figure’’ in the theory of pure hypothetical syllogisms, then the expression “first figure” must have also been used in this connection. The question now arises as to what extent a decision on whether Boethius was dependent on the older Peripatetics is possible on the basis of these reports. We may point to the fact that in Boethius’ system the following may be found: ( 1 ) hypothetical syllogisms in which the first premiss is a conditional. ( 2 ) hypothetical syllogisms in which the first premiss is a disjunction. (3) syllogisms that bear a close relation to the pure hypothetical syllogism. I n Boethius’ system, the first type form the first class of deductive forms (Cf. fj 22 infra), the second form the eighth class of deductive forms (Cf. fj 36 infra) and the third the fourth, fifth and sixth class of deductive forms (Cf. fj 28, 5 30, and § 32 infra). Only one type of deduction belonging to the logic of the older Peripatetics does not have a place in the system of Boethius, namely, the hypothetical syllogisms where the first premiss is the negation of a conjunction. The agreements that we can verify in this way are, however, scarcely of such a nature that they justify us in declaring that Boethius was dependent on the older Peripatetics. Nevertheless, one point should be raised: the fact that the pure hypothetical syllogisms, which in the commentaries are expressly identified with the name of Theophrastus, play a significant role in Boethius’ system. Still the possibility that this type of deduction
THE LOGIC O F THE OLDER PERIPATETICS
9
is dealt with in the Stoic logic should not be left out of consideration. We refer here to a technical expression of Stoic logic whose significance has been established by Lukasiewicz and which ca.n be expressed in English as “deductions from two non-simple premisses” (Cf. Lu. p. 118 and p. 129). Pure hypothetical syllogisms might be characterized as deductions from two non-simple premisses. Therefore, it is possible that the Stoics also presented the theory of pure hypothetical syllogisms. We shall find aid in resolving our problem in the remark of Alexander to which we recently referred. I n his work “de syllogismo hypothetico”, Boethius uses the expressions “the first figure”, “the second figure”, and “the third figure” and, in particular, he speaks of sentences of the first, second, and third figures. He regards as sentences of the second figure conjunctive connections of two implications which agree in their antecedents, i.e. whose antecedents consist of the same propositional variables, and as sentences of the third figure conjunctive connections of two implications whose consequents consiets of the same propositional variables; we will find the observations verified in the chapter devoted to the analysis of the principal work (Cf. Q 29 and Q 31 infru). We note that Boethius’ usage of the terms “first figure” and “second figure” agrees with that of Theophrastus and not that of Alexander. The three expressions “first”, “second”, and “third figure” recall the Aristotelean syllogistics. On the basis of an analysis of the Aristotelean syllogisms, we can indeed see an agreement between the three figures of the pure hypothetical syllogism and the three Aristotelean figures of the categorical syllogism. If we keep this agreement in mind, Alexander’s method of naming seems more appropriate. It can, therefore, hardly be assumed that someone not following Theophrastus’ example would define the second and third figure in such a way as to agree with Theophrastus. We have seen that Boethius defines these concepts the way Theophrastus did. We may therefore conclude that Boethius followed Theophrastus’ example.
10
9
THE SOURCES
7.
THE LOGIC
OF
or
”DE SYLLOGISMO
HYPOTHETICO”
CHRYSIPPOS
The Stoic propositional logic has been described by Lukasiewicz in his article “Zur Geschichte der Aussagenlogik”. It seems expedient to refer to this description rather than go back to the sources. In the Stoic logic we can distinguish: ( 1) Wndemonstrable inference schemes. ( 2 ) Derived inference schemes. The undemonstrable inference schemes can be listed. There are exactly five such inference schemes laid down by Chrysippos. These schemes are described in the treatise of Lukasiewicz but it seems justified to repeat them here. Our presentation agrees with that given by Lukasiewicz, except that instead of the letters which Lukasiewicz uses for propositional variables we will use ordinal numbers in order to approach the original text more closely. The five undemonstrable inference schemes of Chrysippos can be represented in the following way:
I. If the first, then the second. But the first, Therefore, the second. 11. If the first, then the second. But not the second, Therefore, not the first. 111. Not both the first and the second. But the first, Therefore, not the second. IV. Either the first or the second. But the first, Therefore, not the second. V. Either the first or the second. But not the first, Therefore, the second. (Cf. Lu. p. 117). As far as the derived inference schemes are concerned, it should be remarked that, according to the testimony of Cicero, they were
THE LOGIC
OF CHRYSIPPOS
11
very many (Cf. Lu. p. 118). Among these two are known; one can find them in Lukasiewicz’s work (Cf. Lu. p. 117-118). It seems unmistakable that this Stoic logic was not the model of Boethius. Further evidence against the dependence of Boethius on the Stoics stems from the fact that Boethius mentions the Peripatetics, but not the Stoics. The following must also be taken into consideration: If Boethius used Chrysippos’ system as model, it could not have occurred that he would dispense with one of the undemonstrable inference schemes unless,he recognized that it were superfluous in the Stoic system. But in Boethius’ work we do not find the third undemonstrable inference scheme of Chrysippos and it may not be assumed that Boethius came to the conclusion that it was superfluous in the Stoic system. This indicates that Boethius did not intend Chrysippos’ logic as model. It should not be denied that the logical systems of the Stoics and Boethius agree in important points. I n particular, one can find in Boethius inference schemes related to the first, second, fourth and fifth undemonstrable inference schemes of Chrysippos ; the inference scheme belonging to the first and eighth classes of inference schemes (Cf. infra, 5 22 and 5 36). We can however explain this agreement by assuming that both the system of Chrysippos and that of Boethius were connected with the logic of the older Peripatetics. That Boethius was influenced by the older Peripatetics seems sufficiently proved. That there are important resemblances between the Stoic and the Peripatetic logic is shown by the fact that Ioannes Philoponos refers to the Peripatetics and Stoics and compares the terminology of the two schools when he is concerned with hypothetical syllogisms (Cf. CAG XI11 2, p. 242, 14-p. 243, 10). It can also be noticed that Scholz, in his History of Logic, points out that Theophrastus and Eudemos anticipate the Stoic theory of hypothetical and disjunctive syllogisms (Cf. Sch. p. 32). All these facts suggest acceptance of the view that the work of Boethius was not directly influenced by the propositional logic of Chrysippos.
12
THE SOURCES OF “DE SYLLOGISMO
HYPOTHETICO”
5 8. LATINPRESENTATIONS OF PROPOSITIONAL
LOGIC IN ANTIQUITY
We shall consider three Latin presentations of propositional logic. The oldest of the three is a section of Cicero’s Topics. We have already referred indirectly to this work when we spoke of Boethius’ commentary on the Topics (Cf. $ 1 supra). Of the three presentations, Cicero’s is the only one directly available to us. Of the other two we must depend on references from Cassiodorus and Isodorus Hispalensis. Our principal source is a section of a work by Cassiodorus. From Cassiodorus’ reports we can establish the following : (1) Marius Victorinus wrote a book entitled “de syllogismis hypotheticis” (on hypothetical syllogisms) (Cf. Pr. I p. 661, note 3). We recall that according to Samuel Brandt the correct title of Boethius’ work was “de hypotheticis syllogismis” (Cf. $ 1 supra). If we accept this view the titles of the books of Marius
Victorinus and Boethius are almost identical ; this suggests that their contents were similar. (2) Tullius Marcellus portrayed in a work of seven books the theory of categorical and hypothetical syllogisms ; the fourth and fifth books of this work were devoted to the theory of hypothetical syllogisms (Cf. Pr. I , p. 664, note 16). The report from the work of Isodorus Hispalensis concerns only Marius Victorinus and not Tullius Marcellus and is taken word for word from Cassiodorus (Cf. Pr. I, p. 661, note 3). There is no doubt that the work of Marius Victorinus antedates that of Boethius since we know that Marius Victorinus lived in the fourth century. On the other hand, it appears impossible to date the work of Tullius Marcellus with respect to that of Boethius. As far as we know, Tullius Marcellus is mentioned only once in ancient literature, namely, in Cassiodorus. Prom this we can conclude only that the book of Tullius Marcellus existed at the time that Cassiodorus wrote. As the evidence now stands, we can only point out that it is possible that the work of Tullius Marcellus was earlier than Boethius. I n Cicero’s book, seven inference types are named (Cf. Ci.
THE DOCUMENT OF THE SCHOOL OF AMMONIOS
13
13, 54 to 14, 57). On the basis of an analysis of these seven inference types we can see that they are very closely connected with the five undemonstrable syllogisms of Chrysippos, as Prantl has correctly recognized (Cf. Pr. I , p. 524). We can show this connection by pointing out that the first, second, fourth and fifth syllogisms of Chrysippos are identical with the corresponding inference types of Cicero ; the third of Chrysippos’ syllogisms corresponds to the third and sixth inference type of Cicero; only the seventh is not exemplified in the Stoic syllogisms. Seven hypothetical inference types, which are identical in essentials to those of Cicero, can be found in the description of Dialectics given by Cassiodorus (Cf. Pr. I, p. 663, note 13). Since Cassiodorus refers here to the work of Marius Victorinus and Tullius Marcellus, we must assume that these seven inference types were included in these works. It is t o be mentioned that our opinion here agrees with that of Prantl, who says, “Whoever is acquainted with the way that Cassiodorus compiles cannot doubt for a moment that this enumeration is borrowed from Victorinus” (Cf. Pr. I, p. 663, note 13). We believe that we can establish the following: The theory of the seven hypothetical inference types that formed the basis of the Latin presentations of propositional logic could not have been Boethius’ model. I n “de syllogismo hypothetico” we do not find the two inference types that correspond to the third undemonstrable syllogism of Chrysippos ; the seventh inference type which is characteristic of Latin presentations of propositional logic is also missing. We recall that Boethius remarks that he did not draw the theory represented in his principal work from Latin writers (Cf. 3 5 supra) ; this remark is supported by the facts to which we have referred.
9. THE DOCUMENT OF THE SCHOOL OF AMMONIOS The Greek document of the school of Ammonios Hermeiu was published by Maximilianus Wallies in 1899; Wallies included it among the scholia to Ammonios’ commentary on the first book of Aristotle’s Analytics (Cf. CAG IV 6, p. 67-69).
0
14
THE SOURCES OF ”DE SYLLOCISMO HYPOTHETICO”
Prantl mentions the title of the document, which at the time that he wrote his history of logic was not yet published, in a note and adds, “From Ammonios’ style we may conclude that we have here perhaps the earliest pattern of the work of Boethius” (Cf. Pr. I, p. 657, note 168). Thus, Prantl suspected that the Greek document with which we are concerned was the source of Boethius’ work; if only for this reason, we should be obliged to include this document in our considerations. It would be advantageous tointroduce a short expression as a name for this document. Since Wallies remarks in his preface that the scholia which he is publishing as an appendix to Ammonios’ commentary appear to be exerpts from the notes of a follower of Ammonios (Cf. CAG I V 6, p. VIII), we may be permitted to call it “the Ammonian”. If we wish to establish the period of the Ammonian document we must first establish when Ammonios lived. We are enabled to do this by Ammonios’ naming Proclos as his teacher (Cf. CAG IV 5, p. 1, 8)’ while we have established that Proclos lived in the fifth century A.D. If we adopt this procedure, we obtain the result that the Ammonian document and Boethius’ work were approximately contemporaneous. This is to be interpreted in such a way that it is not impossible that the Ammonian document was already in existence when Boethius composed “de syllogismo hypothetico”. The Ammonian document is, it seems, a fragment, since at its beginning it is announced that eight points are to be mentioned with respect to the hypothetical syllogism (Cf. CAG IV 6, p. 67, 41), while only six are mentioned in the text. The point referred to as the sixth in the Ammonian document is essential (Cf. CAG IV 6, p. 68, 2 3 ) . Here exactly five hypothetical syllogisms or inference schemes are mentioned. On the basis of an analysis of these syllogisms it can be shown that they are identical with the Chrysippos’ five undemonstrable syllogisms (Cf. CAG I V 6, p. 68, 23-32). I n summary we may remark that if we compare the theory presented in the Ammonian document and that found in Latin presentations of propositional logic with the theory of Chrysippos,
THE DOCUMENT OF THE SCHOOL OF AMMONIOS
15
we may say that the first and third theories are more closely related than the second and third since where the Stoic logic has five undemonstrable syllogisms, the Ammonian document speaks of five hypothetical syllogisms while the writings of the Latin authors give seven hypothetical inference types. We have shown that the Stoic logic could not have been the model of Boethius’ works (Cf. 4 7 supra). Since the logic represented in the Ammonian document is derived from the Stoic logic, this document cannot be regarded as a source of Boethius’ work. We may add that Prantl’s surmise referred to at the beginning of these considerations has been refuted by the facts.
I1
THE EFFECTS OF BOETHIUS’ PROPOSITIONAL LOGIC IN THE EARLY SCHOLASTIC PERIOD
9
10. GENERALREMARKS
It is generally recognized that the Boethius’ logical writings became significant in the development of medieval logic (Cf. Sch., p. 37). We will observe what may be established in this connection with respect to the two works of Boethius that we are considering. We can start with a conclusion taken from a treatise of Martin Grabmann, a reliable medieval scholar: “We find none (of Boethius’ logical monographs) mentioned before the end of the eleventh century. I n the eleventh and especially in the first half of the twelfth century Boethius’ logical monographs were in more general use” (Cf. Gr., p. 10). We gather from this that Boethius’ logical writings, namely, “de syllogismo hypothetico” and the commentary on Cicero’s Topics, were in circulation in the eleventh and the first half of the twelfth century in hand-written form. I n this connection we mention the fact that in the logical literature of this period the influence of Boethius is clearly noticeable.
8
11.
THE LOGIC
OF
ST. GALLEN
We mention firstly the treatise entitled “de syllogismis” which is connected with the name of Notker Labeo. Since we are here concerncd with a work which proceded from the school of St. Gallen we will permit ourselves to refer to it as “the Logic of St. Gallen”. The 13th chapter of this work entitled “de ordine modorum” (on the ordering of modes of inference) is essential for our purposes (Cf. Not., p. 605-613). I n this chapter seven inference types are mentioned. At the beginning of the chapter it is mentioned that the inference type which appears first is also the first in Boethius. The presentation
THE DIALECTIES OF PETER ABELARD
17
of the Logic of St. Gallen is closely connected with Boethius’ commentary on Cicero’s Topics. As we will show, Boethius expresses the third inference type differently in his commentary than Cicero did in his Topics (Cf. 9 40 infru), and thus also differently than they are expressed by later authors such as Marius Victorinus, Martianus Capella and Cassiodorus. The Logic of St. Gallen agrees with Boethius’ commentary in these points (Cf. Boe., p. 825, 51, and Not,, p. 609, 20-22). This shows clearly that Boethius’ commentary was the model of the 13th chapter of the Logic of St. Gallen. 12.
TEE
DIALECTICS
OF
PETER ABELARD
We now refer to the main logical work of Peter Abelard, “Dialectics”, published incomplete by Victor Cousin in 1836. More exactly, we are concerned with the fourth part of this work called in the Cousin edition “de propositionibus et syllogismis hypotheticis”. (On hypothetical propositions and syllogisms). (Cf. Ab. C., p. 434 and Ab. G., p. 605). That Abelard repeats the theory that can be found in the principal work of Boethius is obvious and undeniable. We will show some evidence for this. I n the Cousin edition the relation of the work of the medieval author to Boethius’ work is revealed since the author indicates for each chapter what the corresponding parts of the book “de syllogismo hypothetico” are. Prantl begins his analysis of this part of the work of Abelard with the remark that the whole content of Boethius’ “de syllogismo hypothetico” is here repeated (Cf. Pr. 11, p. 203). Bernhard Geyer, in characterizing the fourth part of Abelard’s Dialectics, declares, “We are concerned with a paraphrase of Boethius’ “de syllogismo hypothetico” (Cf. Ab. G., p. 605). We’remark further that in the article “Aussagenlogik im Mittelalter” (Erkenntnis, Vol. 7, no. 3), this author has compared the presentation of Peter Abelard and Boethius; we refer to this work.
18
EFFECTS THE EARLY SCHOLASTIC PERIOD
13. UNEDITED COMMENTARIES
ON “DE SYLLOGISMO HYPOTHETICO”
Finally, we will mention that there are a number of commentaries on the work “de syllogismo hypothetico”, which may be found in written manuscripts dating from the 12th century and which have never been published. A description of these commentaries may be found in the treatise of Martin Grabmann referred to in $ 10.
3
14. SUMMARY OF
RESULTS
In summary we believe to have established the following : Each of the two forms of propositional logic constructed by Boethius were influential in the early scholastic period. The propositional logic of the commentary on Cicero’s Topics had its greatest effect on the logic of St. Gallen. The propositional logic of the book “de syllogismo hypothetico” is continued in the fourth part on Peter Abelard’s Dialectics. That there was a lively interest in this work is attested by the many commentaries handed down to us in handwritten form.
I11 CHOICE OF METASCIENCE AND METALANGUAGE $ 15. INTRODUCTION OF THE CONCEPTS
“METASCIENCE”
AND “META-
LANGUAGE”
I n according with existing usage we now introduce the expressions “metalanguage” and “metascience” (Cf. Ta., p. 22). We regard the term “syntax language” as synonymous with “metalanguage”. The two expressions designate relations ; the first a relation between languages and the second a relation between sciences. The converse of these relations is designated by the expressions “object language” and “object science”; i.e. if x is the metalanguage with respect to y, y is the object language with respect to x ; the concept object science is determined in the same way. The essence of the concept metalanguage can be expressed as follows : If x is the metalanguage with respect to y, x contains expressions which designate the expressions of y. Furthermore, if x is the metalanguage with respect to y, x can be either a different language than y or the same language as y. The concept metascience is closely connected to the concepts truth and falsity. We can express this connection by saying: Science x is called the metascience with respect to y if the proof that sentences of y are true or false as the case may be can be carried out in 5. There is obviously a connection between the concepts metascience and metalanguage. It may be said of any science that it is represented in a certain language. If x is the metascience with respect to y, then the language in which x is represented is the metalanguage with respect to the language in which y is represented. We will now refer to a law which is of great significance to the history of the sciences. Science as a whole and the individual sciences in particular are conceived of as being in progressive
20
CHOICE OF METASCIENCE A N D M E T A L A N G U A G E
development. If the history of science x at time t is written, the form that science x has achieved at time t serves as metascience. To illustrate this law we notice that in a modern history of mathematics, modern mathematics serves as the metascience. I n each of the two books of Boethius, i.e. “de syllogism0 hypothetico” and the commentary on Cicero’s Topics, a science is represented. These sciences are here designated object sciences. Modern logistic will serve as the metascience with respect to these sciences. The language in which these two object sciences is represented is a form of Latin created by Boethius. As metalanguage we will use a language which contains the language of logistic.
3
16.
THE
SYSTEMS
OF
MATERIAL
IMPLICSTION
AND
STRICT
1MPLICATION
We have indicated that we will here regard modern logistic as o w metascience; we must also tell which of the modern representations of logistic we have in mind. There are two systems to be considered; we will call them briefly “the system of material implication” and “the system of strict implication”. The system of material implication is represented in the inathematical logic of the Principia Mathematica; in this connection we are concerned especially with the section entitled “The Theory of Deduction”. The system of strict implication is represented in the work of Lewis and Langford entitled “Symbolic Logic”. The sixth chapter is essential to our purpose. We note that this chapter was composed by Lewis. It may seem strange that we regard it necessary to consult two differing systems of modern logic, but we have considerable reason for doing this. It seems to us that we must consider two possibilities in the interpretation of the inference schemes of Boethius ; firstly, there is the possibility of interpreting certain of the implications that appear in these inference schemes as material implications,on the other hand, there is the possibility of considering them as strict implications. Therefore, in testing, we will use both the system of material implication and the system of strict implication as a basis.
DESCRIPTION O F THE METALANGUAGE
0
17.
DESCRIPTION O F THE
21
METALANGUAQE TO BE USED I N THE
INVESTIGATIONS TO FOLLOW
We will now consider the language to be used as metalanguage more closely. I n constructing the principles of the metalanguage we will use the teachings of the Polish logisticians Jan Lukasiewicz and Alfred Tarski. We will point out in advance of the presentation what ideas we have borrowed and from which of the two representatives of logistic they are taken. From Lukasiewicz we borrowed the idea of parenthesis-free notation, or, in other words, the principle that the functors are always to precede their argument (Cf. Lu., p. 125-126). From Tarski we borrowed the idea that every expression of the object language is correlated to two expressions of the metalanguage, namely (1) An expression which may be called the translation of the correlated expression of the object language into the metalanguage. (2) An expression which is the name of the correlated expression of the object language (Cf. Ta., p. 28).
In the two books of Boethius we find logical formulae; we have already indicated that these formulae are our principal concern (Cf. 5 2 supra). Our metalanguage must therefore be of such a nature as to make it possible to translate into this language every logical formula which occurs in the two books of Boethius. In the formulae presented by Boethius we find expressions that are to be designated propositional variables. “Propositional variable” is an expression of our metalanguage which is a general name for certain expressions of the object language. We can distinguish two different systems of propositional variables, one simple and one extended. The simple system has only two propositional variables, viz. the two expressions “hoc est” (this is) and “illud est” (that is). We call attention to the fact that not the simple signs “hoc” and “illud” but the complex signs “hoc est” and “illud est” are to be regarded as propositional variables.
22
CHOICE OF METASCIENCE A N D METALANGUAGE
Sometimes Boethius uses the simple expressions “hoc” and
“illud” in place of the complex expressions “hoc est” and “illud est”; in these cases we will regard the simple signs “hoc” and
“illud” as propositional variables. The expression “hoc est” (or “hoc”) may be termed the first propositional variable and the expression “iltud est” (or “illud”) the second propositional variable. This simple system of propositional variables agrees in essentials with the one used by Cicero in the Topics. It is to be noticed that Cicero uses the simple signs “hoc” (this) and “illud” (that) as propositional variables (Cf. Ci. 14, 56-57). Boethius uses the simple system of propositional variables in two places : (1) A t the beginning of “de syllogismo hypothetico” (Cf. Boe., p. 608). (2) In the fifth book of the commentary on Cicero’s Topics immediately following Cicero’s presentation (Cf. Boe., p. 831-833). The extended system consists of four distinct propositional variables viz. the following four expressions : “a est” or “b est” or << c est” or “ d est” or
“est a” “est b” “est c” “est d”.
In the formulae which Boethius constructs the two simple signs of which the expressions are composed ordinarily appear in the second order. We stress again that the complex signs and not their elements are to be regarded as propositional variables. Boethius uses the extended system of propositional variables only in “de syllogismo hypothetico” ; this system completely replaced the simple one. I n order to explain this fact we note that a large number of the formulae that Boethius presents in “de syllogismo hypothetico” cannot be expressed if only two distinct propositional variables are allowed, since in many of these formulae three or even four propositional variables occur.
DESCRIPTION O F THE METALANGUAGE
23
In addition to the propositional variables, there occur only signs that may be designated functors ; we will enumerate them and, at the same time, introduce their names. These names belong to our metalanguage : ( 1 ) The sign of negation: “mn” (not). (2) The signs of implication: “si” (if) and “cum” (when). ( 3 ) The sign of disjunction: “aut . . . aut” (either . . . or). (4) The signs of conjunction: “ a t p i ” (but), “at” (but), “autem” (but) and “et” (and). ( 5 ) The sign of inference : “igitur” (therefore). (6) The modal signs: “necesse est” (it is necessary) and “contingit” (it is possible). Boethius says that the two conjunctions “si” and ‘Lcum”,i.e. the two signs of implication, are synonymous (Cf. Pr. I, p. 702, note 143). We believe we may assume that Boethius used two different signs for implication out of stylistic considerations only. It could, for example, be considered poor style if the same sign was immediately repeated and this can be avoided if one has two synonymous signs. This can be shown by an example. I n “de syllogismo hypothetico” we find the following formula: “si cum sit a, est 6 , est
G”
(Cf. Boe., p. 621). I n this formula the two signs of implication occur in immediate succession. If there were only one sign of implication, the same sign would have to occur twice in immediate succession. We remark that we mll not copy this characteristic of Boethius’ language in our metalanguage; it will be our principle to keep the number of signs as small as possible. We believe we may also regard the four signs of conjunction as synonymous. I n “de syllogismo hypothetico” we find some places where expressions are interpretable as conjunctions, although no sign of conjunction occurs (Cf. Boe., p. 626, p. 628, p. 629 and p. 676). It seems permissable to add a sign of conjunction, e.g. “et”, a t these places.
24
CHOICE OF METASCIENCE A N D IKETALANGUAGE
The two modal signs are not synonymous; however, this will not oblige us to use two modal signs in our metalanguage. Boethius puts the modal signs before “mse u” and “non esse u” (Cf. Boe., p. 612). Strictly speaking, these expressions are not identical with the expressions “est u” and “non est a”. We believe we may disregard this difference in our metalanguage and proceed as if the expressions “est a” and “non est u” were at these places instead of “esse a” and “non esse a”. We will now proceed to indicate how Boethius’ logical formulae are to be translated into our metalanguage. We distinguish arguments and functors and adopt, as a syntactical rule, that functors are always to precede their arguments. The following may be arguments : ( 1 ) propositional variables. (2) well formed expressions constructed from functors and propositional variables. Our language contains four distinct propositional variables, viz. the four letters “p”, “q”, “r” and “s”. The use of these letters as propositional variables is recommended by the fact that in modern logistic, especially in the works named in § 16, they are used in this sense. As the sign of negation we will use the upper-case letter “ N ” ; this agrees with the practise of the Polish logisticians. (Cf. Lu., p. 126; Ta., p. 23). We state, as a syntactical rule, that the sign of negation is a functor which always takes one argument. On the basis of the two syntactical rules laid down, we can establish that the following expressions are well-formed :
“Np” and “ N N p ” . As the sign of implication, we use the lower-case letter “c”. We note that Lukasiewicz used the corresponding upper-case letter “G” as the sign of implication (Cf. Lu., p. 176). We have already indicated in Q 16 that there are two possibilities for the interpretation of Boethius’ inference schemes : certain implications that occur in them are to be understood either as
DESCRIPTION O F THE METALANGUAGE
25
material or as strict implications. This holds analogously for the logical formulae that are not inference schemes. We can refer to Boethius’ logical formula stated above. In that formula, the synonymous signs “gi” and “cum” can both be interpreted either as signs of material or strict implication. The sign “c” is ambiguous and can therefore be identified neither with a sign of material, nor with a sign of strict, implication. The use of the sign “c” in the sense referred to recommends itself because, on one hand, it is reminiscent of “G” and yet is distinct from it. That the sign of implication of the object language is ambiguous can be regarded as a fault of this language; it is however obviously not a fault of our metalanguage that the expression used to translate that sign is ambiguous. With respect to the sign of implication we also lay down a syntactical rule, namely, that the sign of implication takes two arguments, preceding the first of the two arguments while the second argument immediately follows the first. We can now identify the following as well-formed :
“cpq” and “cNpq”.
As sign of disjunction we use a lower-case letter “a”. The sign (‘a” is ambiguous, like the sign “c” is. The sign “a” is, on the one hand, a sign of material exclusive disjunction and, on the other, a sign of strict exclusive disjunction. It is possible to construct a system where both of the expressions “material exclusive disjunction” and “strict exclusive disjunction” receive legitimite definitions. We will however restrict ourselves t o indicating them by referring to an example. If the sentence ‘(Socratesis either well or ill” is interpreted as a material exclusive disjunction it says “it is not the case that Socrates is both well and ill and i t is also not the case that Socrates is not well and not ill”; when the same sentence is interpreted as a strict exclusive disjunction, it says “it is impossible that Socrates is both well and ill and it is also impossible that Socrates is not well and not ill”. I n addition, it should be remarked that the sentence “Socrates
26
CHOICE OF METASCIENCE AND METALANGUAGE
is either well or ill” was used by Peter Abelard as an example of disjunction (Cf. Ab. C., p. 441-442) and that Boethius himself in “de syllogismo hypothetico” presents the incomplete sentence “is either ill or well” as an example of disjunction (Cf. Boe., p. 636). The sign of disjunction, like the sign of implication, takes two arguments and immediately precedes the first of the two arguments. It is easy to see that the two expressions “apq” and “aNpq” are well-formed. As sign of conjunction, we will use the Latin word “et”. We note that this word is not ambiguous as are the signs of implication and disjunction but has a single meaning. The sign of conjunction also takes two arguments and immediately precedes the first one. In logistic literature, the expression “logical product” is in general use (Cf. PM, p. 6 and L.a.L., p. 123). It seems therefore appropriate to state that if ‘9’’ and “u” are any arguments whatever, the expression “et t ZL” represents the logical product of “t” and “u”. We have borrowed the conjunction “et” from the object language, but we have changed what might be called its syntax. It is not the case in Latin that the functor “et” is placed before the arguments; on the contrary, it is placed between its arguments. We may however say that the syntactical rule that we have adopted is not too different from Latin syntax and that this syntax has a place for it, since Latin also uses as sign of conjunction the two membered conjunction “et . . , et” and in this case at least a part of the functor precedes the two arguments. We have also borrowed the sign of inference, the Latin word “igitur” from the object language; here too the syntax of the word is changed. The word “igitur” is in our language a functor which takes two arguments and immediately precedes the first. As opposed to this, the word “igitur” of the object language is placed immediately after the first or second sign of that sentence which may be called the conclusion.
DESCRZPTION OF TEJT METALANGUAUE
27
I n logical formulae the functor “igitur” can occur only at the beginning; in this it differs from all the other functors. This rule represents a serious limitation; we should like to point out why this limitation seems desirable. We will construct no logical formulae in which the functor “igitur” occurs any place but at the beginning of the formula. If we lay down rules for our language which did not prohibite the functor “igitur” at any place but a t the beginning, possibilities would occur of which we do not make use; and that contradicts the principle of parsimony which we would like to follow. If we will be allowed to state the maxim we will follow in a short and impressive form, me may say: what we do not use, we will not possess. A logical formula is called an inference scheme, if and only if, it begins with the functor “igitur”. It seems convenient to illustrate here the same inference scheme in the object language and the metalanguage; thus it will be possible to compare the original with the translation. We will choose as an example the first inference scheme of “de syllogismo hypothetico”. The inference scheme is stated in the object language as follows:
“si est a, est b, atqui est a , est igitur b” (Cf. Boe., p. 615). This expression corresponds to the following formula of the metalanguage :
“igitur et cpqpq” ; the first argument of the functor “igitur” is
“et cpqp” ; the second argument of this functor is
38
CHOICE O F METASCIENCE A N D M E T A L A N G U A G E
the first argument of the functor “et” is “Cpp;
the second argument of this functor is
,,P”; the first argument of the functor ((c” is
((P” and the second argument of this functor is “G!”.
From this analysis we can see that the formula is well-formed. In order to tra,nslate the two modal signs of the object language into our metalanguage, we introduce a sign borrowed from Lewis’ system of strict implication. This sign is a rhombus so placed that one of the two diagonals is horizontal, and, as a result, the other is vertical. Following the usage of C. West Churchman in his article “On finite and infinite modal systems”, we call this sign “the modal operator” (Cf. Ch., p. 77). The modal operator is a functor which, like the sign of negation, always takes a single argument and precedes this argument. The modal operator will serve as the translation of the expression “contingit” (it is possible). The expression “contingit e8se u” of the object language corresponds t o the expression
“OP” of the metalanguage and the expression (‘necesse est a s e u” of the object language corresponds to the expression
“NONp” of the metalanguage. As an analysis of this expression, we note: the argument of the first sign of negation is the expression
“ONp’’;
DESCRIPTION OF TEE METALANGUAGE
29
the argument of the modal operator is the expression
‘“P” and the argument of the second sign of negation is the expression
“P”. We can easily see that the whole expression is well formed. This finishes the statement of the principles of our metalanguage.
IV ANALYSIS OF “DE SYLLOGISM0 HYPOTHETICO”
8
18. RULESFOR
CORRELATING
INFERENCE
SCEIEMES AND PRO-
POSITIONAL FORMS
We note that the concept inference scheme has been defined in 17 (Cf. supra). Inference schemes are those and only those formulae
that begin with the functor “igitur”. A propositional form will mean a logical formula that is not an inference scheme. This definition implies, that a propositional form begins with some sign other than the sign of inference. Both the inference schemes and the propositional forms will be divided into classes and will be ordered within each class. The ordering of the propositional forms can be regarded as primitive with respect to the ordering of inference schemes. In the ordering of the inference schemes, each inference scheme is correlated to a certain propositional form. The correlation of the inference schemes to the propositional forms can be made in several ways, i.e. there are several possible rules of correlation. We will lay down the rule of correlation which Boethius generally followed and which we shall call “the regular rule of correlation”, namely The inference scheme 5 will be correlated with that sequence of signs, which is a propositional form, which is a part of the inference scheme x, and whose first sign is the third sign of the inference scheme.
3
19.
DIVISIONOF
THE PROPOSITIONAL
FORMS AND INFERENCE
SCHEMES I N T O E I G H T CLASSES
We have indicated in 8 4 that the essential contents of the logic of Boethius are the inference schemes. The propositional forms are taken into consideration only as parts of inference schemes.
ORDERING PRINCIPLES
31
The propositional variables can be regarded as the simplest propositional forms. These propositional forms are an exception since although they occur as parts of inference schemes, they are nevertheless not correlated with inference schemes by Boethius. It has been mentioned that Boethius also constructed in his system propositional forms that do not occur as parts of inference schemes in his logic and consequently are not correlated to inference schemes. This seems to contradict the statement that in Boethius’ logic propositional forms are taken into consideration only as parts of inference schemes. Nevertheless we believe we can hold t o our interpretation. Boethius constructed the propositional forms we are here considering only to investigate whether inference schemes are correlated to them and he believed to have established that this is not the case. Thus it also holds in this case that consideration of the inference schemes is decisive in the construction of propositional forms. We shall see that from the standpoint of modern logic inference schemes can be correlated also to these propositional forms. The inference schemes, like the propositional forms, will be divided into eight classes. We will determine these classes by the rule that two inference schemes will belong to the same class, if and only if the propositional forms to which they are correlated are elements of the same class. tj 20.
ORDERING PRINCIPLES
FOR
PROPOSITIONAL FORMS
AND
INFERENCE SCHEMES
Like Boethius we will order the elements of each class of propositional forms lexicographically ; the ordering of propositional forms, together with additional conventions will establish the ordering of the corresponding inference schemes. I n agreement with the text, we will also order the classes of propositional forms and the classes of inference schemes, so that we can, for example, speak of the first class of propositional forms and of the first class of inference schemes. It will be noticed that since we have, on the one hand, ordered the elements of the classes and, on the other hand, the classes
32
ANALYSIS OF “DE SYLLOGISM0 HYPOTHETICO”
themselves, it is easily possible to order all the elements of all the classes.
8
21.
THEFIRST
CLASS OF PROPOSITIONAL
FORMS
There are four elements of the first class of propositional forms, namely (1)
CPq
(2)
CPNq
(3)
CNPP cNpNq. (Cf. Pr. I, p. 705, note 155).
(4)
These propositional forms occur here in the same order as in the original. We have used the expression “lexicographic” in 3 20; we should like to explain that this concept is borrowed from set theory (Cf. H., p. 46). An order can be called lexicographic if it satisfies the following conditions 1) The sequence of signs consisting of the sign of negation and the propositional variable following it are considered as a single sign, i.e. as occupying a single place; 2 ) The propositional variable x precedes the sequence of signs whose first sign is the sign of negation and whose second sign is the propositional variable x itself in the alphabet, e.g. the sign “p” precedes the sign “Np” in the alphabet. We can easily see that in a lexicographical ordering propositional form ( 1 ) does precede propositional forms (2), (3) and (4), and, similarly, propositional form (2) precedes propositional forms (3) and (a), and, finally, propositional fornz (3) precedes propositional form (4).
8
THE FIRST CLASS OF INFERENCE SCHEMES Every propositional form is correlated with two inference schemes so that to the four propositional forms of the first class there are 22.
FIRST CLASS OF INFERENCE SCHEMES
33
correlated eight inference schemes; these inference schemes are the elements of the first class of inference schemes. Of the two inference schemes which are correlated to the same propositional forms, one can always be designated as the first and one as the second; we will not state the general rule which determines this order. We will group these inference schemes, placing all of the first inference schemes, and all of the second, together. There are four inference schemes in the first group:
Igitur et cpqpq Igitur et cpNqpNq Igitur et cNpqNpq Igitur et cNpNqNpNq There are also four inference schemes in the second group: (1’)
Igitur et cpqNqNp
(2’)
Igitur et cpNqqNp
(3’)
Igitur et cNpqNqp
(4’)
Igitur et cNpNqqp. (Cf. Pr. I, p. 705, note 155).
It is to be noted that inference scheme (1) is identical with the inference scheme stated in Q 17 and that it is there shown how to analyse this inference scheme (Cf. supra, p. 27-28). The analysis of the other inference schemes involves greater difficulty only in so far as in these cases at least two expressions occur that are not simple propositional forms, but instead consist of a sign of negation and a propositional variable. The inference schemes are correlated with the analogously numbered propositional forms by the regular rule of correlation. We will show this with respect to inference scheme (1) and (1’). The sequence of signs ‘(cpq’, is a propositional form which forms a part of inference schemes (1)
34
ANALYSIS OF “DE SYLLOGISMO
HYPOTIFETICO”
and (1’)and the first sign of this sequence is the third sign of the two inference schemes ; thus these inference schemes are correlated with that sequence on the basis of the regular rule of correlation (Cf. $ 18). This sequence is however identical with propositional form (1) (Cf. 8 21); therefore the two inference schemes are correlated with propositional form (1) by the regular rule of correlation. With respect to inference schemes ( 2 ) and ( 2 0 , (3) and (30, (4) and (4’)the correlation can be proved analogously. We now note that the eight inference schemes that we have presented are correct. We define: ‘‘x is a correct inference scheme” as: there is an expression y such that: 1) y is a sentence of the metascience. 2 ) y agrees with x. To establish under what circumstances a sentence would belong to the metascience would be a matter of further investigation. We will limit ourselves to explaining that all sentences provable from the axioms of a science may be regarded as belonging to that science. The concept of agreement of sentences is related to the concept of translation and represents an equality relation. We will say that one sentence agrees with an other if according to certain translation rules, one can be obtained from the other. We will now consider one of the eight inference schemes, namely inference scheme (l),and show that this formula is correct. We recall that we are considering two systems of modern logistic as metasciences, namely, the system of material implication and the system of strict implication (Cf. 8 16). We will consider the two cases separately, using first the system of material implication and then the system of strict implication as our metascience. a) Both the functor “c” and the €unctor “igutw” of our language correspond to the sign of implication of the language of Principia Mathematica, a sign which is introduced by definition 1.01 (Cf. P M I, p. 94). We note that the sign of implication which corresponds to our functor “igitzcr” is in no way differentiated from other signs
FIRST CLASS OF INFERENCE SCHEMES
36
of implication in the language of Principia Mathematica. It can be discovered, since this sign of implication is surrounded by the greatest number of dots. I n translating formulae of our language into the language of Principia Mathematica, it should be noted that the syntax of the languages differs; in our language functors which take two arguments are placed before the arguments, while in the language of Principia Mathematica, they me placed between the arguments. Thus we see that the following sentence is the translation of inference scheme (1) into the language of Principia Mathematica : (a) PIq*P-I*q The formula (a) is a sentence belonging to the system of material implication, our present metascience, since this formula is provable in it. The proof of this formula can be carried out by using the principle of identity (formula 2.08) and the principle of importation (formula 3.31) (Cf. PM I, p. XII). We may therefore say: there is an expression y such that: 1) y is a sentence belonging t o the system of material imphcation, 2 ) y agrees with inference scheme (1). We have thus proved the correctness of inference scheme (1) for the case in which the system of material implication is used as metascience. b) Both the functor “c” and the functor “igitur” of our language correspond to that sign of the language of the system of strict implication whose meaning is determined by definition 11.02 (Cf. L. a. L., p. 124). The syntax of the language of strict implication agrees with the syntax of the language of Principia Mathematica in that functors which take two arguments are placed between the arguments. The translation of inference scheme (1) in the language of strict implication is : (b)
p
l ) Following Oskar Becker in Be., we will use the sign “ <” as the sign of strict implication (Cf. Be., p. 500).
36
ANALYSIS OF “DE SYLLOGISMO HYPOTHETICO”
Formula (b) is provable in the system of strict implication. We will not present the full proof, but will instead limit ourselves to the remark that we use a formula which closely resembles formula (b), namely
p.p
(c)
and that formula (c) is an axiom of the system of strict implication (Cf. L.a.L., p. 125). We may thus say that there is a sentence of the system of strict implication that agrees with inference scheme (1). It is therefore proved that our inference scheme is correct if the system of strict implication is used as metascience. I n conclusion we note that the proofs of the correctness of the other seven inference schemes may be carried out in an analogous manner.
3
23.
THE SECOND
CLASS O F PROPOSITIONAL FORMS
There are eight elements of the second class of propositional €orms. The eight propositional forms occur in lexicographic order in the text. They are as follows: CPCP
cpcqA’r
cpdqr cpcATqNr cNpcqr cNpcqNr cNpcNqr cNpcNqNr (Cf. Pr. I, p. 706, note 157). We should not neglect to mention that in “de syllogismo hypothetico” the presentation of these eight propositional forms is accompanied by detailed explanations in which each of the eight propositional forms is considered (Cf. Boc., p. 618-620). We will
SECOND CLASS O F INFERENCE SCHEMES
37
not take up the discussions in detail, limiting ourselves instead to the remark that we regard them as not properly speaking a part of our theory. We note furthermore that the presentation of the theory of Boethius appearing in Prantl’s History of Logic does not consider these discussions.
$ 24. THE SECOND
CLASS O F INFERENCE SCHEMES
To each of the eight propositional forms, two inference schemes are correlated. Consequently, sixteen inference schemes are correlated to the eight propositional forms of the second class; these sixteen inference schemes are the elements of the second class of inference schemes. We may again term one of the two inference schemes correlated with the same propositional form the first, and the other the second, inference scheme, and gather the first inference schemes and the second inference schemes into groups. The inference schemes of the first group are as follows: (5)
igitur et cpcqrpcqr
(6)
igitur et CpcqNrpcqNr
(7)
igitur et cpcNqrpcNqr
(8)
igitur et CpcNqNrpcNqNr
(9)
igitur et cNpcqrNpcqr
(10)
igitur et CNpcqNrNpcqNr
(11)
igitur et cNpcNqrNpcNqr
(12)
igitur et cNpcNqNrNpcNqNr
The inference schemes of the second group are as follows: (5’)
igitur et cpcqrcqNrNp
(Sf 1
igitur et cpcqNrcqrNp
(7’)
igitur et cpcNqrcNqNrNp
(Sf)
igitur et CpcNqNrcNqrNp
(9’)
igitur et CNpcqrcqNrp
38
(107
(11’) (127
ANALYSIS OF “DE SYLLOGISMO
HYPOTHETICO”
igitur et CNpcqNrcqrp igitur et cNpcNqrcNqNrp igitur et CNpcNqNrcNqrp (Cf. Pr. I, p. 706, note 157).
These inference schemes may be analysed on the basis of the syntactical rules set up in Q 17. We wish in passing to make a remark significant to the judging of the editions. Inference schemes (7) and (8) are correctly presented in the first edition. Errors occur in both places in the Basel edition. At the end of inference scheme (7) the expression “cum igitur non sit b, non est c” occurs where the expression
“cum igitur non sit b, est G” should; and at the end of inference scheme (8) the expression cum igitur non sit b, est c” occurs where the expression “cum igitur non sit b, non est c7’ should (Cf. Boe., p. 620). This gives the impression that the expressions were simply switched, and Prantl, in quoting the passage, simply corrects it (C€. Pr. I, p- 706, note 157). I n the Migne edition the same errors occur. This is noted because it supports our contention that the Migne edition is a reprint of the Basel edition (Cf. supra 2 p. 2 ) . The relation between the inference schemes of the second class and the propositional forms of the second class is determined by the regular rule of correlation. We will now show that the eight inference schemes of the first group are correct but the inference schemes of the second group are incorrect. We will verify this remark by considering one element of each group ; the elements to be considered will obviously be inference schemes (5) and (5’). 66
SECOND CLASS O F INFERENCE SCHEMES
39
a) If we translate inference scheme (5) into the language of Principia Mathernatica, we obtain the following formula : (a)
p.3.q3r:p:3.q3r
Formula (a) is a provable formula of the system of material implication. This can be proved in several ways. I n this connection the simplest is as follows: starting with formula (a) of Q 22 (Cf. supra, p. 35), we note that formula (a) of 5 24 follows if “q” is replaced by the expression “q 3 r”. As B result of these considerations, we see that inference scheme (5) may be termed correct if the system of material implication is used as metascience. b) If inference scheme (5) is translated into the language of the system of strict implication, we obtain the following formula : (b)
p.<.q
This sentence is provable in the system of strict implication, if in formula (b) of $ 22 (Cf. supra, p. 35) “q” is replaced by the expression “q < r”. We conclude that inference scheme (5) may also be regarded as correct if the system of strict implication is used as metascience. a‘) If the inference scheme (5‘) is translated into the language of Principia Mathernatica, we obtain the following formula : (a’)
p.3 .q3r
:qr)-T
: 3 ., u p
I n order to show that this formula does not belong to and is not provable in the system of material implication, we will proceed as follows : we consider the case where (‘p” is replaced by a true proposition, “q” by a false proposition and “8’by any proposition whatever. It can easily be seen that in this event the expression obtained from (a’) takes the truth value “false”. We note that if the formula (a’) belonged to the system of material implication, no sentence obtainable from this formula by replacement of the propositional variables by propositions could take the truth value “false”.
40
ANALYSIS OF “DE SYLLOGISMO
RYPOTHETICO”
We conclude that the formula (a’) does not belong to the system of material implication and is not provable in it. Summarizing, we may say that inference scheme ( 5 ’ ) is incorrect if the system of material implication (i.e. the system of Principia Mathematica) is used as metascience. b’) If inference scheme ( 5 ’ ) is translated into the system of strict implication, we obtain the following formule : (b’)
p<.q
:.<.-p
We believe that the following sentence holds: if formula (b’) is provable in the system of strict implication, formula (a’) must be provable in the system of material implication. In this connection, we refer to a remark appearing in the book of Lewis and Langford to the effect that all theorems proved up to that point in the book hold if the sign of strict implication is replaced uniformly by the sign of material implication (Cf. L.a.L., p. 141). If formula (b’) belonged to the system of strict implication and was provable in this system, it would be among those theorems for which it was stated that they had already been proved; it should therefore also hold if the sign of material implication is replaced uniformly by the sign of strict implication, i.e. as a result of such transformation it should lead to a formula provable in the system of Principia Mathematica. We have already showed that formula (a’)is not provable in the system of material implication. We conclude therefore that formula (b’) is not provable in the system of strict implication. We may therefore say that formula ( 5 ’ ) may be called incorrect if the system of strict implication is used as metascience. It seems appropriate to add a remark concerning the inference schemes of the second group. Assume that in each of these formulae the third sign of implication were replaced by the functor “et”. As result of this operation we obtain from formula ( 5 ‘ ) the following formula : (5”)
igitur et cpcqr et qNrNp
SECOND CLASS O F INFERENCE SCHEB5S
41
The formulae, which result from the application of this operation, may be said to be correct if the system of Principia Mathematica is used as metascience. If the formula ( 5 ” ) is translated into the language of Principia Mathematica, we obtain the formula: (a7 p . 3 . q 3 r : q . ~ r : 3 . ~ p
As is easily seen, this formula differs from formula (a’)only in that the sign of conjunction has replaced the third occurrence of the sign of implication. The formula (a”)can obviously be proved; the proof depends on the fact that the expression “q . N r” is equivalent with the negation of the expression
“q 3 r” If formula ( 5 ” )is translated into the system of strict implication, we obtain the following formula:
p . < . q < r : q - r :. < . - p It should be noticed that formula (b”) cannot be proved analogously to formula (a”); the expression (b”)
‘(q
N
r’7
cannot be substituted for the negation of the expression
<‘ q < 7” That this substitution is indeed not permissible can be shown as follows: According to the explicit statement of Lewis the formula
- ( P N d - (.P
-((P
N
r”
42
ANALYSIS OF “DE SYLLOGISMO HYPOTFLETICO”
for the negation of the expression
“q r” is not permissible. We have however only shown that formula (b”)cannot be proved analogously to formula (a”); we have not proved that it cannot be proved at all. Since we do not want to investigate this point at length, we cannot determine whether formula (5”) is correct if the system of strict implication is used as metascience. THE THIRD CLASS OF PROPOSITIONAL FORMS There are eight elements of the third class of propositional forms. We believe we can here limit ourselves to stating that propositional form which occurs first in lexicographic order, namely § 26.
(13)
ccPqr (Cf. Pr. I, p. 708, note 158).
Here too there are added in “de syllogism0 hypothetico” detailed explanations which we will not consider since they do not belong t o the theory (Cf. Boe., p. 621-622).
5
THE THIRD CLASS OF INFERENCE SCHEMES To each of the eight propositional forms of the third class there correspond two inference schemes ; the sixteen inference schemes correlated with the propositional forms of the third class form the third class of inference schemes. The inference schemes of the third class can be divided into a first and second group in an analogous fashion to the way we have divided those of the first and second classes. We will present only the first inference scheme of the first group and the first inference scheme of the second group: 26.
(13)
igitur et ccpqrcpqr
(13’)
igitur et ccpqriVrepNq (Cf. Pr. I, p. 708, note 158).
The relation between the inference schemes and propositional
FOURTH CLASS OB’ PROPOSITIONAL FORMS
43
forms are again determined by the regular rule of correlation. The inference schemes of the first group are correct; the inference schemes of the second group however are correct only if the system of material implication is used as metascienee l). If, in inference scheme (13’), the third sign of implication is replaced by the functor “et”, we obtain the following inference scheme : (13”)
igitur et ccpqrNr et pNq
Inference scheme (13”) is correct if the system of material implication is used as metascience. These assertions can be proved analogously to the corresponding assertions of 0 24.
4 27.
THE ITOURTHCLASS
OF PROPOSITIONAL FORMS
Boethius calls the eight propositional forms that constitute the fourth class “sentences of the first figure” (Cf. Boe., p. 623). They are as follows:
et cpqcqr et cpqcqNr et cpNqcNqr et cpNqcNqNr et cNpqcqr et cNpqcqNr et cNpNqcNqr et cNpNqcNqNr (Cf. Boe., p. 623). We note the following fact. Starting with propositional form (21), it is possible to construct the class of propositional forms which contains not only eight, but sixteen, elements. In constructing the l) That formula (13‘) is to be regarded as correct in the above mentioned case was called to my attention by M r . Norman M. Martin, the translator of this treatise.
44
ANALYSIS OF “DE SYLLOGISMO
HYPOTHETICO”
propositional forms, Boethius restricted himself to those cases where the propositional variable “q” which appears twice in every propositional form, is either negated at both occurrences or unnegated a t both occurrences. If we do not obey this restriction and therefore regard it as possible that in the same propositional form a propositional variable may occur negated at one place and unnegated a t an other, we obtain eight additional propositional forms. Boethius did not consider these eight propositional forms, since, it seems he believed that no inference schemes could be correlated with them. § 28.
THE FOURTH
CLASS OF INFERENCE SCHEMES
Two inference schemes are correlated with each of the propositional forms of the fourth class; these inference schemes form the fourth class of inference schemes. As previously, we divide the inference schemes of the fourth class into two groups, and indicate the first elements of the first and second groups respectively; they are as follows: (31)
igitur et et cpqcqrpr
(21’)
igitur et et cpqcqrNrNp (Cf. Pr., p. 710, note 159)
We will analyse inference scheme (21). The first argument of the functor “igitur” in inference scheme (21) is the expression (a)
66
et et cpqcqrp”,
and the second argument is the expression (b)
66
97
r .
The first argument of that functor “et” which occurs immediately at the beginning of (a) is the expression (0)
“ e t cpqcqr”,
and the second argument is the expression
(d)
?”-
FOURTH CLASS OF INFERENCE SCHEMES
45
The first argument of the functor “et” which occurs immediately at the beginning of (c) is the expression (e)
c‘cpq7y ,
and the second argument is the expression <<
cqr”. (f) The next step reduces expressions (e) and (f) to the simplest expressions i.e. the arguments of the functors are propositional variables. We have thus analysed inference scheme (21). The relation between the propositional forms and the inference schemes is, as in the previous cases, determined by the regular rule of correlation. If formulae (21) and (21‘) are translated into the language of Principia Mathematica, we obtain the following formulae :
(69
p 3 q . q 3 r : p : 3 .r
(g‘)
p3q.q3r:-r:3.-p
We consider formulae (g) and (g’) provable. From this we conclude that both the inference schemes of the first and those of the second group are correct if the system of material implication is used as nietascience. If inference schemes (21) and (21’) are translated into the system of strict implication, the following formulae are obtained : (h)
(h’1
p < q . q < r : p :. < . r p < q . q < r : - r :. < . - p
We note that formulae (h) and (h‘) cannot be proved in the system of strict implication analogously to the way formulae ( 8 ) and (g’) are proved in the system of Principia Mathematica. The proof in the system of Principia Mathematica depends on the formula known as the principle of importation (Cf. P.M. I, p. 110). If we consider this principle translated into the language of the system of strict implication, it appears as follows :
p < . q
< :pq.< . r
46
ANALYSIS OF “DE SYLLOGISMO HYPOTHETICO”
a s far as we can see this formula does not appear in the book of Lewis and Langford; it is especially to be noted that formula 14.26 (Cf. L.a.L., p. 138) is not identical with our formula. We therefore are unable to decide whether formulae (h) and (h’) are provable; consequently we are unable to decide whether the inference schemes of the first and second groups are correct if the system of strict implication is used as metascience. It seems useful to consider a class of inference schemes similar to the inference schemes just treated, although not identical with them; let us call them the coordinate of the fourth class of inference schemes. We will divide the coordinate class into two groups and present the first elements of each group, namely, the following inference schemes : (21 a)
i g i t u r et cpqcqrcpr
(21 a’)
igitur et cpqcqrcNrNp
We note that the first argument of the functor “igitur” in inference scheme (21 a) is the expression “et cpqcqr”
and the second argument is the expression cpr”. (<
It is obvious that inference schemes (21) and (21 a) are similar although not identical and the same holds for inference schemes (21’) and (21 a’). I n order to correlate the coordinate class with the propositional forms of the fourth class, a new rule of correlation must be constructed; we may obtain this rule by changing the regular rule of correlation in one respect. According to the regular rule of correlation, inference scheme x is correlated with the propositional form whose first sign is identical with the third sign of the inference scheme (Cf. supra, $ 18); by the new rule an inference scheme x will be correlated with the propositional form whose first sign agrees with the second sign of inference scheme x. We will, in the following, refer to this rule as the modified rule of correlation.
FOURTH CLASS O F INFERENCE SCHEh5ZS
47
It can be easily seen that inference schemes (21 a) and (21 a’) are correlated with propositional form (21) by the modified rule of correlation. It can be shown that the inference schemes of the first and second groups of the coordinate class are correct both if the system of Principia Mathematica and if the system of strict implication is used as metascience. If inference scheme (21 a) is translated into the language of Principia Mathematica the following formula is obtained :
p 3 q . q 3 r . 3 .p 3 r ; this is formula 3.33, which is proved in Principia Mathematica (Cf. P.M. I, p. 112). If the same inference scheme is translated into the language of the system of strict implication, the following formula is obtained :
p
this is formula 11.6 which is introduced as an axiom (Cf. L.a.L., p. 124-125). I n investigating the logic of the older Peripatetics we noted the fact that Theophrastus coined the concept which designated the class of pure hypothetical syllogisms; according to the testimony of Alexander of Aphrodisias he introduced the expression “syllogisms by analogy” for it (Cf. supra, 5 6). It can be shown that the inference schemes that belong to the coordinate class of the fourth class should be included among these syllogisms. The distinguishing property of pure hypothetical syllogisms is that both premisses and the conclusion are conditional sentences, i.e. implications (Cf. supra, $ 6 ); it is easy to see that the inference schemes which belong to the subclass possess this property. I n the commentaries of Alexander of Aphrodisias and Ioannes Philoponos we find inference schemes presented which are regarded as pure hypothetical syllogisms; it is to be assumed that these inference schemes were constructed by Theophrastus (Cf. CAG I1 1, 326, 20-328,7; CAG XI11 2, 302, 6-23).
48
ANALYSIS OF “DE SYLLOGISMO HYPOTHETICO”
We find, in particular, in the commentary of Alexander of Aphrodisias the following formula :
~i
td
A,
td
B, E;
td
B, td I’,~i ciea td A, td r.
(Cf. CAG I1 1, 326, 22-23). I n order to interprete this formula and to translate it into our language we note the following: a) The three expressions “td A”, “ t d B” and “td P’ are the first, second and third propositional variables, and correspond to the propositional variables “p”, “p” and “r”. b) The Greek conjunction “E?’ is the sign of implication. c) That two expressions are to be conjoined by a logical conjunction is expressed in this language by placing the expressions next to each other. d) The Greek participle “ciea” means therefore and corresponds to our functor “igitur”. If we consider these four remarks we can easily see that the Greek inference scheme is identical with inference scheme (21 a). We recall, that Boethius a t the beginning of “de syllogismo hypothetico” mentions Theophrastus and thereby indicates that the theory presented here is to be associated with that of Theophrastus (Cf. supra, $ 5 ) . It is therefore to be assumed that Boethius used as source a work in which the inference scheme of Theophrastus, which can be identified with the inference scheme (21 a), was presented. We suspect that Boethius constructed inference scheme (21) by modifying inference scheme (21 a) and that he was not conscious of the difference of the schemes (21) and (21 a).
S
29.
THE FIFTH
CLASS OF PROPOSITIONAL
FORMS
There are sixteen elements in the fifth class of propositional forms; Boethius calls them sentences of the second figure (Cf. Boe., p. 626). Following Boethius’ text, we will speak of the first and second subclasses ; the first subclass consists of those sentences which Boethius calls non-homogenous sentences (propositiones non-
FIFTH CLASS OF PROPOSITIONAL FORMS
49
aequimodi), and the second those which he calls homogenous sentences (propositiones aequimodij (Cf. Boe., p. 626). The following eight propositional forms belong to the first subclass of the first class: (29)
et cpqcNpr et cpqcNpNr et cpNqcNpr et cpNqcNpNr et cNpqcpr et cNpqcpNr et cNpNqcpr et cNpNqcpNr (Cf. Boe., p. 626).
The following eight propositional forms belong to the second subclass of the fifth class: (37)
et cpqcpr
(38)
et cpqcpNr et cpNqcpr et cpNqcpNr et cNpqcNpr et cNpqcNpNr et CNpNqcNpr et CNpNqcNpNr (Cf. Boe., p. 628).
While Boethius used the functor “et” in presenting the propositional forms of the fourth class, he appears to have neglected to do so in the presentation of the propositional form of the fifth class. Interpretation is thereby made more difficult. In the editions the propositional forms of the second subclass of the fifth class are presented in a very inconvenient way. The signs are broken up in
50
ANALYSIS OF “DE SYLLOGISMO HYPOTHETICO”
such a fashion and the punctuation marks are so placed that one gets the impression, that one part of (42) forms a unity together with (41) and another part of (42) forms a unity together with (43);this confusion is obviously due to the fact that the functor “et” is suppressed in the propositional forms. We note that an error occurring in the edition of 1499 is also found in the editions of 1570 and 1860. This fact seems to show that the editors did not understand that part of the text essential to us, namely, the logical formulae. 30.
THE FIFTH
CLASS OF INFERENCE
SCHEMES
Two inference schemes are correlated by Boethius to each of the propositional forms of the first subclass of the fifth class; Boethius believed that no inference schemes can be correlated with the propositional forms of the second subclass (Cf. Boe., p. 626 and p. 629). The inference schemes correlated with the propositional forms of the first subclass form the fifth class of inference schemes. We will, as before, divide the inference schemes of the fifth class into two groups and indicate the first element of the first and the first element of the second group; they are as follows: (29)
igitur et et CpqcNprNqr
(29’)
igitur et et cpqcNprXrq
The relationship between the inference schemes and the propositional forms are determined by the regular rule of correlation. With the respect to correctness, the inference schemes of the fifth class are to be judged analogously to those of the fourth class. As in the fourth class of inference schemes, we may here also construct a coordinate class. We indicate the first elements of the two groups of the coordinate class; they are as follows: (29 a)
igitur et cpqcNprcNqr
(29 a‘)
igitur et cpqcNprcNrq
FIFTH CLASS OB INFERENCE SCHEMES
51
As can easily be seen, inference schemes (29 a ) and (29 a’) are correlated with propositional form (29) by the modified rule of correlation. Inference schemes (29 a) and (29 a’) are pure hypothetical syllogisms; we may therefore assume that they were known to Theophrastus. We note that these inference schemes are referred to in the commentary of Alexander of Aphrodisias (Cf. CAG I1 1, p. 327, 16-20). We can aIso regard this as evidence that Boethius was not conscious of the difference between the inference schemes of a class and the corresponding inference schemes of the coordinate class. Boethius’ presentation depends on the conception that every inference scheme consists of three parts; Boethius calls the first part “propositio” (primary premiss), the second “assumptio” (subsidiary premiss) and the third “conclusio” (conclusion) (Cf. Boe., p. 614 and Pr. I, p. 704). To make this clear we add that in the inference scheme the primary and subsidiary premisses are connected by the functor “et” and the conjuntion of primary and subsidiary premisses is the first argument of the functor “igitur”, while the conclusion is the second argument of this functor. Both inference scheme (29) and (29 a) fall under this general theory; it is to bs noted however that different formulae serve as primary premiss, subsidiary premise and conclusion in the two cases. I n (29) the primary premiss is t.he expression “et cpqcNpr”, the subsidiary premiss is the expression
“W, and the conclusion is the expression L<
r ,,.
On the other hand, in (29 a) the primary premiss is the expression
%pq’,, the subsidiary premiss is the expression
“cNpr”
52
ANALYSIS OF “DE SYLLOGISM0 HYPOTHETICO”
and the conclusion is the expression
“cNqr”. It is obvious that Boetbius intended to construct inference schemes whose primary premiss would be one of the sentential forms belonging to the first subclass of the fifth class of propositional forms. This conception is based on the general characterization of the inference schemes (Cf. Boe., p. 626). Yet where the sixteen inference schemes are listed, a different conception is followed ; the above mentioned propositional forms are no longer the primary premisses of inference schemes, but instead serve as the first arguments of the functor “igitur” (Cf. Pr. I, p. 712, note 160). We may say that in Boethius’ presentation the inference schemes of the coordinate class of the fifth class have inadvertently taken the place of the corresponding inference schemes of the fifth class. We will now consider a problem whose treatment we have delayed till now. We have already remarked that in Boethius’ opinion no inference schemes can be correlated to the propositional forms of the second subclass. We now have the alternative: either there is in fact no possibility of correlating inference schemes t o those propositional forms or Boethius failed to see the possibility of doing so. We can show that if the system of Principia Mathematica is used as metascience, this problem is decidible. The following inference scheme is correlated with propositional form (37) by the regular rule of correlation:
igitur et et cpqcprp et qr If this inference scheme is translated into the language of Principia Mathematica the following inference scheme is obtained : p3q.p3y:p:3.q.r This formula is provable in the system of Principia Mathematica; it can be derived from formula 3.43 (Cf. P.M. I, p. 113) by the use of the principle of importation. It is thus shown that propositional form ( 3 7 ) may be correlated with a correct inference scheme. We may now say that Boethius failed to see the possibility of
SIXTH CLASS O F PROPOSITIONAL FORMS
53
correlating inference schemes with the propositional forms of the second subclass; modern logic shows that there is such a possibility.
5
THE SIXTH CLASS OF PROPOSITIONAL FORMS There are again sixteen elements of the sixth class of propositional forms; in agreement with Boethius, they may be termed sentences of the third figure (Cf. Boe., p. 629). We again assign to the first subclass those propositional forms that Boethius called non-homogenous and to the second subclass those that are called homogenous sentences (Cf. Pr. I, p. 714, note 161). The following eight propositional forms belong to the first subclass of the sixth class: 31.
(45)
et cqpcrNp et cqpcNrNp et cNqpcrNp et cNqpcNrNp et cqNpcrp et cqNpcNrp et cNqNpcrp et cNqNpcNrNp
(Cf. Boe., p. 629). The following eight propositional forms belong to the second subclass of the sixth class: (53)
et cqpcrp
(54)
et CqpcNrp
(55)
et cNqpcrp
(56)
et cNqpcNrp
(57)
et cqNpcrNp
(58)
et cqNpcNrNp
54
ANALYSIS OF “DE SYLLOGISMO
IIYPOTHETICO”
et cNqNpcrNp
et CNqNpcNrNp (Cf. Boe., p. 631). We note that the principle of lexicographic ordering is not strictly followed within the two subclasses. If we wished to restore the lexicographic ordering in the first subclass we must exchange the positions of propositional forms, (47) and (49) and also of propositional forms (48) and (50). In the original text propositional forms (45), (46), (47) and (48) and propositional forms (49), ( 5 0 ) , (51) and (52) each constitute a group ; within these groups, the order is lexicographic. If we wish to restore lexicographic order in the second subclass, the positions of propositional forms (55) and (57) and of propositional forms ( 5 6 ) and (58) must be exchanged.
Q 32. THE SIXTH
CLASS OF INFERENCE SCHEMES
Here too, according Boethius’ opinion, two inference schemes are correlated with each of the propositional forms of the first subclass, but no inference schemes are to be correlated with the propositional forms of the second subclass. The inference schemes correlated with the propositional forms of the first subclass constitute the sixth class of inference schemes. The theorems concerning the sixth class of inference schemes are analogous to those stated with respect to the fifth class (Cf. supra, 3 30) ; we will therefore restrict ourselves mentioning the following points. The first propositional form of the first subclass is correlated with the following inference schemes : (45) (45’)
igitur et et cqpcrNpqNr igitur et et cqpcrNprNq (Cf. Pr. I, p. 714, note 161).
The following inference schemes belong to the coordinate class : (45 a)
(45 a’)
igitur et cqpcrNpcqNr igitur et CqpcrNpcrNq
SEVENTH CLASS O F PROPOSITIONAL FORMS
55
These inference schemes are correlated with propositional form (45) by the modified rule of correlation. Both inference schemes are
referred to: in the commentary of Alexander of Aphrodisias (Cf. CAG I1 1, p. 327, 7-13). An inference scheme can be correlated to the first propositional form of the second subclass which would be, when translated into the language of Principia Mathematica :
q3p.r3p:qvr:3.p This formula is provable in the system of Principia Mathematica; it can be derived from formula 3.44 (Cf. P.M. I, p. 113) by the use of the principle of importation. But this formula cannot be represented as simply in the language of Boethius as in the language of Principia Mathematica since Boethius’ language has no sign for non-exclusive disjunction.
Q 33. THE SEVENTH
CLASS OF PROPOSITIONAL FORMS
There are sixteen elements in the seventh class of propositional forms. Boethius presents them in lexicographic order. We will restrict ourselves to indicating that element which is the first in this ordering, namely, the following propositional form :
ccpqcrs (Cf. Boe., p. 632). We will indicate how this formula may be analysed: the first argument of the first sign of implication in (61) is the expression LLcp”’, and the second argument is the expression
‘‘crd ’. It can easily be seen that formula (61) is well formed. We note that detailed explanations are added to the presentation of the propositional forms of the seventh class (Cf. Boe., p. 632634) ; we need not consider these explanations for the same reason that we did not consider the explanations added to the presentation of the propositional forms of the second and third classes.
56
3
ANALYSIS OF “DE SYLCOUISMO HYPOTHETICO”
34. THE SEVENTH
CLASS OF INFERENCE
SCHEMES
Two inference schemes are correlated with each of the propositional forms of the seventh class. Consequently, thirty-two inference schemes are correlated with the sixteen propositional forms of the seventh class; these thirty-two inference schemes are the elements of the seventh class of inference schemes. We distinguish in each case a first and a second inference scheme and gather the first and the second inference schemes in two groups. The following two inference schemes are correlated with the first propositional form of the seventh class : (61)
i g i t u r et ccpqcrscpqcrs
(61’)
i g i t u r et ccpqcrscrNscpNq
(Cf. Pr. I, p. 715, note 162). We will indicate how this inference scheme is to be analysed: The first argument of the functor “igitur” in (61) is the expression (a)
‘< et ccpqcrscpq”,
a,nd the second argument is the expression r
(b) the first argument of the functor “et” in (a) is the expression 2
LC
ccpqcrs”, (c) and the second argument is the expression (d)
‘‘cpq”.
Continuation of this analysis will yield no further difficulties. The first argument of the functor cLigitur”in (61’)is the expression (el
“et ccpqcrscrNs”
and the second argument is the expression (f
“cpNq” ;
the first argument of the functor “et” in (e) is the expression (8)
“ccpqcrs”,
EIGHTH CLASS OF PROPOSITIONAL FORMS
57
and the second argument is the expression (h)
“crNs”;
here, too, continuation of the analysis will yield no difficulty. One can easily see in both cases that the formulae are well formed. The relationship between the propositional forms and the inference schemes is here too determined by the regular rule of correlation. The inference schemes of the first group (that to which (61) belongs) are correct; the inference schemes of the second group (that to which (61’) belongs) are incorrect. These theorems may be proved analogously to the corresponding theorems in 24 and in f3 26. We will limit ourselves to the following remarks. If formula (61’) is translated into the language of Principia Mathematica, the following formula is obtained :
p3q.3.r3s :r3-s
:3.p3-q
We consider the case where a true proposition is substituted for “p” and “q” and the false proposition for “r” and “s”. It can easily be seen that in this case an expression will result which takes the truth value “false”. However, such a formula cannot belong to the system of Principia Mathematica. It follows that formula (61‘) is incorrect if the system of Principia Mathematica is used as metascience. If, in formula (Sl’), the fourth and fifth signs of implication are replaced by the functor “et”, the following formula results: igitur et ccpqcrs et rNs et pNq It can be shown that this formula may be called correct if the system of Principia Mathematica is used as metascience. 35. THE EIGHTH CLASS OF PROPOSITIONAL FORMS There are four elements of the eighth class of propositional forms. They are as follows: (77)
am
5s
ANALYSIS OF “DE SYLLOGISMO RYPOTHETICO”
(78)
aPNq
(79)
aNP4
aNPNq We recall that in $ 17 we introduced the sign “a” as a sign of disjunction and that it is, on one hand, the sign of material exclusive disjunction and on the other, the sign of strict exclusive disjunction (Cf. supra). Furthermore, we note that in the original text, the four elements appear in an order different from that of our presentation; (80) follows (77), then (78) and finally (79) (Cf. Boe., p. 636). Boethius did not adhere strictly to the principle of lexicographic order in this case. (80)
8
36.
THE EIGHTH
CLASS OF INFERENCE SCHEMES
Boethius correlates four inference schemes with propositional form (77) and two with each of the other propositional forms; the ten inference schemes correlated with the propositional forms of the eighth class form the eighth class of inference schemes. The following inference schemes are correlated with propositional form (77): (77)
igitur et apqpNq
(77’)
igitur et apqNpq
(77”)
igitur et apqNqp
(77‘‘,)
igitur et apqqNp
(Cf. Boe., p. 636). We will not state the inference schemes correlated with the second, third and fourth propositional forms. The irregularity consisting of the fact that four inference schemes are correlated with the first propositional form and only two to the others is due to the fact that in the first case the disjunction is considered as exclusive, while in the other three cases is considered as non-exclusive. It must be said that our presentation of the four propositional
EI(fHTH OLASS OF INFERENCE SCHEMES
69
forms of the eighth class is not quite correct. The sign “a” is defined as the sign of exclusive disjunction. Strictly speaking, it has this meaning in propositional form (77), while in (78),(79) and (80) it is to be interpreted as the sign of the non-exclusive disjunction. All of the inference schemes which constitute the eighth class are correct. We will limit ourselves to showing this for inference scheme (77). a) If we attempt t o translate formula (77) into the language of Principia Mathematica, we are faced with a diEculty that this language has no sign of exclusive disjunction. This difficulty can be overcome if we notice that the exclusive disjunction whose first member is “p” and whose second member is “q” can be represented by the following expression : IP~P-“P~”4! The non-exclusive disjunction ‘‘p v q” thus becomes the exclusive disjunction when it is oonjuctively connected with the non-exclusive disjunction “ N p v q”. If we translate inference scheme (77) into the language of Principia Mathematica we obtain the following formula :
-
pvq.-pv-q:p:3.-q and this formula is provable in the system of Principia Mathematica. b) If we attempt to translate inference scheme (77) into the language of the system of strict implication, we encounter an analogous difficulty; here too the difllculty is not unsurmountable. The non-exclusive disjunction of p and q can be represented in the language of the system of strict implication as follows:
- -
0 (=P 4) For clarity, we add the remark that this expression means the same as “not p and not q is impossible” or “it is impossible that not I, and not q”. It should be mentioned that in those works of C. I. Lewis in which the older presentation of the system of strict implication may be found, the strict logical sum is defined in this way (Cf. L., p. 293). The strict exclusive disjunction is a conjunction of two strict
60
ANALYSIS OF “DE SYLLOGISMO HYPOTHETICO”
non-exclusive disjunctions and can be stated by the following expression : 0 (- P ” 2 ) 0 (MI This new expression means the same as “it is impossible that not p and not q and it is also impossible that p and q”. The meaning of this determination can be clarified by an example borrowed from Boethius’ text. If we interpret the sentence ‘(Socratesis either ill or well” as a strict exclusive disjunction, we obtain the following meaning : “it is impossible that Socrates is not ill and is not well and it is also impossible that he is ill and well” (Cf. Boe., p. 636). If we translate inference scheme (77) into the language of the system of strict implication, the following formula is obtained:
--
~o(~~”q).”o(Pq):P:.<.”q a.nd this formula is provable in the system of strict implication.
3
37.
THE THEORY
OF MODALITY
Indications of the theory of the modalities can be found in “de syllogismo hypothetico”. This theory may be considered propositional logic since the variables which occur are propositional variables (Cf. supra, 0 1). Boethius points out that there are two sorts of simple or predicative propositions (Propositiones simplices, praedicativae), namely, those that have no modal signs (praeter modum) and those that have modal signs (cum modo). Of the propositions that have modal signs those connected with the concept of necessity or with that of possibility are mentioned as being logically important (Cf. Boe., p. 611). Boethius constructs examples of both types of propositions. As examples of propositions having no modal signs, he states the following: “it is day” ‘(Socrates is a philosopher” ; As examples of propositions having modal signs, we have: “it is necessary that fire is hot”
THE THEORY OF MODALITY
61
“it is possible that the Trojans were beaten by the Greeks” (Cf. Boe., p. 611). It is convenient to introduce a short name for those expressions that Boethius considers as having modal signs and which he distinguishes as being logically important. We will call them “modal expressions”. The essential thing concerning modal expressions is that when they are translated into our metalanguage, they result in expressions in which the modal operator is an element (Cf. supra, 6 17). Boethius constructs six distinct modal expressions ; these six expressions may be represented in our metalanguage as follows :
Since both signs which occur as functors, viz. the sign of negation and the modal operator, are functors which take a single argument, the analysis of these expressions is extraordinarily easy (Cf. supra 5 17) ; it can easily be seen that the six expressionsare well formed. The expressions ( l ) , (2) and (3) constitute the first group, the expressions (a), (5) and (6) the second group. Within these groups (1) and (4) are the first, (2) and ( 5 ) the second and (3) and (6) the third members. Boethius considered the first members of both groups affirmative. The second member of the first group he terms the necessary negation (necessarianegatio), the third the negation of the necessary (necessarii negatio). The second member of the second group he terms the contingent negation (contingens negatio), and the third member the negation of the contingent (contingentis negatio) (Cf. Boe., p. 612).
62
ANALYSIS
OF
“DE
SYLLOGISMO
HYPOTHETICO”
The theorems stated by Boethius with respect to the six modal expressions may be expressed as follows : (a) first and third members of both groups cannot both be false nor both be true; (b) the first and second members of the first group can both be false; (c) the first and second members of the second group can both be true (Cf. Boe., p. 612).
It is apparent that we can supplement this system of three theorems by the following two theorems: (d) the first and second members of the first group cannot both be true; (e) the first and second members of the second group cannot both be false. Boethius did not explicitely state theorems (d) and (e); they should however be regarded as belonging to his theory. Boethius calls the first and second members of the first group contraries (contrariae), indicating that these sentences cannot both be true. He does not say that the first and second members of the second group are contraries since that they both can be true; by analogy, we obtain the result that they cannot both be false (Cf. Boe., p. 612). These indications of the theory of the modalities show important points of agreement with the modern logic of the modalities. In his treatise, “Zur Logik der Modalitaten”, Oskar Becker showed that it is possible to construct a calculus of modalities in which there are exactly six basic or irreducible modalities (Cf. Be., p. 507 and 505). I n our metalanguage these six modalities may be expressed as follows: (7)
P
(8)
NP
(9)
NOP “OP
(10)
THE THEORY OF MODALITY
63
NN 0 Np (Cf. Be., p. 510). I n order to compare the system of Boethius with this modern system, we must recall that in addition to propositions with modal signs Boethius recognizes propositions without modal signs also. Boethius notes that propositions without modal signs are affirmative or negative (Cf. Boe., p. 612). We are therefore justified in saying that Boethius recognized the two basic modalities represented by expressions (7) and (8). We can establish the following: (9) is identical with (6), (11) is identical with (l), (12) is identical with (3). Further identifications can be made by using the principle of the eliminibility of double negations. Under this condition we have: (9) is identical with (2), (10) is identical with (4), (12) is identical with ( 5 ) . I n summary, we note: none of the six basic modalities fail to appear in Boethius’ system. Finally, we will show that the theorems stated by Boethius in this connection and presented by us above are represented in the modern presentation of the logic of modalities, i.e. they are correct. We shall however not attempt to prove each of the theorems represented; instead we will select those theorems for which it is obvious that a counterpart exists in the modern presentation. According to theorem (a), the expressions
“ N 0N,” and
“ N N O Np” cannot both be false.
64
ANALYSIS OF “DE SYLLOGISMO HYPOTHETICO”
This corresponds to the following theorem of the modern presentation : “ p is either necessary or possibly false (not necessary)” (Cf. Be., p. 511). By theorem (a), the expressions
“0P>’ and
“ N 0p” cannot both the false. This corresponds to the following theorem of the modern presentation : “ p i s either possible (not impossible) or impossible” (Cf. Be., p. 511). By theorem (b), the expressions
“ N 0Np” and
“ N 0N N p ” can both be false. This corresponds to the following remark concerning the modern presentation : ”it is obviously not the case that ( ‘ p is either necessary or impossible” ” (Cf. Be., p. 511). In the case just considered it is not easy to recognize that the modern theorem says the same thing as Boethius’ theorem; it is however in this case so important that we cannot fail to consider it. To make this clear, we note the following. The expression “ N 0Ny” can be read: “not p is not possible”; and this is synonymous with: ‘ p is necessary”. The expression “ N 0NNy” can be read: “not not p is not possible” ; and this means : “ p is impossible”. On the basis of this interpretation it is possible to state Boethius’ theorem in the following form: “the expressions “ p is necessary” and “ p is impossible” can both be false”.
THE THEORY OB MODALITY
65
This theorem means the same as the theorem: “it is not the case that : “ p is either necessary or impossible” ”. It is thus shown that Boethius’ theorem is synonymous with the modern theorem which we have regarded as its counterpart. We have thus shown that there is in fact considerable agreement between those theorems of Boethius represented by us and the results of the modern logic of modalities.
V ANALYSIS OF A SECTION OF THE BOETHIUS’ COMMENTARY ON CICERO’S TOPICS $ 38. THE THREE
ENUMERATIONS O F THE
SEVEN
CONDITIONAL
SYLLOGISMS
We now turn to the consideration of the form of propositional logic to be found in Boethius’ commentary on Cicero’s Topics. A t the beginning of the fifth book of this commentary, Boethius notes that he has treated all the hypothetical syllogisms in an other book; he obviously has “de syllogismo hypothetico” in mind (Cf. Boe., p. 823). The exposition which follows this remark covers more than the first half of the fifth book of the commentary; it constitutes that part of the commentary that is of interest to us here (Cf. supra, $ 1). I n order to determine this section more precisely one can best indicate its beginning and its end. It begins with the words “de omnibus quidem hypotheticis syllogismis” (Cf. Boe., p. 823) and continues to the place immediately preceding the following words of Cicero, “proximus est locus” (Cf. Boe., p. 934). Boethius notes that Cicero mentioned some modi (inference types). From the exposition that follows, it is to be assumed, that Boethius identifies the modi that Cicero mentioned with the system of the seven conditional syllogisms (Cf. Boe., p. 823). By conditional syllogisms we understand inference schemes. At the place which Boethius has in mind, Cicero enumerates seven inference schemes. Boethius quotes this place in the fifth book of his commentary (Cf. Boe., p. 817). We will call the quotation of this place from Cicero’s Topics in Boethius’ commentary “the quotation”. In the text of the commentary as given by the editions we find the seven conditional syllogisms enumerated three times. The
FIRST GROUP OF CONDITIONAL SYLLOGISMS
67
first and the second enumerations precede the quotation, while the third follows it (Cf. Boe., p. 831-833). It may be mentioned that the second enumeration agrees so closely with the first, that it may be called a duplication of the first. Propositional variables are used only in the third enumeration of the seven conditional syllogisms; the system of propositional variables which we called the simple system is used (Cf. supra, § 17). I n all three enumerations each of the conditional syllogisms is illustrated by an example. These examples are expressions related to the inference schemes; like the inference schemes, they contain functors and always contain a sign which can be identified with the functor “igitar” ; they contain however no propositional variables, instead having simple, i.e. atomic, sentences. The examples of conditional syllogisms which Boethius gives with the first and second enumerations, are extremely simple and the two sequences agree almost completely member for member. We will quote these examples in English; in this translation the English word “therefore” occurs instead of the functor “igitur” It seems desirable to divide the seven conditional syllogisms into four groups; we will divide them in such a way that the first and second modi constitute the first group, the third modus constituhs the second group, the fourth and fifth modi the third group and finally the sixth and seventh modi form the fourth group.
0
39.
THE FIRST
GROUP OF CONDITIONAL SYLLOGISMS
The following inference schemes are the two elements of the first group. (1)
igitur et cpqpq
(1’)
igitur et cpqNqNp
These two inference schemes are presented by Boethius in the third enumeration (Cf. Boe., p. 831-832). The following sentence is given as an example of (1): “if it is day, it is light, it is day, therefore it is light”;
68
ANALYSIS OF A SECTION OF THE “COIKNENTARY”
and the following sentence is given as example of (1’): “if it is day, it is light, but it is not light, therefore it is not day” (Cf. Boe., p. 825 and p. 827).
It is undeniable that inference schemes (1) and (1’) are identical with inference schemes presented in “de syllogismo hypothetico”, viz. the inference schemes of the first class with the same numbering (Cf. supra, 3 2 2 ) . $ 40. THE SECOND GROUP O F CONDITIONAL SYLLOGISMS The following four inference schemes constitute the second group : (1)
igitur et NcpNqpq
(2)
igitur et NcNpqNpNq igitur et NcNpNqNpq igitur et NcpqpNq
Inference scheme (1) may be analyzed in the following way. The first argument of the functor “igitur” in (1)is the expression (a)
“et NcpNqp”,
the second argument is the expression c r >>.
(b) qY the first argument of the functor “et” in (a) is the expression (c)
“NcpNq”
the second argument is the expression
the argument of the first sign of negation in (c) is the expression the first argument of the sign of implication in (e) is the expression
69
SECOND G R O U P OF CONDITIONAL SYLLOGISMS
the second argument is the expression
“Nq”; (g) the argument of the sign of negation in (g) is the expression
“q”. This analysis shows that inference scheme (1) is well formed. This can be shown by analogous considerations in the case of the other three inference schemes. There is no presentation of the inference schemes using propositional variables in Boethius’ text. We do, however, find sentences which may be regarded as examples of these inference schemes. We quote these examples in English and with an order and numbering that make it easy to see of which inference scheme a sentence is an example. (h)
I. “It is not the case that if it is day it is not light, but it is day, therefore it is light”. 11. “It is not the case that if it is not light, it is day, but it is not light, therefore it is not day”.
111. “It is not the case that if it is not day, it is not night, but it is not day, therefore it is night”.
IV. “It is not the case that if he is awake, he snores, but he is awake, therefore he does not snore”. (Cf. Boe., p. 825-826
and p. 827).
We note in passing that the fourth example is not strictly speaking on a part with the other three. It therefore seems permissible to replace this example with the following: IV‘.
“It is not the case that if it is day, it is night, but it is day, therefore it is not night”.
70
ANALYSIS OF A SECTION OF THE “COMMENTARY”
The inference schemes of this group are original with Boethius and not simply drawn from the explicated text, i.e. not drawn from Cicero’s Topics. Boethius was stimulated to construct these inference schemes by Cicero’s text ; he modified the inference types of Cicero and did so consciously. Boethius notes that he introduced the conjunction “si”, i.e. the sign of implication, in the type of inference (“nos idcirco causalem conjunctionem apposuimus eam, quae est si” Boe., p. 830); and we can make this more precise by saying Boethius substituted the sign of implication for the functor “et” in Cicero’s types of inference. Such a transformation appears possible from a syntactic point of view since the sign “et” and the sign of implication are similar functors, i.e. they both take two arguments. It is of interest t o investigate how these types of inference are to be regarded from the standpoint of modern logic. We will limit ourselves to testing formula ( 1 ) ) since the exposition may be carried out analogously for the other inference schemes. a) If (1) is translated into the language of Principia Mathematica, the following formula is obtained : (i)
- -
(PI 4 ) - P - I* 4
By permissible transformations, we can obtain :
(k)
P.P-3.P and formula (k) is provable since it is identical with formula 3.27 (Cf. P.M. I, p. 112). This shows that inference scheme (1) is correct. We note here the following fact. If inference scheme (1) is interpreted in the way mentioned, it may be said to contain a superfluous expression or, more precisely, instcad of the expression
“et NcpNqp”
we could substitute the simpler expression
“NcpNq”.
If one considers the example instead of the inference scheme we can say: instead of the expression
THIRD G RO UP OF CONDITIONAL SYLLOGISMS
71
“it is not the case that if it is day, it is not light, but it is day” we could have the simpler expression “it is not the case that if it is day, it is not light”. We will indicate with respect to the example why the simpler expression could be substituted for the more complex. The sentence “it is day” can be derived from the sentence “it is not the case that if it is day, it is not light”; therefore the conjunction of the two sentences does not say more than the first sentence alone. b) If inference scheme (1) is translated into the language of the system of strict implication, the following formula is obtained:
-
. p : < .q We suspect that formula (1) is not provable. To support this, we note : the expression ‘“ ( p < q)” (1)
( p <-q)
-
is synonymous with: “it is possible that both p and q ” ; we cannot however derive “q” from “it is possible that both p and q”; nor can “q” be derived from “it is possible that both p and q” and “p”. We can thus conclude that if the implication sign in inference scheme (1) is interpreted as a sign of strict implication, the resulting formula is incorrect. It is recalled that the logic of St. Gallen borrowed the third inference type from Boethius’ book (Cf. supra, § 11). This shows that it is not easy to see that there is something dubious in this inference type.
THE THIRD GROUP OF CONDITIONAL SYLLOGISMS The following inference schemes belong to the third group:
§ 41.
igitur et apqpNq igitur et apqNpq
(Cf. Boe., p. 833).
72
ANALYSIS OF A SECTION OF “COMMENTARY”
The following sentences serve as examples : I. “it is day or it is night, but it is day, therefore, it is not night”. 11. “It is day or it is night, but it is not day, therefore it is night” (Cf. Boe., p. 826, p. 827 and p. 833). Inference schemes (1) and (2) are identical with inference schemes of the eighth class, viz. (1) with (77) and (2) with (77’) (Cf. supra,
9
36).
$ 42. THE FOURTH
GROUP OF CONDITIONAL SYLLOGISMS
The following inference schemes belong to the fourth group: (1)
igitur et N et pqpiVq igitur et N et pqNpq (Cf. Boe., p. 833).
The following sentences serve as examples : I. “It is not the case that it is day and night, but it is night, therefore it is not day”. 11. “It is not the case that it is night and day, but it is not night, therefore it is day”. (Cf. Boe., p. 833). Whether the system of material implication or that of strict implication is used as metascience, inference scheme (1) is correct, but inference scheme (2) is incorrect 2). We note that the sentences which serve as examples do not conflict with interpreting the inference schemes in terms of the 1) That formula (1) is correct if the system of strict implication is used as metascience has been detected by the translator.
SOLUTION O F THE LAST DIFFICULTY
73
system of strict implication. Thus we may say that sentence I is synonymous with : “it is not possible that it is both day and night, but it is night, therefore it is not day”; and with this interpretation sentence I may indeed be called an example of inference scheme (1).
$ 43. SOLUTION OF THE LAST DIFFICULTY Boethius assumes that there are exactly seven conditional syllogisms. I n our presentation ten inference schemes occurred, since the first group contains two, the second group four and the third and fourth group each two inference schemes. We may solve the apparent contradiction by noting that Boethius obviously considers only the first of the four elements of the second group a conditional syllogism. We have thus come to the end of our presentation of Boethius’ propositional logic.
APPENDIX BY NORMAN M. MARTIN
1.
9
24
Formula (b’) is indeed unprovable in the system of strict implication, as is indicated. If one uses the following four-valued matrices (due to Dr William T. Parry)
P
r 1 2 3 4
1 2 3 1 2 3 2 2 4 3 4 3 4 4 4
4 4 4 4 4
-pop 4 3 2
1 1 1
1
3
P
75
APPENDIX
- --
By the principles of adjunction and inference, we obtain:
“0 (f) which is equivalent to
(P“ P )
-
(9) (P
p.q“r:<:po.qowT
th)
. - .- - . -
This is equivalent by definition 11.02 to
0[p q
ti)
r :
( p0 .q 0
-
r)J
“ p . q r : . ( p 0 q 0 r)” is by 12.5 equivalent to “ - ( P O .qO - r ) : q - r : . p 7 ’ ; hence by the principle of substitution of equivalents (L.a.L., p. 125), (i) is equivalent to
- -
( p 0 . q 0 r ) :q r : . < p which by definition 17.01 and the elimination of double negations yields (b”). (j)
N
N
Q 26
2.
If (1 3‘) is translated into the language of Principia Mathematica, it yields : (a’)
p3q.3r :-r :.3.p3-q
Substituting “ p 3 q” for “p” and “r’, for “q” in 2.16, we get: (b’)
p3qQ3r:3:.-r3::(p3q)
By the principle of importation we get tc’)
p3qp.3r::r:.3.-(p3q)
By 2.51 and the principle of the syllogism, we obtain (a’): Translating (13’) into the system of strict implication, we obtain : (d’)
p
<.p
<-q
76
APPENDIX
which is not provable since it takes the value 4 if p takes 1, q 2 and r 4 (Cf. note I). If “ p < gy’ is substituted for “r” in (d’), we get
(4
p
:.<.p<-q
which is inconsistent with Lewis’ system since by use of 12.1, the principle D (L.a.L., p. 182), and 20.01 we can obtain
(3%d [- ( P <
(f’)
-
!I)P
<
-
21
which contradicts 12.9. If ( 13ii) is translated into the language of Principia Mathematica we get: :. 3 . p . - q
p 3 4 . 3 r :-r
(a”)
Since “ p 3 q” is equivalent to (b”)
N
(p.
-
( p . N q)” this is equivalent to
“-
q ) :3 r : .
N
r : .3. p
.
N
q
which is consequence of the correct inference scheme (1’). If (13”) is translated into the system of strict implication we obtain :
p < q . < r : - r :. < p - . g
(c”)
By the use of the matrices mentioned in note 1, if p takes 1, q 2 and r 4, (c”) will take the value 4, and hence is not provable. If we substitute “ N p” for “q” and “ p < p” for 3’’ we get (d”) p
< - p . < .p <
- p :
-
(p <-p)
: . < .p
--
P
This is equivalent to
p<-p.<.p
(e’0
but this together with 20.01 leads to the contradictory result
(3P?2) (P
(f”) 3.
Q
-
-
4 :P * 4 )
2s
(h) and (h‘) are provable in the system of strict implication as
77
APPENDIX
the author seems to suggest. The proof depends on the rearrangement of the terms of the antecedent by 12.5, from the equivalent form
p . p < q : q < r :. < r
(i)
which is provable by the use of 19.6, 11.6 and 11.7. From (h), we can proof (h’) by theorem 12.6. f 30
4.
If (29) is translated into the language of the system of strict implication, we obtain :
(4
p
:-q:.
This is provable with the use of 11.6, substituting “p” and “ N p” for “q” giving (b)
-q
<- p . - p
“-
q” for
One can derive (a) from (b) by the use of 12.43 and 19.6. (b) is an instance of (21) (Cf. note 3). (29’) can be derived from (29) by 12.6. If the inference scheme which can be correlated with propositional form (37) is translated into the language of the system of strict implication, we obtain :
p < q . p < r : p :. < q r This is equivalent to
p.p
75
6.
APPENDIX
$ 34
If (61’) is translated into the language of the system of strict implication, we obtain
p < q . <.r <s : r < - s :
<.p <-q
If p and q take 1 and r and s take 4, this formula takes the value 4 and hence is unprovable (Cf. note 1).
-
If we substitute ( p p)” for both “p” and “q” and ‘ L pN p” for both “8’and “s”, we obtain: “ N
-(p-p).<.-(p-p):<:p-p.<.p-p:.p-y .<.-(p-p)::<:-(p-p).<.p-p
If we substitute “ p
-
p” for both “p” and “q” in 12.41, we get:
”(p-p).<.-(pNp):<
:p-p.<.p-p
By the use of 12.1, 11.02, and 18.12, we obtain: p-1). <--((P-P) By the principles of adjunction and inference and 12.9, we get:
P-P which is contradictory. 7.
p
37
79
APPENDIX
By the use of the principle of equivalence, 12.3 and 12.15, (a 1) can be shown to be equivalent to (a 2), (a 3) to (a 4), (b) to ( c ) and (d) to (e). Substituting “0N p” for “p” in 12.9, we obtain a formula equivalent to (a 1) ; substituting “0p” for “p” in the same theorem, we get an equivalent of (a 3). By the use of 19.77, 20.01 and principle C (L.a.L., p. 182), we can prove an equivalent of (b). Substituting “ N 0p Q p” for “p“ and “ p p” for “q” in 12.44, we obtain: N
-
N
-op“o“p.<.p-p:<:“(p”p).< (-
0P
0 -PI
By use of 18.41, 18.42, 19.68, and 11.1, we obtain -oP-o--P.<.PNp yielding (e) by 12.9, and the principle of inference. 8.
5
40
Pormula (1) is indeed unprovable; if p takes the value 1 and q the value 2, the formula takes the value 4 (Cf. note 1). 9.
$ 42
If the system of strict implication is used as metascience, formula (1) is correct, but (2) is incorrect. Translated, they read :
(1) is equivalent to
p.p3-q:<-q which is an instance of 14.29. If, in (2), p and p take the value 4, (2) takes the value 4 (Cf. note 1).
