Hypothetical Syllogisms Research Articles

О СООТНОШЕНИИ ВИЗАНТИЙСКОЙ И ЗАПАДНОЙ ЛОГИЧЕСКИХ ТРАДИЦИЙ

The article explores a little-known work of Byzantine literature of the 13th century, a textbook on log-ic by Nikephoros Blemmydes, an outstanding figure in a special period in the history of Byzantium, the period of the Nicene Empire, which arose after the conquest of Constantinople by the Latins. Nikephoros Blemmydes (1197—c. 1272) was a philosopher, rhetorician, theologian and teacher. A treatise Epitome logica in Greek is analyzed in the present article in order to identify the sources of this work and its influence on the development of logic in the East and in the West. For a detailed analysis in this article, one of the main sections of scholastic logic is chosen—the doctrine of the categorical syllogism. The presentation of this doctrine by Nikephoros Blemmydes has its peculiarities. First, only three figures with 14, and not 19, correct modes are presented. Second, the presentation is accompanied by Greek lettering and graphic images of the modes of syllogism. In the treatises by Latin authors such a graphic development of the doctrine of categorical syllogism has not been found by us. However, similar patterns were discovered by us in George of Trebizond (1395, Crete—approx. 1473, Rome). George of Trebizond was one of those Greeks who left the de-caying Byzantium and came to the West, enriching it with the culture of the East and becoming the first representatives of the Italian Renaissance. His “Dialectics”, written in 1439–1440, is believed to have become the first textbook on logic in the history of the Renaissance. He wrote it in Latin, but used, as we can see from the text, along with Latin textbooks, mainly Greek manuscripts. Based on the study of the “Logic” of Nikephoros Blemmydes, the conclusion is drawn that the treatise by Nikephoros Blemmydes is least likely to bear traces of Western influence (textbooks by W. Sherwood, Peter of Spain, etc.). It is suggested that the scholastic doctrine of categorical and hypo-thetical syllogisms was developed by peripatetics and Alexandrian neoplatonists, inherited by Byz-antine logicians, and only later underwent its final development in the West.

고대 후기의 가언삼단논법

고대 후기의 가언삼단논법

A arte Luliana e a recepção medieval dos Tópicos de Aristóteles

The aristotelian opposition between demonstrative and probable arguments is reflected in the medieval commentaries that distinguish between dialectical syllogisms, used by the art of persuasion, and the demonstrative syllogisms based on necessary principles, whose knowledge is immediate and per se notum. In a manuscript of Ars demonstrativa, attributed to ramon Llull, there is a remark about the usage of probable arguments, although they are based on necessary principles. about that, Llull added that naming this work as «demonstrative» might sound slightly «scandalous». However, Llull seems to attest a dilution of the distinction between probable arguments and demonstrative principles, which is possibly related to the Boethian reading of the Topics. Boethius defines a list of differentiae which are the terms that link, in the syllogisms, a premise with a necessary conclusion or a general principle. The medieval commentators of the De Topicisdifferentiis realized that these dialectical places are also applicable to hypothetical syllogisms, which are compounded by conditional premises. Medieval masters made lists of differentiae that include terms such as: opposition, likeness, relation, superior and inferior, authority and trassumptio (the Latin term for metaphor). ramon Llull shows an indirect reception of the De topicis differentiis of Boethius in the figure T of his Ars, as well as a specific application of the argumentative places in the theological dialogue with other religions. Thus, the evidence of the faith articles, although they are “first principles” and true for themselves, is not accessible to the intellect unless it is proved by probable arguments.

Alexander of Aphrodisias on syllogistic reasoning

The article deals with ancient ideas on the nature of syllogistics on the example of Empire's official Peripatetic philosopher, Alexander of Aphrodisias. We interpret Alexander's position on the syllogistic form as a theory of constant function. Alexander offers a conjunctive and purely formal understanding of the nature of syllogistic necessity. This approach to the modal properties of assertoric judgments differs from Theophrastus’ ontological position, who believed that modal characteristics of assertoric premises are determined by looking to the state-of-affairs to which they refer. Also, the paper examines Theophrastus’ legacy of hypothetical syllogisms related to Alexander. Stoic and Peripatetic versions are also compared against the background of Alexander's logical amalgamation. The article elucidates late “Peripatetic conservatism” regarding the hypothetical syllogistics.

아리스토텔레스의 삼단논법과 보에티우스의 삼단논법

In this paper, we discuss the syllogisms from both fronts : Aristotle and Boethius. We mainly focus on the differences with respect to categorical and hypothetical syllogisms in Aristotle and Boethius. Regarding Aristotle’s works on logic, it is not unusual to claim that Aristotle extensively worked on categorical syllogisms. In Prior Analytics, Aristotle gave proofs for many valid moods. However we cannot find a similar treatment for hypothetical syllogism in his works. Thus, it might be a reason for his students to improve Aristotle’s theory of syllogisms by filling the gaps in hypothetical syllogisms. As Boethius already emphasized in the Book I of De Hypotheticis Syllogismis, the Peripatetic School only initiated the start. The remaining and detailed treatise was due to Boethius himself. The impression we got from Boethius’s and Aristotle’s works on syllogisms is that the categorical syllogism is almost entirely developed by Aristotle himself, whereas the extensive and the detailed study of the hypothetical syllogism was carried out by Boethius, although he was not the first figure in the history who gave concise expositions on hypothetical syllogism. Boethius’s translations of syllogisms in medieval era were the primer resource and textbook for Aristotelian logic during the early Medieval ages. Therefore, the scholars of those times had no other chance but to work from Boethius’s commentaries. So, the evident properties of Aristotelian figures were not then visible. Boethius wrote several textbooks in logic. A detailed investigation might yield significant information on Boethius’s logic compared with the influence of Aristotle on Boethius.

Lexique historique de la langue scientifique arabe ed. by Roshdi Rashed

Reviewed by: Lexique historique de la langue scientifique arabe ed. by Roshdi Rashed Thérèse-Anne Druart Roshdi Rashed, editor. Lexique historique de la langue scientifique arabe. Foreword by Abdullah Bin Mohammed Al Salmi. Hildesheim: Georg-Olms-Verlag, 2017. Pp. liv + 971. Cloth, €98.00. This beautifully laid out dictionary will be very useful to anyone seriously interested in Arabic philosophy. Philosophers in Islamic lands often wrote classifications of the sciences or used examples taken from various scientific disciplines, particularly mathematics, that can be somewhat puzzling for scholars in philosophy. For instance, in his autobiography, Avicenna tells us that his father sent him to a vegetable seller to study Indian calculation. I always wondered what exactly Indian calculation was and finally found the answer to this puzzle in this book (195–97): it was a system of calculation on a tablet with dirt that allowed for easily deleting intermediate calculations. Besides, considering the close link between mathematics and logic, Roshdi Rashed, who focuses mainly on mathematics and philosophy of mathematics, has asked Ahmad Hasnaoui to collaborate on issues of logic. Hence, some entries, such as that on demonstration (burhân, 54–57), highlight both similarities and differences between the mathematical and philosophical conceptions of demonstration. The book begins with a study in English of the translation of the Greek heritage and the development of scientific language in Arabic, followed by a study in French of the constitution of the Arabic mathematical language. The entries that follow are ordered according to the Arabic roots, but people who do not read Arabic can find what they are looking for by consulting the index of terms, based on French. For instance, any philosopher interested in knowing more about the conception of the syllogism could find the appropriate entry (697–702) and then, if he or she looks to the following pages, would discover that the very same Arabic term (qiyâs) has different technical meanings in mathematics and pharmacology. Entries provide the Greek background in function of the texts that were translated from Greek into Arabic; and they then often provide quotations from various medieval authors in both Arabic and French. The entries end with a list of correlated terms in both French and Arabic. Besides, any "Arabist" puzzled by a technical passage in a specific text may well be lucky and find a translation of it with an explanation of the technical terms if he or she consults the index of treatises. Up to fairly recently, logic has been a much neglected field in Arabic philosophy. One is, therefore, very pleased to find that the entry for 'syllogism' divides into 1. definition and delimitation; 2. categorical syllogistic; and 3. hypothetical syllogistic—thus presenting a synthetic approach. But the entry does not neglect the historical evolution and shows how, for instance, Avicenna thought Aristotle's definition of the syllogism could also apply to hypothetical syllogisms, whereas Averroes did not think so. The same entry explains that Aristotle had only three figures of the syllogism while Avicenna and Averroes discussed the possibility of a fourth figure, which Averroes attributes to Galen, though rejecting it as non-natural. Yet, it indicates that, from the twelfth century on, several logicians will adopt this fourth figure attributed to Galen. The amount of information gathered in this book is very impressive. As Rashed rightly points out, the best medieval Arabic dictionaries did not pay much attention to scientific vocabulary and the scientific meanings of a term often differ markedly from the ordinary meaning(s). Therefore, the painstaking work of determining how mathematicians, astronomers, logicians, and the rest defined their own technical terms, renders a great service to readers of medieval texts. Rashed focuses on the transfer from Greek into Arabic, but a similar work needs to be done about the transfer from Arabic into Latin. [End Page 174] Thérèse-Anne Druart The Catholic University of America Copyright © 2018 Journal of the History of Philosophy, Inc.

Propositional syntax and semantics induced knowledge re-structuring in a fuzzy logic network for ad hoc reasoning

Traditional approaches to fuzzy reasoning usually employ Generalized Modus Ponens, Generalized Modus Tollens and Generalized Hypothetical Syllogisms to derive fuzzy inferences from a given set of fuzzy observations/facts and fuzzy if-then rules. However, the above approaches occasionally fail to generate inferences, although semantically plausible fuzzy inferences follow as a natural consequence of the given set of rules and fuzzy facts/observations. This paper serves to fill this void. It extends propositional syntax and semantics in the context of fuzzy reasoning to transform/re-structure individual rules to a suitable form (by negating and transposing propositions from antecedent to consequent and vice-versa), so as to get the propositions present in the antecedent (consequent) instantiated (refuted) by positive (negative) facts to enable the rules for forward (backward) firing. Next fuzzy compositional rule of inference is used to perform forward/backward reasoning. Consequently, when the rules are embedded in a fuzzy Petri net like structure, the rules apparently fire randomly in forward or backward direction without maintaining any topological order in transition-firing. Such random firing of rules in distributed parts of the network is referred to as ad hoc reasoning. Here, we transform such randomly ordered forward/backward firing sequence of transitions in the network into a fixed topological order of transition-firing in forward direction only by replacing backward firing transitions into equivalent forward firing transitions. The proposed method of automated reasoning has successfully been applied in diagnostic application of an electronic rectifier circuit, and the results are appealing.

Morality is necessary for happiness

An argument for the eponymous conclusion is given through a series of hypothetical syllogisms, the most basic of which is as follows: morality is necessary for self-respect; self-respect is necessary for happiness; therefore, morality is necessary for happiness. Some of the most obvious objections are entertained and rejected.

Denying Conditionals: Abaelard and the Failure of Boethius' Account of the Hypothetical Syllogism

AbstractBoethius' treatise De Hypotheticis Syllogismis provided twelfth-century philosophers with an introduction to the logic of conditional and disjunctive sentences but this work is the only part of the logica vetus which is no longer studied in the twelfth century. In this paper I investigate why interest in Boethius acount of hypothetical syllogisms fell off so quickly. I argue that Boethius' account of compound sentences is not an account of propositions and once a proper notion of propositionality is available the argument forms accepted by Boethius are seen to be incoherent. It was Peter Abaelard who first understood the nature of propositionality and propositional connectives and used this to criticise Boethius' claims in De Hypothetics Syllogismis. In place Boethius' confusion Abaelard offered a simple and correct account of the hypothetical syllogism.

The Development of Modus Ponens in Antiquity : From Aristotle to the 2nd Century AD

Abstract'Aristotelian logic', as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as 'hypothetical syllogisms'. However, Aristotle did not discuss such arguments, nor did he call any arguments 'hypothetical syllogisms'. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them 'hypothetical syllogisms'; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle's logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called 'hypothetical syllogisms'? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle's dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle's logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories.

A GREEK PARALLEL TO BOETHIUS' DE HYPOTHETICIS SYLLOGISMIS

AbstractIn this paper I present the text, a translation, and a commentary of a long anonymous scholium to Aristotle's Analytics which is a Greek parallel to Boethius' De Hypotheticis Syllogismis, but has so far not been recognized as such. The scholium discusses hypothetical syllogisms of the types modus ponens and modus tollens and hypothetical syllogisms constructed from three conditionals ('wholly hypothetical syllogisms'). It is Peripatetic, and not Stoic, in its theoretical approach as well as its terminology. There are several elements of early Peripatetic hypothetical syllogistic preserved in it, and there is a large number of close parallels to Boethius' De Hypotheticis Syllogismis which we find in no other source. It is very likely that there was a Greek source from which both the scholium and large parts of Boethius' De Hypotheticis Syllogismis are ultimately derived.

Why the order of the figures of the hypothetical syllogisms was changed

In chapter 6 of Alcinous’ Handbook of Platonism we find a discussion of categorical, hypothetical, and mixed syllogisms. Alcinous distinguishes three figures of the hypothetical syllogism, and illustrates each figure with a syllogism based on an argument from Plato. Here he remarks in passing that most people called the second hypothetical figure the third and that some called the third figure the second. We may assume that those who called the third figure the second and those who called the second the third were the same. In a parallel passage, Alexander of Aphrodisias advocates the same ordering of figures of hypothetical syllogisms as Alcinous, and reports that Theophrastus, in the first book of his Analytics, had the second and third figure in reverse order. Combining these passages, we can infer that at the turn of the second century A.D. there existed two different views on the ordering of the figures of the hypothetical syllogisms, of which one goes back to Theophrastus, whereas the other presumably was the result of a later change.

Wholly Hypothetical Syllogisms

AbstractIn antiquity we encounter a distinction of two types of hypothetical syllogisms. One type are the 'mixed hypothetical syllogisms'. The other type is the one to which the present paper is devoted. These arguments went by the name of 'wholly hypothetical syllogisms'. They were thought to make up a self-contained system of valid arguments. Their paradigm case consists of two conditionals as premisses, and a third as conclusion. Their presentation, either schematically or by example, varies in different authors. For instance, we find 'If (it is) A, (it is) B; if (it is) B, (it is) C; therefore, if (it is) A, (it is) C'. The main contentious point about these arguments is what the ancients thought their logical form was. Are A, B, C schematic letters for terms or propositions? Is 'is', where it occurs, predicative, existential, or veridical? That is, should 'Α στι' be translated as 'it is an A', 'A exists', 'As exist' or 'It is true/the case that A'? If A, B, C are term letters, and 'is' is predicative, are the conditionals quanti ed propositions or do they contain designators? If one cannot answer these questions, one can hardly claim to know what sort of arguments the wholly hypothetical syllogisms were. In fact, all the above-mentioned possibilities have been taken to describe them correctly. In this paper I argue that it would be mistaken to assume that in antiquity there was one prevalent understanding of the logical form of these arguments even if the ancients thought they were all talking about the same kind of argument. Rather, there was a complex development in their understanding, starting from a term-logical conception and leading to a propositional-logical one. I trace this development from Aristotle to Philoponus and set out the deductive system on which the logic of the wholly hypothetical syllogisms was grounded.

Cicero, On Invention 1.51–77 Hypothetical Syllogistic and the Early Peripatetics

Abstract: In On Invention, Cicero discusses both induction and deduction. In regard to the latter, Cicero presents a controversy between those who advocate a five-part analysis of deductive reasoning and those who prefer three parts. The issue is not practical or pedagogical, but conceptual in nature. Cicero himself prefers analysis into five parts, and rather confusingly he presents the argument of the advocates of five parts as if it were his own. The argument is striking in that it makes elaborate use of mixed hypothetical syllogisms in order to argue for five parts. Cicero claims that the five-part analysis has been preferred by all who take their start from Aristotle and Theophrastus. A survey of what Theophrastus is reported to have said concerning the hypothetical syllogism renders Cicero's claim intelligible. That is not to say that Theophrastus himself advocated a five-part analysis. Most likely the association with him derives from his known interest in hypothetical syllogistic. Later rhetoricians who identified themselves with the Peripatos made the cormection with the founders of the school, thereby gaining authority for a controversial analysis.

Aristotle and Logical Theory by Jonathan Lear

BOOK REVIEWS 'i.77 learning, at least, to heed the prohibitions that are so necessary, so grounding a feature of any articulated moral system. But Nozick seems uninterested in the interdictory and writes for the most part as though some kind of ideal utilitarianism were true. People are concerned with realizing values and with growth in value, and perhaps with seeing to it that value-seeking and -creation become cosmic activities. I have to quote, though I feel embarrassed to do so, and can scarcely believe that it is written by the author of the book's earlier sections, Nozick's peroration. We can envision a humanistic philosophy, a self-consciously artistic one, sculpting ideas, value, and meaning into new constellations, reverberative with mythic power, lifting and ennobling us by its content and by its creation, leading us to understand and to respond to value and meaning-to experience them and attain them anew (647). This reminds us of something, perhaps a little of Teilhard de Chardin at his most vapid, perhaps of H. G. Wells of the period of God the Invisible King, m013t of all, I fear, of the literary lady in chapter 34 of Martin Chuzzlewit, who saluted the presentation of a Program to a Hominy as follows. " Mind and matter," said the lady in the wig, " glide swift into the vortex of immensity. Howls the sublime, and softly sleeps the calm Ideal, in the whispering chambers of Imagination. To hear, sweet it is. But then, outlaughs the stern philosopher, and saith to the Grotesque, 'What ho! arrest for me that Agency. Go, bring it here! ' And so the vision fadeth." J. M. CAMERON 389 Dundas St. London, Ontario, Canada Aristotle and Logical Theory. By JONATHAN LEAR. Cambridge: Cambridge University Press, 1980. Pp. xi + 123. $17.50. Aristotle is "not only the father of logic, but also the (grand-) father of metalogic "; his logical programme is " one of the triumphs of human thought"· The book is intended to be "an interpretation of that programme " (ix). It consists of six chapters, dealing respectively with "Syllogistic consequence" (1-14), "Completeness and compactness" (15-33), " Hypothetical syllogisms " (34-53), " Invalid inferences" (5475 ), " Invalid proofs " (with " A note on ignorance", 76-97), " Proof by refutation" (98-114). A Bibliography (115-120) and two Indices (121123 ) are added. In every chapter the author endeavors to show-and often 278 BOOK REVIEWS with success-that Aristotle had either the modern problem in question or a problem analogous to it. The Index locorum gives an insight into the author's main interests: the most frequently quoted Aristotelian chapters are (in that order): An.Pr. A 23, Met r, An. Pr.a 22,An.Pr.A 4, B 14, B 16, and A 44. On the other hand, An.Pr.A 8-22 are not mentioned, not even A 15. The book contains a number of analyses which would merit a thorough examination. This will not be offered here. Instead some remarks will be made on two points which may be of interest to Neo-Scholastics and render the work of Mr. Lear sympathetic to them. 1. The study is completely a-historical, and indeed in two ways. First, the Organon is considered as forming one block, without any sort of evolution and change occurring, e.g. between the Topics and the Prior Analytics. Solmsen is not even mentioned, nor are other scholars who did work on the problem, e.g. Minucci. Now, while this is a method often used by the Neo-Scholastics, it is hardly defensible. And once the possibility of changes occurring in Aristotle is admitted, the results of the inquiry might become very different from what they are. This is one way in which the work is a-historical. It is also a-historical in another way: the author is not interested in the interpretations supplied by his predecessors, except some of those who wrote recently in English or have been translated into English. One striking example of this laek of interest is the fact that Theophrastus is not mentioned. Now it is true that he did not have the privilege of an education at Cambridge. But, since he was a...

Aristotle and Logical Theory.

Preface 1. Syllogistic consequence 2. Completeness and compactness 3. Hypothetical syllogisms 4. Invalid inference 5. Invalid proofs Appendix: a note on ignorance 6. Proof by refutation Bibliography Indices.

Dialectic in the eleventh and twelfth centuries: garlandus compotista

Dialectic is a standard and important part of the logica vetus (or old logic) in medieval philosophy. It has its ultimate origins in Aristotle's Topics,its fundamental source in Boethius's De topicis differentiis,and its flowering in its absorption into fourteenth-century theories of consequences or conditional inferences. The chapter on Topics in Garlandus Compotista's logic book is the oldest scholastic work on dialectic still extant. In this paper I show the differences between Boethius's Theory of Topics and Garlandus's in order to illustrate the role of Topics in early scholastic logic. I argue that for Garlandus Topics are warrants for the inference from the antecedent to the consequent in a conditional proposition and that he is interested in Topics because of his overriding interest in hypothetical syllogisms. I conclude by discussing briefly the relationship between Garlandus's use of Topics and twelfth-century accounts.

Effects of Structure and Semantics on the Solution of Pure Hypothetical Syllogisms

Effects of Structure and Semantics on the Solution of Pure Hypothetical Syllogisms

Medieval Logic: An Outline of Its Development from 1250 to ca. 1400 by Philotheus Boehner, O. F. M., and: De Puritate Artis Logicae by Walter Burleigh

258 BOOK REVIEWS Medieval Logic: An Outline of Its Development from 11250 to ca. 1400. By PHILOTHEUS BoEHNER, 0. F. M. Chicago: University of Chicago Press, 195~. Pp. 148. $3.00. De Puritate Artis Logicae. By WALTER BuRLEIGH. Edited by Philotheus Boehner, 0. F. M. St. Bonaventure: The Franciscan Institute and Louvain: E. Nauwelaerts, 1951. Pp. 131. $1.50. The first work is a booklet designed to promote harmony and understanding between scholastic and modern logicians. It is the author's contention that many modern scholastics do not have a genuinely scholastic logic since they do not consider logic as an essentially formal science. And the reason for this is that they are not acquainted with their own tradition, namely, genuine scholastic logic of the 13th, 14th and 15th centuries wherein there is found a logic containing many elements of a strictly formal logic, so much so, that these elements appear to be definite anticipations, if not the exact doctrines, of modern logic. Avoiding the vague phrases, "traditional logic" and "classical logic," Fr. Boehner indicates precisely what he means by " nco-scholastic logic," " Aristotelian logic " and " modem logic," before stating and substantiating his conclusions. This is done in the Introduction. The body of the work is divided into three parts; Part One: Elements of Scholastic Logic; Part Two: Important Contributions of Scholastic Logic; and Part Three: Systems of Scholastic Logic. Part One is divided into two chapters: I, The Legacy of Scholastic Logic, and II, New Elements of Scholastic Logic. For the legacy of scholastic logic he has chosen Albert the Great's logical works as his source. There are eleven works cited, paraphrases on the six works of the Organon plus paraphrases on Boethius's works on division, categorical syllogisms, and hypothetical syllogisms. The other two works are Albert's treatment of the material found in Porphyry's Isagoge and Gilbert de la Poree's Six Principles. These works are merely listed and summarily described with regard to their contents. The new elements which Fr. Boehner lists in chapter two are the small tracts added by later scholastics which he divides into five groups: 1. De Syncategorematibus, ~- De proprietatibus terminorum , 3. De insolubili, 4. De obligatione, 5. De consequentiis. Part Two contains three chapters: I, The Syncategoremata As Logical Constants; II, The Theory of Supposition; and III, The Theory of Consequences . In the first chapter. Fr. Boehner indicates clearly the distinction between categorematic and syncategorematic words and the significative and suppositional dependence of the latter upon the former. He sees in this distinction a definite parallelism with the modern distinction between the constants and variables of logical discourse. A splendid quotation from Albert of Saxony shows that the latter considered the syncategoremata as formal elements which do not change. This is another indication of the BOOK REVIEWS ~59 formalized· character of medieval logic. In the second chapter Fr. Boehner briefly schematizes the theories of Peter of Spain, William Ockham, and Walter Burleigh on supposition. He likens the theory of supposition to the modern functional calculus. In chapter three the author treats of the notion and division of consequences according to William Ockham and Albert of Saxony. He also gives a brief survey of the consequential rules found in Ockham which have corresponding theorems of the propositional calculus. He gives the rule in translation, followed by the Latin text; then he explains and gives an instance of the rule; finally, he places it in symbolization. Part Three contains only one chapter in which the author merely indicates the content of five medieval logicians' works by schematizing a division of their most important logical opus. The five logicians summarized in this manner are Peter of Spain, William Ockham, John Buridan, Walter Burleigh, and Albert of Saxony. Two appendices contain a few sophismata of Albert of Saxony as well as a major portion of his rules of supposition. The second work is believed to be the first edited text of the shorter of two texts bearing the name De Puritate Artis Logicae. It contains Burleigh 's treatment of the general rules of consequences and a much longer discussion (eighty per cent of the text) on the syncategoremata. The publication of this text...