This paper investigates whether Aristotle was aware of the fourth figure, and if he was aware of the fourth figure, why he excluded it from his system. Various commentators have explained why this figure does not exist in the system, so this paper compiles ane examines these arguments through a certain logical frame. By inquiring into why the fourth figure was not included in his logical system, the paper considers whether (...) class='Hi'>logical factors may explain this exclusion. (shrink)

Perhaps the strangest controversy in the history of logic is that over the fourth figure of the syllogism. There was never any argument as to what syllogisms are valid, but merely as to how they should be arranged. Aristotle had divided syllogisms into figures according to whether the middle term was subject of one premiss and predicate of the other, or predicate of both premisses, or subject of both. Theophrastus and Eudemus subdivided the first figure into those (...) moods in which the middle term was subject of the major premiss and those moods in which the middle was predicate of the major premiss. The latter moods were termed indirect. Galen said they constituted a separate figure; and so the controversy started. It has continued to the present with many writers objecting to the fourth figure, generally on grounds that it does not represent a genuine process of thought, and even ascribing this objection to Aristotle. The issue has been joined by writers who believe that it does represent a genuine process of thought. (shrink)

Syllogism is a form of logical argument allowing one to deduce a consistent conclusion based on a pair of premises having a common term. Although Aristotle was the first to conceive and develop this way of reasoning, he left open a lot of conceptual space for further modifications, improvements and systematizations with regards to his original syllogistic theory. From its creation until modern times, syllogism has remained a powerful and compelling device of deduction and argument, used by a variety (...) of figures and assuming a variety of forms throughout history. The Aftermath of Syllogism investigates the key developments in the history of this peculiar pattern of inference, from Avicenna to Hegel. Taking as its focus the longue durée of development between the Middle Ages and the nineteenth century, this book looks at the huge reworking scientific syllogism underwent over the centuries, as some of the finest philosophical minds brought it to an unprecedented height of logical sharpness and sophistication. Bringing together a group of major international experts in the Aristotelian tradition, The Aftermath of Syllogism provides a detailed, up to date and critical evaluation of the history of syllogistic deduction. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:New Light from Arabic Sources on Galen and the Fourth Figure of the Syllogism NICHOLAS RESCHER The Problem of the Origin of the Fourth Figure FLYING IN THE FACE of the long-standing tradition--going back in Europe to Renaissance times--which credits Galen of Pergamon with the origination of the fourth syllogistic figure, recent authorities have almost to a man evinced doubt about Galen's claim (...) to this innovation. Heinrieh Scholz speaks of "the Galenian syllogistic figure which has been attributed to him, probably wrongly." i j. W. Stakelum concludes his careful discussion of the matter with the categorical assertion that "Galen did not teach the fourth figure." ~ I. M. Bochenski--following Lukasiewicz (see below)--says that the fourth figure "was only ascribed to Galen by a misunderstanding." : William Kneale (also following Lukasiewicz) writes that "it is easy to see how misunderstanding.., by some Arabian philosopher... could have given rise to the tradition that Galen added a fourth figure to Aristotle's syllogistic theory." 4 The currently received account of the matter is that of Jan Lukasiewicz who, in his book on Aristotle's Syllogistic (1951; 2d ed., 1957), presents the known reports about Galen and the fourth figure as follows (I list them in the order in which they came to light) : 1) Several passages in Averroes' (d. 1198) Middle Commentary on Prior Analytics credit Galen with introducing a fourth figure (ad Anal. Pr. I, 5; I, 8; I, 23). These passages, known in Europe through a Renaissance Latin translation of a Hebrew version, provided the basis upon which Zabarella in his work, De Quarta Syllogismorum Figura (in his Opera [Leiden: 1587], pp. 41-53), popularized the "Galenian figure" in European logic. 2) In 1844 Minoides Mynas published in the Preface to his edition of Galen's EisagSg~ dialektik~ an anonymous Greek fragment (late, perhaps 6th century) which states that certain "later scholars" transformed the indirect moods of the first figure added by Theophrastus and Eudeffms into a new fourth figure, citing Galen as the originator of this doctrine.~ 1Abriss der Geschichte der Logik (Miinster: 1931), trans, into English by K. F. Leidecker, ConciseHistory of Logic (NewYork: 1961);p. 38ofthe Englishversion. "Why 'Galenian' Figure?" The New Scholasticism, XVI (1942),289-296. s Formale Logik (Miinchen-Freiburg:1956; 2nd. ed., 1962), trans, into English by Ivo Thomas, A History of Formal Logic (Notre Dame: 1961);p. 142of the Englishversion. Williamand Martha Kneale, The Developmentof Logic (Oxford: 1962),p. 184. Alsoprinted in Karl Kalbfieisch, ~ber Galen s Einleitung in die Logik, 2~. Supplementband der Jahrbi~cherfi~r klassische Philologie (Leipzig:1897), p. 707. This datum casts some 28 HISTORY OF PHILOSOPHY 3) A Greek fragment was found around 1858 by Karl Prantl in a logical work of the Byzantine scholar Ioannes Italus (llth century) which says that Galen taught the existence of a fourth figure, adding sarcastically that he (Galen) thought thus to appear cleverer than the older logical commentators (presumably Theophrastus and Eudemus are intended), but fell far short.6 4) In 1899 Maximilian Wallies published7 an anonymous Greek scholium (of perhaps the 6th or the 7th century) on Ammonins' commentary on Prior Analytics which states that Galen "says in his Apodictic that there are four figures, because he looks at the compound syllogisms consisting of four terms (i.e., with three premisses )" rather than the simple (three-term, two-premiss) syllogisms of Aristotle. (The scholiast goes on to explain at some length how such compound syllogisms can be sorted into four groups.) It is readily seen that, given these data, two possibilities stand before us: a) Galen did actually invent the traditional fourth figure. Reports (1)-(3) are correct, and the anonymous scholiast of (4) was put on the wrong track by the fact that Galen also said that compound syllogisms of four terms can be classed into four groups. b) Galen did not invent the traditional fourth figure. Only after his time when once the indirect moods of the first figure had--somehow--become systematized into a separate "figure," was this credited to Galen by a mistake along the lines described by the anonymous scholiast. Thus (4) alone is right, and (1)-(3) are mistaken. Which of... (shrink)

This article investigates the nature of Aristotelian syllogistics and shows that the categorical syllogism is fundamentally about showing the connection, in the premises of the syllogism, between the major and minor terms as stated in the conclusion. It discusses how this is important for the use of the syllogism in scientific demonstration. The article then examines modern deductive logic with an eye to they way in which it contrasts with Aristotelian syllogistics. It shows howmodern logic is about making (...) necessary connections between each proposition by means of external or second order rules. In the syllogism, on the other hand, the necessity between the premises as a whole unit and the conclusion is based on the internal middle term. The article concludes with a discussion of Günther Patzig’s claim that Aristotelian syllogisms are best thought of as tautological propositions. If this were the case, then the differences asserted to exist between syllogistic and modern logic would not hold. However, it is shown that Patzig’s assimilation of syllogistics to modern deductive logic is illegitimate. (shrink)

The follow-up research of Aristotle’s syllogism has different approaches. The traditional syllogism follows Aristotle’s conceptual system and hopes to make improvements within Aristotle’s theory. Mathematical logic proposes a new conceptual system to accurately interpret Aristotle’s syllogism. Lei Ma puts forward an extended syllogism whose conceptual system is different from Aristotelian logic and mathematical logic. He thinks that Aristotle’s syllogism and traditional syllogism have tedious figures, moods, and reasoning rules, which are difficult for us to memorize. It is a theoretical (...) conclusion of the human reasoning process but does not conform to the actual human thinking process. Ma’s syllogism is called substitution logic, which mainly concerns the substitution characteristics of a human thinking process, and summarizes the substitution rules in the reasoning process. Substitution logic appropriately describes the actual human reasoning process, thus inspiring us to establish a unified scientific theory of thinking and carry out normative research on the unity of abstract thinking and imaginative thinking. Substitution logic may be applied to the field of artificial intelligence, making artificial intelligence closer to the reality of human thinking. It seems that the research direction of substitution logic will give birth to human-like AI systems and promote the revolutionary transformation of AI research. (shrink)

This paper introduces the reader to the medieval Hebrew tradition of logic by considering its treatment of Aristotelian syllogistic. Starting in the thirteenth century European Jews translated Arabic and Latin texts into Hebrew and produced commentaries and original compendia.Because they stood culturally and geographically at the cross-roads of two great traditions they were influenced by both.This is clearly seen in the development of syllogistic theory, where the Latin tradition ultimately replaces, though never entirely, its Arabic counterpart.Specific attention is devoted (...) to the debate about the so-called Galenian fourth figure.In medieval Hebrew logic one finds both defenders and detractors of the figure, the former appearing towards the beginning of the period in question.With the ascendancy of scholastic logic the fourth figure virtually disappears from Hebrew texts. (shrink)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows from (...) the premises: there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

Although most natural law ethical theories recognize moral absolutes, there is not much agreement even among natural law theorists about how to identify them. The author argues that in order to understand and determine the morality (or immorality) of a human action, it must be considered in relation to the organized system of human practices within which it is performed. Such an approach, he argues, is to be found in the natural law theory of Thomas Aquinas, especially once it is (...) recognized that the logical structure of Aquinas's ethical theory is basically that of an Aristotelian science. In order to depict this structure and to explain how it bears upon the analysis of action, the author investigates a number of issues that have attracted the attention of Thomistic and Aristotelian scholarship. He examines the nature of practical reason, its relationship with theoretical reason, the derivation of lower from higher ethical principles, the incommensurability of human goods, the relationship between will and intellect, and the principle of double effect. The book will be useful to students and scholars interested in ethics, especially from an Aristotelian and/or Thomistic perspective. One appendix reproduces the Leonine text of the De malo (question 6), with facing English translation. Another appendix provides facing Latin text and English translation of the Summa Theologiae I-II (question 94, article 2). ABOUT THE AUTHOR: Kevin L. Flannery, S.J., author of many works on the history of logic--particularly Aristotelian logic--is dean of the faculty of philosophy and professor of the history of ancient philosophy at the Pontifical Gregorian University in Rome. PRAISE FOR THE BOOK: ""I know of no more careful and detailed study of Thomas's principle that the realms of theoretical and practical rationality are parallel. This principle is indeed central to an understanding of Thomas Aquinas's ethical thought, and Father Flannery examines its ramifications most perspicaciously. His book constitutes a valuable and important addition to the Thomistic literature. Father Flannery's writing is lucid, if not always extremely engaging, and his arguments are well supported by references to, and quotations from, the Aristotelian and Thomistic texts upon which they are based.""--The Medieval Review ""It would be hard to overstate the importance of this book at the present juncture of Thomistic studies.""--Ralph McInerny, University of Notre Dame ""... of interest and of great value mostly to specialists in Aquinas's ethics or moral theory more generally. Flannery presents complex matters clearly, and his explanations of the logical presuppositions of Aquinas's moral thought are always illuminating.""--Jean Porter, Theological Studies ""This is a significant work that belongs on the bookshelf of any serious student of Aquinas.... An important contribution to Thomistic studies.""--Irish Theological Quarterly ""This scholastic text will particularly appeal to thinkers interested in Aristotelian mathematics and logic and their practical integration into Thomistic natural law ethics. However, all natural law scholars must address the questions of context and method raised by Flannery's insights and his careful, precise textual analyses."" -- Beverly Whelton, Review of Metaphysics. (shrink)

Compare two conceptions of validity: under an example of a modal conception, an argument is valid just in case it is impossible for the premises to be true and the conclusion false; under an example of a topic-neutral conception, an argument is valid just in case there are no arguments of the same logical form with true premises and a false conclusion. This taxonomy of positions suggests a project in the philosophy of logic: the reductive analysis of the modal (...) conception of logical consequence to the topic-neutral conception. Such a project would dispel the alleged obscurity of the notion of necessity employed in the modal conception in favour of the clarity of an account of logical consequence given in terms of tractable notions of logical form, universal generalization and truth simpliciter. In a series of publications, John Etchemendy has characterized the model-theoretic definition of logical consequence as truth preservation in all models as intended to provide just such an analysis. In this paper, I will argue that Aristotle intends to provide an account of a modal conception of logical consequence in topic-neutral terms and so is engaged in a project comparable to the one described above. That Aristotle would be engaged in this sort of project is controversial. Under the standard reading of the Prior Analytics, Aristotle does not and cannot provide an account of logical consequence. Rather, he must take the validity of the first figure syllogisms (such as the syllogism known by its medieval mnemonic ‘Barbara’: A belongs to all B; B belongs to all C; so A belongs to all C) as obvious and not needing justification; he then establishes the validity of the other syllogisms by showing that they stand in a suitable relation to the first figure syllogisms. I will argue that Aristotle does attempt to provide an account of logical consequence—namely, by appeal to certain mereological theorems. For example, he defends the status of Barbara as a syllogism by appeal to the transitivity of mereological containment. There are, as I will discuss, reasons to doubt the success of this account. But the attempt is not implausible given certain theses Aristotle holds in semantics, mereology and the theory of relations. (shrink)

The question as to what makes a perfect Aristotelian syllogism a perfect one has long been discussed by Aristotelian scholars. G. Patzig was the first to point the way to a correct answer: it is the evidence of the logical necessity that is the special feature of perfect syllogisms. Patzig moreover claimed that the evidence of a perfect syllogism can be seen for Barbara in the transitivity of the a-relation. However, this explanation would give Barbara a different (...) status over the other three first figure syllogisms. I argue that, taking into account the role of the being-contained-as-in-a-whole formulation, transitivity can be seen to be present in all four first figure syllogisms. Using this wording will put the negation sign with the predicate, similar to the notation in modern predicate calculus. (shrink)

This chapter contains sections titled: Knowledge and Analysis The Relation Between Prior and Posterior Analytics Syllogistic Interpretations of Aristotle's Syllogistic Logic Knowledge of Facts Aristotelian Causes Demonstration Principles Definitions and Demonstrations Necessity Science and Dialectic Fallibility Applicability Readings of Aristotle's Theory of Science Epistemological Status of the Analytics Bibliography.

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. (shrink)

We want to show that Aristotle’s general conception of syllogism includes as its essential part the logical concept of necessity, which can be understood in a causal way. This logical conception of causality is more general then the conception of the causality in the Aristotelian theory of proof (“demonstrative syllogism”), which contains the causal account of knowledge and science outside formal logic. Aristotle’s syllogistic is described in a purely intensional way, without recourse to a set-theoretical formal semantics. (...) It is shown that the conclusion of a syllogism is justified by the accumulation of logical causes applied during the reasoning process. It is also indicated that logical principles as well as the logical concept of causality have a fundamental ontological role in Aristotle’s “first philosophy”. (shrink)

Aristotle's modal syllogistic has been an object of study ever since the time of Theophrastus; but these studies have been somewhat desultory. Remarkably, in the 1990s several new lines of research have appeared, with series of original publications by Fred Johnson, Richard Patterson and Ulrich Nortmann. Johnson presented for the first time a formal semantics adequate to a de re reading of the apodeictic syllogistic; this was based on a simple intuition linking the modal syllogistic to Aristotelian metaphysics. Nortmann (...) developed an ingenious de dicto analysis. Patterson articulated the links between the modal syllogistic and the metaphysics, using an analysis which strictly speaking is neither de re nor de dicto. My own studies in this field date from 1976, when my colleague Peter Roeper and I jointly wrote a paper "Aristotle's apodeictic syllogisms" for the XXIInd History of Logic Conference in Krakow. This paper contained the disjunctive reading of particular affirmative apodeictic propositions, which I still favour. Nonetheless, I did not consider that paper's results decisive or comprehensive enough to publish, and my 1981 book The Syllogism contained no treatment of the modal syllogism. The paper's ideas lay dormant till 1989, when I read Johnson's and Patterson's initial articles. I began publishing on the topic in 1991. Gradually my thoughts acquired a certain comprehensiveness and systematicity, till in 1993 I was able to take a semester's sabbatical to write up a draft of this book. (shrink)

Aristotle founds his modal syllogistic, like his plain syllogistic, on a small set of ?perfect? or obviously valid sylligisms. The rest he reduces to those, usually by means of modal conversion principles. These principles are open to more than one reading, however, and they are in fact invalid on one traditional reading (de re), valid on the other (de dicto). It is argued here that this way of framing the contrast is not Aristotelian, and that an interpretation involving modal (...) copulae allows us to see how these principles, and the modal system as a whole, are to be understood in light of close and precise connections to Aristotle's essentialist metaphysics. (shrink)

In 1951 Lukasiewicz [[sic]] linked Aristotle's Prior Analytics with modern formal logic. This book attempts to analyze Aristotle's syllogistic theory in the light of Lukasiewcz's work and the whole tradition of classic interpretations of Aristotle's logic. The first of the book's five chapters shows that for Aristotle the syllogism is basically a relationship of terms couched in conditional form; a relationship of variables rather than concrete terms; and a relationship that sees S linked with P not by the copula but (...) by less metaphysical expressions. The classical S is P form is not Aristotelian at all but a much later renovation. The second chapter maintains that Aristotle distinguishes "relative necessity" from "absolute necessity." Aristotle himself ignores his own distinction at times and thus creates logical difficulties for himself. The oft-fought sea battle is fought again in this chapter. Chapter three deals with the controversial notions of "perfect" and "imperfect" syllogisms. The author asserts that by "perfect syllogism" Aristotle means self-evident syllogisms that serve as axioms for deducing other proofs. Chapters four and five deal with the figures of the syllogism and the syllogism as a deductive system respectively. Patzig concludes that Aristotle's syllogistic theory is purely formal in its structure and can only be properly understood if it is not tied to a metaphysics and converted into a philosophical logic. He denies that it is a universal logic, but rather sees it as a specialized theory of relations.--J. J. R. (shrink)

Not until the early 1920’s was it possible to distinguish Aristotelian or traditional logic from Aristotle’s own ancient logic. We can now recognize many aspects of his logical investigations that are themselves modern, in the sense that modern logicians are making discoveries that Aristotle had already made or had anticipated. Here we gather five salient features of Aristotle’s logical investigations that reveal a striking philosophical modernity: 1) Aristotle took logic to be that part of epistemology used to (...) establish knowledge of logical consequence; 2) Prior Analytics is a metalogical treatise on the syllogistic deduction system; 3) Aristotle recognized the epistemic efficacy of certain elemental argument patterns, and he explicitly formulated them as rules of natural deduction in corresponding sentences; 4) Prior Analytics is a proof-theoretic treatise in which Aristotle describes a natural deduction system and demonstrates certain of the logical relationships among syllogistic deduction rules ; and finally, 5) Aristotle worked with a notion of substitution sufficient for distinguishing logical syntax and semantics. In this connection he also distinguished validity from deducibility sufficiently well to note the completeness of his logic. (shrink)

The paper uses the theory of generalized quantifiers to discuss existential import and its implications for Aristotelian logic, namely the square of opposition, conversions and the assertoric syllogistic, as well as for more recent generalizations to intermediate quantifiers like “most”. While this is a systematic discussion of the semantic background one should assume in order to obtain the inferences and oppositions Aristotle proposed, it also sheds some light on the interpretation of his writings. Moreover by applying tools from modern (...) formal semantics to the investigation of classical Aristotelian logic and its extensions, we combine different approaches to the logic of quantification. We will present variants of quantifiers that are associated to the four corners of the square of opposition with and without existential import and discuss their role for the logical square, conversions and the syllogistic. It will turn out that there is no way to ascribe existential import that validates all inferences and relations which one is willing to hold in Aristotelian logic. Two options, however, provide reasonable results. Existential import should either be ascribed only to affirmative statements or only be ascribed to universal quantification. The former option is preferable for a mere reconstruction of the classical Aristotelian logic while the latter option is more attractive if Aristotelian logic is generalized to intermediate quantifiers. (shrink)

_ Source: _Volume 26, Issue 2, pp 149 - 173 In the fourth treatise of his legal-theological work _Kitāb al-Anwār wa-al-Marāqib_, Yaʿqūb al-Qirqisānī analyzes a criticism of the Aristotelian syllogism and its epistemological foundations. Qirqisānī defends Aristotelian logic by quoting a passage from an unknown commentary on Aristotle in which the Aristotelian theory of syllogism is explicated. This paper focuses on the historical, theological, and philosophical meanings of the criticism of the syllogism in Qirqisānī’s discussion and (...) analyzes his interpretation of the syllogism as a source of knowledge that should be applied in the realm of legal reasoning and in the interpretation of biblical law. (shrink)

This article investigates the nature of Aristotelian syllogistics and shows that the categorical syllogism is fundamentally about showing the connection, in the premises of the syllogism, between the major and minor terms as stated in the conclusion. It discusses how this is important for the use of the syllogism in scientific demonstration. The article then examines modern deductive logic with an eye to they way in which it contrasts with Aristotelian syllogistics. It shows howmodern logic is about making (...) necessary connections between each proposition by means of external or second order rules. In the syllogism, on the other hand, the necessity between the premises as a whole unit and the conclusion is based on the internal middle term. The article concludes with a discussion of Günther Patzig’s claim that Aristotelian syllogisms are best thought of as tautological propositions. If this were the case, then the differences asserted to exist between syllogistic and modern logic would not hold. However, it is shown that Patzig’s assimilation of syllogistics to modern deductive logic is illegitimate. (shrink)

Aristotle's syllogistic theory, as developed in his Prior Analytics, is often regarded as the birth of logic in Western philosophy. Over the past century, scholars have tried to identify important precursors to this theory. I argue that Platonic division, a method which aims to give accounts of essences of natural kinds by progressively narrowing down from a genus, influenced Aristotle's logical theory in a number of crucial respects. To see exactly how, I analyze the method of division as it (...) was originally conceived by Plato and received by Aristotle. I argue that, while Plato allowed that some divisions fail to rigorously investigate the essence, he began a program continued by Aristotle (and others in antiquity and the middle ages) of seeking norms for division that would apply in any domain whatsoever. This idea of a rigorous, general method was taken up and developed by Aristotle in his syllogistic. Aristotle also used Plato's conception of predication as parthood in his semantics for syllogistic propositions. As part of my argument, I prove that a semantics based on Platonic divisional structures is sound and complete for the deduction system used in the literature to model Aristotle's syllogistic. (shrink)

In the introductory first chapter the author states his conviction that Aristotle’s theory of learning, at the center of which stands the apodeictic syllogism, is inadequate because partial. Chapter 2 is a balanced survey of Aristotle’s syllogistic, which does not serve the purpose of discovery, but is intended to turn into science knowledge already acquired. All learning proceeds from preexisting knowledge which is structured by demonstration. Next Bauman turns to Plato’s theory of learning as present in the Meno: learning is (...) remembering. His account of the Socratic method of making a disciple see the truth by pertinent questions is excellent. He warns us to avoid extreme interpretations of ἀνάμνησις. According to Aristotle, Plato confounded the process of remembering with induction. The process of learning consists in the passing from opinion to knowledge. Aristotle’s theory as explained in the Posterior Analytics widely differs from Plato’s. In chapter 5, Bauman examines whether the syllogism based on a universal major is a petitio principii, a charge laid against it ever since antiquity and repeated by John Stuart Mill. For Mill, induction is the only form of real inference, but Bauman shows that Mill’s position is not correct. Aristotle sets up a method which allows us to understand why things are the way they are and to pass on knowledge. In chapter 6, Bauman compares Aristotle’s method with what teachers usually do and with how textbook authors conceive the passing on of knowledge. Some previous knowledge is necessary for all teaching. Aristotle focused on how the sciences should be taught, showing how conclusions are implicitly contained in principles or can be deduced from them. Bauman thinks that this is restricting learning far too much and remains unsatisfactory. He disagrees with Aristotle’s statement that all scientific knowledge is expressible in propositions in the indicative mood. The syllogism cannot be the only mode for all teaching. This difficulty, however, can be solved when one keeps in mind what strictly scientific knowledge in the Aristotelian sense is, and that there are other forms of knowledge such as opinion, factual knowledge, and dialectic. Furthermore one should also consider that the syllogism is basic to all forms of reasoning, even if there are syllogisms of different types. A printing mistake on page 191, line 13 should be corrected.—Leo J. Elders, Institute for Philosophy, “Rolduc,” Kerkrade, The Netherlands. (shrink)

The rediscovery of Aristotle’s works on syllogisms in the Latin world, especially the Sophistici Elenchi and then the Prior Analytics, gave rise to sophisticated views on the nature of syllogistic form and syllogistic matter in the thirteenth century. It led to debates on the ontology of the syllogism as studied in the Prior Analytics, i.e. the syllogism made of letters and the four logical constants a/e/i/o, with deep consequences on the definition of logic as a universal method for all (...) sciences and as a science itself. (shrink)

I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic (...) is a fragment of my system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle. (shrink)

In Prior Analytics A7 Aristotle points out that all valid syllogistic moods of the second and third figures as well as the two particular moods of the first figure can be reduced to the two universal first-figure moods Barbara and Celarent. As far as the third figure is concerned, it is argued that Aristotle does not want to say, as the transmitted text suggests, that only those two valid moods of this figure whose premisses are both (...) universal statements are directly reducible to Barbara and Celarent, but rather that it is those four valid moods of this figure whose respective minor premisses are universal statements of which this is true. It is shown that in order to carry this sense the transmitted text has to be corrected by inserting just one word, which seems to have dropped out. (shrink)

It is fashionable now-a-days to regard Aristotle’s logic as being the skeleton in the closet of Aristotelian philosophy. As Miss Anscombe has acidly remarked, “Aristotle himself … misconceived the importance of the categorical syllogism, supposing that the theory of it gave him the key to the nature of scientific knowledge. He expresses this view in what I find his worst book: Book I of the Posterior Analytics.”.

The rediscovery of Aristotle’s works on syllogisms in the Latin world, especially the Sophistici Elenchi and then the Prior Analytics, gave rise to sophisticated views on the nature of syllogistic form and syllogistic matter in the thirteenth century. It led to debates on the ontology of the syllogism as studied in the Prior Analytics, i.e. the syllogism made of letters and the four logical constants a/e/i/o, with deep consequences on the definition of logic as a universal method for all (...) sciences and as a science itself. (shrink)

In 1928, Friedrich Solmsen argued that Aristotle'sPosterior Analyticswas largely composed before thePrior Analytics. Ross rejected Solmsen's position in 1939, and a rather lengthy series of rebuttals and counter-attacks between the two scholars followed. Quite recently, Jonathan Barnes has revived this issue with arguments in favour of something very close to Solmsen's thesis: that Aristotle first developed a theory of demonstration (‘apodeictic’) before he had worked out the syllogistic, and that thePosterior Analyticswas originally conceived against this background. Subsequently, when Aristotle formulated (...) a syllogistic, he is supposed by Barnes to have revised or added to the contents of thePosterior Analyticsso as to make syllogistic the logic of Aristotelian science. Thus, Barnes says: ‘the syllogism is in fact an incidental adjunct to the theory of demonstration: the theory can be formulated without reference, explicit or implicit, to Syllogistic, and itcould havebeen discovered by someone who knew nothing whatever about the Syllogism’ (pp. 33–4). (shrink)

The article examines the relevance of Aristotle’s analysis that concerns the syllogistic figures. On the assumption that Aristotle’s analytics was inspired by the method of geometric analysis, we show how Aristotle used the three terms, when he formulated the three syllogistic figures. So far it has not been appropriately recognized that the three terms — the major, the middle and the minor one — were viewed by Aristotle syntactically and predicatively in the form of diagrams. Many scholars have misunderstood Aristotle (...) in that in the second and third figure the middle term is outside and that in the second figure the major term is next to the middle one, whereas in the third figure it is further from it. By means of diagrams, we have elucidated how this perfectly accords with Aristotle's planar and graphic arrangement. In the light of these diagrams, one can appropriately capture the definition of syllogism as a predicative set of terms. Irrespective of the tricky question concerning the abbreviations that Aristotle himself used with reference to these types of predication, the reconstructed figures allow us better to comprehend the reductions of syllogism to the first figure. We assume that the figures of syllogism are analogous to the figures of categorical predication, i.e., they are specific syntactic and semantic models. Aristotle demanded certain logical and methodological competence within analytics, which reflects his great commitment and contribution to the field. (shrink)

In this article I explain three puzzling features of Simplicius’ use of syllogistic reconstructions in his commentary on Aristotle’s Physics: (1) Why does he reconstruct Aristotle’s non-argumentative remarks? (2) Why does he identify the syllogistic figure of an argument but does not explicitly present its reconstruction? (3) Why in certain lemmata does he present several reconstructions of the same argument? Addressing these questions, I argue that these puzzling features are an expression of Simplicius’ assumption that formal reasoning underlies Aristotle’s (...) prose, hence they reflect his attempt to capture as faithfully as possible Aristotle’s actual mode of reasoning. I show further that, as a consequence of this seemingly descriptive use of syllogistic reconstructions, logic serves Simplicius not only as an expository and clarificatory tool of certain interpretations or philosophical views, but also motivates and shapes his exegetical stances and approach. (shrink)

It is well-known that the Arab philosophers of the Aristotelian tradition, like some of their Alexandrian predecessors, attached rhetoric and poetics to logic, and supported this inclusion by the idea that the principal poetic procedure - that is, essentially, metaphor - is a kind of syllogism: the poetic syllogism. However, until now, no texts prior to those of Avicenna had been identified which render the structure of this syllogism explicit. In the present contribution, we present and translate a passage (...) from al-Fārābī which shows that for him, the poetic syllogism is an incorrect syllogism of the second figure - a different interpretation from that which will be given by Avicenna. (shrink)

The first part of the book examines language as a tool of logic, and deals with Alfarabi's analysis of the meanings of various terms as they are used in logic and philosophy.

This book explores the reaction of tenth-century Arab philosopher Abu Nasr Alfarabi to the logical works of Aristotle. From numerous short treatises the author develops a systematic and comprehensive topical survey of Alfara bi's logical writings. The book is divided into two major parts: language as a tool of logic and logic as a tool with which to analyze language. The first five chapters deal with Alfarabi's analysis of the meanings of various terms as they are used in (...) logic and philosophy. Alfarabi refutes the Arab grammarians who claimed that Arab logicians were building a language within a language and shows that the philosophical meanings of terms are in fact their most original and essential meanings. The final chapter deals with Alfarabi's analysis of certain aspects of the Arabic language and demonstrates that Arabic, like any natural language, conforms to universal logical structures of which natural languages are only a concrete expression. (shrink)

Although the theory of the assertoric syllogism was Aristotle's great invention, one which dominated logical theory for the succeeding two millenia, accounts of the syllogism evolved and changed over that time. Indeed, in the twentieth century, doctrines were attributed to Aristotle which lost sight of what Aristotle intended. One of these mistaken doctrines was the very form of the syllogism: that a syllogism consists of three propositions containing three terms arranged in four figures. Yet another was that a syllogism (...) is a conditional proposition deduced from a set of axioms. There is even unclarity about what the basis of syllogistic validity consists in. Returning to Aristotle's text, and reading it in the light of commentary from late antiquity and the middle ages, we find a coherent and precise theory which shows all these claims to be based on a misunderstanding and misreading. (shrink)

The author discusses what he deems an oversight in prior's article on lukasiewicz's book "aristotle's syllogistic". He thinks prior missed lukasiewicz's exposure of the "symbolic logicians' fairy tale" which is the attempt to fit aristotle's logic into the boolean and russellian systems by "the lopping and stretching of inconvenient limbs." he concludes that lukasiewicz has "broken the ice that had begun to form" on traditional logic and that logic did not begin in the nineteenth century.

The History of Philosophical and Formal Logic introduces ideas and thinkers central to the development of philosophical and formal logic. From its Aristotelian origins to the present-day arguments, logic is broken down into four main time periods: Antiquity and the Middle Ages The early modern period High modern period Early 20th century Each new time frame begins with an introductory overview highlighting themes and points of importance. Chapters discuss the significance and reception of influential works and look at historical (...) arguments in the context of contemporary debates. To support independent study, comprehensive lists of primary and secondary reading are included at the end of chapters, along with exercises and discussion questions. By clearly presenting and explaining the changes to logic across the history of philosophy, The History of Philosophical and Formal Logic constructs an easy-to-follow narrative. This is an ideal starting point for students looking to understand the historical development of logic. (shrink)

"However that may be, Aristotelian syllogistic concerned itself exclusively with monadic predicates. Hence it could not begin to investigate multiple quantification. And that is why it never got very far. None the less, the underlying grammar of Aristotle's logic did not in itself..

This comment calls attention to the nature of the Aristotelian and classical logics, and the difficulty of representing their judgments and inferences by means of Venn diagrams. The meaning of ‘all’ in the different calculi produces problems. A second problem is that the specification of existence in Venn diagrams for statements and arguments cannot be restricted to a single class, overlooked by Wiebe. This problem is further complicated by his adoption of classical (Renaissance) syllogistic, which is inconsistent. Aristotle’s term (...) logic is consistent. So also is the medieval extension, though the inclusion of singular premisses renders it less perspicuous though more flexible. (shrink)

This comment calls attention to the nature of the Aristotelian and classical logics, and the difficulty of representing their judgments and inferences by means of Venn diagrams. The meaning of ‘all’ in the different calculi produces problems. A second problem is that the specification of existence in Venn diagrams for statements and arguments cannot be restricted to a single class, overlooked by Wiebe. This problem is further complicated by his adoption of classical (Renaissance) syllogistic, which is inconsistent. Aristotle’s term (...) logic is consistent. So also is the medieval extension, though the inclusion of singular premisses renders it less perspicuous though more flexible. (shrink)

I examine the theoretical difficulties of Aristotle’s syllogism and the traditional syllogism. I propose a more unified ordinary thinking logic different from the syllogism. I show that the new logic based on the substitution of thinking elements can be used to describe the reasoning process of human minds more properly, bypassing rigid figures, moods and cumbersome rules of the syllogism. I also show that the new logic combines the categorical inference with relation and modal inferences, expanding the scope of the (...) syllogism so that more complex quantification inferences can be captured. I conclude that the substitution of thinking elements is the basic characteristics of human thinking, and the substitutions can be further applied not only to all research fields of abstract and image thinking but also to practical fields of action methodology. (shrink)

AbstractŁukasiewicz introduced a new methodological approach to the history of logic. It consists of the use of modern formal logic in the research of the history of logic. Although he was not the first to use formal logic in his historical research, Łukasiewicz was the first who used it consistently and formulated it as a requirement for a historian of logic. The aim of this paper is to present Łukasiewicz's contribution and the history of its formulation. In addition, the paper (...) will present its reception. There are two main themes in Łukasiewicz’s work on the history of logic. He discussed Stoic logic and Aristotle’s syllogistic. The second part of the paper focuses on current reception of Łukasiewicz’s contribution and on the question of whether there is a difference between knowledge about his contribution to Stoic logic and to Aristotle’s syllogistic, given that it has been hypothesised that Łukasiewicz’s contribution is currently more known among Aristotelian scholars than among those who focus on Stoic logic. Based on the papers that have appeared in the last five years at Web of Science and Scopus this hypothesis has not been confirmed. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several (...) attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)

Aristotle was the founder not only of logic but also of modal logic. In the Prior Analytics he developed a complex system of modal syllogistic which, while influential, has been disputed since antiquity--and is today widely regarded as incoherent. Combining analytic rigor with keen sensitivity to historical context, Marko Malink makes clear that the modal syllogistic forms a consistent, integrated system of logic, one that is closely related to other areas of Aristotle's philosophy. Aristotle's modal syllogistic differs significantly from modern (...) modal logic. Malink considers the key to understanding the Aristotelian version to be the notion of predication discussed in the Topics--specifically, its theory of predicables and the ten categories. The predicables introduce a distinction between essential and nonessential predication. In contrast, the categories distinguish between substantial and nonsubstantial predication. Malink builds on these insights in developing a semantics for Aristotle's modal propositions, one that verifies the ancient philosopher's claims of the validity and invalidity of modal inferences. While it acknowledges some limitations of this reconstruction, Aristotle's Modal Syllogistic brims with bold ideas, richly supported by close readings of the Greek texts. (shrink)

This paper examines the quest for the quantification of the predicate, as discussed by W.S. Jevons, and relates it to the discussion about universals and particulars between Plato and Aristotle. We conclude that the quest for the quantification of the predicate can only be achieved by stripping the syllogism from its metaphysical heritage.

In the Organon Aristotle describes some deductive schemata in which inconsistencies do not entail the trivialization of the logical theory involved. This thesis is corroborated by three different theoretical topics by him discussed, which are presented in this paper. We analyse inference schema used by Aristotle in the Protrepticus and the method of indirect demonstration for categorical syllogisms. Both methods exemplify as Aristotle employs classical reductio ad absurdum strategies. Following, we discuss valid syllogisms from opposite premises (contrary and contradictory) (...) studied by the Stagerian in the Analytica Priora (B15). According to him, the following syllogisms are valid from opposite premises, in which small Latin letters stand for terms such as subject and predicate, and capital Latin letters stand for the categorical propositions such as in the traditional notation: (i) in the second figure, Eba,Aba ` Eaa (Cesare), Aba, Eba ` Eaa (Camestres), Eba, I ba ` Oaa (Festino), and Aba,Oba ` Oaa (Baroco); (ii) in the third one, Eab,Aab ` Oaa (Felapton), Oab,Aab ` Oaa (Bocardo) and Eab, Iab ` Oaa (Ferison). Finally, we discuss the passage from the Analytica Posteriora (A11) in which Aristotle states that the Principle of Non-Contradiction is not generally presupposed in all demonstrations (scientific syllogisms), but only in those in which the conclusion must be proved from the Principle; the Stagerian states that if a syllogism of the first figure has the major term consistent, the other terms of the demonstration can be each one separately inconsistent. These results allow us to propose an interpretation of his deductive theory as a broad sense paraconsistent theory. Firstly, we proceed to a hermeneutical analysis, evaluating its logical significance and the interplay of the results with some other points of Aristotle’s philosophy. Secondly, we point to a logical interpretation of the Aristotelian syllogisms from opposite premises in the antilogisms method proposed by Christine Ladd-Franklin in 1883, and we also present a logical treatment of the Aristotelian demonstration with inconsistent material in the paraconsistent logics Cn, 1 _ n _!, introduced by da Costa in 1963. These two issues seem having not yet been analysed in detail in the literature. (shrink)