Provides comprehensive coverage of logical structure as well as the techniques of formal reasoning.

'De syllogismis' is the fifth treatise of John Buridan's 'Summulae dialecticae', a textbook he wrote for his logic course in the Faculty of Arts at the University of Paris. 'De syllogismis' contains material related to Aristotle's Analytica Priora and Boethius's 'De hypotheticis syllogismis'. The textbook discusses inferences involving not only propositions de inesse, but also propositions featuring oblique, reduplicative and infinite terms. Buridan displays a keen interest in modal inferences and inferences involving propositional attitudes. Buridan's De syllogismis continues along the (...) lines of his nominalist conception of the relations between mind, language and reality. (shrink)

This book traces the development of formal logic from its origins inancient Greece to the present day. The authors first discuss the work oflogicians from Aristotle to Frege, showing how they were influenced by thephilosophical or mathematical ideas of their time. They then examinedevelopments in the present century.

I consider the proper interpretation of the process of ecthesis which Aristotle uses several times in the Prior analytics for completing a syllogistic mood, i.e., showing how to produce a deduction of a conclusion of a certain form from premisses of certain forms. I consider two interpretations of the process which have been advocated by recent scholars and show that one seems better suited to most passages while the other best fits a single remaining passage. I also argue that ecthesis (...) for Aristotle really means ?setting out? the case to be proved using letters. Aristotle?s remarks about the use of letters in mathematical proofs suggest that he had some understanding of rules equivalent to universal generalization and existential instantiation; the ?proofs through ecthesis? are so called because they rest on the latter rule, with which use of letters is involved in a special way. (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)

Jan Lukasiewicz's treatise on Aristotle's Syllogistic, published in the 1950s, has been very influential in framing contemporary understanding of Aristotle's logical systems. However, Lukasiewicz's interpretation is based on a number of tendentious claims, not least, the claim that the syllogistic was intended to apply only to non-empty terms. I show that this interpretation is not true to Aristotle's text and that a more coherent and faithful interpretation admits empty terms while maintaining all the relations of the traditional square of opposition.

"One of the most careful and intensive among the introductory texts that can be used with a wide range of students. It builds remarkably sophisticated technical skills, a good sense of the nature of a formal system, and a solid and extensive background for more advanced work in logic.... The emphasis throughout is on natural deduction derivations, and the text's deductive systems are its greatest strength. Lemmon's unusual procedure of presenting derivations before truth tables is very effective." --Sarah Stebbins, The (...) Journal of Symbolic Logic. (shrink)

Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaśkowski in 1934. This article traces the development of natural deduction from the view that these founders embraced to the widespread acceptance of the method in the 1960s. I focus especially on the different choices made by writers of elementary textbooks—the standard conduits of (...) the method to a generation of philosophers—with an eye to determining what the ‘essential characteristics’ of natural deduction are. (shrink)

Of course we all know now that mathematics has proved that logic doesn't really make sense, but Etchemendy (philosophy, Stanford Univ.) goes further and challenges the received view of the conceptual underpinnings of modern logic by arguing that Tarski's model-theoretic analysis of logical consequences is wrong. He may have found the soft underbelly of the dead horse. Annotation copyrighted by Book News, Inc., Portland, OR.

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

This volume is the first annotated translation in any language of the entire text of the Summulae de dialectica, by the Parisian master of arts John Buridan (1300-1358). One of the most influential works in the history of late medieval philosophy, the Summulae is Buridan's systematic exposition of his nominalist philosophy of logic. Buridan's doctrine spread rapidly and for some two hundred years was dominant at many European universities. His work is of increasing interest today not only to historians of (...) medieval philosophy but also to modern philosophers, several of whom find in Buridan's ideas important clues to problems of contemporary philosophy. Gyula Klima provides a substantial introduction to Buridan's life and work and discusses his place in the history of logic. Through extensive notes Klima assists philosopher and medievalist alike to read Buridan with understanding and insight. Those with a philosophical interest in the relations among the structures of language, thought, and reality will find much to ponder in the Summulae. (shrink)

The Division of the Canon is an ancient Pythagorean treatise on the relationship between mathematical and acoustical truths. Euclidean in style, sectional in nature, and essentially Pythagorean, the Division has been susceptible to quotation since antiquity and has attracted the attention of many musicologists, classicists, mathematicians, and historians of science.

During the years 800-1200 A.D., Arabic scholars studied many of the works of Greek philosophy, and recorded their interpretations. Significant Arabic interpretations of Aristotle's Prior Analytics, the key work of his logical Organon, however, have remained largely unavailable in the West. The recent discovery of several Arabic manuscripts in Istanbul revealed the “Short Commentary on Prior Analytics” by the medieval Arabic philosopher al-Farabi. Nicholas Rescher here presents the first translation of this work in English, and supplements this with an informative (...) introduction and numerous explanatory footnotes. (shrink)

Terence Parsons presents a new study of the development and continuing value of medieval logic, which expanded Aristotle's basic principles of logic in important ways. Parsons argues that the resulting system is as rich as contemporary first-order symbolic logic.

An account of the role and the nature of logic in imperial stoic philosophy which challenges the prevailing orthodoxy and presents a novel interpretation of this crucial period of ancient philosophy.