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)

Late ancient Platonists and Aristotelians describe the method of reasoning to first principles as "analysis." This is a metaphor from geometrical practice. How far back were philosophers taking geometric analysis as a model for philosophy, and what work did they mean this model to do? After giving a logical description of analysis in geometry, and arguing that the standard (not entirely accurate) late ancient logical description of analysis was already familiar in the time of Plato and Aristotle, I argue that (...) Plato, in the second geometrical passage of the "Meno" (86e4-87b2), is taking analysis as a model for one kind of philosophical reasoning, and I explore the advantages and limits of this model for philosophical discovery, and in particular for how first principles can be discovered, without circularity, by argument. (shrink)

Notes on recent books about Aristotle, including Jonathan Barnes, Truth etc.

The present book is the English version of a monograph 'Die aristotelische Syllogistik', which first appeared ten years ago in the series of Abhand 1 lungen edited by the Academy of Sciences in Gottingen. In the preface to the English edition, I would first like to express my indebtedness to Mr. J. Barnes, now fellow of Oriel College, Oxford. He not only translated what must have been a difficult text with exemplary precision and ingenuity, but followed critically every argument and (...) check ed every reference. While translating it, he has improved the book. Of those changes which I have made on Mr. Barnes' suggestion I note only the more important ones on pages 4, 12, 24sq, 32, 39, 6lsq, and 158. Since the second edition of the German text appeared in 1963 some further reviews have been published, or come to my notice, which I have 2 been able to make use of in improving the text of this new edition. I must mention here especially the detailed critical discussions of my results and arguments published by Professor W. Wieland in the Philosophische Rundschau 14 (1966), 1-27 and by Professor E. Scheibe in Gnomon 39 (1967), 454-64. Both scholars, while agreeing with the main drift and method of my interpretation, criticise some of my results and disagree with some of my arguments. It would not be possible to discuss these technical matters here with the necessary thoroughness. (shrink)

David Charles presents a major new study of Aristotle's views on meaning, essence, necessity, and related topics. These interconnected views are central to Aristotle's metaphysics, philosophy of language, and philosophy of science, and are also highly relevant to current philosophical debates. Charles aims to reach a clear understanding of Aristotle's claims and arguments, to assess their truth, and to evaluate their importance to ancient and modern philosophy.

What is the relation between a whole and its parts? The metaphysics of structure and composition is much discussed in modern philosophy; now Verity Harte provides the first sustained examination of Plato's rich but neglected discussion of the topic, and shows how it can illuminate current debates. This book is an invaluable resource both for scholars of Plato and for modern metaphysicians.

Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and (...) every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. (shrink)

I provide a survey of the contents of the works belonging to Aristotle's Organon in order to define their nature, in the light of his declared intentions and of other indications (mainly internal ones) about his purposes. No unifying conception of logic can be found in them, such as the traditional one, suggested by the very title Organon, of logic as a methodology of demonstration. Logic for him can also be formal logic (represented in the main by the De Interpretatione), (...) axiomatized syllogistic (represented in the main by the Prior Analytics) and a methodology of dialectical and rhetorical discussion. The consequent lack of unity presented by those works does not exclude that both the set of works called Analytics and the set of works concerning dialectic (Topics and Sophistici Elenchi) form a unity, and that a certain priority is attributed to the analytics with respect to dialectic. (shrink)

In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of the system is deducible (...) by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (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)

How does Aristotle think about sentences like ‘Every x is y’ in the Prior Analytics? A recently popular answer conceives of these sentences as expressing a mereological relationship between x and y: the sentence is true just in case x is, in some sense, a part of y. I argue that the motivations for this interpretation have so far not been compelling. I provide a new justification for the mereological interpretation. First, I prove a very general algebraic soundness and completeness (...) result that unifies the most important soundness and completeness results to date. Then I argue that this result vindicates the mereological interpretation. In contrast to previous interpretations, this argument shows how Aristotle’s conception of predication in mereological terms can do important logical work. (shrink)

By a thorough study of the Posterior Analytics and related Aristotelian texts, Richard McKirahan reconstructs Aristotle's theory of episteme--science. The Posterior Analytics contains the first extensive treatment of the nature and structure of science in the history of philosophy, and McKirahan's aim is to interpret it sympathetically, following the lead of the text, rather than imposing contemporary frameworks on it. In addition to treating the theory as a whole, the author uses textual and philological as well as philosophical material to (...) interpret many important but difficult individual passages. A number of issues left obscure by the Aristotelian material are settled by reference to Euclid's geometrical practice in the Elements. To justify this use of Euclid, McKirahan makes a comparative analysis of fundamental features of Euclidian geometry with the corresponding elements of Aristotle's theory. Emerging from that discussion is a more precise and more complex picture of the relation between Aristotle's theory and Greek mathematics--a picture of mutual, rather than one-way, dependence. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (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)

Among Plato's works, the Statesman is usually seen as transitional between the Republic and the Laws. This book argues that the dialogue deserves a special place of its own. Whereas Plato is usually thought of as defending unchanging knowledge, Dr Lane demonstrates how, by placing change at the heart of political affairs, Plato reconceives the link between knowledge and authority. The statesman is shown to master the timing of affairs of state, and to use this expertise in managing the conflict (...) of opposed civic factions. To this political argument corresponds a methodological approach which is seen to rely not only on the familiar method of 'division', but equally on the unfamiliar centrality of the use of 'example'. The demonstration that method and politics are interrelated transforms our understanding of the Statesman and its fellow dialogues. (shrink)

This volume contains a translation into clear modern English of an unjustly neglected work by Sextus Empiricus, together with introduction and extensive commentary. Sextus is our main source for the doctrines and arguments of ancient Scepticism; in Against the Ethicists he sets out a distinctive Sceptic position in ethics.

Some philosophers argue that false speech and false belief are impossible. In the Sophist, Plato addresses this 'falsehood paradox', which purports to prove that one can neither say nor believe falsehoods. In this book Paolo Crivelli closely examines the whole dialogue and shows how Plato's brilliant solution to the paradox is radically different from those put forward by modern philosophers. He surveys and critically discusses the vast range of literature which has developed around the Sophist over the past fifty years, (...) and provides original solutions to several problems that are so far unsolved. His book will be important for all who are interested in the Sophist and in ancient ontology and philosophy of language more generally. (shrink)

There is a mystery at the heart of Plato's Parmenides. In the first part, Parmenides criticizes what is widely regarded as Plato's mature theory of Forms, and in the second, he promises to explain how the Forms can be saved from these criticisms. Ever since the dialogue was written, scholars have struggled to determine how the two parts of the work fit together. Did Plato mean us to abandon, keep or modify the theory of Forms, on the strength of Parmenides' (...) criticisms? Samuel Rickless offers something that has never been done before: a careful reconstruction of every argument in the dialogue. He concludes that Plato's main aim was to argue that the theory of Forms should be modified by allowing that forms can have contrary properties. To grasp this is to solve the mystery of the Parmenides and understand its crucial role in Plato's philosophical development. (shrink)

In this paper we provide an interpretation of Aristotle's rule for the universal quantifier in Topics Θ 157a34–37 and 160b1–6 in terms of Paul Lorenzen's dialogical logic. This is meant as a contribution to the rehabilitation of the role of dialectic within the Organon. After a review of earlier views of Aristotle on quantification, we argue that this rule is related to the dictum de omni in Prior Analytics A 24b28–29. This would be an indication of the dictum’s origin in (...) the context of dialectical games. One consequence of our approach is a novel explanation of the doctrine of the existential import of the quantifiers in dialectical terms. After a brief survey of Lorenzen's dialogical logic, we offer a set of rules for dialectical games based on previous work by Castelnérac and Marion, to which we add here the rule for the universal quantifier, as interpreted in terms of its counterpart in dialogical logic. We then give textual evidence of the use of that rule in Plato's dialogues, thus showing that Aris... (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)

David Bronstein sheds new light on Aristotle's Posterior Analytics--one of the most important, and difficult, works in the history of western philosophy--by arguing that it is coherently structured around two themes of enduring philosophical interest: knowledge and learning. He argues that the Posterior Analytics is a sustained examination of scientific knowledge, an elegantly organized work in which Aristotle describes the mind's ascent from sense-perception of particulars to scientific knowledge of first principles. Bronstein goes on to highlight Plato's influence on Aristotle's (...) text, and argues against current orthodoxy that Aristotle is committed to the Socratic Picture of inquiry, according to which one should seek what a thing's essence is before seeking its demonstrable attributes and their causes. (shrink)

While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics (...) on the one hand and computerized formal proofs on the other hand. Along the way, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed. (shrink)

Plato famously promised to complement the Sophist and the Statesman with another work on a third sort of expert, the philosopher--but we do not have this final dialogue. Mary Louise Gill argues that Plato promised the Philosopher, but did not write it, in order to stimulate his audience and encourage his readers to work out, for themselves, the portrait it would have contained. The Sophist and Statesman are themselves members of a larger series starting with the Theaetetus, Plato's investigation of (...) knowledge, and the whole series relies on the Parmenides, the second part of which presents a philosophical exercise, introduced as the first step in a larger philosophical program. Gill contends that the dialogues leading up to the missing Philosopher, though they reach some substantive conclusions, are philosophical exercises of various sorts designed to train students in dialectic, the philosopher's method; and that a second version of the Parmenides exercise, closely patterned on it, spans parts of the Theaetetus and Sophist and brings the philosopher into view. This is the exercise about being, the subject-matter studied by Plato's philosopher. Plato hides the pieces of the puzzle and its solution in plain sight, forcing his students (and modern readers) to dig out the pieces and reconstruct the project. Gill reveals how, in finding the philosopher through the exercise, the student becomes a philosopher by mastering his methods. She shows that the target of Plato's exercise is internally related to its pedagogical purpose. (shrink)

This paper (1) criticizes Patzig's explanation of Aristotle's reason for calling his first figure syllogisms perfect syllogisms, i.e. the transitivity relation: it can only be used for Barbara, not for the other three moods. The paper offers (2) an alternative interpretation: It is only in the case of the (perfect) first figure moods that we can move from the subject term of the minor premiss, taken to be a predicate of an individual, to the predicate term of the major premiss. (...) This contention is supported (i) by Aristotle's wording of the dictum de omni et nullo and (ii) by Aristotle's use of a formula which puts the minor term in the first position when he first states Barbara and Celarent. (shrink)

Critical discussion of J.M.E. Moravcsik's paper on Plato's method of division.

Reading through the Corpus Aristotelicum we come across a group of expressions meant to indicate predicative relations, which lead us to think that Aristotle connected predication to a part-whole relation. He frequently calls the ‘εἴδη’, “species”, ‘μέρη’, “parts”, of their genera. More generally, the universal is said to contain that of which it is true. In a parallel way, what is contained by something is also what is under something else. Again, it is quite common for him to consider the (...) species as partaking of their genera. Although the relation of partaking is normally used to qualify the way in which species are related to their genera, sometimes also the relation between particulars and species or even genera is expressed in terms of partaking. (shrink)