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)

Ever since ?ukasiewicz, it has been opinio communis that Aristotle's modal syllogistic is incomprehensible due to its many faults and inconsistencies, and that there is no hope of finding a single consistent formal model for it. The aim of this paper is to disprove these claims by giving such a model. My main points shall be, first, that Aristotle's syllogistic is a pure term logic that does not recognize an extra syntactic category of individual symbols besides syllogistic terms and, second, (...) that Aristotelian modalities are to be understood as certain relations between terms as described in the theory of the predicables developed in the Topics. Semantics for modal syllogistic is to be based on Aristotelian genus-species trees. The reason that attempts at consistently reconstructing modal syllogistic have failed up to now lies not in the modal syllogistic itself, but in the inappropriate application of modern modal logic and extensional set theory to the modal syllogistic. After formalizing the underlying predicable-based semantics (Section 1) and having defined the syllogistic propositions by means of its term logical relations (Section 2), this paper will set out to demonstrate in detail that this reconstruction yields all claims on validity, invalidity and inconclusiveness that Aristotle maintains in the modal syllogistic (Section 3 and 4). (shrink)

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th (...) century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

In Prior Analytics 1.27–30, Aristotle develops a method for finding deductions. He claims that, given a complete collection of facts in a science, this method allows us to identify all demonstrations and indemonstrable principles in that science. This claim has been questioned by commentators. I argue that the claim is justified by the theory of natural predication presented in Posterior Analytics 1.19–22. According to this theory, natural predication is a non-extensional relation between universals that provides the metaphysical basis for demonstrative (...) science. (shrink)

In Posterior Analytics 1.3, Aristotle advances three arguments against circular proof. The third argument relies on his discussion of circular proof in Prior Analytics 2.5. This is problematic because the two chapters seem to deal with two rather disparate conceptions of circular proof. In Posterior Analytics 1.3, Aristotle gives a purely propositional account of circular proof, whereas in Prior Analytics 2.5 he gives a more complex, syllogistic account. My aim is to show that these problems can be solved, and that (...) Aristotle’s third argument in 1.3 is successful. I argue that both chapters are concerned with the same conception of circular proof, namely the propositional one. Contrary to what is often thought, the syllogistic conception provides an adequate analysis of the internal deductive structure of the propositional one. Aristotle achieves this by employing a kind of multiple-conclusion logic. (shrink)

TheGenerales Inquisitiones de Analysi Notionum et Veritatumis Leibniz’s most substantive work in the area of logic. Leibniz’s central aim in this treatise is to develop a symbolic calculus of terms that is capable of underwriting all valid modes of syllogistic and propositional reasoning. The present paper provides a systematic reconstruction of the calculus developed by Leibniz in theGenerales Inquisitiones. We investigate the most significant logical features of this calculus and prove that it is both sound and complete with respect to (...) a simple class of enriched Boolean algebras which we call auto-Boolean algebras. Moreover, we show that Leibniz’s calculus can reproduce all the laws of classical propositional logic, thus allowing Leibniz to achieve his goal of reducing propositional reasoning to algebraic reasoning about terms. (shrink)

In Prior Analytics 1.15 Aristotle undertakes to establish certain modal syllogisms of the form XQM. Although these syllogisms are central to his modal system, the proofs he offers for them are problematic. The precise structure of these proofs is disputed, and it is often thought that they are invalid. We propose an interpretation which resolves the main difficulties with them: the proofs are valid given a small number of intrinsically plausible assumptions, although they are in tension with some claims found (...) elsewhere in Aristotle’s modal syllogistic. The proofs make use of a rule of propositional modal logic which we call the possibility rule. We investigate how this rule interacts and coheres with the core elements of the modal syllogistic. (shrink)

Das griechische Wort syllogismos besteht aus der Vorsilbe syl- und dem Wort logismos. Platon verwendet es gelegentlich im Sinne von ›Überlegung‹. Aristoteles prägt den technischen Begriff des ›Syllogismos‹ im Sinne von ›gültiger deduktiver Schluss‹.

This paper offers a close reading of Aristotle’s account of categories in Topics I.9. It consists of four sections. The first argues that the categories introduced in Topics I.9 are different from those introduced in Categories 4. In particular, the first category of Topics I.9, the category of essence , is different from the firstcategory of Categories 4, the category of substance .The second section contains the main proposal of this paper: I shall argue that the category of essence is (...) characterized by Aristotle as including all terms which are the subject of some essential predication, i.e., all terms which possess an essence. In Topics I.9, Aristotle accepts essential predications such as ‘whiteness is colour’ as well as ‘man is animal’. Thus, the category of essence includes non-substance terms such as ‘whiteness’ as well as substance terms such as ‘man’. The third section proposes a solution to a problem concerning the status of differentiae in the Topics. The problem ist hat, according to the Topics, differentiae are predicated in the essence but do not signify essence . The solution will be that differentiae function as the predicate of essential predications but do not belong to the category of essence.The fourth section investigates the nine non-essence categories of Topics I.9 . I shall argue that these include paronymic non-substance terms such as ‘white’ and ‘coloured’, whereas the corresponding nominal forms such as ‘whiteness’ and ‘colour’ belong to the category of essence. (shrink)

Aristoteles gilt als der Begründer der formalen Logik. Sein ›Organon‹ hat in beispielloser Weise die Entwicklung der westlichen Logik beeinflusst. Das logische Werk des Aristoteles wurde zunächst von seinem Schüler Theophrast fortgeführt, teilweise erweitert und modifiziert.

Modalitäten wie Notwendigkeit, Möglichkeit oder Unmöglichkeit spielen eine bedeutende Rolle in den Schriften des Aristoteles – als Gegenstand theoretischer Untersuchung ebenso wie als begriffliches Instrumentarium.

Unter der Bezeichnung ›Organon‹ werden traditionell sechs Abhandlungen des Aristoteles zusammengefasst, die als sein logisches Werk gelten: Kategorien, De interpretatione, Analytica priora, Analytica posteriora, Topik, Sophistici elenchi. Die Zusammenfassung dieser Schriften zu einer Werkgruppe geht nicht auf Aristoteles zurück, sondern auf antike Kommentatoren und Editoren, vermutlich auf den Peripatetiker Andronikos von Rhodos im 1. Jh. v. Chr. Von einigen neuplatonischen und arabischen Kommentatoren wurden auch die Rhetorik und Poetik zum ›Organon‹ gezählt.