Ancient Greek and Roman Logic

Imagine a dog tracing a scent to a crossroads, sniffing all but one of the exits, and then proceeding down the last without further examination. According to Sextus Empiricus, Chrysippus argued that the dog effectively employs disjunctive syllogism, concluding that since the quarry left no trace on the other paths, it must have taken the last. The story has been retold many times, with at least four different morals: (1) dogs use logic, so they are as clever as humans; (2) (...) dogs use logic, so using logic is nothing special; (3) dogs reason well enough without logic; (4) dogs reason better for not having logic. This paper traces the history of Chrysippus's dog, from antiquity up to its discussion by relevance logicians in the twentieth century. (shrink)

This paper intends to explain key differences between Aristotle’s understanding of the relationships between nous, epistêmê, and the art of syllogistic reasoning(both analytic and dialectical) and the corresponding modern conceptions of intuition, knowledge, and reason. By uncovering paradoxa that Aristotle’s understanding of syllogistic reasoning presents in relation to modern philosophical conceptions of logic and science, I highlight problems of a shift in modern philosophy—a shift that occurs most dramatically in the seventeenth century—toward a project of construction, a pervasive desire for (...) rational certainty, and a general insistence on the reducibility of the sciences. The major motivation of this analysis is my intention to show that modern attempts to reduce science/epistêmê to a single science/method of inquiry occlude dialectical and ethico-political dimensions of “reason” and, hence, also impoverish philosophy’s critical capacities. (shrink)

Original and penetrating, this book investigates of the notion of inference from signs, which played a central role in ancient philosophical and scientific method. It examines an important chapter in ancient epistemology: the debates about the nature of evidence and of the inferences based on it--or signs and sign-inferences as they were called in antiquity. As the first comprehensive treatment of this topic, it fills an important gap in the histories of science and philosophy.

Celebrando o papel de Newton da Costa na história da paraconsistência, este trabalho examina o uso e abuso da deliberada auto-contradição. Iniciado por Parmênides, desenvolvido por Platão, e continuado por Cícero, uma antiga tradição filosófica usava deliberadamente discursos paraconsistentes para revelar a verdade. Nos tempos modernos, o decisionismo tem usado uma deliberada auto-contradição contra a revelação Judaico-Cristã. DOI:10.5007/1808-1711.2010v14n1p1.

This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the same properties (...) of the natural deduction counterpart ? that is, Corcoran's system. Moreover, the first-order logic framework that we work with allows us to understand how complicated the semantics of the syllogistic is in providing us with examples of bizarre, unexpected interpretations of the syllogistic rules. Finally, we provide a first attempt at finding the structure of that semantics, reducing the search to the characterization of the class of models of T RD. (shrink)

Aristoteles Περὶ ἑρμηvείας Erklärungen Griechisch-deutsch ins Deutsche übersetzt von Gottfried Scherer -/- In seiner Schrift Erklärungen handelt Aristoteles von Namen, die alles, vieles oder Einzelnes bezeichnen, von Prädikatswörtern, die bestimmende oder zufällige Eigenschaften angeben, und von Worten und deren Kombinationsmöglichkeiten. Dabei untersucht er konkret die Struktur von Aussagesätzen, die er von anderen Redensarten abgrenzt, und stellt fest, dass sie etwas als existent und zutreffend, als wahr oder falsch bezeichnen. Er gibt an, wann mehrwertige Aussagen entstehen und beschreibt, was für Gattungsnamen (...) und was für Eigennamen gilt. Er definiert lakonisch, was ein Mensch im Unterschied zu anderen Lebewesen ist, und unterscheidet verschiedene Möglichkeiten, Namen und Prädikatswörter zu negieren. Ausführlich diskutiert er die Frage, ob logisch richtige Kombinationen von Bejahungen und Verneinungen Tatsachen präjudizieren oder ob Tatsachen unabhängig von Logik entstehen - Tatsachen und deren Feststellung werden ausdrücklich nicht erörtert, weil sich darüber nicht streiten lasse. Er gibt Hinweise darauf, wie notwendige, hypothetische und wirksame Bestimmungen zu verwenden seien und beschreibt Möglichkeiten, solche Aussagen schlüssig zu benutzen. Seine formalen Überlegungen demonstriert er an Beispielen, in denen er sich auch auf das Zeitgeschehen bezieht. -/- Mit dieser Übersetzung, die sich weitgehend an der Bekkerschen Ausgabe der Aristotelischen Schriften orientiert, soll ein lesbarer Text vorgelegt werden, der systematisch versucht, die griechischen Wortbedeutungen lebendig werden zu lassen. Auf eine Interpretation des Textes im Sinne traditioneller oder moderner Logiken oder auf eine an den Massengeschmack der 20er und folgender Jahre angepasste oder an andere Zeitgeister angelehnte Verdeutschung wurde bewusst verzichtet. Die Übersetzung möchte dem Leser Lust machen, selber zu denken - und sich intensiver mit Aristoteles zu beschäftigen. -/- . (shrink)

We develop a modified system of standard logic, Augmented Standard Logic (ASL), and we employ ASL in an effort to show that, contrary to prevailing opinion, both Aristotle and Diodorus presented impressive arguments, having valid structures and highly plausible premisses, in their famous fatalism debate. We argue that ASL, which contains standard logic and a full system of modal and temporal logic emanating from a modicum of primitives, should not only enable one to appreciate the sophisticated philosophizing which characterized this (...) ancient debate, but should prove to be quite useful in application to contemporary issues. (shrink)

This volume presents 27 essays on logic in ancient philosophy by Jonathan Barnes, one of the most admired philosophers of his generation.

Truth, etc. is a wide-ranging study of ancient logic based upon the John Locke lectures given by the eminent philosopher Jonathan Barnes in Oxford. The book presupposes no knowledge of logic and no skill in ancient languages: all ancient texts are cited in English translation; and logical symbols and logical jargon are avoided so far as possible. Anyone interested in ancient philosophy, or in logic and its history, will find much to learn and enjoy here.

The Introduction to philosophy written by Porphyry at the end of the second century AD is the most successful work of its kind ever to have been published. Porphyry's aim was modest, but he gave highly influential treatments of a number of perennial philosophical questions. Jonathan Barnes presents a complete new English translation, preceded by a substantial introduction and followed by an invaluable commentary, the first to be published in English and the fullest for a century, whose primary aim is (...) to analyse and assess the philosophical theses and arguments which the Introduction puts forward. (shrink)

The most accessible and comprehensive guide to Aristotle currently available.

ABSTRACT: English translation of the 2nd/3rd century Peripatetic Philosopher's Alexander of Aphrodisias commentary on Aristotle's non-modal syllogistic, i.e. on one of the most influential logical texts of all times. -/- Volume includes introduction on Alexander of Aphrodisias and the early commentators, translation with notes and comments, appendices with a new translation of Aristotle's text, a summary of Aristotle's non-modal syllogistic and textual notes.

In this paper I hope to show that Geach misunderstands the nature of Plato's argument in the Euthyphro and more importantly the reasoning behind the dialectical strategy adopted by Socrates. Furthermore I shall argue that Geach's reading of the Euthyphro engenders serious difficulties, that stand in the way of understanding the manner in which Plato construes the problem of determining the nature of, and relationship between universal and particulars, which is of great significance because it is precisely this problem, in (...) relation to piety, that is central to the Euthyphro. In the appendix I shall sketch the outline of an argument in order to justify the assumption that the theory of Forms is present in the Euthyphro. (shrink)

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians.

ABSTRACT: Alexander of Aphrodisias’ commentaries on Aristotle’s Organon are valuable sources for both Stoic and early Peripatetic logic, and have often been used as such – in particular for early Peripatetic hypothetical syllogistic and Stoic propositional logic. By contrast, this paper explores the role Alexander himself played in the development and transmission of those theories. There are three areas in particular where he seems to have made a difference: First, he drew a connection between certain passages from Aristotle’s Topics and (...) Prior Analytics and the Stoic indemonstrable arguments, and, based on this connection, appropriated at least four kinds of Stoic indemonstrables as Aristotelian. Second, he developed and made use of a specifically Peripatetic terminology in which to describe and discuss those arguments – which facilitated the integration of the indemonstrables into Peripatetic logic. Third, he made some progress towards a solution to the problem of what place and interpretation the Stoic third indemonstrables should be given in a Peripatetic and Platonist setting. Overall, the picture emerges that Alexander persistently (if not always consistently) presented passages from Aristotle’s logical œuvre in a light that makes it appear as if Aristotle was in the possession of a Peripatetic correlate to the Stoic theory of indemonstrables. (shrink)

ABSTRACT: This paper discusses ancient versions of paradoxes today classified as paradoxes of presupposition and how their ancient solutions compare with contemporary ones. Sections 1-4 air ancient evidence for the Fallacy of Complex Question and suggested solutions, introduce the Horn Paradox, consider its authorship and contemporary solutions. Section 5 reconstructs the Stoic solution, suggesting the Stoics produced a Russellian-type solution based on a hidden scope ambiguity of negation. The difference to Russell's explanation of definite descriptions is that in the Horn (...) Paradox the Stoics uncovered a hidden conjunction rather than a hidden existential sentence. Sections 6 and 7 investigate hidden ambiguities in 'to have' and 'to lose' (including inalienable and alienable possession) and ambiguities of quantification based on substitution of indefinite plural expressions for indefinite or anaphoric pronouns, and Stoic awareness of these. Section 8 considers metaphorical readings and allusions that add further spice to the paradox. (shrink)

ABSTRACT: The 3rd BCE Stoic logician "Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,952." After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F. Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is why Hipparchus’ (...) logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic. OPEN ACCESS LINK. (shrink)

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Content: 1. Pre-Aristotelian Logic 1.1 Syntax and Semantics 1.2 Argument Patterns and Valid Inference 2. Aristotle 2.1 Dialectics 2.2 Sub-sentential Classifications 2.3 Syntax and Semantics of Sentences 2.4 Non-modal Syllogistic 2.5 Modal Logic 3. The early Peripatetics: Theophrastus and Eudemus 3.1 Improvements and Modifications of Aristotle's Logic 3.2 Prosleptic Syllogisms 3.3 Forerunners (...) of Modus Ponens and Modus Tollens 3.4 Wholly Hypothetical Syllogisms 4. Diodorus Cronus and Philo the Logician 5. The Stoics 5.1 Logical Achievements Besides Propositional Logic 5.2 Syntax and Semantics of Complex Propositions 5.3 Arguments 5.4 Stoic Syllogistic 5.5 Logical Paradoxes 6. Epicurus and the Epicureans 7. Later Antiquity. (shrink)

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with a focus on issues that may be of interest to contemporary logicians and covering important topics in Post-Aristotelian logic that are frequently neglected (such as Peripatetic hypothetical syllogistic, the Stoic axiomatic system of propositional logic and various later ancient developments).

ABSTRACT: This paper discusses the Stoic treatment of fallacies that are based on lexical ambiguities. It provides a detailed analysis of the relevant passages, lays bare textual and interpretative difficulties, explores what the Stoic view on the matter implies for their theory of language, and compares their view with Aristotle’s. In the paper I aim to show that, for the Stoics, fallacies of ambiguity are complexes of propositions and sentences and thus straddle the realms of meaning (which is the domain (...) of logic) and of linguistic expressions (which is the domain of linguistics), but also involve a pragmatic element; that the Stoics believe that the premises of the fallacies, when uttered, have only one meaning and are true, and thus should be conceded; that hence there is no need for a mental process of disambiguation in the listeners; that Aristotle, by contrast, appears to assume that the premises always have all their meanings, and accordingly recommends that the listeners explicitly disambiguate them, which presupposes a process of mental disambiguation. I proffer two readings of the Stoic advice that we ‘be silent’ when confronted with a fallacy of ambiguity in dialectical discourse, and explicate how each leads to an overall consistent interpretation of the textual evidence. Finally, I demonstrate that the method advocated by the Stoics works in all cases of fallacies of lexical ambiguity. (shrink)

The ‘Dialectical school’ denotes a group of early Hellenistic philosophers that were loosely connected by philosophizing in the — Socratic — tradition of Eubulides of Megara and by their interest in logical paradoxes, propositional logic and dialectical expertise. . Its two best known members, Diodorus Cronus and Philo the Logician, made groundbreaking contributions to the development of theories of conditionals and modal logic. Philo introduced a version of material implication; Diodorus devised a forerunner of strict implication. Each developed a system (...) of modal notions that satisfies the basic logical requirements laid down by modern standard modal theories. In antiquity, Diodorus Cronus was famous for his so called Master Argument, which aims to prove that only the actual is possible. (shrink)

ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; Stoic basic principles of propositional logic; 4. (...) Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)

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

ABSTRACT: This paper traces the evidence in Galen's Introduction to Logic (Institutio Logica) for a hypothetical syllogistic which predates Stoic propositional logic. It emerges that Galen is one of our main witnesses for such a theory, whose authors are most likely Theophrastus and Eudemus. A reconstruction of this theory is offered which - among other things - allows to solve some apparent textual difficulties in the Institutio Logica.

ABSTRACT: This paper collects the evidence in Ammonius' surviving works for elements of a propositional logic, coming to the conclusion that Ammonius had a theory of hypothetical syllogisms in the tradition of Aristotle and the Peripatetics, with Platonic elements mixed in, and using some Stoic elements, but not a propositional logic in the narrower sense as we find it in Stoic logic.

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

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

ABSTRACT: At the turn of the second century AD there existed two different views on the ordering of the figures of the (wholly) hypothetical syllogisms. One goes back to Theophrastus, whereas the other (adopted e.g. by Alexander of Aphrodisias and Alcinous) seems to have been the result of a later change. This reversal of the order of figures has so far not received a satisfactory explanation. In this paper I show how it came about.

ABSTRACT: Summary presentation of the surviving logic theories of Philo the Dialectician (aka Philo of Megara) and Diodorus Cronus, including some general remarks on propositional logical elements in their logic, a presentation of their theories of the conditional and a presentation of their modal theories, including a brief suggestion for a solution of the Master Argument.

ABSTRACT: A detailed presentation of Stoic logic, part one, including their theories of propositions (or assertibles, Greek: axiomata), demonstratives, temporal truth, simple propositions, non-simple propositions(conjunction, disjunction, conditional), quantified propositions, logical truths, modal logic, and general theory of arguments (including definition, validity, soundness, classification of invalid arguments).

ABSTRACT: A detailed presentation of Stoic theory of arguments, including truth-value changes of arguments, Stoic syllogistic, Stoic indemonstrable arguments, Stoic inference rules (themata), including cut rules and antilogism, argumental deduction, elements of relevance logic in Stoic syllogistic, the question of completeness of Stoic logic, Stoic arguments valid in the specific sense, e.g. "Dio says it is day. But Dio speaks truly. Therefore it is day." A more formal and more detailed account of the Stoic theory of deduction can be found (...) in S. Bobzien, Stoic Syllogistic, OSAP 1996. (shrink)

ABSTRACT: In this paper I argue (i) that the hypothetical arguments about which the Stoic Chrysippus wrote numerous books (DL 7.196) are not to be confused with the so-called hypothetical syllogisms" but are the same hypothetical arguments as those mentioned five times in Epictetus (e.g. Diss. 1.25.11-12); and (ii) that these hypothetical arguments are formed by replacing in a non-hypothetical argument one (or more) of the premisses by a Stoic "hypothesis" or supposition. Such "hypotheses" or suppositions differ from propositions in (...) that they have a specific logical form and no truth-value. The reason for the introduction of a distinct class of hypothetical arguments can be found in the context of dialectical argumentation. The paper concludes with the discussion of some evidence for the use of Stoic hypothetical arguments in ancient texts. (shrink)