The Megarians, as well as their Stoic heirs, are known to have been fatalists or logical determinists in the following, very broad sense of these terms: with respect to at least certain classes or kinds of nontautologous propositions, they held that the mere truth of a proposition entails its necessity. This paper explores, in a very tentative fashion, the relation between several versions of logical determinism and two passages in the Aristotelian corpus, one of which is specifically directed against (...) Megarian doctrine and the other of which is an argument purporting to establish a thesis concerning the necessity of future events which Aristotle wished to reject but which the Megarian logician Diodorus Cronus apparently accepted. (shrink)

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. (...) Further volumes will follow, including Mediaeval and Renaissance Logic and Logic: A History of its Central. In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality. Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredly international enterprise, whose research programmes reach to every corner of the learned world. Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value. Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas. (shrink)

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich (...) contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness. (shrink)

The first monograph on Stoic modal logic. Part 1 discusses the Stoic notion of propositions (assertibles, axiomata): their definition; their truth-criteria; the relation between sentence and proposition; propositions that perish; propositions that change their truth-value; the temporal dependency of propositions; the temporal dependency of the Stoic notion of truth; pseudo-dates in propositions. Part 2 discusses Stoic modal logic: the Stoic definitions of their modal notions (possibility, impossibility, necessity, non-necessity); the logical relations between the (...) modalities; modalities as properties of propositions; contingent propositions; the relation between the Stoic modal notions and those of Diodorus Cronus and Philo of Megara; the role of ‘external hindrances’ for the modalities; the temporal dependency of the modalities; propositions that change their modalities; the principle that something possible can follow from something impossible; the interpretations of the Stoic modal system by B. Mates, M. Kneale, M. Frede, J. Vuillemin and M. Mignucci are evaluated. -/- For a shorter, updated, English version of Part 1 of the book see S. Bobzien, ‘Stoic Logic’, in K. Algra et al. (eds), The Cambridge History of Hellenistic Philosophy, Cambridge 1999, 92-157. For a shorter, updated, English version of Part 2 of the book see S. Bobzien 'Chrysippus' Modal Logic and its Relation to Philo and Diodorus', in K. Doering / Th. Ebert (eds) Dialektiker und Stoiker (Stuttgart 1993) 63-84. (shrink)

Essays on Aristotle's Sea-Battle, Lazy Argument, Argument Reaper, Diodorus' Master Argument -/- The book is devoted to the ancient logical theories, reconstruction of their semantic proprieties and possibilities of their interpretation by modern logical tools. The Ancient arguments are frequently misunderstood in modern interpretations since authors usually have tendency to ignore their historical proprieties and theoretical background what usually leads to a quite inappropriate picture of the argument’s original form and mission. Author’s primary intention was to draw attention to the (...) complexity of some historical arguments and to the theoretical context in which arguments were created, circulated, developed, and finally tuned. Four well-known ancient arguments – with a common central subject related to the future contingencies problem – are reconstructed from available historical sources: “The Sea Battle”, which is drawn from Aristotle’s treatise De Interpretatione; two arguments, usually ascribed to the Stoics, “The Lazy Argument” and “The Reaper”; “The Master Argument” of the Megarian philosopher Diodorus. Arguments are linguistically and semantically detaily analyzed, formally presented by reflecting some relevant corresponding hypotheses based on physical or logical theories of their ancient authors, and finally covered by appropriate logical tools familiar to a modern reader. Two appendices are added at the closing part of the book. One covering some assumptions relevant for understanding of rival streams in ancient theories of meaning related to the nature of names and naming; the other is devoted to the ancient understanding of logical proposition and attempts to find an adequate Latin translation of the Greek delicate philosophical term “ἀξίωμα”. (shrink)

‘Puzzles’, ‘word games’, ‘logical anomalies’, whatever we call them, they perplex us and challenge our familiar patterns of reasoning. One of these puzzles, among many others, originated from the mind of an ancient Megarian logician, Eubulides of Miletus, and endures to the modern day.1 Its name, ‘sorites’, can be traced to the Greek word soros, meaning ‘heap.’ The answer to whether one grain of sand ‘is a heap’ or ‘is not a heap’ seems quite simple: it is not a (...) heap. However, as we add grains to the one, at what future point does the non-heap become a heap? Our decision is fraught with uncertainty. Are the objects or the language we are using to describe them vague? In academic philosophy, the ancient Greek puzzle has gained the status of a paradox, as philosophers apply stoic and modern logic to these propositions considered to have vague predicates. The current debate has developed quite serious and wide-ranging implications, such as whether sorites issues provide adequate grounds for abandoning our standard ontology (or our understanding of what really exists), 2 and (germinating into another discipline) whether vagueness in the language of legal rules can generate disagreement as to whether there are right answers to questions of law. 3 Several unique solutions to the paradox have been proposed, yet all suffer from specific inadequacies that might, upon further reflection, disappear in the event that we endeavour to produce a synthesis. When we say that we are attempting to derive a synthesis, we seek to combine elements of two solutions for the sake of creating a new (and hopefully, though not necessarily, a better) one. The paradox of the sorites, or ‘the heap’, appeals to us because it challenges our assumption that we may categorically describe clear cases and their negations, yet fail to distinguish the borderline ones. I will demonstrate that some combinations of solutions are easy fits while others are extremely weak and awkward as attempts at resolving the paradox. Nonetheless, the process of synthesis produces three noteworthy combinations. (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)

We argue that the extant evidence for Stoic logic provides all the elements required for a variable-free theory of multiple generality, including a number of remarkably modern features that straddle logic and semantics, such as the understanding of one- and two-place predicates as functions, the canonical formulation of universals as quantified conditionals, a straightforward relation between elements of propositional and first-order logic, and the roles of anaphora and rigid order in the regimented sentences that express multiply (...) general propositions. We consider and reinterpret some ancient texts that have been neglected in the context of Stoic universal and existential propositions and offer new explanations of some puzzling features in Stoic logic. Our results confirm that Stoic logic surpasses Aristotle’s with regard to multiple generality, and are a reminder that focusing on multiple generality through the lens of Frege-inspired variable-binding quantifier theory may hamper our understanding and appreciation of pre-Fregean theories of multiple generality. (shrink)

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1973.

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)

After comparing Fr. 6 with the literature of its day, Mansfeld concludes that Parmenides' poem is not a polemic against Heracleitus. Rather, the poem reflects an opinion of the low estate of human knowledge not uncommon in that day. This does not, of course, preclude any influence of Heracleitus on the poem. In a second chapter, Mansfeld analyzes the argument of Fr. 3 as a disjunctive syllogism and argues that Parmenides is the founder of a tradition of logic continued (...) by Zeno, the Megarians, and the Stoics. The final chapters are devoted to a discussion of the Doxa and the central place of the Proem in interpreting the poem as a whole. The book contains an extensive bibliography, topical index, and index of Greek texts.—O. J. L. (shrink)

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).

This chapter contains sections titled: Introduction The Stoic Logical System Conclusion Bibliography.

The wider topic to which the content of this paper belongs is that of the relationship between formal logic and real argumentation. Of particular potential interest in this connection are held to be substantive arguments constructed by philosophers reputed equally as authorities in logical theory. A number of characteristics are tentatively indicated by the author as likely to be encountered in such arguments. The discussion centers afterwards, by way of specification, on a remarkable piece of argument quoted in Cicero’s (...) dialog On Divination and ascribed to Stoic thinkers. The Stoics’ formal theory of inference is summarily referred to in this context, with special emphasis on their basic deductive schemata (‘indemonstrables’), some of them recognizable as links in the overall structure of the quoted argument. The main lines of Cicero’s criticism of the Stoic argument are next commented upon, with emphasis on his implied view as to the requirements of a good argument. Towards the end of the paper, a few considerations are added on the changes in the prevailing style of argumentation conspicuous in the three famous Roman Stoics. (shrink)

This book uncovers and examines the confusion in antiquity between Aristotle's hypothetical syllogistic and Stoic logic, and offers a fresh perspective on the ...

reasons for the disappreciation as well as for the rehabilitation of Stoic logic; it is found in I. M. Bochenski's Ancient Formal Logic (Amsterdam, 1951), and it clearly portrays the difference in attitude of the logicians of the twentieth century towards the Stoic logical system.

In this paper I would like to argue that Stoic logic is a kind ofrelevant logic rather than the classical logic. To realize this purpose I willtry to keep as close as possible to Stoic calculus as expressed with the helpof their arguments.

This paper is an attempt to reject the classical interpretation of Stoic ethics as virtue ethics. The typical assumptions of this interpretation, that virtue is the supreme good and that happiness can be reduced to virtue, are questioned. We first lay out the conceptual framework of Stoic philosophy and present an outline of their reduction of happiness to virtue. The main part of the paper provides an argument for reinterpretation of virtue as rationality. In the last part of (...) the paper – more speculative – we argue against the possibility of reduction of happiness to virtue. The main point of the argument is the conceptual incompatibility of the two notions. (shrink)

A. A. Long; VI*—The Logical Basis of Stoic Ethics, Proceedings of the Aristotelian Society, Volume 71, Issue 1, 1 June 1971, Pages 85–104, https://doi.org/10.10.

An overview of the ancient philosophers and their philosophical system (divided into the fields of logic, physics, and ethics) comprising the living, organic, enduring, and evolving body of interrelated ideas identifiable as the Stoic perspective.

This paper deals with Aristotelian and Stoic logic. In the first part the author writes about the history of logic and shows, why Stoic logic had not been studied properly from the Middle Ages up to the beginning of the 20th century, when an increasing interest in the study of Stoic logic is visible. The paper describes the character of Aristotelian and Stoic logic respectively. Stoic logic is first introduced (...) as a system of propositional logic. On this basis a complementarity between the two logical systems of Antiquity is stated. The ways to support the thesis that Stoic logic involved some features of predicative logic are shown at the end of the first part. The ways in which the two rival logical systems were perceived by the authors of the first centuries A. D., namely Galen and Boethius, is described in the second part. Galen is seen as the first to develop an objective synthesis of the two systems. In his Eisagogé dialektiké he tries to show the goals of both rival logical systems. Boethius, in his De hypotheticis syllogismis, is also combining the Aristotelian and the Stoic logic. However, his synthesis is different from Galen's because Boethius probably forms the synthesis unknowingly and indirectly through compiling his confused sources. (shrink)

I argue that the Stoic logic is explosive. The claim applies to the Stoics' syllogistic in the strictest sense, because there is a provable syllogism which qualifies as a principle of explosion. It applies also to the general consequence operation, in the sense that every sentence is derivable from any pair containing both a sentence and the negation of the sentence. Finally, it applies to the connective of implication (conditional), in the sense that any conditional is derivable, providing (...) its antecedent is a conjunction of a sentence and the negation of the sentence. All three claims allow weakening, i.e., additions of extra premises to an inference or extra conjuncts to the antecedent of an implication, respectively. Consequently, no concept of relevance, let alone paraconsistency or connexivity is applicable to the Stoic logic; in particular, the Stoics' connective of implication is either material (Boolean) or formal (strict). (shrink)

ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental (...) rules which establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out. (shrink)

Literature on the Stoa usually concentrates on historical accounts of the development of the school and on Stoicism as a social movement. In this 1977 text, Professor Rist's approach is to examine in detail a series of philosophical problems discussed by leading members of the Stoic school. He is not concerned with social history or with the influence of Stoicism on popular beliefs in the Ancient world, but with such questions as the relation between Stoicism and the thought of (...) Aristotle, the meaning and purpose of such Stoic paradoxes as 'all sins are equal', and the philosophical interrelation of Stoic physics and ethics. There are chapters on aspects of Stoic logic and on the thought of particular thinkers such as Panaetius and Posidonius, but ethical problems occupy the centre of the stage. (shrink)

This paper shows that, for the Hertz–Gentzen Systems of 1933, extended by a classical rule T1 and using certain axioms, all derivations are analytic: every cut formula occurs as a subformula in the cut’s conclusion. Since the Stoic cut rules are instances of Gentzen’s Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a “relevance criterion” and of two “balance criteria”, and (...) hence that a particular derivable sequent has no derivation that is “normal” in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T1 rule and the instances of Cut with just two antecedent formulae in the first premiss. OPEN ACCESS. (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.

After Plato's Forms, and Aristotle's substances, the Stoics posited the fundamental reality of lekta - the meanings of sentences, distinct from the sentences themselves. This volume analyses the resulting unique, complex, and consistent cosmic view in which lekta are the keystones of the structure of reality: they are all there is to say.

Lives of the stoics (Zeno, Aristo, Herillus, Cleanthes, Sphaerus, Chrysippus) on philosophy -- Logic and theory of knowledge -- Perception, knowledge, and sceptical attack -- The stoic-academic debate and Cicero's testimony -- Conceptions and rationality -- Physics -- Theology -- Bodily and non-bodily realities -- Structures and powers -- The soul -- Fate -- Ethics -- The general account in Diogenes Lartius -- The account preserved by Stobaeus -- The account in Cicero on goals -- Other evidence for (...) stoic ethics -- Passions and the goal : criticism within the stoic school and the evidence of Galen -- A critique from the academic-peripatetic point of view -- Pyrrhonist critique of basic ethical concepts -- Later stoic ethics : a sampler -- Musonius Rufus -- Seneca -- Epictetus. (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)

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)