Stoics: Logic

In this essay, I argue that Frege plagiarized the Stoics --and I mean exactly that-- on a large scale in his work on the philosophy of logic and language as written mainly between 1890 and his death in 1925 (much of which published posthumously) and possibly earlier. I use ‘plagiarize' (or 'plagiarise’) merely as a descriptive term. The essay is not concerned with finger pointing or casting moral judgement. The point is rather to demonstrate carefully by means of detailed evidence (...) that there are numerous (over a hundred) and extensive parallels both in formulation and --more importantly-- in content between the Stoics and Frege, parallels so plentiful that one would be hard pressed to brush them off as coincidence. These parallels include several that appear to occur in no other modern works that were published before Frege’s own and were accessible to him. Additionally, a cluster of corroborating historical data is adduced to support the suggestion, showing how easy it would have to been for Frege to plagiarize the Stoics. This (first) part of the essay is easy to read and vaguely entertaining, or so I hope. (shrink)

Since Mates’ seminal Stoic Logic there has been uncertainty and debate about how to treat the term anapodeiktos when used of Stoic syllogisms. This paper argues that the customary translation of anapodeiktos by ‘indemonstrable’ is accurate, and it explains why this is so. At the heart of the explanation is an argument that, contrary to what is commonly assumed, indemonstrability is rooted in the generic account of the Stoic epistemic notion of demonstration. Some minor insights into Stoic logic ensue.

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)

Here I propose an interpretation of the ancient Stoic psychological theory on which (i) the concepts that an adult human possesses affect the content of the perceptual impressions (φαντασίαι αἰσθητικαί) she forms, and (ii) the content of such impressions is exhausted by an ‘assertible’ (ἀξίωμα) of suitable complexity. What leads the Stoics to accept (i) and (ii), I argue, is their theory of assent and belief formation, which requires that the perceptual impression communicate information suitable to serve as the content (...) of belief. In arguing for (i), I reject a rival interpretation on which conceptualization occurs subsequently to the formation of a perceptual impression. In arguing for (ii), I deny that perceptual impressions have two kinds of content: one formulated in an assertible, the other sensory, featuring independently of this assertible. I explore the implications of (i) and (ii) for the Stoic theory of emotions, expertise, and rationality, and argue that they shed new light on the workings of impression, assent, and belief. (shrink)

At the heart of the Stoic theory of modality is a strict commitment to bivalence, even for future contingents. A commitment to both future truth and contingency has often been thought paradoxical. This paper argues that the Stoic retreat from necessity is successful. it maintains that the Stoics recognized three distinct senses of necessity and possibility: logical, metaphysical and providential. Logical necessity consists of truths that are knowable a priori. Metaphysical necessity consists of truths that are knowable a posteriori, a (...) world order according to certain metaphysical principles and natures that god crafts within the constraints of matter. Finally, what is providentially necessary is what occurs according to the chain of fate, but only once it is in process or past. -/- The method of the paper is a close reading of Diogenes Laertius 7.75, adducing broad textual evidence along the way, to show that the Stoic theory of modality embraces Philonian possibility, both that which is capable of being true as a matter of logical consistency, and that which is possible according to the bare fitness of the entity. What differentiates the Stoics from Philo is their additional commitment to possibility as opportunity, resisting the collapse of determinism into necessity. (shrink)

Chrysippus often talks as if there is a third option when we might expect that two options in response to a question are exhaustive. Things are true, false or neither; equal, unequal, or neither; the same, different, or neither.. and so on. There seems to be a general pattern here that calls for a general explanation. This paper offers a general explanation of this pattern, preserving Stoic commitments to excluded middle and bivalence, arguing that Chrysippus employs this trichotomy move when (...) he wishes to argue that apparent contradictories are only contraries, and wishes to endorse a third option. This general explanation of the pattern of trichotomies sheds light on a number of interpretive puzzles, including on Chrysippus's response to the paradox of the cone. The purpose of these trichotomies is also discussed, and it is suggested that they originate from the dialectical context in which philosophical problems were posed. (shrink)

Rather than considering ordinary language as deficient and incapable of grasping the structure of reality, the Stoics set out a theory, based on their notion of a lekton, by which ordinary language is a reflection of the structure of lekta which themselves are constitutive of reality.

The principate of Augustus coincided with a surge of interest in the short Aristotelian treatise which we now entitle Categories, contributing to its later installation at the outset of the philosophical curriculum and its traditional function as an introduction to logic. Thanks in part to remarks made by Plutarch and Porphyry , the origin of this interest has often been traced to Andronicus of Rhodes: his catalogue and publication of the Aristotelian corpus began with the Categories and may have drawn (...) fresh attention to a previously obscure treatise. But the later Neoplatonic sources name several other philosophers who also discussed the Categories and played an important role in crafting its interpretation during the first centuries of our era. For example, the Neoplatonist Simplicius discusses the views of Stoics and Platonists who questioned the Categories' value as a treatment of grammar or ontology, while others defended its usefulness as an introduction to logic. These early debates, as these later sources suggest, exercised a lasting influence on the shape of subsequent philosophy and philosophical education within and beyond the Aristotelian tradition. (shrink)

The paper deals with the possible readings of The Reaper Argument premisses. Some conjectures related to the Stoics’ alleged proof of the argument are discussed.

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)

The Lazy Argument, as it is preserved in historical testimonies, is not logically conclusive. In this form, it appears to have been proposed in favor of part-time fatalism (including past time fatalism). The argument assumes that free will assumption is unacceptable from the standpoint of the logical fatalist but plausible for some of the nonuniversal or part-time fatalists. There are indications that the layout of argument is not genuine, but taken over from a Megarian source and later transformed. The genuine (...) form of the argument seems to be given in different form and far closer to Megarian logical fatalism and its purpose is not to defend laziness. If the historical argument has to lead to a logically satisfactory solution, some additional assumptions and additional tuning is needed. (shrink)

Argues that Seneca distinguishes two modes of philosophical learning understood as concept formation: fortifying accretion and critical weeding. Progress is achieved by alternating between the two modes. A reading of Epistula moralis 102 illustrates the two types of philosophical discourse Seneca employs for each of the two modes: dialectical argumentation and high-minded “big talk,” very often in a style alluding to and evocative of Plato.

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)

Demonstrates the sophistication of Seneca’s Stoicism by setting his contributions within the context of his school. Seneca’s contributions to physics, metaphysics, logic, determinism, theodicy and eschatology are set within a systematic reconstructions of Stoic positions. Ample documentation of sources and scholarship as well as the thematic, handbook-like structure allow for this book to be used as a look-up tool and introduction to the Stoic cosmos and the place of humans within it. -/- There are a number of new readings and (...) interpretations, e.g. concerning the corporeality of the principles; mixture (krasis di’ holôn); the elements and pneuma; identity; the relation of God and humans; Seneca’s attitude to logic; Seneca’s philosophy of time; the role of lekta in defining reality and action; Stoic and Senecan theodicy; Seneca’s contribution to the determinism debate. (shrink)

The philosophers and scholars of the Hellenistic world laid the foundations upon which the Western tradition based analytical grammar, linguistics, philosophy of language, and other disciplines probing the nature and origin of human communication. Building on the pioneering work of Plato and Aristotle, these thinkers developed a wide range of theories about the nature and origin of language which reflected broader philosophical commitments. In this collection of nine essays, a team of distinguished scholars examines the philosophies of language developed by, (...) among others, Plato, Aristotle, Epicurus, the Stoics, and Lucretius. They probe the early thinkers' philosophical adequacy and their impact on later theorists. With discussions ranging from the Stoics on the origin of language to the theories of language in the twelfth and thirteenth centuries, the collection will be of interest to students of philosophy and of language in the classical period and beyond. (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)

ABSTRACT: This paper traces the earliest development of the most basic principle of deduction, i.e. modus ponens (or Law of Detachment). ‘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)

The article confronts one of the ἄποpοι λόγοι discussed in ancient Eristic-Stoic logic: the famous “cuckold” (κερατίνης), where an interrogator has his respondent to admit to have been or still be cuckolded. The source of the problem is a principle of dialectics related to the principle of the excluded-middle according to which a question admits only a positive or a negative answer. To the question “Have you ceased to be cuckolded?” both answers seem to presuppose that the respondent has been (...) cuckolded. To solve the problem one can either deny the dialectical principle of the excluded-middle or (as Stoic logic does) introduce the idea that negation refers not simply to the predicate (as in Aristotelian logic) but to the entire proposition (by denying the existence of the subject, for example). (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 ...

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)

Bobzien presents the definitive study of one of the most interesting intellectual legacies of the ancient Greeks: the Stoic theory of causal determinism. She explains what it was, how the Stoics justified it, and how it relates to their views on possibility, action, freedom, moral responsibility, moral character, fatalism, logical determinism and many other topics. She demonstrates the considerable philosophical richness and power that these ideas retain today.

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: 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)

The article confronts one of the ἄποpοι λόγοι discussed in ancient Eristic-Stoic logic: the famous “cuckold” (κερατίνης), where an interrogator has his respondent to admit to have been or still be cuckolded. The source of the problem is a principle of dialectics related to the principle of the excluded-middle according to which a question admits only a positive or a negative answer. To the question “Have you ceased to be cuckolded?” both answers seem to presuppose that the respondent has been (...) cuckolded. To solve the problem one can either deny the dialectical principle of the excluded-middle or (as Stoic logic does) introduce the idea that negation refers not simply to the predicate (as in Aristotelian logic) but to the entire proposition (by denying the existence of the subject, for example). (shrink)

Stoic work on ambiguity represents one of the most innovative, sophisticated and rigorous contributions to philosophy and the study of language in western antiquity. This book is both a comprehensive survey of the often difficult and scattered sources, and an attempt to locate Stoic material in the rich array of contexts, ancient and modern, which alone can guarantee full appreciation of its subtlety, scope and complexity. The comparisons and contrasts which this book constructs will intrigue not just classical scholars, and (...) philosophers, but also logicians, theoretical linguists, communication theorists and historians of grammar and of literary theory. The Stoics on Ambiguity is designed to be intelligible to readers with no Greek or Latin. (shrink)

This monograph discusses the sources for ancient propositional logic, mainly in Sextus Empiricus and Diogenes Laertius bk. VII. It is argued that most of the sources in Sextus which have hitherto been taken to be sources for Stoic logic either do not report Stoic logic at all or report pre-Chrysippean Stoic logic. These texts report (in the first case) a group labelled the Dialecticians whose most prominent members were Diodorus Cronus and Philo or else (in the second case) early Stoic (...) logicians heavily influenced by the Dialecticians. The texts discussed concern the theory of signs, the theory of proof and the classifications of propositions and of arguments. (shrink)