This paper discusses the reports in Diogenes Laertius and in Sextus Empiricus concerning the classification of propositions. It is argued that the material in Sextus uses a source going back to the Dialectical school whose most prominent members were Diodorus Cronus and Philo of Megara. The material preserved in Diogenes Laertius, on the other hand, goes back to Chrysippus.

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)

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)

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: 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: 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.

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)

The early Greek Stoics were the first philosophers to recognize the object of normal human perception as predicative or propositional in nature. Fundamentally we do not perceive qualities or things, but situations and things happening, facts. To mark their difference from Plato and Aristotle, the Stoics adopted phantasia as their word for perception.

The Stoics are traditionally regarded as the founders of propositional logic. However, this is not entirely correct. They developed a theory of inference from signs (omens). And their theory became a continuation of the logical technique of Babylonian divination (in particular, of Babylonian medical forecasting). The Stoic theory was not so much propositional logic as it was a technique of propositional logic for databases consisting of IF-THEN expert rules. In the Babylonian divination, each event (...) has a positive or negative value and all events are connected to each other. The Stoics also developed this idea and proposed a special modal logic in which logical determinism is considered an axiom. The paper reconstructs the sign-inference of the Stoics, as well as their modal logic. In particular, two Stoic squares of oppositions are proposed (for signs and for modal operators), which differ markedly from Aristotle’s square. (shrink)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient (...) class='Hi'>logic texts. A renaissance in ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

In this paper we give a detailed comparison of the key elements of Frege’s formal language of thought and apparently similar views in Stoic formal logic. That is, we compare their views on the following topics: connectives, negation, simple sentences, propositional content, predicates and their incompleteness, and quantifications. We show that in most of these cases the similarities between Frege’s views and the Stoic views are only superficial. Frege’s views are far more systematic, better developed and can in (...) no case directly or fully be traced back to Stoic ideas. Furthermore, we show that Prantl, pace Bobzien 2021, is in most of these cases not a reliable source with respect to Stoic logic and that it is very unlikely that Frege’s views on logic were influenced by Prantl’s interpretation of Stoic formal logic. (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)

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 (...) class='Hi'>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)

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: 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.

In this extended 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. -/- (Corrigendum: At school, Frege likely had seven, not ten, years of intensive Latin and four years of intensive ancient Greek courses.). (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)

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: 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.

Discusses Boethius's logical monographs: his treatises on division, on categorical syllogisms, and most importantly, his works on the theory of topical argument and on hypothetical syllogisms. The theory of topics, as developed in late antiquity and known almost entirely through Boethius, concerns the devising of arguments that rest on obvious general principles but are not, in their basic formulation, formally valid deductions. In his work on hypothetical syllogisms, Boethius seems to take account of Stoic propositional logic, but in (...) fact, he does not properly grasp its principles. This chapter shows how Boethius's logical mongraphs fit with the Aristotelianizing approach of his commentaries. (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).

In this paper I argue that the Stoic theory of signs as reported by Sextus Empiricus in AM and in PH belongs to Stoic logicians which precede Chrysippus. I further argue that the PH-version of this theory presupposes the version in AM and is an attempt to improve the older theory. I tentatively attribute the PH-version to Cleanthes and the AM-version to Zeno. I finally argue that the origin of this Stoic theory is to be found in the Dialectical school (...) (probably Philo of Megara) whose theory of signs has been preserved in Pseudo-Galen, Historia philosopha cap. 9. (shrink)

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 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)

Logic is an ancient discipline that, ever since its inception some 2500 years ago, has been concerned with the analysis of patterns of valid reasoning. Aristotle first developed the theory of the syllogism (a valid argument form involving predicates and quantifiers), and later the Stoics singled out patterns of propositional argumentation (involving sentential connectives). The study of logic flourished in ancient times and during the middle ages, when logic was regarded, together with grammar and rhetoric (the (...) other two disciplines of the trivium), as the foundation of humanistic education. (shrink)

This paper aims to explain the distinction between syllogisms and subsyllogisms in Stoic logic. Subsyllogisms replace at least one premise in a syllogism with a premise that is, according to Galen and Alexander, equipollent to the respective syllogistic premise. This equipollence is not synonymy of meaning between two linguistic expressions, but obtains between two propositions when they are true or false by the same standard. Subsyllogistic premises are simple propositions that are equipollent to the non-simple premises of the respective (...) syllogisms. For subsyllogistic premises that replace syllogistic premises consist of predicates combining with cases rather than of connectives or negations governing propositions. Because of this difference in their premises, subsyllogisms have a different logical form than syllogisms and are not formally valid. (shrink)

Being mainly concerned with the origins and development of formal logic, current “histories of logic” often devote scarce, if any, space to logic in the early modern period. In standard narratives, emphasis is put, on one side, on Aristotle’s Organon and on the Stoics’ logic of propositions, and on the other side, on the development of mathematical logic from Boole and Frege on. The picture often emerging from such reconstructions represents early modern philosophers—net of their (...) criticisms of Aristotelian syllogism— as largely estranged from the discipline. It is true that, the medieval period’s concern with semantic and inference issues has been accounted... (shrink)

ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and modal theorems, and to make (...) clear the exact relations between them; moreover, to elucidate the philosophical reasons that may have led Chrysippus to modify his predessors’ modal concept in the way he did. It becomes apparent that Chrysippus skillfully combined Philo’s and Diodorus’ modal notions, with making only a minimal change to Diodorus’ concept of possibility; and that he thus obtained a modal system of modalities (logical and physical) which fit perfectly fit into Stoic philosophy. (shrink)

An examination of a particular passage in Cicero's De fato—Fat. 13–17—is crucial to our understanding of the Stoic theory of the truth-conditions of conditional propositions, for it has been uniquely important in the debate concerning the kind of connection the antecedent and consequent of a Stoic conditional should have to one another. Frede has argued that the passage proves that the connection is one of logical necessity, while Sorabji has argued that positive Stoic attitudes toward empirical inferences elsewhere suggest that (...) that cannot be the right interpretation of the passage. I argue that both parties to the debate have missed a position somewhere between them which both renders a connection between antecedent and consequent that is not merely empirical and makes sense of the actual uses to which the Stoics put the conditional. This will be an account which grounds the connection between antecedent and consequent in a prolêpsis, a special kind of concept which plays a special epistemological role for the Stoics, especially in grounding scientific explanations. My contention will be that Stoic conditionals are true when there is a conceptually necessary connection between antecedent and consequent such that the former explains the latter via a prolêpsis. (shrink)

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated (...) by Herbrand’s Induction-Axiom Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

At Varro LL VI.56 and SE M 8.275-276, we find reports of the Stoic view that children and articulate non-rational animals such as parrots cannot genuinely speak. Absent from these testimonia is the peculiar case of the superficiality of the actor’s speech, which appears in one edition of the unstable text of PHerc 307.9 containing fragments of Chrysippus’ Logical Investigations. Commentators who include this edition of the text in their discussions of the Stoic theory of speech do not offer a (...) univocal account of the superficiality of the parrot’s, the child’s, and the actor’s speech. In this paper, I offer a reconstruction of the Stoic account of genuine and superficial speech and show that not only is there an account of superficial speech that univocally explains the superficiality of the speech of parrots, children, and actors, but that this account challenges traditional assumptions about the entities at the heart of the Stoic theory of language—lekta. It will turn out that genuine speech is the expression of a lekton by way of performing a speech act, and that this account of superficial speech can be used to explain other linguistic phenomena that are of interest to the Stoics, such as sentences in insoluble sophisms and sentences containing demonstratives that do not refer to anything in the subject term. Importantly, my reconstruction shows, against the near consensus view of lekta, that lekta do not primarily explain what makes an utterance meaningful. Rather, they primarily explain what makes an utterance an instance of genuine speech. (shrink)

The Stoic philosopher Chrysippus wrote extensively on the liar paradox, but unfortunately the extant testimony on his response to the paradox is meager and mainly hostile. Modern scholars, beginning with Alexander Rüstow in the first decade of the twentieth century, have attempted to reconstruct Chrysippus? solution. Rüstow argued that Chrysippus advanced a cassationist solution, that is, one in which sentences such as ?I am speaking falsely? do not express propositions. Two more recent scholars, Walter Cavini and Mario Mignucci, have rejected (...) Rüstow's thesis that Chrysippus used a cassationist approach. Each has proposed his own thesis about Chrysippus? solution. I argue that Rüstow's view is fundamentally correct, and that the cassationist thesis gains greater plausibility when viewed in light of a passage in Sextus Empiricus? Adversus mathematicos that the previous commentators have ignored, and when understood within the broader context of Stoic logical theory and philosophy of language. I close with a brief remark on the significance of Chrysippus? work for the modern debate on the semantic paradoxes. (shrink)

The standard account of the history of logic, in its crudest outline, runs as follows. Logic as a discipline was invented by Aristotle with his creation of syllogistic theory, which, despite the emergence of propositional logic in the work of the Stoics, refinements by mediæval logicians, and Leibniz's sketches of a characteristica universalis, dominated philosophy until the middle of the nineteenth century, when Boole's work on the "algebra of logic" and Frege's Begriffsschrift finally absorbed it (...) into a larger framework. Boolean and Fregean logic were rivals until the beginning of the twentieth century, when the superior power of the Fregean system, as developed by Peano and Russell, won the day, and modern logic as we know it began to exert its hegemony. However, as with any crude account, there are glaring omissions and anomalies that threaten its usefulness even as a first approximation, and Peirce's role in the story has perhaps been the most seriously underplayed of all. Whilst rooted in the Boolean tradition, Peirce also—admittedly slightly later, but nevertheless independently of Frege—invented a notation for quantifiers: a notation, indeed, that influenced modern notation to a far greater extent than Frege's own cumbersome two-dimensional script. Furthermore, Peirce's logic of relations was a significant development in its own right, and both this and the stirrings of the model-theoretic approach in his work was to feed into the mainstream tradition through the work of Schröder, Löwenheim, Skolem, and Tarski, in particular. (shrink)

The roots of the Lvov-Warsaw School can be traced back to Aristotle himself. But in later times we better put them into thinking GW Leibniz and who somehow inherited many of these ways of thinking, such as the philosopher and mathematician Bernhard Bolzano. Since he would pass the key figure of Franz Brentano, who had as one of his disciples to Kazimierz Twardowski, which starts with the brilliant Polish school of mathematics and philosophy dealt with. Among them, one of the (...) most interesting thinkers must be Jan Lukasiewicz, the father of many-valued logic.Jan Łukasiewicz began teaching at the University of Lvov, and then at Warsaw, but after World War II must to continue in Dublin. Some questions may be very astonishing in the CV of Łukasiewicz. For instance, that a firstly Polish Minister of Education in Paderewski cabinet, into the new Polish Republic, and also Rector for two times at Warsaw University, was awarded with a Doctorate ‘Honoris Causa’ in spring 1936, at University of Münster, into the maximum of effervescency of Nazism in Germany. The explanation must be their good relation with a very good friend, the former theologian, and then logician, Heinrich Schölz, who was the first Chairman of Mathematical Logic in German universities.Łukasiewicz firstly studied Law, and then Mathematics and Philosophy in Lvov. His doctoral supervisor was Kazimierz Twardowski, and in 1902 he obtain his Ph. D. title with a very special mention: ‘sub auspiciis Imperatoris’. Also he received a doctorate ring with diamonds from the Kaiser of the Austro-Hungarian Empire, Franz Joseph I.From 1902, Łukasiewicz was employed as a private teacher, and also as a desk in the Universitary Library of Lvov. So it was until 1904 when he obtained a scholarship to study abroad. He defends his ‘Habilitationschrift’ in 1906, entitled “Analysis and construction of the concept of cause”. This permits to give university courses. His first lectures were on the Algebra of Logic, according to the recent translation to Polish of this book of the French logician Louis Couturat.Between 1902 and 1906, Lukasiewicz continued his studies in the universities of Berlin and Leuwen. In 1906, by his ‘Habilitationschrift’, he obtain the qualification as university professor at Lvov. And the, in 1911, he was appointed as associate professor in his ‘alma mater’.Jan Łukasiewicz was also very active in historical research on logic, giving a new and up-to-date interpretation of Aristotle’s syllogism and of the Stoics’ propositional calculus. According to Scholz, the better pages on history of logic are due to him. And also, as Arianna Betti says, “Jan Lukasiewicz is first and foremost associated with the rejection of the Principle of Bivalence and the discovery of Many-Valued Logic.”The discovery of MVL by Lukasiewicz was in 1918, a little earlier than Emil Leon Post. According to Jan Wolenski, “although Post’s remarks were parenthetical and extremely condensed, Lukasiewicz explained his intuitions and motivations carefully and at length. He was guided by considerations about future contingents and the concept of possibility”. So, he introduces, firstly, three-valued logic, then four-valued logic, generalized to logics with an arbitrary finite number of veritative values, and finally, to logics with a countably infinite-valued number of such values.Very noteworthy is his treatment of the history of logic in the light of the new formal logic. Thus, not only he addressed the issue of future contingents departing from Aristotle, but also put in value logic of the Stoics, at least so far taken. In fact, Heinrich Scholz said, rightly, that Lukasiewicz had written the most lucid pages on the history of logic. (shrink)

The present paper investigates the question as to how and for what purposes the Middle Platonic author of the Anonymous Commentary on Plato’s Theaetetus uses Aristotelian and Stoic syllogistic in his interpretation of the Platonic text. This investigation shows that the commentator employs Aristotelian categorical syllogistic as an exegetical tool for reconstructing arguments in the Platonic text, enabling him not only to uncover doctrinal statements that are in his view hidden in the Platonic text, but also to dissociate Plato from (...) unwelcome propositions. By contrast, the commentator uses Stoic hypothetical syllogistic as a polemical tool for constructing ad hominem arguments against the Stoics. More precisely, the author exploits the Stoic type of deductive reasoning to draw anti-Stoic conclusions from premises that are accepted by the Stoics, and in doing so, he manages not only to refute Stoic doctrines, but also to corroborate the corresponding Platonic theories. (shrink)

The twentieth-century logical mainstream, derived from works by Łukasiewicz and Scholz, pictures the history of logic for the most part as the prehistory of Boolean–Fregean mathematical logic. Particularly, with respect to classical propositional calculus, the Stoic logic has been pictured as an early stage of it and Aristotle's or the Peripatetics' logic as a theory that assumes it. Although it was not emphasised, it follows that the ancient logics contain the principle of explosion. In the (...) endmost quarter of the twentieth century, a competitive view began to spread to the effect that the ancient systems of logic were paraconsistent or relevantistic. In the twenty-first century, the latter view prevails and has every chance of becoming a new orthodoxy. It is claimed that although in Łukasiewicz's argument for the classicality of ancient logics, there are gaps, it may be demonstrated that the ancient logics contain the principle of explosion. (shrink)

One of the most intriguing claims of Stoic logic is Chrysippus's denial of the modal principle that the impossible does not follow from the possible. Chrysippus's argument against this principle involves the idea that some propositions are ?destroyed? or ?perish?. According to the standard interpretation of Chrysippus's argument, propositions cease to exist when they are destroyed. Ide has presented an alternative interpretation according to which destroyed propositions persist after destruction and are false. I argue that Ide's alternative interpretation as (...) well as some versions of the standard interpretation conflict with Stoic doctrines about the nature of propositions. I propose another version of the standard interpretation based on Frede's account of the Stoic theory of the proposition. I hold that this version of the standard interpretation both escapes Ide's objections and is consistent with Stoic logic and philosophy of language. (shrink)

The anonimous scholiumOn all forms of syllogism was copied in 1884 from the Paris Codex 2064 by E. Richter. In 1899 M. Wallies published it in the preface to Ammonius' commentary on the Prior Analytics of Aristotle. There appear in that scholium, apart from the complex figure of Galenos, other characteristic forms of inference.Among these forms I found five so-called non-demonstrable stoic syllogisms, three modifications of the law of transposition of which the third is not mentioned by the authors of (...) Princ pia Mathematica, and a modification of the form of inference known as Euclid's law. This form of inference was applied by Euclid in mathematics and by Saccherius in syllogistics; it is mentioned for the first time by Cardan in a treatise of 1570 and later by Clavius in his commentary of 1574 on the Elements of Euclid and in the commentary on Theodosius'Sphaerica of the year 1586.In 1658 Erhard Weigel made the first attempt at refuting the logical law of Euclid as formulated by Cardan and Clavius and in 1686 James Bernoulli tried to prove it. (shrink)

Adopting a central feature of Stoic epistemology, Descartes treated belief as action that might be undertaken wisely or rashly, and enunciated a method for avoiding false belief, a discipline of the will “to include nothing more in my judgments than what presented itself to my mind with such clarity and distinctness that I would have no occasion to put it in doubt”.1 He called such acts of the will “affirmations”, i.e., acts of accepting sentences or propositions as true.

This paper examines Avicenna's remarks and consideretions on conditional propositions and the oppositional relations between them. İt also tries to detect his ancient and Arabic logic after him. İt aims to show that Avicenna's theory of conditionals has original aspects different from that of ancient logicans and that the improved this theory significantly. İt also aim at showing that Avicenna's theory is essentially Peripatetic, not Stoic.

This Companion provides a comprehensive guide to ancient logic. The first part charts its chronological development, focussing especially on the Greek tradition, and discusses its two main systems: Aristotle's logic of terms and the Stoic logic of propositions. The second part explores the key concepts at the heart of the ancient logical systems: truth, definition, terms, propositions, syllogisms, demonstrations, modality and fallacy. The systematic discussion of these concepts allows the reader to engage with some specific logical and (...) exegetical issues and to appreciate their transformations across different philosophical traditions. The intersections between logic, mathematics and rhetoric are also explored. The third part of the volume discusses the reception and influence of ancient logic in the history of philosophy and its significance for philosophy in our own times. Comprehensive coverage, chapters by leading international scholars and a critical overview of the recent literature in the field will make this volume essential for students and scholars of ancient logic. (shrink)

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)

This is a very rewarding topic when we consider the history of dialectical logic, although it may seem to be unrewarding if it is formal logic that is under discussion. We are familiar with Hegel's hypercritical writings on this subject, with his evaluation of the use of symbols in cognition as "empty play," and with his — to say the least — skeptical reaction to early attempts at the mathematization of logic. He characterized the construction of calculi (...) as "reduction" of logic to a mechanical level lower than that it had already attained. Leibniz's feat in the realm of "logic" was condemned by the great dialectician, and a subsequent experiment in the improvement of logic through mathematics, undertaken by a follower of Leibniz, Ploucquet, was described by Hegel as a transformation of the process of inference into an "utterly contentless" manipulation of sentences . The fact is, however, that Ploucquet anticipated Lambert and Hamilton in constructing a table of propositions in which not only subjects but predicates were quantified, thus opening the road to the future algebraization of logic. (shrink)