History of Logic from Aristotle to Gödel (www.historyoflogic.com)

by Raul Corazzon | e-mail: rc@ontology.co

Annotated Bibliography on the Ancient Stoic Dialectic (Second Part: H - Z)

Contents of this Section

Bibliography

  1. Hájek, Alan. 2009. "Two Interpretations of Two Stoic Conditionals." Logical Analysis and History of Philosophy no. 12:206-221

    "Four different conditionals were known to the Stoics. The so-called ‘first’ (Philonian) conditional has been interpreted fairly uncontroversially as an ancient counterpart to the material conditional of modern logic; the ‘fourth’ conditional is obscure, and seemingly of little historical interest, as it was probably not held widely by any group in antiquity. The ‘second’ (Diodorean) and ‘third’ (Chrysippean) conditionals, on the other hand, pose challenging interpretive questions, raising in the process issues in philosophical logic that are as relevant today as they were then.This paper is a critical survey of some modern answers to four of the most tantalizing of these questions; the issues that I will discuss arise out of interpretations of the Diodorean and Chrysippean conditionals as expressions of natural law, and as strict implications.I will reject these interpretations, concluding with my own proposal for where they should be located on a ‘ladder’ of logical strength." (p. 206)

  2. Hamelin, Octave. 1902. "Sur la logique des Stoïciens." Année Philologique no. 12:13-26

    Repris dans Cahiers philosophiques 2017/4 (n° 151), pp. 127-136.

  3. Hay, William. 1969. "Stoic Use of Logic." Archiv für Geschichte der Philosophie no. 51:145-157

    "To sum up. I began by reporting briefly the present widely held opinion that Stoic logic was a logic of propositions. I reminded us that in twentieth-century logic, the logic of propositions, consists of rules governing inferences according to their sentence-connectives and that it by no means exhausts the rules of logic. Rather propositional functions or predicates are added to that, and in turn many-place predicates are added. Some investigators have supposed that Stoic logic was confined to a logic of propositions. That restriction may be suggested by the concentration of the Stoics on singular propositions as those which express what exists most clearly and by their claim that all inferences depend on their logic. If, however, the Stoics had no more logic than the logic of propositions, they had no way of accounting for believing (much less for knowing) non-simple propositions in conditional or disjunctive forms, so that such non-simple propositions would be useful in inference.Evidence was introduced that the Stoics had and used a rule of instantiation in conditional propositions. This led us to see a use for their rules about the three kinds of simple propositions, those with indefinite subjects,tis,ti, 'someone,' 'something;' those with definite subjects, demonstrative articles such asoutos,touto, 'this one', 'that thing' and those with intermediate subjects, 'Socrates', 'Dion',anthropos, 'a man'.There is further evidence that the Stoics claimed to be able to rephrase universal propositions of the Peripatetic form as conditional propositions with indefinite subjects. Some philosophers from other schools acknowledged that the conditionals followed from the standard universal. There was disagreement about the converse. The charge was made that the Stoics failed to acknowledge eternal forms and that they replace them by things which existed in the mind only, or rather since they were corporealists in the body of the knower only. Another paper would be required to discuss the place of these grasps in the Stoic account of knowledge and of ethics, for action involves how I take things." (pp. 155-156)

  4. Hirzel, Rudolf. 1879. "De Logica Stoicorum." In Satura Philologa. Hermanno Sauppio Obtulit Amicorum Conlegarum Decas, 61-78. Berlin: Weidemann.

  5. Hitchcock, David. 2005. "The Peculiarities of Stoic Propositional Logic." In Mistakes of Reason. Essays in Honour of John Woods, edited by Peacock, Kent A. and Irvine, Andrew D., 224-242. Toronto: University of Toronto Press

    "Aristotle, the founder of logic, nowhere defines the concepts of argument and of validity. He simply uses them in his definition of a syllogism as ‘an argument in which, certain things being posited, something other than those things laid down results of necessity through the things laid down.’(1)(...)Between Aristotle, writing in the fourth century BCE, and Boole, writing more than two millennia later,(3) only one logician published a system of logic. That was Chrysippus (c. 280–207 BCE), the third head of the Stoic school. Chrysippus’s system of propositional logic was dominant for 400 years, until bits of it were eventually absorbed into a confused amalgamation with Aristotle’s categorical logic, a bowdlerization nicely described by Speca.(4) For centuries the system from which these surviving bits were extracted was forgotten. Only the careful work of such scholars as Mates,(5) Frede,(6) Hülser,(7) and Bobzien(8)One can show that a conjunction follows from its conjuncts, but not has allowed us to appreciate once again the achievement of Chrysippus.Despite its rigour and soundness, the system is oddly incomplete.One can show that a conjunction follows from its conjuncts, but not that either conjunct follows from the conjunction. One can detach the antecedent from a conditional, but not put it back on; in other words, there is no deduction theorem, no rule of ‘conditional proof’ or ‘ifintroduction.’ One can show what follows from an exclusive disjunction and the affirmation or denial of one of its disjuncts, but not what the exclusive disjunction follows from. Further, there is no evidence that anybody ever tried to extend the system.Why was Stoic propositional logic so incomplete? I shall argue that many of its peculiarities can be explained by the rather restrictive accounts of argument and of validity that Chrysippus adopted as the foundation of his system. The omissions from the system were not accidental oversights, or not just accidental oversights, but were dictated by the requirement that everything demonstrable in the system be a valid argument. With much more complex and restrictive accounts of argument and of validity than those adopted by Woods (*) in his reconstruction of Aristotle’s earlier logic, Chrysippus was forced into a much more restrictive formal system than contemporary classical propositional logic." (pp. 224-225)1 Topics I.1.100a25–27. Cf. Aristotle’s Sophistical Refutations 164b27–165a2 and Prior Analytics I.1.24b18–20. Here and elsewhere, translations are my own. For Aristotle’s works, I use the version of the Oxford Classical Texts, unless otherwise indicated.3 George Boole, The Mathematical Analysis of Logic (Cambridge: Macmillan, Barclay & Macmillan, 1847).4 Anthony Speca, Hypothetical Syllogistic and Stoic Logic, Philosophia Antiqua Volume 87 (Leiden: Brill, 2001).5 Benson Mates, Stoic Logic (Berkeley and Los Angeles: University of California Press, 1953).6 Michael Frede, Die stoische Logik (Göttingen: Vandenhoeck & Ruprecht, 1974).7 Karlheinz Hülser, Die Fragmente zur Dialektik der Stoiker, 4 vols. (Stuttgart-Bad Cannstatt: Friedrich Frommann, 1987–8). Cited henceforth as ‘FDS’ followed by the fragment number.8 Susanne Bobzien, ‘Stoic Syllogistic,’ Oxford Studies in Ancient Philosophy 14 (1996): 133–92. Susanne Bobzien, ‘The Stoics,’ in The Cambridge History of Hellenistic Philosophy, Kempe Algra, Jonathan Barnes, Jaap Mansfeld, and Malcolm Schofield, eds. (Cambridge: Cambridge University Press, 1999), 92–157.[(*) John Woods, Aristotle’s Earlier Logic (Oxford: Hermes Science, 2001).]

  6. Hossenfelder, Malte. 1967. "Zur Stoischen Definition von Axioma." Archiv für Begriffsgeschichte no. 11:238-241.

  7. Hülser, Karlheinz. 1983. "The Fragments on Stoic Dialectic: A New Collection." In Meaning, Use, and Interpretation of Language, edited by Bäuerle, Rainer, Schwarze, Christoph and Stechow, Arnim von, 235-249. Berlin: Walter de Gruyter

    "For the Stoics dialectic was the discipline where they developed their theory of cognition and of language as well as some kind of grammar und formal logic. All those topics were formed into a system and a lot of remarkable statements made about them. Hence, Stoic dialectic had much influence, and founded the western tradition of systematic linguistic theory. But the original writings of the Stoics are, nevertheless, lost. Thus, in order to study the origins of systematic linguistic thought, we have to collect the testimonies and fragments on Stoic dialectic from many scattered sources, i.e. from later authors who mentioned, reported or criticized Stoic ideas. In the last centuries this task was performed by different scholars. I only mention Rudolf T. Schmidt (1) and -- above all -- Hans v. Arnim whose 'Stoicorum veterum fragmenta' is the famous standard collection of fragments on all the three parts of Stoic philosophy up to now (2). With regard to Stoic dialectic Prof. U. Egli came up with the idea that it would be worth the trouble to collect the fragments once again. He applied for a research program, sponsored by the 'Deutsche Forschungsgemeinschaft (DFG)', and asked me to realize what he had in mind, the result of which is a new collection of fragments on Stoic dialectic, the subject of this paper.The formal data of this new collection are the following ones: The collection which amounts to 1257 fragment-numbers (plus ca. 70 additional a-numbers) embraces about 1800 texts, the greatest part of which is quoted in Greek or Latin as well as translated into German; various commentaries are inserted. All this comes to 978 crowded typewritten pages. Superadded are some indices and an introduction by the editor. The book is entitled Die Fragmente zur Dialektik der Stoiker - zusammengestellt, ins Deutsche übersetzt und teilweise kommentiert - von K. Hülser (the abbreviation of which will be FDS), and is forthcoming: In 1982 it is published in 8 volumes within the publications of the 'Sonderforschungsbereich 99' at the University of Konstanz (Fed. Rep. of Germany). This edition, though it has a small number of copies and no ISBN-number, serves its purpose as a citable on for the time being (available in the library of the University of Konstanz), but will be replaced by a more 'genuine' one as soon as possible. [*]As for the kind and the content of the new collection, three approaches will be offered in the following. The first one starts from the function of collections of fragments in general; it will explain why v. Arnim's collection is insufficient and a new one necessary, and consequently it leads to certain requirements concerning FDS. The second approach, then, starts from the arrangement of fragments in FDS and will show some systematic aspects of Stoic dialectic connected with it. The third one eventually is centered on the problem of intended completeness; in some cases this aim, being unterstood systematically, leads to interesting results though it widens the concept of fragments." (pp. 235-236)(1) R. T. Schmidt, Stoicorum grammatica, Halle 1839; repr. Amsterdam 1967. A German translation with an introduction and some additional notes by K. Hülser was published in Braunschweig / Wiesbaden 1979, completed by a bibliography on Stoic dialectic by U. Egli.(2) H. v. Arnim, Stoicorum veterum fragmenta Vol. IV (Indices, by M. Adler), Leipzig 1903-1905, 1924; repr. Stuttgart 1964.[* The definitive edition is: Die Fragmente Zur Dialektik Der Stoiker, Stuttgart: Frommann-Holzboog 1987-88, 4 volumes.]

  8. ———. 1992. "Sextus Empiricus und die Stoiker." Elenchos. Rivista di Studi sul Pensiero Antico no. 13:233-276.

  9. Iacona, Andrea. 2023. "Valid Arguments as True Conditionals." Mind no. 132:428–451

    Abstract: "This paper explores an idea of Stoic descent that is largely neglected nowadays, the idea that an argument is valid when the conditional formed by the conjunction of its premises as antecedent and its conclusion as consequent is true. As will be argued, once some basic features of our naïve understanding of validity are properly spelled out, and a suitable account of conditionals is adopted, the equivalence between valid arguments and true conditionals makes perfect sense. The account of validity outlined here, which displays one coherent way to articulate the Stoic intuition, accords with standard formal treatments of deductive validity and encompasses an independently grounded characterization of inductive validity."

  10. Ierodiakonou, Katerina. 1990. Analysis in Stoic Logic, University of London, London

    Unpublished dissertation (a PDF version can be downloaded from British Library Document Supply Service).Abstract: "This thesis focusses on the notion of analysis in Stoic logic, that is to say on the procedure which the Stoic logicians followed in order to reduce all valid arguments to five basic patterns. By reconsidering the uses of its Aristotelian homonym and by examining the evidence on the classification of Stoic arguments, I distinguish two methods of Stoic analysis and I discuss their rules: (i) the analysis of non-simple indemonstrables, which constitutes a process of breaking up an argument by means of general logical principles ; and (ii) the analysis of (yllogistic) arguments, which replaces demonstration and is effected by employing standard well-determined rules. The ancient sources provide us with concrete examples illustrating the first type of analysis; however, there is no single text that reports the exact procedure of analysing (syllogistic) arguments. Modern scholars have reconstructed in different ways this type of Stoic analysis; I deal with all of them separately and show that the proposed reconstructions are insightful but historically implausible. Based on the textual materiel concerning the notion of analysis not only in its Stoic context but also in some other of its uses, and especially in mathematical practice, I suggest an alternative reconstruction of the Stoic method of reducing valid arguments to the basic indemonstrables."

  11. ———. 1990. "Rediscovering Some Stoic Arguments." In Greek Studies in the Philosophy and History of Science, edited by Nicolacopoulos, Pantelis, 133-148. Dordrecht: Kluwer.

  12. ———. 1993. "The Stoic Indemonstrables in the Later Tradition." In Dialektiker und Stoiker. Zur Logik der Stoa und ihrer Vorläufer, edited by Döring, Klaus and Ebert, Theodor, 187-200. Stuttgart: Franz Steiner

    "The Stoic logical system is commonly presented as based on five types of argument called “indemonstrables” (άναπόδεικτοι), and undoubtedly, our knowledge of them is more certain than that of any other aspect of Stoic logic (M. 8.224-227; P. 2.157-158; D.L. 7.80-81 =FDS 1131,1128,1036). Fundamental though it may have been, however, even this part of Chrysippean doctrine has not escaped significant development overtime. In fact, there are brief remarks in our main sources that explicitly suggest the existence of alternative lists of indemonstrable arguments accepted by different groups of Stoic logicians:είσί δέ καί άναπόδεικτοί τινες, τω μή χρή£ειν άποδείξεως, άλλοι μεν παρ' άλλοις, παρά δέ τω Χρυσίππω πέντε (D.L. 7.79);πολλούς* μέν αναπόδεικτους όνειροπολουσιν, πέντε δέ τούτους μάλιστα εκτίθενται (?. 2.157 = FDS 1128).Fortunately enough, some interesting passages, also belonging to the later tradition, elaborate extensively on the criticisms and alterations, which the standard list has undergone in the centuries following its introduction. The present paper concentrates on these texts and points out only a few of the issues raised by these accounts of unorthodox developments of the doctrine of the five basic indemonstrables. More specifically, I intend to focus on two alternative lists of indemonstrable arguments that differ from Chrysippus’ list in adding extra types of inference as well as in objecting to the usefulness and application of others." (p. 187)ReferencesFDS = K. Hülser, Die Fragmente zur Dialektik der Stoiker. 4 Bde. Stuttgart/Bad Cannstatt 1987-1988

  13. ———. 2006. "Stoic Logic." In A Companion to Ancient Philosophy, edited by Gill, Mary Louise and Pellegrin, Pierre, 505-529. Malden: Blackwell

    "Conclusion. As I indicated at the beginning of the chapter, it was only towards the middle of the twentieth century that Stoic logic began to be studied on its own merits and not as an appendix to Aristotle's syllogistic. To a great extent it was the revival of interest in the logical contributions of the Stoics that convinced scholars to investigate more carefully the other parts of Stoic philosophy, namely ethics and physics. The literature on Stoic logic that has since been published has managed to reconstruct a logical calculus, which still surprises us with its sophistication and its similarities to modern systems of logic. At the same time, though, it also has become clear that we should not fail to take seriously into account what differentiates Stoic logic from its modern counterparts. For only in this way can we get a better understanding of how the history of logic has evolved in close connection to the other parts of philosophy, and more importantly, only in this way do we have a chance to appreciate the peculiar features and insights of ancient logic." (p. 527)

  14. ———. 2014. "The Stoic System: Logic and Knowledge." In The Routledge Companion to Ancient Philosophy, edited by Warren, James and Sheffield, Frisbee, 438-454. New York: Routledge.

  15. ———. 2019. "Dialectic as a subpart of Stoic philosophy." In Dialectic after Plato and Aristotle, edited by Bénatouil, Thomas and Ierodiakonou, Katerina, 113-133. Cambridge: Cambridge University Press.

  16. Imbert, Claude. 1980. "Stoic Logic and Alexandrian Poetics." In Doubt and Dogmatism. Studies in Hellenistic Epistemology, edited by Schofield, Malcolm, Burnyeat, Myles and Barnes, Jonathan, 183-216. Oxford: Clarendon Press.

  17. Jedan, Christoph, and Strobach, Nico. 2002. Modalities by Perspective. Aristotle, the Stoics and a Modern Reconstruction. Sankt Augustin: Academia Verlag.

  18. Jennings, Raymond Earl. 1994. The Genealogy of Disjunction. New York: Oxford University Press

    See Chapter 10 Stoic Disjunction, pp. 252-275.

  19. Kahn, Charles H. 1969. "Stoic Logic and Stoic LOGOS." Archiv für Geschichte der Philosophie no. 51:158-172

    '"I turn now to the principal claim of Professor Hay's paper (*): that the logic of the Stoics was not exclusively a logic of propositions but that it included arguments whose major premiss was, in effect, a universally quantified conditional, "(x) (If Ax, then Bx)," instead of the ordinary conditional composed of two self-contained sentences "If A, then B." Hay brings evidence of three sorts to bear in favor of this thesis. (1) First of all, there are the logical and historical considerations already alluded to: how could the Stoics have claimed to reduce all valid arguments, including the Aristotelian syllogism, to their five undemonstrated schemata, if they did not have some device equivalent to quantification"? (2) Secondly, there is the question of the epistemic function of logic: where the major premiss is a conditional such asIf Plato lives, then Plato breathes interpreted truth-functionally, and I am able to draw the conclusionPlato breathes, how could I be in a position to know or believe the conditional premiss without already knowing or believing the conclusion? (For the truth of the conditional depends upon the truth of the consequent in this case, since the antecedent is taken as true.) But the epistemic problemwill not arise in this form if the major premiss may be universally quantified. I do not need to know that Socrates breathes - I do not need to know anything about Socrates at all - in order to agree that if anything is alive, that same thing (or animal) breathes. (3) Furthermore, Hay calls our attention (and apparently for the first time) to several decisive texts in which the Stoics make theoretical use of generalized conditionals of the form 'If anyone is born under the Dog Star, he will not die at sea.' Finally (4) Hay suggests that the Stoic motive for the alleged reformulation of universal propositions as conditionals was their desire to avoid positing essences or classes or universals of any sort.I am inclined to believe that Hay's principal thesis is correct, at least in principle; but it raises new problems almost as serious as those it solves. First of all, did the Stoics realize that they were introducing quantification when they offered a conditional compounded in this way of two indefinite propositions? If so, this seems to defeat their claim that all valid arguments could be reduced to their five undemonstrated forms. But if they didnot see this, they were poorer logicians than Aristotle at a crucial point they will have set up a propositional calculus only at the cost of distorting the facts concerning quantification. We seem to be faced with a dilemma. Either Stoic logic is based solely on the propositional connectives, and then it is epistemically sterile. (This appears to be Mueller's view.) Or else it involves generalized conditionals and a rule of instantiation, but then it is defective as logic since we are left without any account of the quantified conditional. (a) I suspect that the latter is likely to be true, and that by formulating indefinite conditionals to achieve generality, and then instantiating for a definite, ostensibly indicated subject, the Stoics believed that they could in fact do without quantification, i. e. without any theory involving 'all' and 'none.' " (pp. 163-164)(*) [Stoic Use of Logic, 1969](a) I have oversimplified in order to put the problem sharply. It is worth noting that the decisive text fromDe Fato is explicitly meta-linguistic: "If G (a generalized conditional) is true, then C (an ordinary conditional) is also true" (see Hay, note 15). Therefore arguments making use of such a rule of instantiation will be valid but not necessarily reducible to one of the five undemonstrated schemata (compare the examples in Mates, p. 64 and p. 65 n. 32). In the Symposium discussion in St. Louis several suggestions were made for reconstructing the Stoic generalized conditional without quantification theory, as the meta-linguistic representation for a "bundle of individual conditionals" (Quine,Methods of Logic, p. 13), much as an axiom schema may represent an infinite set of individual axioms. I leave it to others to decide how far such a suggestion can be worked out systematically.

  20. Kidd, Ian G. 1989. "Orthos Logos as a Criterion of Truth in the Stoa." In The Criterion of Truth. Essays Written in Honour of George Kerferd, edited by Huby, Pamela and Neal, Gordon, 137-149. Liverpool: Liverpool University Press.

  21. Kneale, William, and Kneale, Martha. 1962. The Development of Logic. Oxford: Clarendon Press

    Reprinted 1975 with corrections.See Chapter III:The Megarian and the Stoics pp. 113-176.

  22. Labarge, Scott. 2002. "Stoic Conditionals of Necessity and Explanation." History and Philosophy of Logic no. 23:241-252

    Abstract: "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 aprolêpsis, 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 aprolêpsis."

  23. Leeman, Anton Daniël. 1954. "Posidonius the Dialectician in Seneca's Letters." Mnemosyne no. 7:233-240

    "In this article, which is the fourth and last of a series of papers on some aspects of the relations between Posidonius and Seneca's later epistles, I shall discuss a number of dialectical passages in these epistles, their philosophical source - Posidonius, as I hope to establish - and finally, the philosophical attitude of this source (1).In my examination of Seneca's conception of dialectics, I have already stressed his scepticism and distrust of this way of arguing (III). In his earlier letters especially he appears to regard dialectics as a waste of time. More than once he indicates how, in his opinion,the sophisms of the dialecticians ought to be treated, viz. with the help of striking examples from history and every-day life, in the form of rhetorical sententiae. This opinion, however, does not keep him from paying more and more attention to dialectical disputations.He apparently records them from technical, philosophical sources." (p. 17)(1) The previous articles are to be found in this periodical, S. IV, Vol, IV, 1951, p. 175 sqq. (The Epistolary Form of Sen. Ep. 102) [indicated as I]; Vol. V, 1952, p. 57 sqq. (Seneca and Posidonius: Philosophical Commentary on Sen. Ep. 102, 3-19) [indicated as II]; Vol. VI, 1953, p. 307 sqq. (Seneca's Plans for a Work "Moralis philosophia" and. their Influence on his later Epistles) [indicated as III].

  24. Lefebvre, René. 2007. "Représentation et évidence : les stoïciens face à leurs adversaires de l'Académie." Elenchos.Rivista di Studi sul Pensiero Antico no. 28:337-367.

  25. Löbl, Rudolf. 1986. Die Relation in der Philosophie der Stoiker. Amsterdam: Rodopi

    Inhaltsübersicht: Literaturangaben 7; Einleitung 13; Teil I: 17; A. Physis 19; B. Logos 62; Teil II: 111; A. Die äusseren Relationen 113; B. Die inneren Relationen 129; C. Die transcendentale Relationen 134; Excursus: Zu Physik 141-150.

  26. Long, Anthony Arthur. 1978. "Dialectic and the Stoic Sage." In The Stoics, edited by Rist, John M., 101-124. Berkeley: University of California Press

    Reprinted in: A. A. Long, Stoic Studies, Cambridge: Cambridge University Press, 1996, pp. 85-106.

  27. López-Astorga, Miguel. 2016. "Pseudo-Conditionals and Causal Assertibles in Stoic Logic." Principia no. 3:417-426.

  28. Luhtala, Anneli. 2000. On the Origin of Syntactical Description in Stoic Logic. Münster: Nodus Publikationen

    Contents: Acknowledgments 9; 1. Introduction 11; 2. Ancient Grammar 16; 3. Truth, Meaning and Existence 30; 4. Aristotle 40; 5. The Stoics; 6. Apollonius Dyscolos 146; 7. General conclusions 193; Bibliography 197; Index Nominum 209-214."This study examines the dialectical origin of syntactical description in our traditional grammar. Two famous texts take pride of place in containing the first descriptions of a 'clause' in Greek literature, namely Plato's Sophist and Aristotle's Peri hermeneias. These descriptions arose in the context of a more general inquiry into the nature of truth and language which gave rise to the first speculations on the form of the logical proposition in Greek Antiquity. By establishing as the unit of propositional analysis a combination of two linguistic items, Onoma (`name', 'noun') and rhema (`verb', 'predicate') these philosophers laid the foundation for the doctrine of the parts of speech which later constituted the core of ancient grammar. Their concern was to establish the two functional constituents of the proposition, roughly the subject and the predicate, by means of which true and false statements could be made. The object of their concern -- the minimal statement consisting of a noun and a verb -- came to figure as the point of departure for syntactical analysis when it began to be pursued in independent grammatical treatises. In the grammar of Apollonius Dyscolus (2nd century A.D.), which is our first extant grammatical treatise on syntax, syntactical description proceeds from the minimal self-sufficiency (autoteleia) of the linguistic expression. But the description of the minimal sentence by Apollonius bears witness to the distinctly Stoic origin of the notion of self-sufficiency." (p. 11)

  29. Łukasiewicz, Jan. 1967. "On the History of the Logic of Propositions." In Polish Logic 1920-1939, edited by McCall, Storrs, 66-87. Oxford: Oxford University ress

    Originally published in Polish as: "Z historii logiki zdan", Przeglad Filozoficzny, 37, 1934; translated by the author in German as:" Zur Geschichte der Aussagenlogik", Erkenntnis, 5, 1935, pp. 111-131.Translated in English also in: Ludwik Borowski (ed.), Jan Łukasiewicz, "Selected Works", Amsterdam: North-Holland, 1970, pp. 197-217.

  30. ———. 1967. "Philosophical Remarks on Many-Valued Systems of Propositional Logic." In Polish Logic 1920-1939, edited by McCall, Storrs, 40-65. Oxford: Oxford University Press

    Originally published in German as: "Philosophische Bemerkungen zu mehrwertighen Systemen des Aussagenkalküls", Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, 23, 1930.Translated in English also in: Ludwik Borowski (ed.), Jan Łukasiewicz, Selected Works, Amsterdam: North-Holland 1970, pp. 153-178.

  31. Mates, Benson. 1949. "Stoic Logic and the Text of Sextus Empiricus." American Journal of Philology no. 70:290-298

    "The text of Sextus Empiricus contains a number of corrupt places which can easily be corrected by reference to a few technical terms and elementary concepts of Stoic logic. It is the aim of the present paper to prove this assertion with respect to a certain class of cases and, in so doing. to show that any future editor of Sextus ought to have a clear understanding of Stoic logic." (p. 290)

  32. ———. 1953. Stoic Logic. Berkeley: University of California Press

    Second revised edition 1961.Contents: I. Introduction 1; Chapter I. § 1: The problem § 2: Stoic authors to be considered §3: Sources for Stoic logic; Chapter II. Signs, sense, and denotation 11; § 1: Exposition of the Stoic theory § 2: Comparison with modern theories; Chapter III. Propositions, truth, and necessity 27; § 1: Propositions § 2: Truth § 3: Necessity and Possibility; Chapter IV. Propositional connectives 42; § 1: Implication § 2: Disjunction § 3: Conjunction and the other logical connectives § 4: The interdefinability of the connectives; Chapter V. Arguments 58 § 1: Definition and Classification § 2: The five basic types of Undemonstrated Argument § 3: The Principle of Conditionalization § 4: The analysis of non-simple arguments § 5: Invalid arguments; Paradoxes; Chapter VI. Evaluations of Stoic logic 86; § 1: The judgments of Prantl and Zeller § 2: The confusion about συννεμένον § 3: Conclusion; Appendix A. Translations 95; Appendix B. Glossary 132; Bibliography 137; Indices -141-148."There are those who cannot write history without praising and blaming.Such persons, if they are favorably impressed by the newer studies of ancient and medieval logic, will feel that just as Prantl and Zeller praised Aristotelian logic and disparaged that of the Stoics, so now we should praise the Stoic logic and condemn the Aristotelian.There are also those who cannot write history without embracing certain huge generalizations which are supposed to make history intelligible.This sort of investigator will admit that Prantl, Zeller, and many others have been greatly misled by one such generalization, according to which Hellenistic times were times of decadence and decay in all branches of learning, and especially in philosophy. But he will conclude nothing more than that Prantl, Zeller, and the others have embraced the wrong generalization.In other words, he would propose that we excogitate another (and presumably better) hypothesis about Greco-Roman history and then proceed as before.Both tendencies are inimical to honest historical writing. There is no reason whatever to believe that an adequate history of logic or of anything else will have the relatively simple structure of a novel. The great generalizations, which are supposed to make the chaos of events intelligible, are, at best, of heuristic value. It must be remembered that they require more evidence than would be required for the support of any of the particular conclusions to be deduced from them. Hence, any conclusion based on such a generalization either can be established without the generalization or else is not warranted at all. But these grand views may well have no heuristic value either, for, as is amply demonstrated by the comments of Prantl and Zeller on Stoic logic, they sometimes have the effect of blinding the scholar to facts which he would otherwise be able to see.It is difficult to understand how any historian motivated by a desire to discover the truth (rather than by the desire to tell a good story) could share the satisfaction of Prantl and Zeller over the loss of the Stoic writings." (pp. p3-94)

  33. Mau, Jürgen. 1957. "Stoische Logik. Ihre Stellung gegenüber der Aristotelischen Syllogistik und dem Modernen Aussagenkalkül."

  34. Meyer, Herman. 1956. "Le prolongement de la logique stoïcienne dans la logique contemporaine." Revue Philosophique de la France et de l'Étranger no. 146:387-392

    "La logique contemporaine formalise des expressions de la forme suivante : << Il existe un x tel que ... >>, et << tous les x tels que ... >>. Les Stoïciens, en revanche, n'avaient pas poussé la théorie logique au delà des énoncés non analysés. C'est-à-dire que, dans leur logique, des expressions particularisées ou généralisées n'interviennent pas. Cependant, la logique des énoncés non analysés se trouve toujours à la base de la logique des mathématiques. Nous essayerons de brossér une esquisse visant à présenter les découvertes des logiciens stoïciens et à montrer de quelle façon, en se prolongeant au travers des siècles, elles figurent toujours parmi les thèmes principaux de la logique théorique contemporaine." (pp. 387-388)

  35. Mignucci, Mario. 1965. Il significato della logica stoica. Bologna: Patron

    Indice: Avvertenza 7; Introduzione 9; Cap. I - LO STATUS QUAESTIONIS. 1. Le interpretazioni del Prantl e dello Zeller 17; 2. La rivalutazione della logica stoica 19; 3. L’interpretazione del Lukasiewicz 29; 4. Le posizioni successive al Lukasiewicz 33; 5. Discussione dell’interpretazione del Lukasiewicz 40; Cap. II - LA CONCEZIONE DELLA LOGICA. 1. La rappresentazione 67; 2. La conoscenza intellettuale 80; 3. La definizione di esprimibile 88; 4. L’incorporeità degli esprimibili 96; 5. La logica come scienza filosofica 103; 6. La natura della dialettica 109; Cap. III - LA DOTTRINA DELLE PROPOSIZIONI. 1. La definizione di proposizione 119; 2. La polemica della scuola megarica sulla validità del condizionale 130; 3. La concezione stoica del condizionale 139; 4. Le proposizioni congiuntive e disgiuntive 148; Cap. IV. LA TEORIA DEGLI ARGOMENTI. 1. La definizione di argomento 157; 2. Gli argomenti anapodittici 166; 3. La teoria degli anapodittici e la sillogistica aristotelica 178; Bibliografia 191; INDICI. Luoghi citati 201; Nomi di persona 209-212.

  36. ———. 1967. "Il problema del criterio di verità negli stoici antichi." In Posizione e criterio del discorso filosofico, edited by Giacon, Carlo, 145-169. Bologna: Patron.

  37. ———. 1988. "The Stoic Notion of Relatives." In Matter and Metaphysics. Fourth Symposium Hellenisticum (Pontignano, August 21-28, 1986), edited by Barnes, Jonathan and Mignucci, Mario, 129-221. Napoli: Bibliopolis

    "The fragments of the Stoics which are explicitly concerned with a theory of relations are few, scattered and difficult to interpret. The largest of them is preserved in Simplicius' commentary on the Categories (165.32 ff.;SVF ii 403) and it expounds an important distinction which the Stoics made between two kinds of relatives. This doctrine is attributed to the Stoics, but no representative of the school is mentioned. Echoes of it are reflected in some sceptical arguments reported by Sextus Empiricus (M VIII 455-456) and Diogenes Laertius (IX 87-88) (1). Besides, there are some related passages in the scholia on Dionysius Thrax's Ars grammatica which are supposed to go back to Apollonius Dyscolus (II century A.D.), where, although the Stoics are not explicitly named, Stoic material is believed to be used and referred to (2). There is also a text of Sextus (M VIII 453-454; SVF II 404) in which a general definition of relatives is attributed by him to the Dogmatists and reasons can be given for saying that his Dogmatists must be identified with the Stoics. Finally, some passages in which the name of Chrysippus is tied to questions which are supposed to concern our problems are difficult to interpret and on closer inspection they reveal themselves not to pertain to the theory of relatives (3).In the face of this complicated situation in our sources, I will examine first Simplicius' passage, trying to disentangle it from spurious connections with other parts of the Stoic doctrine which have generated more than one misunderstanding of it. Secondly, I will inquire to what extent a possibly general definition of relatives implied in Simplicius' distinction is consistent with the statements reported by other sources, in order to determine whether Simplicius' report can be inserted in a coherent framework.This sketch of the plan of our inquiry shows that we confer a central role on Simplicius' passage, and this assumption might be disputed, since Simplicius is a late authority and no Stoic master of the first generation is mentioned in it. We will discuss these problems later. Whatever their solution might be, it must be pointed out that Simplicius' text is almost the only one in which a relevant aspect of the Stoic doctrine of relatives is expounded and discussed. The other sources are much vaguer and mostly concerned with a general characterization of the notion of relative. Therefore, it is difficult in this situation not to confer a special position on Simplicius passage." pp. 129-1301) These texts are not found in von Arnim's collection. They will be discussed in section VIII.(2) These passages too are not in von Arnim. We will examine them later (cf. sections XI-XII).(3) I am thinking especially of three passages we will consider later, namely Varro De lingua latina X 59 (SVF a 155); Plutarch, De Stoicorum repugnantibus 1054EF (SVF II 550); Aulus Gellius Noctes atticae VII 1, 1-6 (SVF II 1169): cf. sections XIV and XV.

  38. ———. 1993. "The Stoic Themata." In Dialektiker und Stoiker. Zur Logik der Stoa und Ihrer Vorläufer, edited by Döring, Klaus and Ebert, Theodor, 217-238. Stuttgart: Franz Steiner

    Reprinted in Andrea Falcon, Pierdaniele Giaretta (eds.), Ancient Logic, Language, and Metaphysics: Selected Essays by Mario Mignucci, New York: Routledge 2020, Part I, Essay III."One of the classical problems of Aristotle's syllogistic is how to interpret the reduction of imperfect to perfect syllogisms. Second- and third-figure syllogisms are imperfect, that is, not self-evident syllogisms, and they are shown to be valid by linking them to the perfect ones, namely the syllogisms of the first figure. (1) This link is normally interpreted as a deduction of imperfect syllogisms from perfect ones by means of some implicit rules of inference and some previously stated theorems mainly concerning the conversions of propositions. Aristotle does not make explicit the arguments by which imperfect syllogisms are derived from perfect syllogisms, and scholars have proposed different ways to formalize these deductions. (2) What is common to all these views is that they impose something on the text, that is, a formalization of an informal reasoning that is only sketched.It is the Stoics who produced a sort of formalization of a metalinguistic theory of deduction, although the Aristotelians, as usual, claimed that Aristotle had already thought of everything and that the Stoics merely copied him by adding a few useless details.(3) The Stoics presented five types of arguments, which they called ‘άναπόδβικτοι’, "indemonstrables." Among them were a version of Modus Ponens, a version of Modus Tollens, and three other elementary inferences of conjunction and disjunction." (p. 217)(1) Cf. Patzig (1969) 51 ff.(2) It is sufficient to refer to the reconstruction put forward by Lukasiewicz (1957) 51 ff. and, on the other hand, of the more recent proposal of Corcoran (1974a) 85-131, where natural deduction is used to formalize Aristotle’s proof. The Lukasiewicz approach has been considerably improved by Patzig (1969) 137 ff.(3) l am thinking in particular of a passage from Alexander (in APr. 284.12 ff.; cf. text [G]). We will discuss it in detail later.ReferencesCorcoran, J. (1974a). Aristotle’s natural deduction system. In: Corcoran, J. (1974b) 85-131Corcoran, J. (ed.) (1974b). Ancient logic and its modem interpretations. Dordrecht/ Boston 1974Lukasiewicz, J. (1957). Aristotle’s syllogistic from the standpoint of modem formal logic. Second ed. enlarged, Oxford 1957 (reprint 1963)Patzig, G. (1969). Die Aristotelische Syllogistik. Logisch-philologische Untersuchungen über das Buch A der „Ersten Analytiken“. Göttingen 1969

  39. Milne, Peter. 1995. "On the Completeness of Non-Philonian Stoic Logic." History and Philosophy of Logic no. 16:39-64

    Abstract: "The majority of formal accounts attribute to Stoic logicians the classical truth-functional understanding of the material conditional and exclusive disjunction.These interpretations were disputed, some Stoic logicians favouring modal and/or temporal analyses; moreover, what comes down to us of Stoic logic fails to secure the classical interpretations on purely formal grounds.It is therefore of some interest to see how the non-classical interpretations fare. I argue that the strongest logic we have good grounds to attribute to Stoic logicians is not complete with respect to the non-classical interpretations of disjunction and the conditional."

  40. Mueller, Ian. 1969. "Stoic and Peripatetic Logic." Archiv für Geschichte der Philosophie no. 51:173-187

    "We know that one of the issues dividing the Stoics and the Peripatetics concerned the use of logic. Alexander [of Aphrodisias] (1) insists that only Peripatetic logic is an organon for philosophy, an instrument for making unknown things known through known premisses. Since the Stoics called logic a part of philosophy, they may well have considered their propositional logic a theoretical discipline for which epistemological considerations were irrelevant. This modern attitude seems quite commensurate with the Stoics' presentation of logic. They seem to have been interested in technical devices and formalization for its own sake.I suggest, then, that an important disagreement between the Peripatetic and Stoic logicians concerned the power of their respective logics to represent arguments. The Peripatetic claims were that all scientific proofs are categorical syllogisms and that the inference schemata of the Stoics represented techniques of argument having no place in science. The Stoic reply was that the first claim is false since there are very elementary relational arguments in mathematics which are not syllogisms. Moreover, they pointed out that all conclusive arguments, including categorical syllogisms, could be represented as propositional arguments by a (trivial) technical device. Formally the Stoics held an unassailable position, but they were vulnerable to attack on methodological grounds, since establishing the truth of the premisses of the newly formulated argument seemed to involve making an inference in terms of the old logic. The Peripatetics therefore insisted on the claim, believed for many centuries after them, that their logic was the instrument of science. We do not know the Stoic response to this claim, but it is reasonable to suppose that they retreated to the view that the theory of deductive inference was a technical discipline studied for some ethical end perhaps, but not as the method of scientific discovery." (p. 184)(1) In Analyticorum Priorum 1 ff.

  41. ———. 1978. "An Introduction to Stoic Logic." In The Stoics, edited by Rist, John M. Berkeley: University of California Press

    "The charge of uselessness permeates the ancient literature on Stoic logic. Alexander [of Aphrodisias] is very concerned to defend Aristotelian logic as the tool(organon) of philosophy and science, a means for making unknown things known through known premises. For Sextus no logic is capable of serving these functions. The gist of both men's attack on Stoic logic is that with its arguments there is no way to establish the premises without first establishing the conclusion. The attack is usually made in terms of the first undemonstrable argument and depends upon the truth-functional interpretation of the conditional. Suppose one wishes to prove 'the second' by establishing 'the first' and 'If the first the second.' Then if 'the first' is established, the only way to establish 'if the first the second' is to establish 'the second,' i.e., to establish the conclusion one is trying to prove. Similar objections could be raised against the other undemonstrable arguments. In each case, when the second premise is taken as true, then the obvious truth-functional argument for the first premise requires establishing the truth of the conclusion. There is no way out of this situation, a fact that strongly suggests that Sextus's insistence on applying the truth-functional interpretation to the conditional represents an argumentative device rather than an accurate reflection of standard Stoic doctrine. If the first premise of an undemonstrable argument expresses a stronger than truth-functional connection between its component propositions, there is no reason why the first premise cannot be established independently of the conclusion.Of course, the position I have just ascribed to the Stoics means that philosophically a great deal of weight must be placed on the knowledge of necessary connections between propositions. Many of Sextus's arguments are directed against the possibility of such knowledge. To consider these arguments would take us outside the domain of logic and into epistemology. The point I wish to make is that the Stoics could have claimed universality for their propositional logic without subjecting themselves to attacks on grounds of uselessness. But to what use did the Stoics put their logic? It is tempting to suppose that the Stoics might have treated logic as a technical discipline developed for its own sake. The picture of Chrysippus analyzing innumerable arguments into the undemonstrable points makes it seem certain that to some extent logic was pursued for its own sake. But at least some Stoics thought of logic as more than a self-sufficient technical discipline.(…)The most important inferences from signs would be those based on he first undemonstrable syllogism. Questions about the viability of inferences from sign to thing indicated or commemorated would almost certainly end up as questions about the connection asserted to hold in the first premise, i.e., as questions of metaphysics or epistemology. One cannot expect logic to settle such questions, nor is there any reason to think the Stoics expected it to. The thrust of their logic was to provide a framework in which questions of inferential validity could be settled and questions that fell outside of logic, e.g., whether sweat implies the existence of pores, made precise. It seems fair to say that Stoic achievement in this area remained unparalleled until the time of Leibniz." (pp. 22-25)

  42. ———. 1979. "The Completeness of Stoic Propositional Logic." Notre Dame Journal of Formal Logic no. 20:201-215

    "In this paper I wish to pursue in more detail the question of the completeness of Stoic propositional logic. I shall bring out certain anomalies in Becker's [1957] argument which obscure the precise sense in which his system is complete. The Kneales' system (*) will be shown to be complete in a stronger sense than Becker's but not to be as historically plausible a reconstruction of the Stoic theory. In conclusion I shall suggest a modification of both systems which is historically more plausible than either and also complete in the stronger sense. In the course of the paper I will also discuss other logical and historical points about the systems.I shall take for granted the truth-functionality of the Stoic propositional connectives but disregard interdefinability relationships. I will also formulate the systems of Becker and the Kneales in ways which diverge slightly but unproblematically from their own presentations." (p. 202)(*) William and Marta Kneale, The Development of Logic, Oxford, 1962.

  43. Mühl, Max. 1962. ""Der Logos Endiathetos und Prophorikos" Von der älteren Stoa bis zur Synode von Sirmium 351." Archiv für Begriffsgeschichte no. 7:7-56

    On the history of the distinction between "internal discourse" and "uttered discourse".

  44. Nasieniewski, Marek. 1998. "Is Stoic Logic Classical?" Logic and Logical Philosophy no. 6:55-61

    Abstract: "In this paper I would like to argue that Stoic logic is a kind of relevant logic rather than the classical logic. To realize this purpose I will try to keep as close as possible to Stoic calculus as expressed with the help of their arguments."

  45. Nasti de Vincentis, Mauro. 1981. "Logica scettica e implicazione stoica. A proposito di Adv. Math. VIII 462-481." In Lo scetticismo antico. Vol. II, edited by Giannantoni, Gabriele, 501-532. Napoli: Bibliopolis.

  46. ———. 1984. "Stopper on Nasti's Contention and Stoic Logic." Phronesis.A Journal for Ancient Philosophy no. 29:313-324

    Reply to M. R. Stopper [pseudonym of Jonathan Barnes], "Schizzi Pirroniani", Phronesis, 28, 1983, pp. 265-297.

  47. ———. 1988. "The Third and Fourth Acccount of Conditionals in Sextus Empiricus." In Temi e prospettive della logica e della filosofia della scienza contemporanee. Vol. I: Logica, edited by Cellucci, Carlo and Sambin, Giovanni, 219-226. Bologna: CLUEB.

  48. ———. 1989. "Stoic Implication and Stoic Modalities." In Le teorie delle modalità. Atti del Convegno Internazionale di storia della logica, 258-263. Bologna: CLUEB

    "A new account of Stoic connexive conditional is given, according to which (in order to agree with textual evidence) the truth-conditions for the so-called Chrysippean implication are a function of the modality of the clauses."

  49. ———. 2002. Logiche della connessività. Fra logica moderna e storia della logica antica. Bern: Haupt

    Indice: Premessa 7; Introduzione 11; 1. L'interpretazione classica e le sue varianti 39; 2. Lapars destruens: le difficoltà dell'interpretazione classica 69; 3. Lapars construens: verso una nuova interpretazione 95; 4. Obbiezioni, risposte e conferme 123; 5. Implicazione crisippea e implicazione boeziana 151; 6. Considerazioni conclusive e problemi aperti 173; Appendice: La dottina boeziana dellarepugnantia - Scelta di testi 193; Riferimenti bibiografici 231-232.Recensione di Luca Castagnoli, Elenchos, 25, 2004, pp. 179-192.

  50. ———. 2004. "From Aristotle's Syllogistic to Stoic Conditionals: Holzwege or Detectable Paths?" Topoi. An International Review of Philosophy no. 23:113-137

    "This paper is chiefly aimed at individuating some deep, but as yet almost unnoticed, similarities between Aristotle's syllogistic and the Stoic doctrine of conditionals, notably between Aristotle's metasyllogistic equimodality condition (as stated at Prior Analytics I 24, 41b27-31) and truth-conditions for third type (Chrysippean) conditionals (as they can be inferred from, say, Sextus Empiricus Outlines of Pyrrhonism II 111 and 189). In fact, as is shown in §1, Aristotle's condition amounts to introducing in his (propositional) metasyllogistic a non-truth-functional implicational arrow '', the truth-conditions of which turn out to be logically equivalent to truth-conditions of third type conditionals, according to which only the impossible (and not the possible) follows from the impossible. Moreover, Aristotle is given precisely this non-Scotian conditional logic in two so far overlooked passages of (Latin and Hebraic translations of) Themistius' Paraphrasis of De Caelo (CAG V 4, 71.8-13 and 47.8-10 Landauer). Some further consequences of Aristotle's equimodality condition on his logic, and notably on his syllogistic (no matter whether modal or not), are pointed out and discussed at length. A (possibly Chrysippean) extension of Aristotle's condition is also discussed, along with a full characterization of truth-conditions of fourth type conditionals."

  51. ———. 2006. "Boethiana. La logica stoica nelle testimonianze di Boezio: nuovi strumenti di ricerca." Elenchos no. 27:377-408

    "In view of the importance of Boethius' "In Ciceronis Topica" as a source for Stoic logic, argues for the constitution of an index of divergent readings between the editions of Orelli (Zurich 1833) and Migne, including those omitted by Stangl (1882). Such an index would show that while Orelli's edition is better, sometimes the reading of Migne is to be preferred. Includes considerations on the gradual Stoicization of Aristotelian syllogistics, on Boethius' reliability as a source for Stoic logic, and on the genuine editio princeps of Boethius' "De topicis differentiis" (Rome 1484, rather than Venice 1492."

  52. ———. 2006. "Conflict and connectedness: between modern logic and history of ancient logic." In Logic and philosophy in Italy: Some trends and perspectives, edited by Ballo, Edoardo and Franchella, Miriam, 229-251. Monza: Polimetrica

    "The goal of this paper is to present a self-contained sketch, if not a comprehensive, account, of the Stoic logic of conditional propositions in terms of the related notions of conflict and connectedness. Since what I take those notions to be departs from the standard view in significant ways, I sketch in this Introduction the familiar contours of both the standard view and its main variants and I also present very general reasons that show why we might consider rejecting central aspects of such a view." (p. 229)

  53. Normore, Calvin G. 1991. "Medieval Connectives, Hellenistic Connections; the Strange Case of Propositional Logic." In Atoms, Pneuma, and Tranquillity. Epicurean and Stoic Themes in European Thought, edited by Osler, Margaret J, 25-38. Cambridge: Cambridge University Press.

  54. Nuchelmans, Gabriel. 1973. Theories of Proposition. Ancient and Medieval Conceptions of the Bearers of Truth and Falsity. Amsterdam: North-Holland

    Chapter 4. The Stoic lekton 45; 5. The Stoic axioma 75-88."The Stoic conception of the bearers of truth and falsity centres around the notion of axioma. As an axioma is a species of the genus lekton, I shall first discuss the nature of the lekton. It will be maintained that the word lekton must have had several shades of meaning, although the deplorable state of our sources makes it impossible to reach a high degree of certainty about the exact borderlines between these different nuances and their ascription to definite authors or periods." (p. 45)"As we saw in the foregoing chapter, an axioma is a complete and independent pragma which is expressed in a speech act of asserting. The complete and independent pragma is the thought of an action or passion and its indispensable complements. In so far as this pragma is put into words it is a Iekton; in so far as it is expressed in a speech act of asserting it is an apophanton or axioma, an asserted thought-content. A pragma such as 'Plato liking Dion' can be expressed in different speech acts: for instance, in a yes-or-no question, 'Does Plato like Dion?', in a wish, 'May Plato like Dion', or in an assertion, 'Plato likes Dion'. On the other hand, the same type of speech act, say asserting, may be related to different pragmata; for I may assert many different things. Reflections of this kind must have led the Stoics to a distinction between the generic element of the pragma or Iekton and the specific element of the speech act in which a certain thought is expressed.As a rule, then, an axioma is a thought-content which is in fact asserted. Nevertheless, the Stoics used the name axioma also for the antecedent and consequent of a conditional, although as parts of the composite whole these are not actually asserted. This may be accounted for by the fact that axioma originally meant that which is assumed or taken to be true. Or, as I suggested at the end of 4.2.5, the Stoics may have regarded the antecedent and consequent as potential axiomata, just as they held that a privative assertion of the form 'Un(kind he is)' contains the potential axioma 'Kind he is'. Such an assertable would lie somewhere between the neutral pragma or Iekton and the factually asserted axioma." (p. 75)

  55. O'Toole, Robert R., and Jennings, Raymond. 2004. "The Megarians and the Stoics." In Greek, Indian and Arabic Logic. Handbook of the History of Logic, Vol. 1, edited by Gabbay, Dov and Woods, John, 397-522. Amsterdam: Elsevier.

  56. Orth, Emil. 1959. "Lekton = Dicibile." Helmantica no. 32:221-226

    "L'article est en latin. L'Auteur y explique le sens de lekton, terme stoïcien, en analysant la gnoseologie stoïcienne, sans faire appel aux textes. Il ajoute un bref aperçu de l'histoire du terme où il signale, entre autres choses, qu'Apulée, Peri hermeneias, emploie pronuntiabile pour lekton et Augustin, Principia Dialecticae, 5, P.L., 32, 1411, dicibile; Isidore de Seville, Etymol. 2, 22, 2, dictio. L'article n'est pas conçu comme une recherche philologique, mais comme un exposé théorique." Bulletin Augustinien pour 1959.

  57. ———. 1962. "Stoicorum Lekton = Iudicium, Dicibile." Emerita no. 30:59-61.

  58. Pozzi, Lorenzo. 1974. "Il nesso di implicazione nella logica stoica." In Atti del Convegno di storia della logica (Parma, 8-10 Ottobre 1972), 177-187. Padova: Liviana.

  59. Preti, Giulio. 1956. "Sulla dottrina del Semieion nella logica Stoica." Rivista Critica di Storia della Filosofia no. 10:5-14

    Ristampato col titolo: La dottrina del segno nella logica Stoica in: G. Preti, Saggi filosofici. Storia della logica e storiografia filosofica. Vol. II, Firenze: La Nuova Italia, pp. 3-16.

  60. Repici, Luciana. 1993. "The Stoics and the Elenchos." In Dialektiker und Stoiker. Zur Logik der Stoa und Ihrer Vorläufer, edited by Döring, Klaus and Ebert, Theodor, 253-270. Stuttgart: Franz Steiner

    "1. The problemTo the best of my knowledge, there is no positive definition of elenchos in the surviving evidence on Stoic logic. What we find instead is the well-known definition of irrefutability (άνελεγξία) as that “strength in the logos, so as not to be carried off by it into the contradictory” (ισχύv εν λόγω, ώστε μή άπάγεσθαι ύπ' αύτου εις* τό αντικείμενου) (Diogenes Laërtius 7.47). Yet, an account of the former could be inferred turning upside down the latter. For, if an irrefutable logos is a logos that cannot be carried off into the contradictory because of its strength, conversely a refutable logos might plausibly be a logos that can be carried off into the contradictory because of its weakness. Accordingly, to carry the logos of an interlocutor off into the contradictory (to ‘refute’ it) might mean to reduce it to the contradictory of what it assumes to be the case and hence to self-contradiction. If so, elenchos should not be thought of here in terms of a generic dependence on the intention of those who use it. More specifically, it should be taken as the procedure by which in the Socratic fashion inconsistent and paradoxical consequences are drawn starting from the interlocutor’s own assumptions - from the premisses of his logos.(1) But what would an operation of this sort mean in a philosophical system like the Stoic one? What sort of elenchos would we be faced with? In what follows an attempt is made to look into the matter from the standpoint both of the logical theories of the Stoics and the technical strategies they are said by the ancient sources to have adopted in their arguments." (p. 253)(1) For different viewpoints on the aim of Socratic elenchos, Robinson (1951) 24-28; Vlastos (1983) 27-57, 71-74. For some discussions on Vlastos’ conclusions, Kraut (1983) 59-70 and (1984) 245-79; Brickhouse-Smith (1984)185-95; Polansky (1985) 247-59; loppolo (1985) 151-62; Waterfield (1989) 39-56.ReferencesBrickhouse, Th. C./Smith, N. D. (1984). Vlastos on Elenchus. In: Oxford Studies in Ancient Philosophy 2 (1984) 185-195Huby, P./Neales, G. (eds.) (1989). The criterion of truth. Essays written in honour of George Kerferd. Liverpool 1989loppolo, A. Μ. (1985). Vlastos e l’elenchos socratico. In: Elenchos 6 (1985) 151-162Kraut, R. (1983). Comments on Gregory Vlastos, „The Socratic Elenchus“. In: Oxford Studies in ancient Philosophy 1 (1983) 59-70Kraut, R. (1984). Socrates and the state. Princeton 1984.Polanski, R. Μ. (1985). Professor Vlastos’s analysis of Socratic Elenchus. In: Oxford Studies in Ancient Philosophy 3 (1985) 247-259Robinson, R. (1951). Plato's earlier dialectic. Oxford 1951Vlastos, G. (1983). The Socratic Elenchus. Afterthoughts on the Socratic Elenchus. In: Oxford Studies in Ancient Philosophy 1 (1983) 27-57, 71-74Waterfield, R. (1989). Truth and the Elenchus in Plato. In: Huby, P. /Neales, G. (1989) 35-96

  61. Rist, John M. 1981. "The Importance of Stoic Logic in the Contra Celsum." In Neoplatonism and Early Christian Thought. Essays in Honour of A. H. Armstrong, edited by Blumenthal, Henry Jacob and Markus, Robert Austin, 64-78. London: Variourum.

  62. Roberts, Louis. 1970. "Origen and Stoic Logic." Transactions and Proceedings of the American Philological Association no. 101:433-444

    "A principle of Neoplatonic theurgy was that in working magic spells one must never translate foreign words.(1) Origen, the Christian Neoplatonist of Alexandria, refers to this principle several times.These reference as well as numerous other problems cannot be understood apart from Origen's use of certain principles derived from Stoic logic. How deep in debt Origen was to Stoicism has not yet been determined, but his use of Stoic logic has been almost entirely ignored(.2)In this paper I shall indicate three areas in Origen's works which may have been greatly influenced by Stoic logic, the problem of names, the question of the truth-value of propositions and the use of argument schemes.The thread connecting all three areas is the general Stoic doctrine of truth. I suggest that a good deal more Stoic logic may lurk hitherto undetected in Origen." (p. 433)(1) See E. R. Dodds, "Theurgy and its Relationship to Neoplatonism," Journal of Roman Studies 37 (1947) 63. Sufficient bibliography may be found in this article. The principal classical statement is Iamblichus, De Myst. 7, 5.(2) The chief work on the subject of Stoicism in the Fathers, Michel Spanneut, Le stoïcisme des Pères de l'Église (Paris 1957) ignores Origen completely, treating the period ending at A.D. 230. Other works which treat the problem but ignore the question of logic are: Henry Chadwick, Early Christian Thought and the Classical Tradition (New York 1966); A. H. Armstrong and R. A. Markus, Christian Faith and Greek Philosophy (New York 1960); Henry Chadwick, "Origen, Celsus and the Stoa," The Journal of Theological Studies 48 (1947) 34-48.

  63. ———. 1972. "Lucretius 1.857-58 and Stoic Logic." The Classical World no. 65:215-217

    "Let us consider lines 857-58 of Book 1:at neque reccidere ad nilum res posse neque autemcrescere de nilo testor res ante probatasThe res of line 857 refers to something physical, whereas res in the next line refers to a series of previous proofs or propositions. I suggest that Lucretius is using res in line 858 in the Stoic logical sense of pragmata and that he is consciously doing so. The Stoic usage is given, for example, in a passage of Sextus Empiricus. The sense is "the meaning of the proposition" and it is contrasted with the physical bodies of "the sound" and the "object" or "existing referent."This is the same contrast we find in lines 857-58 above. The first res refers to physical bodies and the second to the lekton which Seneca explains as an enuntiativum quiddam de corpore, quod alii effatum vocant, alii enuntiatum, alii dictum.(5)The enuntiativum or proposition to which res in 858 refers is the series of proofs in 665-75. These lines are in turn a summary of arguments against creation ex nihilo." (p. 216)(5) Ep. mor. 117.13. The larger context reads: Sunt, inquit, naturae corporum, tamquam hic homo est, hic equus. Has deinde sequuntur motus animorum enuntiativi corporum. Hi habent proprium quiddam et a corporibus seductum, tamquam video Catonem ambulantem. Hoc sensus ostendit, animus credidit. Corpus est, quod video, cui et oculos intendi et animum. Dico deinde: Cato ambulat.Non corpus, inquit, est quod nunc loquor, sed enuntiativum . . .

  64. Rüstow, Alexander. 1910. Der Lugner. Theorie, Geschichte, und Auflösung. Leipzig: Teubner

    Nachdruck: New York, Garland, 1987.

  65. Schubert, Andreas. 1993. "Die Stoischen Vorstellungen." In Dialektiker und Stoiker. Zur Logik der Stoa und ihrer Vorläufer, edited by Döring, Klaus and Ebert, Theodor, 271-290. Stuttgart: Franz Steiner.

  66. Sedley, David. 1982. "On Signs." In Science and Speculation. Studies in Hellenistic Theory and Practice, edited by Barnes, Jonathan, Brunschwig, Jacques, Burnyeat, Myles and Schofield, Malcolm, 239-272. Cambridge: Cambridge University Press.

  67. ———. 1982. "The Stoic Criterion of Identity." Phronesis.A Journal for Ancient Philosophy no. 27:255-275.

  68. ———. 1984. "The Negated Conjuction in Stoicism." Elenchos. Rivista di Studi sul Pensiero Antico no. 5:311-316.

  69. ———. 1989. "Le critère d'identité chez les Stoïciens." Revue de Métaphysique et de Morale no. 94:513-533

    Traduction française deThe Stoic Criterion of Identity (1982).

  70. Speca, Anthony. 2001. Hypothetical Syllogistic and Stoic Logic. Leiden: Brill

    Contents: Acknowledgments VII; Abstract IX; Preface XI-XIII; 1. The Aristotelian Background 1; 2. The Greek Commentators on Aristotle 35; 3. Boethius:On hypothetical syllogisms 67; 4. Boethius:On Cicero's Topics 101; References 135; General index 139; Index locorum 141.Abstract: "Aristotle recorded his intention to discuss hypothetical syllogistic fully (An. pr. 50a39), but no such treatment by him has been available since at least A.D. 200, if even it ever existed. The contributions of his successor Theophrastus have also perished, as have those of his followers of the subsequent few centuries. At the same time, almost all of the surviving sources, especially the Greek commentators and Boethius, did not report hypothetical syllogistic accurately.Rather, they conflated it with Stoic logic, which it resembles in some respects, but from which it is significantly different. Modern scholars, who have not appreciated the nature or extent of this conflation, have unintentionally perpetuated the problem. As a result, the original form of hypothetical syllogistic has been misunderstood, and part of the influence of Stoic logic in late antiquity has remained unclear.This book is an account of the conflation of hypothetical syllogistic and Stoic logic. The first chapter is a study of Aristotle's remarks on hypothetical syllogistic, which suggest that it was not a sentential logic such as the Stoics would develop. The second chapter details the conflation as it appears in the Greek commentaries on Aristotle, which consists principally in a confusion between the original Peripatetic division of hypothetical statements and syllogisms, whose criteria are semantic, and the Stoic division of complex propositions and inference schemata, whose criteria are syntactic. The third and fourth chapters focus on Boethius's On hypothetical syllogisms and On Cicero's Topics, in which even further conflation demonstrates that hypothetical syllogistic and Stoic logic had completely ceased to retain their distinct natures by the end of antiquity."

  71. Stakelum, James W. 1940. Galen and the Logic of Propositions. Romae: Angelicum

    "These pages, restrictedly entitled Galen and the Logic of Propositions, originally formed part of an academic dissertation, Galen's Introduction to Dialectic. The threefold purpose of the larger study was to present Galen's Dialectic in a clear light, to examine his doctrine and weigh its importance as to originality or historical precedent, and from these considerations to draw conclusions as to its influence on succeeding generations. The doctrine, scattered throughout the Galenic text, was gathered under five headings: I. Galen's Introductory Remarks; II. Logic of Propositions; III. Aristotelian Term Logic; IV. Other Classes of Syllogisms; V. Applied Logic. Owing to the limited size of the volumes of this series, published under the sponsorship of Father I. M. Bochenski, O. P., it is impossible to publish here the whole result of the inquiry. Accordingly, we have selected for presentation our Introduction -- rearranged as Part One in several short chapters -- and the most important portion of our examination of Galen's Dialectic, dealing with the logic of propositions. The latter section is divided into three parts. A brief conclusion completes the essay.It is traditional to attribute to Galen an eminent position in the field of logic, but rarely do we find specific reasons assigned for this eminence. The composition of this dissertation has, for me, definitely determined Galen's position in the history of logic. It is hoped that it will serve a similar purpose for others."

  72. Stopper, M.R. 1983. "Schizzi Pirroniani." Phronesis.A Journal for Ancient Philosophy no. 28:265-297

    Critical notice of: Gabriele Giannantoni (ed.), Lo scetticismo antico. Atti del convegno organizzato dal Centro di studio del pensiero antico del C.N.R., Roma 5-8 novembre 1980, Napoli: Bibliopolis, 1981.M. R. Stopper is a pseudonym of Jonathan Barnes.

  73. Stump, Eleonore. 1987. "Boethius’s in Ciceronis Topica and Stoic Logic." In Studies in Medieval Philosophy, edited by Wippel, John F., 1-22. Washington The Catholic University of America <br>"Besides what In Ciceronis Topica shows us about Boethius, it is also a valuable source for us as a witness to the thought and culture of his period.(...)"But perhaps the most interesting unexplored section in In Ciceronis Topica is Boethius’s presentation of Stoic logic, and that is the part I want to concentrate on in this paper. Rather than dwell on the main theme of In Ciceronis Topica, the art of discovery and its instruments, the Topics, I want to focus instead on just that small part of Boethius’s commentary in which he relates the Topics to Stoic dialectic or logic and reveals his understanding of that part of Stoicism. We are beginning to understand the ancient art of discovering arguments, and it has garnered increasing attention among contemporary scholars. But as far as I know, Michael Frede (30) is alone among recent historians of philosophy in considering Boethius’s contribution to our understanding of Stoic logic; and even he gives only a brief discussion of a small part of the relevant Boethian text. So in what follows I will concentrate on just the fifth book of In Ciceronis Topica, in which Boethius discusses the so-called undemonstrated modes or argument forms of the Stoics. I need to make plain here that my area of interest and expertise is Boethius and not the Stoics; but I think that in In Ciceronis Topica Boethius sheds some light on a vexing problem which has troubled students of Stoic logic for some time." (pp. 6-7)(30) Michael Frede, Die stoische Logik, (Göttingen: Vandenhoeck and Ruprecht, 1974), pp. 148-53, 160-62.

  74. Tracy, Kevin. 2006. The Development of Dialectic from Aristotle to Chrysippus, University of Pennsylvania

    Available at ProQuest Dissertation Express order number: 3225558.Abstract: "From Aristotle onward, formal logic was an element of ancient Greek dialectic (dialektiké). Aristotle'sPrior Analytics (4th century BCE) is the earliest evidence of a formal logic in antiquity. The evidence for the formal logic of the Stoic philosopher Chrysippus (3rd century BCE) is fragmentary; nonetheless it makes clear that not more than a century or so after Prior Analytics, Chrysippus revolutionized formal logic. The scholarship on Stoic logic has not yet presented the history of dialectic from Aristotle to Chrysippus as an intelligible narrative. Without such a narrative, one cannot explain what, in general, motivated the innovations of Chrysippus, what made Stoic logic coherent as a unified project, or what relationship that project had to earlier work in logic. This dissertation approaches the problem through the presentation and interpretation of the ancient source material. First it describes the logical doctrines of Aristotle, Theophrastus, and the 'Megarics' in such a way as to make clear what questions these predecessors left for Chrysippus. It then describes how Chrysippus addressed these questions. Finally, it uses the resulting narrative to give a detailed account of Stoic formal logic. The dissertation yields five principal conclusions. First, neither the Peripatetics or the 'Megarics' described logical forms of propositional logic; Chrysippus was the first to do so. Second, the guiding aim of Chrysippus' logic was to avoid adopting a semantic stance in describing logical forms and explaining logical relationships. Third, the Stoics distinguished 'valid' (hugies) from 'true' (aléthes), so that sunartésis is a standard for the validity rather than the truth of the Stoic conditional (sunhémmenon). Fourth, the Stoics produced derivations for categorical arguments in their deduction system. Fifth, the Stoic deduction system is roughly analogous to the first-order fragment of Frege's system, except on two points: it most likely was not designed to accommodate the use of polyadic predicates with multiple quantifiers, although the possibility for doing so inheres in its approach to the analysis of propositions, and it uses the 'natural' approach rather than the 'axiomatic' approach of Frege."

  75. Verbeke, Gérard. 1977. "Der Nominalismus Der Stoischen Logik." Allgemeine Zeitschrift für Philosophie no. 3:36-55.

  76. ———. 1991. "Ethics and Logic in Stoicism." In Atoms, Pneuma, and Tranquillity. Epicurean and Stoic Themes in European Thought, edited by Osler, Margaret J, 11-24. Cambridge: Cambridge University Press.

  77. ———. 1996. "Meaning and Role of the Expressible (Lekton) in Stoic Logic." In Knowledge through Signs. Ancient Semiotic Theories and Practices, edited by Manetti, Giovanni, 133-154. Turnhout: Brepols

    "In his critical survey of Stoic dialectic Sextus states that the doctrine of the expressible, which plays an important part in the theory of knowledge, has been repeatedly put into question:(1) the lekton is an incorporeal, together with time, place and empty space, it belongs to the group of incorporeal objects generally accepted by the Stoics. In their opinion incorporeals are not active, they are unable to effect or produce something: yet they are indispensable in view of a coherent understanding of the universe.(2) Within this framework it was agreed that each argument is composed of incorporeal expressibles, since it is a combination of sentences which are considered to be complete lekta.(3) The question however was asked whether expressibles are really necessary, if they are totally ineffective. Even the meaning of the notion is questionable: it is obviously related to language, but it is not a component neither of spoken nor of written language. In other words it is not a verbal utterance and yet it is referred to by linguistic terms.(4) So it seems to have a definite function in the Stoic theory of knowledge." (p. 133)(1) Sextus,M 8. 336. The author states that the existence of expressibles has been heavily discussed: there was no agreement about this issue. In some other passage Sextus even speaks of an unending debate (8. 262). Sextus lived in the second half of the second century and in the beginning of the third A.D.: at that time Stoicism was still very influential. No other philosophical school ever accepted this doctrine, but it was not disregarded: philosophers had to cope with it especially in their dialectic.(2) Sextus,M 8. 262. An incorporeal object could not affect anything, nor could it be affected. For it could only be affected by something corporeal, and that is excluded, since corporeal and incorporeal are not on the same level.(3) Sextus,M 8. 260-261; 8. 404: every proof is composed of incorporeal expressibles. In Sextus' opinion a vicious circle is unavoidable.(4) Sextus,M 8. 264: according to Sextus lekta are signified and among them are also propositions, which are regarded as complete expressibles.

  78. Viano, Carlo Augusto. 1958. "La dialettica stoica." Rivista di Filosofia no. 49:179-227

    Ristampato in: Autori Vari, Studi sulla dialettica, Torino: Taylor, 1969 pp. 63-111.

  79. Virieux-Reymond, Antoinette. 1949. La logique et l'épistémologie des Stoïciens. Leurs rapports avec la logique d'Aristote, la logistique et la pensée contemporaine. Lausanne: Librairie de l'Université.

  80. ———. 1984. "L'originalité de la logique mégaro-stoïcienne par rapport à la logique d'Aristote." Diotima no. 12:172-174.

  81. White, Michael J. 1986. "The Fourth Account of Conditionals in Sextus Empiricus." History and Philosophy of Logic no. 7:1-14

    Abstract: "This paper develops an interpretation of the fourth account of conditionals in Sextus Empiricus's Outlines of Pyrrhonism that conceptually links it with contemporary 'relevance' interpretations of entailment. It is argued that the third account of conditionals, which analyzes the truth of a conditional in terms of the joint impossibility of antecedent and denial of consequent, should not be interpreted in terms of a "relative" incompatibility of antecedent and denial of consequent because of stoic acceptance of the truth of some conditionals of the form p → -p and its converse. Rather, It is suggested, ancient attempts to avoid the so-called paradoxes of implication involve the fourth account of conditionals. I hypothesize that this account is related to stoic attempts to define truth conditions for conditionals in terms of a theory of the concludency (validity) of arguments in opposition to the more common procedure (represented by the first three accounts of conditionals) of specifying truth conditions for conditionals 'semantically' and using those truth conditions in the development of a theory of argument validity."

  82. Zarnecka-Bialy, Ewa. 1979. "Stoic Logic as Investigated by Jan Lukasiewicz." Reports on Philosophy no. 3:27-40.

  83. Zoecklein, Walter O. 1969. The Ontological Commitments in Stoic Logic, University of California, San Diego

    Available at ProQuest Dissertation Express. Order number: 6919703.