INTRODUCTION: ARISTOTLE AND THE TOMORROW'S SEA BATTLE (DE INTERPRETATIONE CHAPTER 9)
"In 1930 Jan Łukasiewicz published the following
account of Aristotle on bivalence:
"The law of bivalence, i.e. the law according to
which every proposition is either true or false, was familiar to
Aristotle, who explicitly characterized a proposition ... as discourse
which is either true or false.
We read in De interpr.
4, 17a2-3, "not every expression
is a proposition, but only those in which there is truth or falsity".
Aristotle, however, does not accept the validity of this law for
propositions dealing with contingent future events. The famous chapter 9
of De interpretatione
is devoted to this matter. Aristotle
believed that determinism would be the inevitable consequence of the law
of bivalence, a consequence he is unable to accept. Hence he is forced
to restrict the law".(1)
Łukasiewicz's account provides the basis for a dramatic
reading of De interpretatione.
In Chapter 4 Aristotle declares bivalence to
be the differentia of propositions, only to discover in Chapter 9 that that
view of the essential nature of propositions entails a determinism under
which "all things are and happen of necessity; accordingly, there will be no
need to deliberate or to take trouble" (18b30-32). Aristotle is thus faced
with a dilemma: He must either accept determinism with its radically
counter-intuitive implications or deny what had appeared to him to be the
essential nature of propositions. According to Łukasiewicz, Aristotle then
simply grasps the second horn of the dilemma, restricting the application of
the law of bivalence to propositions about past events, present events, and such future events as are
naturally necessitated-eclipses, for instance.
Łukasiewicz's interpretation of Aristotle's response to
determinism in De int. 9 has stood, in one
version or another, at the center of the modern controversy that has its
source in his 1930 article. His portrayal of Aristotle as confronted with
the dilemma of either accepting logical determinism or abandoning universal
bivalence, and as then opting for the latter, seems also to be the
oldest interpretation on record. Boethius, in
his commentaries on De int., attributes this
view to the Stoics, along with other people whom he does not identify. Of
those oldest recorded interpreters of De int. 9
Boethius says that they "thought that Aristotle says that contingent
[propositions] about the future are neither true nor false"
(2).
But those Stoics also thought that Aristotle had grasped the wrong horn of
the dilemma. When they themselves faced up to the choice between tampering
with bivalence and accepting determinism, the Stoics declared that "it is
the foundation of logic that
whatever is stated ... is either true or false" (3);
and, true to their principles, they professed determinism.
Łukasiewicz packs an impressive amount of original
historical research on that oldest interpretation into a few paragraphs, but
its status in antiquity is summarized more effectively in recent work by
Richard Sorabji. He says that the interpretation adopted by Łukasiewicz (and
by many others since 1930) "was the interpretation of Aristotle taken by the
Peripatetics (i.e. the Aristotelian school) according to Simplicius . ...
Moreover, the denial of truth value .. was accepted as being a correct view
in itself, necessary for avoiding determinism, by Epicurus, by the Platonist
Nicostratus, and probably by the Aristotelian Alexander of Aphrodisias,
while being rejected by the Stoic Chrysippus, by the Academic Carneades, and
by Cicero"(4).
Recent commentators on De int. 9, whether they
accept or reject the oldest interpretation, have tended to follow Hintikka's
lead in designating it "the traditional interpretation"
(5). Sorabji, for instance, says "I shall refer to this interpretation of
Aristotle as the traditional one, although this is something of a misnomer,
in view of ... the fact that there are rival interpretations just about as
old" (6)."
pp. 65-66
(1) I am quoting from H. Weber's translation of "Philosophische
Bemerkungen zu mehrwertigen Systemen des Aussagenkalkuls" in McCall
(Polish logic 1967: 63-64), but I am supplying detailed references to passages in
ancient authors and translating the Greek and Latin quoted by
Łukasiewicz and left untranslated by Weber.
(2)Op. cit., p. 64, where Łukasiewicz is quoting from Boethius's second
commentary on De interpretatione (ed.
Meiser, Boetii Commentarii in Librum Aristotelis Peri Hermeneias;
Leipzig: Teubner, 1877-1880, 2 vols.), II
208.1-3. See p. 76 below and n. 47 for the remainder of this passage.
(3)Op. cit., p. 65, quoting Cicero,
Acad. Pr. ii 95.
(4) Sorabji (Necessity,
Cause and Blame: perspective on Aristotle's Theory 1980: 92-93). See his thorough documentation of this summary
account in notes 3 and 5 on those pages. It is
not clear to me that the Simplicius passage cited by Sorabji
(Commentary on Aristotle's Categories,
ed. Kalbfleisch, Commentaria in Aristotelem Graeca
(CIAG) 407.6-13 supports his claim about the
Peripatetics. The occurrence of "aphorismenos"
in lines 10-11 suggests that Simplicius may be ascribing to
these Peripatetics the interpretation Boethius presents as a
Peripatetic antidote to the Stoics'
interpretation of Aristotle as having denied bivalence (see pp.
67-68 below).
(5)Hintikka ("The
Once and Future Seafight", Philosophical Review 73,1964: 461-492); revised version in his
Time and Necessity: Studies in Aristotle's
Theory of Modality (Oxford:
Clarendon Press, 1973), pp. 147-178, esp. pp. 148-149.
(6) (6)Sorabji, op. cit., p. 92.
From: Norman Kretzmann - Boethius and
the truth about tomorrow's sea battle - in: L. M. de Rijk, and H. A. G.
Braakhuis (eds.) - Logos and Pragma. Essays on the philosophy of language in
honour of Professor Gabriel Nuchelmans - Nijmegen, Ingenium Publishers, pp.
65-97
Future contingencies: the problem and its
possible solutions (click to enlarge the image)
From: Gerhard Seel (ed.) - Ammonius
and the Seabattle. Texts, commentary and essays - Berlin,
de Gruyter, 2001, p. 37
Editions:
Studies:
Ammonius and the Seabattle. Texts, commentary, and essays. Edited by Seel Gerhard, Schneider Jean-Pierre, and Schulthess Daniel. Berlin: de Gruyter 2000.
Ammonius on Aristotle: De Interpretatione 9 (and 7,1-17).
Greek text established by A. Busse, reprinted from CAG IV/v. English translation by David Blamk, revised by J.-P. Schneider and G. Seel.
Philosophical commentary by Gerhard Seel. Essays by Mario Mignucci and Gerhard Seel.
Contents: Part I. Preliminaries 1; Part II. Gerahrd Seel: Introduction. Future contingencies: the problem and its possible solutions 13; Part III. Ammonius on Aristotle: De Interpretatione 9 (and 7, 1-17). Greek text with an English translation 39; Part IV: Gerhard Seel: Philosophical commentary 131; Part V: Essays. Gerhard Seel: Ammonius' semantics of the assertoric sentence 213; Gerhard Seel: 'In a definite way true'; truth values and their modalisation in Ammonius 234; Mario Mignucci: Ammonius on future contingent propositions 247; Part VI: Bibliography 285; Part VII: Index of names 307; Index of passages cited 310-312.
Anscombe Gertrude Elizabeth Margaret, "Aristotle and the Sea Battle. De Interpretatione Chapter IX," Mind 65: 1-15 (1956).
Revised reprint in: J. M. E. Moravcsik (ed.) - Aristotle: A Collection of Critical Essays - London, Macmillan, 1968, pp. 15-33 and in: The Collected philosophical papers of G.E.M. Anscombe - Vol. I, Oxford, Blackwell, 1981, pp. 44-56
Barreau Hervé. Le Maître Argument de Diodore. Son interprétation traditionnelle, sa signification historique, sa reconstitution contemporaine. Strasbourg: Université Louis Pasteur 1975.
Barreau Hervé. Suite et fin sur le Maître Argument de Diodore. Strasbourg: Université Louis Pasteur 1978.
Bäck Allan, "Sailing through the Sea Battle," Ancient Philosophy 12: 133-151 (1992).
Becker Oskar, "Über den "Kyrieuon Logos" des Diodoros Kronos," Rheinisches Museum für Philologie 99: 289-304 (1956).
Bochenski Joseph. Ancient formal logic. Amsterdam: North-Holland 1951.
Chapter V. The Stoic-Megaric School 77-102.
Bosley Richard, "In Support of an Interpretation of On Int. 9," Ajatus 37: 29-40 (1978).
Boudot Maurice, "L'argument dominateur et le temps cyclique," Études Philosophiques: 271-298 (1983).
Cambiano Giuseppe, "La Scuola Megarica nelle interpretazioni moderne," Rivista di Filosofia 62: 227-253 (1971).
"Rassegna delle interpretazioni della Scuola Megarica a partire dagli inizi dell'Ottocento: dalle ricostruzioni tendenziose di Tennemann e Hegel alle ricerche di Schleiermacher, Deycks, Henne e altri, dalla presentazione negativa della logica megarica dovuta a Prantl e Zeller e combattuta da Grote e Gomperz sino alla storiografia del Novecento, che ha insistito soprattutto negli ultimi decenni sull'importanza della logica Megarica (Łukasiewicz, Mates, Kneale). ma la filosofia Megarica, benchè sia un campo ampiamente esplorato, sembra costituire ancor oggi uno dei problemi piu difficili per la ricerca storica, filologica e filosofica."
Celluprica Vincenza. L'argomento dominatore di Diodoro Crono e il concetto di possibile in Crisippo. In Scuole socratiche minori e filosofia ellenistica. Edited by Giannantoni Gabriele. Bologna: Il Mulino 1977. pp. 55-73
Celluprica Vincenza. Il capitolo 9 del De interpretatione di Aristotele. Rassegna di studi 1930-1973. Bologna: Il Mulino 1977.
Denyer Nicholas, "Time and modality in Diodorus Cronus," Theoria 47: 31-53 (1981).
Döring Klaus, "Gab es eine Dialektische Schule?," Phronesis 34: 293-310 (1989).
"Until recently, students of ancient philosophy have generally believed that Diodorus Cronus was a member of the Megarian school founded by Socrates' pupil Euclides of Megara. In 1977, however, David Sedley claimed that Diodorus should be associated with the so-called Dialectical school. The article argues against this view. The main results are (1) There are no testimonies which suggest that we should distinguish between Megarians and Dialecticians. (2) The so-called Dialectical school never existed; it is a construct of ancient historians of philosophy."
Ebert Theodor. Dialecticians and Stoics on the classification of propositions. In Dialektiker und Stoiker. Zur Logik der Stoa und ihrer Vorläufer. Edited by Döring Klaus and Ebert Theodor. Stuttgart: Franz Steiner 1993. pp. 111-127
Eck Job van, "Another interpretation of Aristotle's De interpretatione IX. A support for the so-called second oldest or "mediaeval" interpretation," Vivarium.An International Journal for the Philosophy and Intellectual Life of the Middle Ages and Renaissance 26: 19-38 (1988).
Frede Dorothea. Aristoteles und die Seeschlacht. Das Problem der contingentia futura in De interpretatione 9. Göttingen: Vandenhoeck & Ruprecht 1970.
Frede Dorothea, "Omne quod est quando est necesse est esse," Archiv für Geschichte der Philosophie 54: 153-167 (1972).
Frede Dorothea, "The sea-battle reconsidered. A defence of the traditional interpretation," Oxford Studies in Ancient Philosophy 3: 31-87 (1985).
"In De Interpretatione 9, Aristotle treats the relationship between the truth and necessity of future contingents differently from those of the past or present, as exemplified by his consideration of the statement "A sea battle will take place tomorrow". While Aristotle clearly believes in the application of the "principle of bivalence" to the present and past, he does limit its application to the future, despite recent claims to the contrary."
Frede Dorothea, "Fatalism and future truth," Proceedings of the Boston Area Colloquium in Ancient Philosophy 6: 195-227 (1990).
Gaskin Richard, "Alexander's Sea Battle: a discussion of Alexander of Aphrodisias De Fato 10," Phronesis.A Journal for Ancient Philosophy 38: 75-94 (1993).
Gaskin Richard. The sea battle and the master argument. Aristotle and Diodorus Cronus on the metaphysics of the future. Berlin: de Gruyter 1995.
Gaskin Richard, "Sea battles, worn-out cloaks, and other matters of interpretation: Weidemann on Aristotle's peri Hermeneias," Archiv für Geschichte der Philosophie 78: 48-59 (1996).
Hartmann Nicolai, "Le concept mégarique et aristotélicien de possibilité. Contribution à l'histoire du problème ontologique de la modalité," Laval Théologique et Philosophique 49: 131-146 (1993).
Traduit par Jean-Pierre Narbonne
Hintikka Jaakko, "The once and future sea fight: Aristotle's discussion of future contingents in De interpretatione IX," Philosophical Review 74: 461-492 (1964).
Revised and reprinted as Chapter VIII in: J. Hintikka - Time and necessity. Studies in Aristotle's theory of modality - New York, Oxford University Press, 1973, pp. 147-178
Hintikka Jaakko, Knuuttila Simo, and Remes Unto, "Aristotle on modality and determinism," Acta Philosophica Fennica 29 (1977).
Judson Lindsay, "La bataille navale d'aujourd'hui. De interpretatione IX," Revue de Philosophie Ancienne 6: 5-37 (1988).
"Réexamen, mené dans la perspective de la philosophie analytique, des arguments discutés par Aristote à propos des futurs contigents et illustrés par l'exemple de la bataille navale. Aristote ne nie pas la "vérité-par-avance", mais il explique plutôt ce qui est erroné dans l'argument nécessitariste. En outre, il ne répond pas dans le De int. IX à l'argument de la vérité future, mais à un argument subtilement apparenté à celui-ci. Enfin il propose une solution qui vaut non seulement pour le problème qu'il discute, mais aussi pour d'autres problèmes posés par l'idée de "vérité-par-avance".
Prior Arthur Norman, "Diodorus and modal logic. A correction," Philosophical Quarterly 8: 226-230 (1958).
Sedley David, "Diodorus Cronus and Hellenistic philosophy," Proceedings of the Cambridge Philological Society 23: 74-120 (1977).
Reprinted in: Terence Irwin (ed.) - Hellenistic Philosophy (Classical Philosophy Collected Papers vol. 8) - New York, Routledge, 1995, pp. 270-315.
"During the last four decades historians of ancient logic have become increasingly aware of the importance of Diodorus Cronus and his pupil Philo as pioneers of the propositional logic which came to flourish in the Stoa. Their direct influence has so far been recognised in two main areas of Hellenistic controversy -- the validity-criteria for conditional propositions, and the definition of the modal terms 'possible' and 'necessary'. But some broader questions have not been satisfactorily answered. What wee Diodorus' own philosophical allegiances and antecedents? What is his place in the history of Greek philosophy? How far-reaching was his influence on the post-Aristotelian philosophers?
There was little chance of tackling these questions confidently until 1972, when Klaus Döring published for the first time the collected fragments of Diodorus, in his important volume Die Megariker. Meagre though they are, these fragments confirm my suspicion that Diodorus' philosophical background has not been fully explored, and also that his influence on the three emerging Hellenistic schools -- the Stoics, Epicureans, and Sceptics -- was far wider than has hitherto been recognised. There has been much discussion as to which earlier philosophers played the most decisive part in shaping Hellenistic philosophy, and the respective claims of the Platonists and of Aristotle have never lacked expert advocacy. In all this, the claims of so obscure a figure as Diodorus have been underrated." p. 74
Seel Gerhard, "Diodore domine-t-il Aristote?," Revue de Métaphysique et de Morale 87: 293-313 (1982).
Sutula John, "Diodorus and the "Master Argument"," Southern Journal of Philosophy 14: 323-344 (1976).
"Diodorus Cronus was a Megaric logician who was reputed to have derived from uncontroversial premises the surprising conclusion that the possible is that which either is or will be the case. Versions of his lost argument have been reconstructed recently by Prior, Hintikka, and Rescher. I analyze and compare these versions and argue that none of them forms a sound argument."
Vuillemin Jules, "Le chapitre IX du De interpretatione d'Aristote. Vers une réhabilitation de l'opinion comme connaissance probable des choses contingentes," Philosophiques 10: 15-52 (1983).
Vuillemin Jules. Nécessité ou contingence: L'aporie de Diodore et les systèmes philosophiques. Paris: Éditions du Minuit 1984.
Vuillemin Jules. Le chapitre IX du De Interpretatione et la connaissance probable. In Aristote aujourd'hui. Études réunies à l'occasion du 2300 anniversaire de la mort du philosophe. Edited by Sinaceur Mohammed Allal. Toulouse: Éditions Erès 1988. pp.
Vuillemin Jules. Necessity or contingency. The master argument. Stanford: Center for the Study of Language and Information 1996.
Vuillemin Jules, "Nouvelles réflexions sur l'Argument Dominateur: une double référence au temps dans la seconde prémisse," Philosophie 55: 14-30 (1997).
"Révision de l'interprétation modale de la seconde prémisse de l'argument dominateur d'Epictète, selon laquelle l'impossible ne suit pas logiquement du possible, développée par l'Auteur dans son ouvrage intitulé Nécessité ou contingence: l'aporie de Diodore et les systèmes philosophiques (1984). L'Auteur reconstruit son analyse de l'argument aristotélicien du De Coelo et aménage le système chrysippéen à la lumière des principes aristotéliciens de la nécessité conditionnelle et de la contraction synchronique de la possibilité."
Weidemann Hermann, "Das sogenannte Meisterargument des Diodoros Kronos und der Aristotelische Möglichkeitsbegriff," Archiv für Geschichte der Philosophie 69: 18-53 (1987).
Weidemann Hermann, "Aristotle, the Megarics, and Diodorus Cronus on the notion of possibilty," American Philosophical Quarterly 45: 131-148 (2008).
White Michael J., "An S5 Diodorean modal system," Logique et Analyse 88: 477-487 (1979).
"This note points out that the fatalism or logical determinism associated with the name of the ancient logician Diodorus Cronus is not captured "within" the contemporary modal-tense logic developed for the representation of the "Diodorean" modalities, i.e., the interpretation of necessity as present-and-always-future truth, possibility as present-or-sometime-future truth. With additional postulates for "eternal recurrence" or cyclical time (tense-logically equivalent), a form of fatalism (FP entails LFP, PP entails LPP) is captured that does not "collapse" the system into propositional logic (P does not entail LP). The result is the Lewis S5 system. While there is no evidence that Diodorus subscribed to the doctrine of eternal recurrence, most of the Stoics, "logical successors" of the Megarian logicians, did adopt such a conception of time."
White Michael J., "Necessity and unactualized possibilities in Aristotle," Philosophical Studies 31: 298 (1980).
Ebert Theodor. Dialektiker und frühe Stoiker bei Sextus Empiricus. Untersuchungen zur Entstehung der Aussagenlogik. Göttingen: Vandenhoeck & Ruprecht 1991.
Cambronne Patrice, "L'universel & le singulier : l' Hymne à Zeus de Cléanthe: notes de lecture," Revues des études anciennes 100: 89-114 (1998).
Goulet-Cazé Marie-Odile, "Un syllogisme stoïcien sur la loi dans la doxographie de Diogène le Cynique à propos de Diogène Laërce VI 72," Rheinisches Museum 215: 214-240 (1982).
"Le syllogisme sur la loi était un syllogisme stoïcien attribué à Diogène le Cynique dans la doxographie que lui consacre Diogène Laërce. Il n'est que l'écho d'un ensemble de syllogismes sur la loi et la cité dont l'auteur est vraisemblablement Cléanthe. La source de Diogène Laërce pourrait être Apollodore de Séleucie."
Amato Giorgio, "Possibilità, necessità e verità nella teoria deterministica di Crisippo," Parola del Passato 53: 241-254 (199).
Barnes Jonathan, "PIQANA SUNHMMENA," Elenchos.Rivista di Studi sul Pensiero Antico 6: 453-467 (1985).
Barnes Jonathan. The catalogue of Chrysippus' logical works. In Polyhistor. Studies in the history and historiography of ancient philosophy presented to Jaap Mansfeld on his sixtieth birthday. Edited by Algra Keimpe, Van der Horst Pieter, and Runia David. Leiden: Brill 1996. pp. 169-184
The catalogue of the works of Chrysippus preserved by Diogenes Laertius (7, 189 ff.) may derive from a handbook by Chrysippus himself ; this makes it a fairly reliable guide to his thought
Bobzien Susanne. Chrysippus' modal logic and its relation to Philo and Diodorus. In Dialektiker und Stoiker. Zur Logik der Stoa und ihrer Vorläufer. Edited by Döring Klaus and Ebert Theodor. Stuttgart: Franz Steiner 1993. pp. 63-84
Bréhier Émile. Chrysippe et l'ancien stoïcisme. Paris: Presses Universitarires de France 1951.
Première édtion: Paris, Alcan, 1910
Brunschwig Jacques. On a book-title by Chrysippus: "On the fact that the Ancients admitted Dialectic along with Demonstrations". In Aristotle and the later Tradition. Edited by Blumenthal Henry and Robinson Howard. Oxford: Clarendon Press 1991. pp. 81-95
"On what can be inferred from the title and from its place in the list of Chrysippus' works at Diogenes Laertius 7, 201; on its relevance to Plutarch's criticism of Stoic consistency at Stoic rep. 1035a."
Castagnoli Luca, "Il condizionale crisippeo e le sue interpretazioni moderne," Elenchos.Rivista di Studi sul Pensiero Antico 25: 353-395 (2004).
Cavini Walter. I sillogismi ipotetici del papiro parigino attribuito a Crisippo. In Studi classici in onore di Quintino Cataudella. Vol I. Catania: Università di Catania. Facoltà di lettere e filosofia 1972. pp. 39-43
Cavini Walter. Chrysippus on speaking truly and the Liar. In Dialektiker und Stoiker. Zur Logik der Stoa und ihrer Vorläufer. Edited by Döring Klaus and Ebert Theodor. Stuttgart: Franz Steiner 1993. pp. 85-109
Celluprica Vincenza. L'argomento dominatore di Diodoro Crono e il concetto di possibile in Crisippo. In Scuole socratiche minori e filosofia ellenistica. Edited by Giannantoni Gabriele. Bologna: Il Mulino 1977. pp. 55-73
Del Mastro Gianluca, "Il PHerc. 1380: Crisippo, Opera logica," Cronache Ercolanesi 35: 61-70 (2005).
"Analisi paleografica di PHerc. 1380 e ricostruzione della subscriptio: si tratta dell'opera "Degli elementi del discorso e della frase" di Crisippo, di argomento logico-dialettico, affine per tema al trattato sulle ambiguità del linguaggio tramandato in PHerc. 307."
Del Mastro Gianluca. Un'opera logica di crisippo nel PHerc. 1380. In Proceedings of the 24th Congress of papyrology. Helsinki, 1-7 August 2004. Edited by Frösén Jaakko, Purola Tiina, and Salmenkivi Erja. Helsinki: Societas Scientarum Fennica 2007. pp. 249-258
Donini Pierluigi, "Crisippo e la nozione del possibile," Rivista di Filologia 101: 333-351 (1973).
Gould Josiah, "Chrysippus: on the criteria for the truth of a conditional proposition," Phronesis.A Journal for Ancient Philosophy 12: 156-161 (1967).
Gould Josiah. The philosophy of Chrysippus. Leiden: Brill 1970.
Hadot Pierre. Liste commentée des oeuvres de Chrysippe (D. L. VII 189-202). In Dictionnaire des philosophes antiques. Tome 2: de Babélyca d'Argos à Dyscolius. Edited by Goulet Richard. Paris: CNRS-Éditions 1994. pp. 336-356
Hershbell Jackson P., "Epictetus and Chrysippus," Illinois Classical Studies 18: 139-146 (1993).
"Chrysippus' works were read and commented on in Epictetus' school, and they influenced Epictetus' discussions of formal logic."
Hunter Graeme, " A Ciceronian critique of Chrysippus," Apeiron 27: 17-23 (1994).
Analysis of Cicero's argument at De fato 6 against Chrysippus' doctrine about future contingency
Idea Harry A., "Chrysippus's response to Diodorus's Master Argument," History and Philosophy of Logic 13: 133-148 (1992).
"Chrysippus claims that some propositions perish, including some true conditionals whose consequent is impossible and antecedent is possible, to which he appeals against Diodorus's Master Argument. Onthe standard interpretation, perished propositions lack truth values,and these conditionals are true at the same time as their antecedent sare possible and consequents impossible. But perished propositions are false, and Chrysipus's conditionals are true when their antecedent and consequent are possible, and false when their antecedent is possible and consequent impossible. The claim of the Master Argument that Chrysippus rejects, then, is stronger than usually supposed."
Irigoin Jean. Sur les propositions négatives. In
Mise en page et mise en texte du livre manuscrit. Edited by Martin Henri Jean and Vezin Jean. Paris: Éditions du Cercle de la Librairie-Promodis 1990. pp. 34-36
La disposition du texte dans le Papirus Paris 2 (= P. Louvre inv. 2326), fragment d'un ouvrage de logique attribué traditionnellement à Chrysippe.
Leib Ethan J., "On the Sorites: toward a better understanding of Chrysippus," Ancient Philosophy 21: 147-159 (2001).
" Chrysippus' response (as recorded by Cicero, Acad. 2, 39) to the challenge of the skeptic has been misrepresented and hence viewed as weaker than it really is. The sorites argument is an effective and important challenge to the Stoa."
Marrone Livia, "Nuove letture nel PHerc. 307 (Questioni Logiche di Crisippo)," Cronache Ercolanesi 12: 13-18 (1982).
Marrone Livia. Il problema dei 'singolari' e dei 'plurali' nel PHerc. 307. In Atti del XVII Congresso Internazionale di papirologia. Edited by Frösén Jaakko, Purola Tiina, and Salmenkivi Erja. Napoli: Centro internazionale per lo studio dei papiri ercolanesi 1984. pp. 419-427
Vol 2
Marrone Livia, "Proposizione e predicato in Crisippo," Cronache Ercolanesi 14: 136-146 (1984).
Marrone Livia. Il Mentitore nel PHerc. 307. In Proceedings of the XVIII International Congress of Papyrology. Athens 25-31 May 1986. Edited by Mandilaras Basil G. Athens: Greek Payrological Society 1988. pp. 271-276
Marrone Livia. L'ambiguità verbale nel PHerc. 307 (Questioni logiche di Crisippo). In Proceedings of the XIX International Congress of Papyrology. Cairo 2-9 September 1989. Edited by Mosallamy Abd Alla Hassan el. Cairo: Ain Shams University, 1992. pp. 261-267
"Un nouvel examen des col. 4-14 de ce papyrus montre que cette oeuvre a bien, comme le veut le genre des zetemata, un caractère aporétique, présentant des propositions dans lesquelles le semainomenon n'est pas exprimé clairement. La variété et la vivacité de ce texte important pour notre connaissance de la dialectique des stoïciens traduit l'existence d'un milieu culturel riche en innovations et en provocations."
Marrone Livia. Gnoseologia stoica nel PHerc 307. In Ercolano 1738-1988. 250 anni di ricerca archeologica. Edited by Dell'Orto Luisa Franchi. Roma: "L'Erma" di Bretschneider 1993. pp. 339-341
Atti del Convegno internazionale Ravello-Ercolano-Napoli-Pompei 30 ottobre-5 novembre 1988
Marrone Livia, "Le Questioni logiche di Crisippo (PHerc. 307)," Cronache Ercolanesi 27: 83-100 (1997).
Critical edition and Italian translation of Chrysippus' work Logika zetemata (Investigations in logic) found in the Herculaneum Papyrus 307
Marrone Livia. Logica nei papiri ercolanesi. In Testimonianze matematiche a Napoli. Edited by Carbone Luciano and Palladino Franco. Napoli: La Città del Sole 1999. pp. 153-163
Atti del Convegno di Napoli, 19 marzo 1991.
Marrone Livia, "La logica degli epicurei e degli stoici: Filodemo e Crisippo," Cronache Ercolanesi 30: 111-118 (2000).
"Profilo dei trattati filodemei di epistemologia, fondati sulla rielaborazione delle lezioni di Zenone Sidonio (PHerc. 1389, 1003, 671, 861, 1065), e di alcuni testi di logica stoica, con particolare attenzione alle "Questioni logiche" di Crisippo di Soli (PHerc. 1022, 307)."
Mignucci Mario. Alessandro di Afrodisia e la logica modale di Crisippo. In Atti del Convegno internazionale di storia della logica. Edited by Abrusci Michele, Casari Ettore, and Mugnai Massimo. Bologna: CLUEB 1983. pp.
Mignucci Mario, "The Stoic analysis of the Sorites," Proceedings of the Aristotelian Society 93: 231-245 (1993).
"Chrysippus' solution of the sorites argument was in keeping with his general view of paradoxes; we must give up some of our common-sense beliefs, such as our belief in the validity of modus ponens."
Nasti de Vincentis Mauro. Chrysippean implication as stoic equivalence. In Atti del Convegno internazionale di storia della logica organizzato dalla Società italiana di logica e filosofia delle scienze (SILFS), San Gimignano, 4-8 dicembre 1982. Edited by Abrusci Michele, Casari Ettore, and Mugnai Massimo. Bologna: CLUEB 1983. pp. 235-240
Nasti de Vincentis Mauro, "La validità del condizionale crisippeo in Sesto Empirico e Boezio (Parte I)," Dianoia 3: 45-75 (1998).
Nasti de Vincentis Mauro, "La validità del condizionale crisippeo in Sesto Empirico e Boezio (Parte II)," Dianoia 4: 11-43 (1999).
Pachet Pierre, "La deixis selon Zénon et Chrysippe," Phronesis.A Journal for Ancient Philosophy 20: 241-246 (1975).
Sedley David, "The negated conjunction in Stoicism," Elenchos.Rivista di Studi sul Pensiero Antico 5: 311-316 (1984).
Vuillemin Jules, "Le carré Chrysippéen des modalités," Dialectica 37: 235-247 (1983).
"A sentence p is necessary, according to Chrysippus if p is true and, p being susceptible of being false, the external circumstances prevent it from being false. This explanation is used in order to construct the modal square or rather the two modal squares. Something is specific in the chrysippean logic: from the necessity of p we may validly conclude that it is impossible that not p, while the converse conclusion is not valid."
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