Mathieu Marion | Université du Québec à Montréal (original) (raw)
Ancient Dialectic by Mathieu Marion
It is first argued that dialectic was a form of regimented debate, which grew out of public debat... more It is first argued that dialectic was a form of regimented debate, which grew out of public debates in Ancient Greece. A set of rules for dialectical bouts is then given and their meaning explained. The transition from oral to written arguments is briefly examined, leading to the formulation of a delimitation problem in Plato's dialogues, as he inserted dialectical arguments within ordinary dialogue contexts, turning them into discussions where one of the participants reasons hypothetically to make the other realize that they are not entitled to their view. Doing so, Plato adjusted dialectic to a variety of dialogue purposes and in order to explore this variety, a study of the early tradition of classifying Plato's dialogues in terms of their 'character' is suggested, the results of which are then compared with types of dialogues in contemporary Argumentation Theory. 4.1 Public Debates and Dialectic as Regimented Debate Ut nihil affirmet ipse, refellat alios G. E. R. Lloyd argued in Magic, Reason and Experience that beliefs about nature were subjected within Ancient Greek culture to the same "radical examination" as political views: they were openly challenged in public debates, where every assumption was liable to be scrutinized. 1 In the opening section of On the Nature of Man, the author describes public debates between contending speakers concerning such beliefs: He who is accustomed to hear speakers discuss the nature of man beyond its relations to medicine will not find the present account of any interest. For I do not say at all that a man is air, or fire, or water, or earth, or anything else that is not an obvious constituent of man; such accounts I leave to those that care to give them. Those, however, who give them have 1 [42, p. 248].
Récit et reconstruction (Panaccio C., 2019a). Mon texte touche à plusieurs aspects de cette contr... more Récit et reconstruction (Panaccio C., 2019a). Mon texte touche à plusieurs aspects de cette controverse, mais son but n'est pas de prendre parti. 2 Voir aussi Libera A. de, 1991, p. 353. 3 J'emploie à dessein la notion « système épistémique », plus large que celle de « théorie » utilisée par Kuhn. Pour une critique des versions radicales du relativisme, voir Boghossian P., 2009. Pour éviter toute confusion, je précise que je ne partage pas ce relativisme
In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristot... more In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristotle's Prior Analytics, with a view to motivate a new approach, which builds upon previous work by Marion & Rückert (2016) on the dictum de omni. This approach sets Aristotle's work within the context of dialectic and uses Lorenzen's dialogical logic, hereby reframed with use of Martin-Löf's constructive type theory as 'immanent reasoning'. We then provide rules of syllogistic for the latter, and provide proofs of e-conversion, Darapti and Bocardo and e-subalternation, while showing how close to Aristotle's text these proofs remain.
In this paper we provide an interpretation of Aristotle’s rule for the universal quantifier in To... more In this paper we provide an interpretation of Aristotle’s rule for the universal quantifier in Topics 157a34–37 and 160b1–6 in terms of Paul Lorenzen’s dialogical logic. This is meant as a contribution to the rehabilitation of the role of dialectic within the Organon. After a review of earlier views of Aristotle on quantification, we argue that this rule is related to the dictum de omni in Prior Analytics A 24b28–29. This would be an indication of the dictum’s origin in the context of dialectical games. One consequence of our approach is a novel explanation of the doctrine of the existential import of the quantifiers in dialectical terms. After a brief survey of Lorenzen’s dialogical logic, we offer a set of rules for dialectical games based on previous work by Castelnérac and Marion, to which we add here the rule for the universal quantifier, as interpreted in terms of its counterpart in dialogical logic. We then give textual evidence of the use of that rule in Plato’s dialogues, thus showing that Aristotle only made explicit a rule already implicit in practice, while providing a new interpretation of ‘epagogic’ arguments. Finally, we show how a proper understanding of that rule involves further rules concerning counterexamples and delaying tactics, stressing again the parallels with dialogical logic.
ABSTRACT: After presenting the rules of Eleatic antilogic, i.e., dialectic, I argue that Zeno was... more ABSTRACT: After presenting the rules of Eleatic antilogic, i.e., dialectic, I argue that Zeno was a practitioner, and, on the basis of key passages from Plato’s Parmenides (127e-128e and 135d-136c), that his paradoxes of divisibility and movement were not reductio ad absurdum, but simple derivation of impossibilities (adunaton) meant to ridicule Parmenides’ adversaries. Thus, Zeno did not try to prove that there is no motion, but simply derived this consequence from premises admitted to by his opponents. I argue further that these paradoxes were devised, in accordance with Eleatic antilogic, following a scheme that included hypotheses and their contradictories, within which the subject is to be treated both “in relation to itself,” and “in relation to other things”
This paper is an interim report of joint work begun in (Castelnérac & Marion 2009) on dialectic f... more This paper is an interim report of joint work begun in (Castelnérac & Marion 2009) on dialectic from Parmenides to Aristotle. In the first part we present rules for dialectical games, understood as a specific form of antilogikê developed by philosophers, and explain some of the key concepts of these dialectical games in terms of ideas from game semantics. In the games we describe, for a thesis A asserted by the answerer, a questioner must elicit the answerer’s assent to further assertions B1, B2,…, Bn, which form a scoreboard from which the questioner seeks to infer an impossibility (adunaton); we explain why the questioner must not insert any of his own assertions in the scoreboard, as well as the crucial role the Law of Non Contradiction, and why the games end with the inference to an impossibility, as opposed to the assertion of ¬A. In the second part we introduce some specific characteristics of Eleatic Antilogic as a method of enquiry. When Antilogic is used as a method of inquiry, then one must play not only the game beginning with a given thesis A, but also the game for ¬A as well as for A & ¬A, while using a peculiar set of opposite predicates to generate the arguments. In our discussion we hark back to Parmenides’ Poem, and illustrate our points with Zeno’s arguments about divisibility, Gorgias’ ontological argument from his treatise On Not-Being, and the second part of Plato’s Parmenides. We also identify numerous links to Aristotle, and conclude with some speculative comments on the origin of logic.
In this paper we propose, along the model of modern dialogical games (in the school of Lorenzen),... more In this paper we propose, along the model of modern
dialogical games (in the school of Lorenzen), a set of structural and particle rules, for dialectical games as played in the Academy at the time of Plato and Aristotle. We argue that dialectical games are consistency-management games, in which, for a given thesis A asserted at the outset by the proponent, the opponent (usually Socrates) argues for the inconsistency of a set of beliefs of the proponent that includes A, and nothing more. We thus criticize Vlastos' interpretation of the Socratic elenchus, according to which Socrates meant to 'refute' the proponent, i.e., to show that A is false and, therefore, that ¬A is true. Finally, we offer some suggestions concerning the point of playing these games and the birth of logic in the context of dialectical games.
Philosophy of Logic by Mathieu Marion
This formulation does not fully render justice to Carroll's own formulation: "Readers of Euclid w... more This formulation does not fully render justice to Carroll's own formulation: "Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that anyone who accepts A and B as true, must accept Z as true" [Carroll 1895, p.278]. From this one may infer that Carroll has in mind a concept of logical consequence, involving a conditional of the form 'If "A" is true and "B" is true, then "Z" is true'. At the time Carroll wrote, one would not be fully aware of the distinction between object-language and metalanguage conditionals.
The relation between logic and knowledge has been at the heart of a lively debate since the 1960s... more The relation between logic and knowledge has been at the heart of a lively debate since the 1960s. On the one hand, the epistemic approaches based their formal arguments in the mathematics of Brouwer and intuitionistic logic. Following Michael Dummett, they started to call themselves `antirealists'. Others persisted with the formal background of the Frege-Tarski tradition, where Cantorian set theory is linked via model theory to classical logic. Jaakko Hintikka tried to unify both traditions by means of what is now known as `explicit epistemic logic'. Under this view, epistemic contents are introduced into the object language as operators yielding propositions from propositions, rather than as metalogical constraints on the notion of inference.
The Realism-Antirealism debate has thus had three players: classical logicians, intuitionists and explicit epistemic logicians. The editors of the present volume believe that in the age of Alternative Logics, where manifold developments in logic happen at a breathtaking pace, this debate should be revisited. Contributors to this volume happily took on this challenge and responded with new approaches to the debate from both the explicit and the implicit epistemic point of view.
Game semantics has almost achieved the status of a paradigm in computer science but philosophers ... more Game semantics has almost achieved the status of a paradigm in computer science but philosophers are slow to take notice. One reason for this might be the lack of a convincing philosophical account of logical games, what it means to play them, for the proponent to win, etc., pointedly raised by Wilfrid Hodges as the ’Dawkins question’. In this paper, I critically examine two available answers: after a brief discussion of an argument by Tennant against Hintikka games, I focus on Lorenzen’s attempt at providing a direct foundation for his game rules in the life-world, showing some of the difficulties inherent to that project. I then propose an alternative based on the theory of assertions developed by Dummett and Brandom.
In this paper, I explore the possibility of replacing the philosophy of the Erlangen school with ... more In this paper, I explore the possibility of replacing the philosophy of the Erlangen school with Brandom's inferentialism to provide a philosophical basis for game semantics.
After sketching an argument for radical anti-realism that does not appeal to human limitations bu... more After sketching an argument for radical anti-realism that does not appeal to human limitations but polynomial-time computability in its definition of feasibility, I revisit an argument by Wittgenstein on the surveyability of proofs, and then examine the consequences of its application to the notion of canonical proof in contemporary proof-theoretical-semantics.
Logic, Epistemology, and the Unity of Science, 2004
Selected Contributed Papers of the 11th International Congress of Logic, Methodology and Philosophy of Science, Krakòw, 1999, 2003
Taking my lead from a quotation from Carnap and through a discussion of Turing’s and Church’s The... more Taking my lead from a quotation from Carnap and through a discussion of Turing’s and Church’s Theses, I argue that it is necessary for rational decision theory not only to take into account the limitations of Turing Machines, but also the intrinsic limitations of digital computing machines themselves. There are problems that are in principle computable by a Turing machine that are nevertheless intractable for digital computing machines, and the latter, limited as they are to polynomial-time (or “feasible”) computations must be seen as the true idealized (human) computers. After pointing out that a Turing Complexity Thesis (due to H. Levesque) could improve Cherniak’s proposals for a “minimal rationality”, I conclude with some brief critical remarks aimed at the current paradigmatic notion of “rationality” underlying decision theory, and the idea that the full classical logic should keep a privileged status in this context.
It is first argued that dialectic was a form of regimented debate, which grew out of public debat... more It is first argued that dialectic was a form of regimented debate, which grew out of public debates in Ancient Greece. A set of rules for dialectical bouts is then given and their meaning explained. The transition from oral to written arguments is briefly examined, leading to the formulation of a delimitation problem in Plato's dialogues, as he inserted dialectical arguments within ordinary dialogue contexts, turning them into discussions where one of the participants reasons hypothetically to make the other realize that they are not entitled to their view. Doing so, Plato adjusted dialectic to a variety of dialogue purposes and in order to explore this variety, a study of the early tradition of classifying Plato's dialogues in terms of their 'character' is suggested, the results of which are then compared with types of dialogues in contemporary Argumentation Theory. 4.1 Public Debates and Dialectic as Regimented Debate Ut nihil affirmet ipse, refellat alios G. E. R. Lloyd argued in Magic, Reason and Experience that beliefs about nature were subjected within Ancient Greek culture to the same "radical examination" as political views: they were openly challenged in public debates, where every assumption was liable to be scrutinized. 1 In the opening section of On the Nature of Man, the author describes public debates between contending speakers concerning such beliefs: He who is accustomed to hear speakers discuss the nature of man beyond its relations to medicine will not find the present account of any interest. For I do not say at all that a man is air, or fire, or water, or earth, or anything else that is not an obvious constituent of man; such accounts I leave to those that care to give them. Those, however, who give them have 1 [42, p. 248].
Récit et reconstruction (Panaccio C., 2019a). Mon texte touche à plusieurs aspects de cette contr... more Récit et reconstruction (Panaccio C., 2019a). Mon texte touche à plusieurs aspects de cette controverse, mais son but n'est pas de prendre parti. 2 Voir aussi Libera A. de, 1991, p. 353. 3 J'emploie à dessein la notion « système épistémique », plus large que celle de « théorie » utilisée par Kuhn. Pour une critique des versions radicales du relativisme, voir Boghossian P., 2009. Pour éviter toute confusion, je précise que je ne partage pas ce relativisme
In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristot... more In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristotle's Prior Analytics, with a view to motivate a new approach, which builds upon previous work by Marion & Rückert (2016) on the dictum de omni. This approach sets Aristotle's work within the context of dialectic and uses Lorenzen's dialogical logic, hereby reframed with use of Martin-Löf's constructive type theory as 'immanent reasoning'. We then provide rules of syllogistic for the latter, and provide proofs of e-conversion, Darapti and Bocardo and e-subalternation, while showing how close to Aristotle's text these proofs remain.
In this paper we provide an interpretation of Aristotle’s rule for the universal quantifier in To... more In this paper we provide an interpretation of Aristotle’s rule for the universal quantifier in Topics 157a34–37 and 160b1–6 in terms of Paul Lorenzen’s dialogical logic. This is meant as a contribution to the rehabilitation of the role of dialectic within the Organon. After a review of earlier views of Aristotle on quantification, we argue that this rule is related to the dictum de omni in Prior Analytics A 24b28–29. This would be an indication of the dictum’s origin in the context of dialectical games. One consequence of our approach is a novel explanation of the doctrine of the existential import of the quantifiers in dialectical terms. After a brief survey of Lorenzen’s dialogical logic, we offer a set of rules for dialectical games based on previous work by Castelnérac and Marion, to which we add here the rule for the universal quantifier, as interpreted in terms of its counterpart in dialogical logic. We then give textual evidence of the use of that rule in Plato’s dialogues, thus showing that Aristotle only made explicit a rule already implicit in practice, while providing a new interpretation of ‘epagogic’ arguments. Finally, we show how a proper understanding of that rule involves further rules concerning counterexamples and delaying tactics, stressing again the parallels with dialogical logic.
ABSTRACT: After presenting the rules of Eleatic antilogic, i.e., dialectic, I argue that Zeno was... more ABSTRACT: After presenting the rules of Eleatic antilogic, i.e., dialectic, I argue that Zeno was a practitioner, and, on the basis of key passages from Plato’s Parmenides (127e-128e and 135d-136c), that his paradoxes of divisibility and movement were not reductio ad absurdum, but simple derivation of impossibilities (adunaton) meant to ridicule Parmenides’ adversaries. Thus, Zeno did not try to prove that there is no motion, but simply derived this consequence from premises admitted to by his opponents. I argue further that these paradoxes were devised, in accordance with Eleatic antilogic, following a scheme that included hypotheses and their contradictories, within which the subject is to be treated both “in relation to itself,” and “in relation to other things”
This paper is an interim report of joint work begun in (Castelnérac & Marion 2009) on dialectic f... more This paper is an interim report of joint work begun in (Castelnérac & Marion 2009) on dialectic from Parmenides to Aristotle. In the first part we present rules for dialectical games, understood as a specific form of antilogikê developed by philosophers, and explain some of the key concepts of these dialectical games in terms of ideas from game semantics. In the games we describe, for a thesis A asserted by the answerer, a questioner must elicit the answerer’s assent to further assertions B1, B2,…, Bn, which form a scoreboard from which the questioner seeks to infer an impossibility (adunaton); we explain why the questioner must not insert any of his own assertions in the scoreboard, as well as the crucial role the Law of Non Contradiction, and why the games end with the inference to an impossibility, as opposed to the assertion of ¬A. In the second part we introduce some specific characteristics of Eleatic Antilogic as a method of enquiry. When Antilogic is used as a method of inquiry, then one must play not only the game beginning with a given thesis A, but also the game for ¬A as well as for A & ¬A, while using a peculiar set of opposite predicates to generate the arguments. In our discussion we hark back to Parmenides’ Poem, and illustrate our points with Zeno’s arguments about divisibility, Gorgias’ ontological argument from his treatise On Not-Being, and the second part of Plato’s Parmenides. We also identify numerous links to Aristotle, and conclude with some speculative comments on the origin of logic.
In this paper we propose, along the model of modern dialogical games (in the school of Lorenzen),... more In this paper we propose, along the model of modern
dialogical games (in the school of Lorenzen), a set of structural and particle rules, for dialectical games as played in the Academy at the time of Plato and Aristotle. We argue that dialectical games are consistency-management games, in which, for a given thesis A asserted at the outset by the proponent, the opponent (usually Socrates) argues for the inconsistency of a set of beliefs of the proponent that includes A, and nothing more. We thus criticize Vlastos' interpretation of the Socratic elenchus, according to which Socrates meant to 'refute' the proponent, i.e., to show that A is false and, therefore, that ¬A is true. Finally, we offer some suggestions concerning the point of playing these games and the birth of logic in the context of dialectical games.
This formulation does not fully render justice to Carroll's own formulation: "Readers of Euclid w... more This formulation does not fully render justice to Carroll's own formulation: "Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that anyone who accepts A and B as true, must accept Z as true" [Carroll 1895, p.278]. From this one may infer that Carroll has in mind a concept of logical consequence, involving a conditional of the form 'If "A" is true and "B" is true, then "Z" is true'. At the time Carroll wrote, one would not be fully aware of the distinction between object-language and metalanguage conditionals.
The relation between logic and knowledge has been at the heart of a lively debate since the 1960s... more The relation between logic and knowledge has been at the heart of a lively debate since the 1960s. On the one hand, the epistemic approaches based their formal arguments in the mathematics of Brouwer and intuitionistic logic. Following Michael Dummett, they started to call themselves `antirealists'. Others persisted with the formal background of the Frege-Tarski tradition, where Cantorian set theory is linked via model theory to classical logic. Jaakko Hintikka tried to unify both traditions by means of what is now known as `explicit epistemic logic'. Under this view, epistemic contents are introduced into the object language as operators yielding propositions from propositions, rather than as metalogical constraints on the notion of inference.
The Realism-Antirealism debate has thus had three players: classical logicians, intuitionists and explicit epistemic logicians. The editors of the present volume believe that in the age of Alternative Logics, where manifold developments in logic happen at a breathtaking pace, this debate should be revisited. Contributors to this volume happily took on this challenge and responded with new approaches to the debate from both the explicit and the implicit epistemic point of view.
Game semantics has almost achieved the status of a paradigm in computer science but philosophers ... more Game semantics has almost achieved the status of a paradigm in computer science but philosophers are slow to take notice. One reason for this might be the lack of a convincing philosophical account of logical games, what it means to play them, for the proponent to win, etc., pointedly raised by Wilfrid Hodges as the ’Dawkins question’. In this paper, I critically examine two available answers: after a brief discussion of an argument by Tennant against Hintikka games, I focus on Lorenzen’s attempt at providing a direct foundation for his game rules in the life-world, showing some of the difficulties inherent to that project. I then propose an alternative based on the theory of assertions developed by Dummett and Brandom.
In this paper, I explore the possibility of replacing the philosophy of the Erlangen school with ... more In this paper, I explore the possibility of replacing the philosophy of the Erlangen school with Brandom's inferentialism to provide a philosophical basis for game semantics.
After sketching an argument for radical anti-realism that does not appeal to human limitations bu... more After sketching an argument for radical anti-realism that does not appeal to human limitations but polynomial-time computability in its definition of feasibility, I revisit an argument by Wittgenstein on the surveyability of proofs, and then examine the consequences of its application to the notion of canonical proof in contemporary proof-theoretical-semantics.
Logic, Epistemology, and the Unity of Science, 2004
Selected Contributed Papers of the 11th International Congress of Logic, Methodology and Philosophy of Science, Krakòw, 1999, 2003
Taking my lead from a quotation from Carnap and through a discussion of Turing’s and Church’s The... more Taking my lead from a quotation from Carnap and through a discussion of Turing’s and Church’s Theses, I argue that it is necessary for rational decision theory not only to take into account the limitations of Turing Machines, but also the intrinsic limitations of digital computing machines themselves. There are problems that are in principle computable by a Turing machine that are nevertheless intractable for digital computing machines, and the latter, limited as they are to polynomial-time (or “feasible”) computations must be seen as the true idealized (human) computers. After pointing out that a Turing Complexity Thesis (due to H. Levesque) could improve Cherniak’s proposals for a “minimal rationality”, I conclude with some brief critical remarks aimed at the current paradigmatic notion of “rationality” underlying decision theory, and the idea that the full classical logic should keep a privileged status in this context.
Ce document est protégé par la loi sur le droit d'auteur. L'utilisation des services d'Érudit (y ... more Ce document est protégé par la loi sur le droit d'auteur. L'utilisation des services d'Érudit (y compris la reproduction) est assujettie à sa politique d'utilisation que vous pouvez consulter en ligne. [https://apropos.erudit.org/fr/usagers/politiquedutilisation/] Cet article est diffusé et préservé par Érudit.
This volume is a collation of original contributions from the key actors of a new trend in the co... more This volume is a collation of original contributions from the key actors of a new trend in the contemporary theory of knowledge and belief, that we call “dynamic epistemology”. It brings the works of these researchers under a single umbrella by highlighting the coherence of their current themes, and by establishing connections between topics that, up until now, have been investigated independently. It also illustrates how the new analytical toolbox unveils questions about the theory of knowledge, belief, preference, action, and rationality, in a number of central axes in dynamic epistemology: temporal, social, probabilistic and even deontic dynamics.
Ce texte, dont la vocation est de rester inédit (et à être révisé périodiquement), est à l'attent... more Ce texte, dont la vocation est de rester inédit (et à être révisé périodiquement), est à l'attention des étudiantes et étudiantes qui abordent l'œuvre de Wittgenstein pour la première fois, dans le souhait qu'il puisse leur être utile. Il contient dans une première partie une explication détaillée des problèmes entourant la constitution et l'accès au Nachlass de Wittgenstein, et ceux concernant les publications qui en ont été tirées. Son aussi abordés les conséquences délétères de tout ces problèmes sur l'interprétation et la traduction de ses œuvres. Le texte contient aussi un aperçu du contenu de ce Nachlass et les liens vers les principaux sites dont celui des archives Wittgenstein à l’Université de Bergen qui donnent accès à l'intégralité du Nachlass. La deuxième partie est une bibliographie exhaustive des publications de Wittgenstein, en allemand, en anglais et en français. (Mars 2023.)
Dernière version téléversée : décembre 2023
Traduction portugaise : M. Marion, Ludwig Wittgenstein. Introdução ao Tractatus Logico- Philosoph... more Traduction portugaise : M. Marion, Ludwig Wittgenstein. Introdução ao Tractatus Logico- Philosophicus, São Paulo, Annablume, 2012.
The book has been reviewed in these journals: Philosophia Mathematica, vol. 10, 2002, pp. 67-88 (... more The book has been reviewed in these journals: Philosophia Mathematica, vol. 10, 2002, pp. 67-88 (J. Floyd) // Mind, vol. 110, 2001, pp. 501-504 (M. Joseph) // Dialogue, vol. 40, 2001, pp. 624-626 (D. Stern) // Philosophical Review, vol. 110, 2001, pp. 286-9 (P. Mancosu) // Slagmark, n. 31, 2001, pp. 152-154 (P. Kjaergaard) // Philosophy of Science, vol. 67, 2000, pp. 533-536 (E. Wynsberg) // Studia Logica , vol. 65, 2000, pp. 432-434 (A. W. Moore) // Philosophy in Review, vol. 20, 2000, pp. 132-134 (J. R. Loftis) // Ruch Filozoficzny, n. 3-4, 1999 (J. Wolenski)
This paper is also reprinted in: P. Mancosu, The Adventure of Reason. Interplay between Philosoph... more This paper is also reprinted in: P. Mancosu, The Adventure of Reason. Interplay between Philosophy of Mathematics, Mathematics and Mathematical Logic, 1900-1940, Oxford, Oxford University Press, 2010, p. 217-231.
Wittgenstein, Goodstein, and the Origin of the Uniqueness Rule for Primitive Recursive Arithmetic... more Wittgenstein, Goodstein, and the Origin of the Uniqueness Rule for Primitive Recursive Arithmetic mathieu marion and mitsuhiro okada 1. Introduction Reuben Louis Goodstein studied mathematics at Cambridge from 1931 until 1935. 1 His work on ordinal notation systems of transfinite ordinal numbers, under the supervision of John Littlewood, is at the basis of the result that bears his name, "Goodstein's theorem." 2 He was also one of Littlewood's students that attended Wittgenstein's lectures. Although there is no reason to believe that Wittgenstein was particularly close to Goodstein, it seems that he nevertheless held him in some degree of esteem. Indeed, when Wittgenstein cancelled his lectures in 1933 and chose instead to dictate The Blue Book to a selected group of students, it included Goodstein, alongside Alice Ambrose, Margaret Masterman and two further mathematics students, H. S. M. Coxeter and Francis Skinner. Goodstein and Skinner had been close friends since their schooldays at St. Paul's, London. 3 This may explain why upon Skinner's death in 1941, Wittgenstein mailed to Goodstein a number of important manuscripts that had been until then in Skinner's possession, including a set of revisions to The Brown Book. 4 1 On Goodstein's life, see Rose 1988. 2 See Goodstein 1944. Goodstein's theorem is a purely number-theoretic statement using implicitly the fact that all strictly decreasing sequences of transfinite ordinal notational systems up to ε 0 are finite. Its importance was only recognized when Laurence Kirby and Jeff Paris showed that it provides a concrete example of Gödel's incompleteness theorem, i.e., a true number-theoretic statement which is not provable within first-order Peano Arithmetic (Kirby & Paris 1982). 3 See Monk 1990, 336. 4 These manuscripts resurfaced in 2000, and are currently held on loan at Trinity College. For detailed information about their content, see Gibson 2010.
This paper aims to connect two of Wittgenstein’s arguments against Logicism. The ‘modality argume... more This paper aims to connect two of Wittgenstein’s arguments against Logicism. The ‘modality argument’ is directed at the Frege/Russell-definition of numbers in terms of one-one correlations. According to this argument, it is only when the Fs and Gs are few in number that one can know that they can be one-one correlated without knowing their numbers. Wittgenstein’s ‘surveyability argument’ purports to show that only a limited portion of arithmetic can actually be proven within Principia Mathematica. For proof-constructions within this system quickly become unsurveyable and thereby loose their cogency. As we shall argue, the role of visualisation in proofs plays a fundamental role in both arguments.
In this paper we provide, paying attention to their context of origin in the Middle Period, an ov... more In this paper we provide, paying attention to their context of origin in the Middle Period, an overview of Wittgenstein’s remarks on contradiction and consistency proofs, which is organized around three major themes: the distinction between ‘absolute’ and ‘relative’ consistency proofs, the idea of ‘hidden’ contradictions, and the idea that a contradiction does no harm, because once derived the rules can be fixed and previous results can be maintained. We argue that Wittgenstein first distinguished between the calculus of number-theoretic equations and the logical calculus, which does not apply to it. We argue that Wittgenstein’s remarks amount to a sustained attempt at developing an alternative to a ‘realist’ view of contradictions, according to which if a set of axioms is inconsistent, but no contradiction is yet found, it means that one is lurking in the set of yet underived consequences, and this contradiction vitiates the calculus like some silent disease. We also provide rejoinders to A. M. Turing and Charles Chihara, who provide criticisms usually considered definitive.
(This paper is reprinted in H. D. Kurz, L. L. Pasinetti & N. Salvadori (eds.), Piero Sraffa: The ... more (This paper is reprinted in H. D. Kurz, L. L. Pasinetti & N. Salvadori (eds.), Piero Sraffa: The Man and the Scholar, London, Routledge, 2008, p. 217-242.) Abstract: After a brief review of facts and hypotheses concerning Piero Sraffa's intellectual exchanges with the philosopher Ludwig Wittgenstein and their content, a brief presentation of some of the basic ideas of Productions of Commodities by Means of Commodities is given, on the basis of which I argue, first, that Sraffa's 'objectivism' in economics is closely related to the 'physicalism' towards which Wittgenstein moved soon after his return to Cambridge and, secondly, that the mathematics of that book are in line with Wittgenstein's constructivist stance, as it is already found in his Tractatus Logico-Philosophicus.
This paper is a follow up of sorts to my earlier piece 'Wittgenstein and Brouwer' (Synthese, vol.... more This paper is a follow up of sorts to my earlier piece 'Wittgenstein and Brouwer' (Synthese, vol. 137 (2003), p. 103-127) where I discussed substantive points of contact between Wittgenstein and intuitionism, e.g., the role of intuition in mathematics, rule-following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic. I then emphasized the role of Brouwer's 1928 lecture in Vienna that Wittgenstein attended and rejected the anachronistic type of analysis that uses his later views to dismiss it as irrelevant in light of Wittgenstein's later views: he was not in command of them in 1928 and it is only exegetical common sense to assume that it was the author of the Tractatus who listened to that lecture. This approach leads to new conclusions concerning the topics just mentioned. In this new paper, I add to this historical evidence (ignored in biographies of Wittgenstein) that Wittgenstein and Brouwer met afterwards, and then discuss a further topic, Wittgenstein's peculiar notion of 'hypothesis', which played, as I argued in my 1998 book, a key role in the 'middle period'. I argue that the notion already appears in Brouwer's lecture.
Synthese, 2000
In this paper, I present a summary of the philosophical relationship between Wittgenstein and Bro... more In this paper, I present a summary of the philosophical relationship between Wittgenstein and Brouwer, taking as my point of departure Brouwer's lecture on March 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not done serious philosophical work for years, if one is to understand the impact of that lecture on him, it is better to compare its content with the remarks on logics and mathematics in the Tractactus. I thus show that Wittgenstein's position, in the Tractactus, was already quite close to Brouwer's and that the points of divergence are the basis to Wittgenstein's later criticisms of intuitionism. Among the topics of comparison are the role of intuition in mathematics, rule following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic. To the memory of Michael Wrigley.
In this paper, elementary but hitherto overlooked connections are established between Wittgenstei... more In this paper, elementary but hitherto overlooked connections are established between Wittgenstein’s remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of the Tractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein’s rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem’s primitive recursive arithmetic and Goodstein’s equational calculus, Wittgenstein represented generality by the use of free variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein’s criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein’s remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum.
When selections from Wittgenstein's manuscripts appeared in 1956 under the title Remarks on the F... more When selections from Wittgenstein's manuscripts appeared in 1956 under the title Remarks on the Foundations of Mathematics, the philosophical community largely agreed that Wittgenstein had been arguing for what amounts to a specific foundational theory called "strict finitism". This interpretation had originated in critical studies of that book by Georg Kreisel (1959) and Sir Michael Dummett (1978). There have been very few dissenters since, such as Robert Fogelin, who has argued (1968) for connections with Brouwer's intuitionism. I have myself contended (1998) that Wittgenstein's Standpunkt is a form of Kroneckerian, free-variable finitism close to that of Skolem and that of the equational calculus of Wittgenstein's own pupil, Louis Goodstein. "Strict finitism", "finitism", and "intuitionism" are, of course, all variants of "constructivism", which is often pictured as a form of "revisionism" in mathematics. The precise nature of Wittgenstein's constructivism is not, however, what interests me here. It is rather the fact that today's received view has it that he was not a constructivist after all, not because he turns out to have been an apologist of "classical" mathematics but because he wished to hold no position in the debates about the foundations of mathematics. One of the oldest claims of this sort comes from the late Gordon Baker and Peter Hacker, who wrote that Wittgenstein's "philosophy of mathematics does not defend a form of 'strict finitism', depsychologized 'intuitionism' or 'constructivism'" and that his purpose was "not to take sides in the debates between rival schools of mathematicians, but rather to question the presuppositions which provided the framework of their debates" (1985, 345).
What Russell and Frege do is to make connexions between English and German words "all", "or", "an... more What Russell and Frege do is to make connexions between English and German words "all", "or", "and", etc. and numerical statements. This clears up a few points. But that we should actually then say,
Philosophiques, 2000
Montréal. Il a pour mission la promotion et la valorisation de la recherche. Érudit offre des ser... more Montréal. Il a pour mission la promotion et la valorisation de la recherche. Érudit offre des services d'édition numérique de documents scientifiques depuis 1998. Pour communiquer avec les responsables d'Érudit : info@erudit.org Article « Wittgenstein et le lien entre la signification d'un énoncé mathématique et sa preuve » Mathieu Marion et Mitsuhiro Okada Note : les règles d'écriture des références bibliographiques peuvent varier selon les différents domaines du savoir.
Distribution électronique Cairn pour Centres Sèvres. © Centres Sèvres. Tous droits réservés pour ... more Distribution électronique Cairn pour Centres Sèvres. © Centres Sèvres. Tous droits réservés pour tous pays. La reproduction ou représentation de cet article, notamment par photocopie, n'est autorisée que dans les limites des conditions générales d'utilisation du site ou, le cas échéant, des conditions générales de la licence souscrite par votre établissement. Toute autre reproduction ou représentation, en tout ou partie, sous quelque forme et de quelque manière que ce soit, est interdite sauf accord préalable et écrit de l'éditeur, en dehors des cas prévus par la législation en vigueur en France. Il est précisé que son stockage dans une base de données est également interdit.
Wittgenstein commented briefly on Heidegger in a conversation in 1929 with Schlick and Waismann a... more Wittgenstein commented briefly on Heidegger in a conversation in 1929 with Schlick and Waismann and in a dictation to the latter from the former in 1932. In this chapter, I set forth one minor historical argument against current, pragmatist readings that lump together Wittgenstein and Heidegger, an argument which involves reconstructing the context of Wittgenstein's remarks to see their intended point. I thus show that Wittgenstein's remarks were prompted by his having read Heidegger's inaugural lecture 'What is metaphysics?' (1929), and only that text. I argue from this that Wittgenstein never saw himself engaged in the sort of metaphysical enterprise he was engaged into and briefly examine his claims in 1932 that Heidegger needs a therapy analogous to psychoanalysis, and that his speaking 'whereof one should remain silent' amounts to a stylistic (hence moral) mistake.
We reconstruct a variation by Friedrich Waismann of Wittgenstein's rule-following argument, based... more We reconstruct a variation by Friedrich Waismann of Wittgenstein's rule-following argument, based on what we call the 'guessing game' (BB: 112 and PI § §151 and 179), and contrast it with Kripke's case of a deviant pupil (PI § §143 and 185). Our reconstruction follows Waismann's reliance on the cause-reason distinction, and it is completed by an explanation of what it means for the 'chain of reasons' to have an end, beyond which one can only appeal to causes. To conclude, we identify the contemporary debate on 'blind reasoning' as an area where Waismann's variation could play a role.
Waismann's writings can be divided into three periods. 2 The fi rst corresponds to his early work... more Waismann's writings can be divided into three periods. 2 The fi rst corresponds to his early work in Vienna under the aegis of Schlick, thus mainly to his collaboration with Wittgenstein on the fi rst drafts of Logik, Sprache, Philosophie, 3 out of which came not only the book itself many years later but also transcriptions of conversations with Schlick and Wittgenstein 4 and numerous dictations reworked by Waismann, now published under the title The Voice of Wittgenstein. The Vienna Circle. 5 Waismann also did at that stage independent work, albeit largely infl uenced by Wittgenstein, on probability and identity. 6 The second period runs roughly from the moment relations with Wittgenstein were severed-towards the end of 1934-to his arrival in Oxford, where he started lecturing in Michaelmas Term 1939. During this period, Waismann published his only book, Einführung in das mathematische Denken 7 but, while he completed his Logik, Sprache und Philosophie and even had it translated in English, plans for publication did not materialize and he chose instead to publish parts of it in Erkenntnis and Synthese. 8 The third period, extending until his death in 1959, saw the publication of a number of papers that established his reputation in England, collected since in How I see Philosophy 9-a volume which contains Waismann's only published piece on causality, 'The Decline and Fall of Causality' (hereafter DFC). 10 Although usually perceived as one of logical positivists, Waismann clearly distanced himself from them in his last writings; the summary of his 1947 lecture at the Socratic Club on 'The Limits of Positivism' being evidence to this. He was also at pains to distance himself from Wittgenstein, as one can see for example from the posthumous piece 1 References are to the page numbers of this edition of the typescript 'Causality'. This is M 13 in Schulte's Catalogue (Schulte 1979). 2 (Quinton 1977, xi-xii), (Schulte 1979, 109), (McGuinness & Schulte 1994, ix). 3 The manuscript Logik, Sprache, Philosophie and an English translation were destroyed during the war. A reconstructed version was published as (Waismann 1976). An English version had already appeared in 1965 which is now in its second edition (Waismann 1997). For details of this story, see (
This is an introduction to the English translation (in the same issue) of the transcript of the d... more This is an introduction to the English translation (in the same issue) of the transcript of the discussion générale which took place on the last morning of the colloquium on “analytic philosophy” at the Abbey of Royaumont, north of Paris, on 8-13 April 1958.
Paris — Wien, 2005
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This paper forms part of my investigations of British philosophy just prior to the advent of anal... more This paper forms part of my investigations of British philosophy just prior to the advent of analytic philosophy. The central arguments of the paper rely on a distinction between subjective idealism, of which Berkeley's system might be a representative, and for which the reality of the object is dissolved behind its representatives in the mind of the subject (e.g., 'sense-impressions' or 'sense-data'), and the objective realism of Bradley, who rejected the subject-object distinction in favour of a monism for which 'reality' is 'spirit'. I argue first that Mill explicitly followed Berkeley and that Russell also shared some key premises with both, while, secondly, Bradley and followers of Reid, from Cook Wilson to Austin, gave arguments against these very premises. That does not make the latter objective idealists, but the common arguments show that one cannot talk of Moore and Russell as having initiated a "revolution" more than simply another chapter in a rather long debate in the theory of knowledge.
http://plato.stanford.edu/entries/wilson/ This is the entry on John Cook Wilson for the Stanford... more http://plato.stanford.edu/entries/wilson/
This is the entry on John Cook Wilson for the Stanford Encyclopedia of Philosophy (extensive revisions, June 2022), with brief biographical notes and an introduction to his contribution to the theory of knowledge, to metaphysics (on universals) and to philosophy of logic, also charting his influence from H. A. Prichard, Gilbert Ryle and J. L. Austin, to J. McDowell and T. Williamson among others.
British Journal For the History of Philosophy, 2000
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf ... more Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden.
Revue d'Histoire des Sciences, 2014
The development of symbolic logic is often presented in terms of a cumulative story of consecutiv... more The development of symbolic logic is often presented in terms of a cumulative story of consecutive innovations that led to what is known as modern logic. This narrative hides the difficulties that this new logic faced at first, which shaped its history. Indeed, negative reactions to the emergence of the new logic in the second half of the nineteenth century were numerous and we study here one case, namely logic at Oxford, where one finds Lewis Carroll, a mathematical teacher who promoted symbolic logic, and John Cook Wilson, the Wykeham Professor of Logic who notoriously opponent it. An analysis of their disputes on the topic of logical symbolism shows that their opposition was not as sharp as it might look at first, as Cook Wilson was not so much opposed to the “symbolic” character of logic, but the intrusion of mathematics and what he perceived to be the futility of some of its problems, for logicians and philosophers alike.
Philosophiques, 2000
One often hears the opinion voiced that Bradley was an early critique of psychologism. In this pa... more One often hears the opinion voiced that Bradley was an early critique of psychologism. In this paper, I investigate that claim, focussing on his Principles of Logic (1883). I define psychologism in the narrow sense as a thesis pertaining to the foundations of logic, and psychologism in the wide sense as a more general thesis concerning the theory of knowledge, and show that Bradley rejected both, although he had little to say on the narrow version. His criticism of the wider version is based on his distinguishing between psychological and logical content and on his defence of the ideality of logical content, before Frege and Husserl. Nevertheless, he still hung to the idea that the latter harks back to ordinary perception. I then review briefly his criticisms of associationism in psychology, to show that he faced some difficulties in trying to avoid lapsing back into psychologism, with an appeal to a distinction between abstract and concrete universals. I conclude with some remarks on the palace of Bradley in the history of British psychology.
Philosophiques, 2000
British Idealism is a philosophical movement that dominated British universities (and those of it... more British Idealism is a philosophical movement that dominated British universities (and those of its empire), for fifty years around the turn from the XIXth to the XXth century, but it went largely unnoticed in the French-speaking world. Condemned by analytic philosophers, these authors were also ignored in their own country, but some of them, notably Bradley and Collingwood, are now enjoying a newly found popularity within the larger trend towards a study of the origins of analytic philosophy. This text is an introduction to British Idealism that plots, in an historical first part, the outlines of its rise, development and decline. In the second part, we provide reasons for further studies of this movement.
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If establishing an hypothesis through its predictions has a logic, so has the conceiving of an hy... more If establishing an hypothesis through its predictions has a logic, so has the conceiving of an hypothesis. N. R. Hanson, Patterns of Discovery 1 'See the value of imagination', said Holmes. […] We imagined what might have happened, acted upon the supposition, and find ourselves justified. Arthur Conan Doyle, Silver Blaze 2 * Les abréviations suivantes seront utilisées dans les notes. Pour les travaux de Jaakko Hintikka :
In this paper, we begin by suggesting an intuitive model of time embodying a notion of temporal d... more In this paper, we begin by suggesting an intuitive model of time embodying a notion of temporal distance that we claim is at work in Gadamer’s hermeneutics, while it is rejected in Collingwood’s theory of interpretation. To show this, after a brief review of the influence of Collingwood on Gadamer and of their disagreement over the possibility of recovering an author’s intention, we examine in turn their answers to the problem of transposition, upon which the philosophy of Dilthey supposedly foundered. We show that Gadamer embraced the idea of temporal distance in his solution, which consisted in claiming that the distance between an author from the past and us is filled in by tradition, which opens access to the text for us, while Collingwood considered explanations of the actions of historical agents, and by extension understanding of a text, in intentional or rational terms. Furthermore, he thought that such explanations are not causal, and that the thoughts involved in them do not stand within the flow of physical time, which is involved in any notion of temporal distance. This is why Collingwood felt entitled to anti-relativistic conclusions about the recovery of authorial intentions, conclusions that prompted Gadamer to claim that “the dimension of hermeneutical mediation which is passed through in every act of understanding” escaped him. We then discuss the underlying notions of time at work in both Gadamer and Collingwood, showing that Ricœur had a better appreciation of the issue, since he saw that Collingwood’s moves parallel, up to a point, Heidegger’s critique of “vulgar time,” albeit with an entirely different result. We also point to the importance in Collingwood’s thinking of his notion of “incapsulation.”
In this paper, we begin by suggesting an intuitive model of time embodying a notion of temporal d... more In this paper, we begin by suggesting an intuitive model of time embodying a notion of temporal distance that we claim is at work in Gadamer's hermeneutics, while it is rejected in Collingwood's theory of interpretation. To show this, after a brief review of the influence of Collingwood on Gadamer and of their disagreement over the possibility of recovering an author's intention, we examine in turn their answers to the problem of transposition, upon which the philosophy of Dilthey supposedly foundered. We show that Gadamer embraced the idea of temporal distance in his solution, which consisted in claiming that the distance between an author from the past and us is filled in by tradition, which opens access to the text for us, while Collingwood considered explanations of the actions of historical agents, and by extension understanding of a text, in intentional or rational terms. Furthermore, he thought that such explanations are not causal, and that the thoughts involved in them do not stand within the flow of physical time, which is involved in any notion of temporal distance. This is why Collingwood felt entitled to anti-relativistic conclusions about the recovery of authorial intentions, conclusions that prompted Gadamer to claim that "the dimension of hermeneutical mediation which is passed through in every act of understanding" escaped him. We then discuss the underlying notions of time at work in both Gadamer and Collingwood, showing that ricoeur had a better appreciation of the issue, since he saw that Collingwood's moves parallel, up to a point, heidegger's critique of "vulgar time," albeit with an entirely different result. We also point to the importance in Collingwood's thinking of his notion of "incapsulation."
Spanish translation of C. Kobayashi & M. Marion, 'Gadamer and Collingwood on Temporal Distance an... more Spanish translation of C. Kobayashi & M. Marion, 'Gadamer and Collingwood on Temporal Distance and Understanding', History and Theory, vol. 50, December Theme Issue, 2011, p. 81-103.
This entry has 7 co-authors, I am responsible only for section 2.
] 1 For the first occurrence of this sort of move, see Rorty (1961). 2 I was able to consult a ty... more ] 1 For the first occurrence of this sort of move, see Rorty (1961). 2 I was able to consult a typed copy of Rhees' notes at the von Wright & Wittgenstein Archives housed in the Department of Philosophy of the University of Helsinki. 3 Sahlin (1997: 65). 4 Schiller was indeed the first in Britain to describe his own philosophy as 'pragmatist' in "Axioms as Postulates" (1902), a paper that G. E. Moore described as "utterly worthless" (1904: 259), while Peirce considered it "most remarkable" (1931-35: 5.414). He figures significantly in the sources to Lady Welby's 'significs', and they are both discussed C. K. Ogden & I. A. Richards' The Meaning of Meaning, e.g., at (1923: 272f.). They also get a
French translation, with an introduction co-authored with Pascal Engel, of all of F. P. Ramsey's ... more French translation, with an introduction co-authored with Pascal Engel, of all of F. P. Ramsey's major papers in philosophy, logic, mathematics and economics.
Montréal. Il a pour mission la promotion et la valorisation de la recherche. Érudit offre des ser... more Montréal. Il a pour mission la promotion et la valorisation de la recherche. Érudit offre des services d'édition numérique de documents scientifiques depuis 1998. Note : les règles d'écriture des références bibliographiques peuvent varier selon les différents domaines du savoir.
Montréal. Il a pour mission la promotion et la valorisation de la recherche. Érudit offre des ser... more Montréal. Il a pour mission la promotion et la valorisation de la recherche. Érudit offre des services d'édition numérique de documents scientifiques depuis 1998. Note : les règles d'écriture des références bibliographiques peuvent varier selon les différents domaines du savoir.
La dernière décennie a vu un essor considérable de la recherche sur les fragments de l'arithmétiq... more La dernière décennie a vu un essor considérable de la recherche sur les fragments de l'arithmétique de Peano. Parallèlement, la mathématique non standard a continué à croître en popularité, surtout après Tintroduction du nouveau langage de la théorie interne des ensembles créé par Edward Nelson 1. Pour des raisons d'espace, je me contenterai de discuter l'ouvrage d'Yvon Gauthier, De la logique interne, en relation avec ces deux domaines florissants. Je compte aussi m'attarder sur les sections dédiées à Frege, Weyl et Wittgenstein. Je ne prétends pas faire justice à la richesse de l'ouvrage de Gauthier, dont j'ai dû passer sous silence bien des aspects. Le mieux que je puisse espérer ici, c'est d'être arrivé à poser quelques bonnes questions. Le lecteur me pardonnera mon ton pédagogique qui allonge le texte par endroit : la littérature de langue française dans ce domaine est quasi-inexistante. * N.D.L.R. Philosophiques reprend ici cette étude critique précédemment publiée dans nos pages et qui, malheureusement, avait été rendue incompréhensible à la suite d'erreurs techniques indépendantes de notre volonté. Le comité de rédaction présente toutes ses excuses à M.
PAPER ACCEPTED. HISTORY AND PHILOSOPHY OF LOGIC. In this paper, we would like to address two pro... more PAPER ACCEPTED. HISTORY AND PHILOSOPHY OF LOGIC. In this paper, we would like to address two problems engendered by the presence of proofs by ecthesis. (1) Do they involve singular or general terms? In the last century, Łukasiewicz notoriously argued that they involve only the latter. However, this issue remains unsettled, as we shall see in the next section. (2) Is ecthesis a separate procedure somewhat external to the theory of inference or is it constitutive of it? While ecthesis is usually treated in the secondary literature as an alternative mode of inference, i.e., as part of Aristotle’s inferential arsenal, so to speak, it is usually considered as not truly pertaining to his theory of inferences. In Robin Smith’s words, “it is virtually redundant”. If this were the case, one wonders why Aristotle did not simply do away with its occurrences, instead of marring his presentation with them. Hence this second problem.
In order to answer both of these problems, we shall propose a new perspective on ecthesis, presenting it as a procedure such that (answering the second question) it will be seen as fully pertaining to Aristotle’s theory of inference, and (answering the first one) as involving both singular and general terms.By M. Crubellier, M. Marion?, Z. McConaughey, S. Rahman
https://www.tandfonline.com/doi/full/10.1080/01445340.2019.1586623