Varieties of Demonstration in Alfarabi

This paper analyzes a classification of different types of demonstration introduced by Alfarabi (d. 950 CE) in his Kitāb al-Burhān (Book of Demonstration). Alfarabi identifies eight combinations of demonstrative syllogisms, grouped in function of the different types of per se relations expressed by their premises and conclusions, where terms are definitionally connected with one another. The list contains a total of thirty-nine moods illustrated by a rich array of examples drawn from various scientific disciplines, including arithmetic, geometry, and natural philosophy. The combinations and moods are discussed extensively by Averroes (d. 1198 CE) in the section of his Epitome of the Organon devoted to the Posterior Analytics and in his Quaesita on logic. Alfarabi’s classification also possibly inspired a simplified taxonomical effort in Avicenna’s (d. 1037 CE) Kitāb al-Burhān.     

Aristotle on Universal Quantification: A Study from the Point of View of Game Semantics

In this paper we provide an interpretation of Aristotle's rule for the universal quantifier in Topics Θ 157^a34–37 and 160^b1–6 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 24^b28–29. 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.     

Aristotle's Prior Analytics and Boole's Laws of Thought

Prior Analytics by the Greek philosopher Aristotle (384?–?322 BCE) and Laws of Thought by the English mathematician George Boole (1815?–?1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle's system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss many other historically and philosophically important aspects of Boole's book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole's contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology.

… using mathematical methods … has led to more knowledge about logic in one century than had been obtained from the death of Aristotle up to … when Boole's masterpiece was published.     

The Different Ways in which Logic is (said to be) Formal

I argue that, in the Prior Analytics, higher and above the well-known ‘reduction through impossibility’ of figures, Aristotle is resorting to a general procedure of demonstrating through impossibility in various contexts. This is shown from the analysis of the role of adunaton in conversions of premises and other demonstrations where modal or truth-value consistency is indirectly shown to be valid through impossibility. Following the meaning of impossible as ‘non-existent’, the system is also completed by rejecting any invalid combinations of terms in deductions or conversions. The notion of impossibility reaches the core of Aristotle's system in the Prior Analytics. On the one hand, the use of adunaton shows that he is following one of the two requisites for demonstrative science formulated in the Posterior Analytics, i.e. to demonstrate that it is impossible for things to be otherwise than stated. On the other hand, that demonstrations through impossibility are rooted in the notion of contradiction supports the claim that Aristotle might have been trained to use this specific procedure in the context of dialectical exercises in the academy. This need not rule out other influences on Aristotle's preferred procedures of proving or counter-proving, but it paves a way to a better understanding of Aristotle's logic under the light of Plato's dialectic.     

Proof by Assumption of the Possible in Prior Analytics, 1.15; How Not to Blend Modal Frameworks

The present paper aims to show that the reconstruction of the formal framework of the proofs in Pr. An. 1.15, as proposed by Malink and Rosen 2013 (‘Proof by Assumption of the Possible in Prior Analytics 1.15’, Mind, 122, 953-85) is due to affront a double impasse. Malink and Rosen argue convincingly that Aristotle operates with two different modal frameworks, one as found in the system of modal logic presented in Prior Analytics 1.3 and 8-22, and one occurring in many of Aristotle’s works, such as the Physics, De Caelo and the Metaphysics. However, they misconstrue the latter framework. More precisely, they misconstrue the domain of significance of what they call the ‘Principle of Necessitation’. As a consequence, bringing the two frameworks into one results into a contradictory modal logic. On the other hand, if the Principle of Necessitation is rectified, the proofs put forward by Malink and Rosen in the same paper are no longer available.     

Dialectic,the Dictum de Omni and Ecthesis

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.     

The Logic of Necessity in Aristotle--an Outline of Approaches to the Modal Syllogistic,Together with a General Account of de dicto - and de re -Necessity

This article investigates the prospect of giving de dicto- and de re-necessity a uniform treatment. The historical starting point is a puzzle raised by Aristotle's claim, advanced in one of the modal chapters of his Prior Analytics, that universally privative apodeictic premises simply convert. As regards the Prior and the Posterior Analytics, the data suggest a representation of propositions of the type in question by doubly modally qualified formulae of modal predicate logic that display a necessity operator in two distinct positions. Can the N-operator occurring in these positions be given a unified semantical treatment (which would justify dispensing with a notational differentiation)? A positive answer, based on a suitably shaped truth condition for N-formulae, is given, and is supported in the final section with an alternative proof theoretically based conception of a property's essential belonging to an individual.     

Minority Report: Reconsidering Jürgen Moltmann's Turn to a Theology of the Cross

According to virtually any set of criteria, Jürgen Moltmann's first two programmatic books, Theology of Hope and The Crucified God, are strikingly different. The differences are due to his turn to a theology of the cross in the intervening years. Yet Moltmann, with the vast majority of scholars concurring, argues that the two works form a continuous argument. This article investigates the problems with the consensus judgement and calls for a revision of the relationship between the two works, based on the nature of the cross–resurrection dialectic inherent to his programme and also on both the necessity and timing of the turn.     

On Magari's concept of general calculus: notes on the history of tarski's methodology of deductive sciences

This paper is an historical study of Tarski's methodology of deductive sciences (in which a logic S is identified with an operator Cn _S , called the consequence operator, on a given set of expressions), from its appearance in 1930 to the end of the 1970s, focusing on the work done in the field by Roberto Magari, Piero Mangani and by some of their pupils between 1965 and 1974, and comparing it with the results achieved by Tarski and the Polish school (?o?, Suszko, S?upecki, Pogorzelski, Wójcicki). In the last section of the paper we will then compare these works with some recent developments in algebraic logic: this will lead to a better understanding of the results of the methodology of deductive science, but at the same time will show some intrinsic limits to such an approach to logic.

Even if Magari's work on diagonizable algebras and universal algebra and Mangani's axiomatization of MV-algebras and results in model theory are rather famous, the articles on closure operators, published in the 1960s, are almost totally unknown outside Italy (mainly because of a linguistic limitation, the papers we analyse having been written and published in Italian). This paper aims to fill the gap in the literature and to enable the international community to get acquainted with this part of Italian logic. The same applies to some works published in Barcelona (in Catalan) at the end of the 1970s, analysed in the last section.     

Robert Kilwardby's Disputational Logic

The article is concerned with the account of Aristotle's theory of disputation given by Robert Kilwardby (d. 1279) in his commentary, composed in Paris during the 1240s, on Aristotle's Prior Analytics. Specifically, I show that Kilwardby covers demonstrative as well as dialectical disputations, and gives an elementary account of the rules governing such disputations, in their adversarial forms as well as in an idealized form where the interlocutors engage in a cooperative activity. I describe the resemblances and the differences between disputations as theorized by Kilwardby and the game of obligationes as understood by some of his contemporaries.     

The Role of Mathematics in Deleuze’s Critical Engagement with Hegel

Abstract

The role of mathematics in the development of Gilles Deleuze’s (1925–95) philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic is demonstrated in this paper by differentiating Deleuze’s interpretation of the problem of the infinitesimal in Difference and Repetition from that which G. W. F Hegel (1770–1831) presents in the Science of Logic. Each deploys the operation of integration as conceived at different stages in the development of the infinitesimal calculus in his treatment of the problem of the infinitesimal. Against the role that Hegel assigns to integration as the inverse transformation of differentiation in the development of his dialectical logic, Deleuze strategically redeploys Leibniz’s account of integration as a method of summation in the form of a series in the development of his philosophy of difference. By demonstrating the relation between the differential point of view of the Leibnizian infinitesimal calculus and the differential calculus of contemporary mathematics, I argue that Deleuze effectively bypasses the methods of the differential calculus which Hegel uses to support the development of the dialectical logic, and by doing so, sets up the critical perspective from which to construct an alternative logic of relations characteristic of a philosophy of difference. The mode of operation of this logic is then demonstrated by drawing upon the mathematical philosophy of Albert Lautman (1908–44), which plays a significant role in Deleuze’s project of constructing a philosophy of difference. Indeed, the logic of relations that Deleuze constructs is dialectical in the Lautmanian sense.     

Completion,reduction and analysis: three proof-theoretic processes in aristotle’s prior analytics

Three distinctly different interpretations of Aristotle’s notion of a sullogismos in Prior Analytics can be traced: (1) a valid or invalid premise-conclusion argument (2) a single, logically true conditional proposition and (3) a cogent argumentation or deduction. Remarkably the three interpretations hold similar notions about the logical relationships among the sullogismoi. This is most apparent in their conflating three processes that Aristotle especially distinguishes: completion (A4-6)reduction(A7) and analysis (A45). Interpretive problems result from not sufficiently recognizing Aristotle’s remarkable degree of metalogical sophistication to distinguish logical syntax from semantics and, thus, also from not grasping him to refine the deduction system of his underlying logic. While it is obvious that Aristotle most often uses ‘sullogimos’ to denote a valid argument of a certain kind, we show that at Prior Analytics A4-6, 7, 45 Aristotle specifically treats a sullogismos as an elemental argument pattern having only valid instances and that such a pattern then serves as a rule of deduction in his syllogistic logic. By extracting Aristotle’s understanding of three proof-theoretic processes, this paper provides new insight into what Aristotle thinks reasoning syllogistically is and, moreover, it resolves three problems in the most recent interpretation that takes a sullogismos to be a deduction     

Dialectic in the eleventh and twelfth centuries: garlandus compotista

Dialectic is a standard and important part of the logica vetus (or old logic) in medieval philosophy. It has its ultimate origins in Aristotle's Topics,its fundamental source in Boethius's De topicis differentiis,and its flowering in its absorption into fourteenth-century theories of consequences or conditional inferences. The chapter on Topics in Garlandus Compotista's logic book is the oldest scholastic work on dialectic still extant. In this paper I show the differences between Boethius's Theory of Topics and Garlandus's in order to illustrate the role of Topics in early scholastic logic. I argue that for Garlandus Topics are warrants for the inference from the antecedent to the consequent in a conditional proposition and that he is interested in Topics because of his overriding interest in hypothetical syllogisms. I conclude by discussing briefly the relationship between Garlandus's use of Topics and twelfth-century accounts.     

Jung’s Psychoid Monism

Jung’s final psychoid theory of archetypes was an additional attempt to find a solution to the philosophical problem of how to relate mind and matter. In the following essay Jung’s solution is summarized by a set of 17 theses, and Jung’s philosophy will be called psychoid monism. According to psychoid monism, what ultimately and primarily is, is the psycho-physically neutral domain of instinctual experience. The origin of this view can be traced back to Post-Kantian German Idealism (Schopenhauer, Schelling, Hölderlin), and a systematization of the view requires a dialectic approach and, in particular, contradiction-tolerant dialectic logic.     

Ockham and Buridan on the Ampliation of Modal Propositions

This paper explores a currently unnoticed argument used by John Buridan to defend his analysis of modal propositions and to reject the analysis of modal propositions of necessity put forward by William of Ockham. First, I explore this argument and, by considering possible responses of Ockham to Buridan, show some of the ways in which Ockham seems to be keeping closer to Aristotle's remarks about modal propositions in Prior Analytics, 18.     

Wittgenstein and the unity of good

This paper discusses the problem of the unity of moral good, concerning the kind of unity that moral good or the concept thereof constitutes. In particular, I am concerned with how Wittgenstein's identification of various complex modes of conceptual unity, and his introduction of a methodology of clarification for dealing with such complex concepts, can help with the problem of unity, as it rises from the moral philosophical tradition. Relating to this I also address the disputed question, whether Wittgenstein regards good as a family‐resemblance concept, and make an attempt to characterize family‐resemblance concepts generally by way of their similarities and differences from certain other complex modes of conceptual unity. I argue that whilst Wittgenstein does regard good as a family‐resemblance concept, in the Philosophical Investigations he seeks to make a more general methodological point. I conclude with a suggestion of how Wittgenstein's methodological points can help us to put into a broader perspective famous criticisms of the moral philosophical tradition by Michael Stocker and Bernard Williams, and how Wittgenstein's methods more generally can help to address the problem of the unity of good. This illustrates one way in which Wittgensteinian methods can help resolve disputes in moral philosophy, where the main approaches, Aristotelian, Kantian and utilitarian theories, seem to be locked in a stalemate. As I argue, this situation has to do with their assumptions about the unity of moral good. The response isn't to reject those theories, however, but to re‐interpret them in a way consistent with the Wittgensteinian methodology.     

The doubled movement of infinity in Kierkegaard and in Sūfism

S?ren Kierkegaard was a very rigorous critic of traditional philosophical thinking and speculative systems. According to his theory it is possible that there is a logic system, but not a system of life. If such a system exists, it can be known only to God. Man can attain the meaning of life only by his own relationship to God. However, this relationship cannot be explained by philosophy because it has to do with a transcendent ‘double movement of infinity’ which takes place between God and the individual. Like philosophy, mysticism cannot explain one's relationship to God. The difference is that philosophy neglects God as the absolute starting point, while mysticism forgets that an individual—after he has experienced divinity—may return to the real world. The self need not disappear in divinity. The dialectic of the relationship between God and man implies that both poles (God and man) are present, thus ‘the infinite difference between God and man’ does not disappear. Since Sūfism is a type of Islamic mysticism, it may be said that a Sūfi cannot witness God's truth if he remains in his union with God. It is therefore relevant to draw some parallels between Kierkegaard's view and a comparable Sūfi view about the human relationship to God.     

Aristotle on a Puzzle about Logical Consequence: Necessity of Being vs. Necessity of Saying

In the Posterior Analytics (I 6, 75a18–27) Aristotle discusses a puzzle which endangers the possibility of inferring a non-necessary conclusion. His solution relies on the distinction between the necessity of the conclusion's being the case and the necessity of admitting the conclusion once one has admitted the premisses. The former is a factual necessity, whereas the latter is meant to be a normative or deontic necessity that is independent of the facts stated by the premisses and the conclusion. This paper maintains that Aristotle resorts to this distinction because he thinks that, as long as it is conceived as a factual relation, logical consequence cannot exist independently of the facts expressed by the premisses and the conclusion. As a corollary, the necessity of such a consequence relation always requires the necessity of these facts. Aristotle holds this factual conception of logical consequence responsible for the puzzle, since it cannot account for valid syllogisms with contingent or false premisses. The alternative conception of necessity is then introduced by him in order to make good this deficiency. The distinction between the necessity of being and the necessity of saying was revived by the Oxford logician E. W. B. Joseph, and taken over by Frank Ramsey in his seminal Truth and Probability, but has not received attention from recent interpreters of Aristotle's logic. This paper, however, argues that, in spite of its intrinsic interest, the distinction bore no significant fruit in Aristotle's logical doctrine.     

Dialektik Und Formalisierung

Dialectics and Formalization. In an appendix to his article Can the concept of dialectic be made clear? (JGPS 27: 131–165) the author critically examined an attempt by D. Wandschneider of reconstructing Hegelian dialectical logic by displaying antinomic structures. The main steps of this reconstruction were formalized within the framework of second order predicate logic. In his reply (JGPS 27: 347–352) Wandschneider argues that formalization is the wrong way of interpreting and judging his reconstruction project on the ground that the understanding of predication is already presupposed by predicate logic and, therefore, cannot be formalized. In the present contribution it is shown that Wandschneider's criticism is based on a fundamentally mistaken conception of formalization and formal logic and that the problems posed by his project remain completely unresolved.     

Pierre de La Ramée et le déclin de la rhétorique

This article provides a basic general introduction to Ramus, and evaluates his role in the history of logic and rhetoric, especially with relation to the study of argumentation. The author agrees with Ong and other historians of logic that Ramus is not to be taken seriously as a logician, and that his undoubted importance in the history of ideas is to be found elsewhere.Ramus advocates a belief in nature, experience and reason, and rejects the reliance on the authority of ancient philosophers, above all Aristotle, though experience does not mean scientific experiment and, paradoxically, includes the example of great philosophers and writers. In the end Ramus is seen as responsible for substituting for ancient classical rhetoric an entirely ornamental rhetoric of figures which was to take over education (with the exception of the Jesuit schools) almost until our own day. This curtailing and diminishing of rhetoric is seen as a degeneration. Ancient five-part rhetoric had been concerned with convincing and persuading: Aristotle distinguished the analytic, scientific reasoning of logic, from dialectic which was based on opinion and probability and had close links with rhetoric; by the time of Cicero and Quintilian, who addressed themselves to jurists and politicians, logic has given way to dialectic. In the Middle Ages and the Renaissance the two become assimilated; the evolution of this is traced here through the thirteenth-century Summulae logicales of Peter of Spain to the fifteenth-century German humanist logician Rudolph Agricola, who influenced Ramus partly through the intermediary of Johann Sturm. Ramus took over their topical theories but restricted them to logic/dialectic, and left rhetoric with little more than tropes and figures. He believed that there was only one method for teaching all the arts, and one dialectic common to them all. The distinction between analytical and dialectical has disappeared, with far-reaching consequences for the study of argumentation. Over the centuries logic has lost its connection with controversy and persuasion. With the development of the post-Cartesian, post-Baconian emphasis on the clarity of scientific discourse, and the mid-nineteenth-century interest in mathematical and formal logic, the process was complete. Argumentation, or the new rhetoric, aims to fill the gap thus created.