Feminists have long been aware that the notion of women is problematic and using it uncritically without further qualifications leads to exclusions. In the article, I argue that the source of these problems lies in the understanding of concepts as static and clearly defined. I deploy Hegel’s idea of syllogism to define dynamic concepts, which I term ‘fractal concepts’ because of their complexity and constant development. In such structures the balance between the universal, the particular and the individual is (...) maintained and, consequently, every woman is essential for the definition of such a non-essentialist notion of women. I propose that an epistemological interpretation of the Hegelian idea of recognition can be used to describe the development of fractal concepts in the confrontation of different perspectives. It also explains the obstruction of this process by the desire for domination and control, which results in the appropriation of group identities by their privileged members. I situate the logical and political framework developed in the paper in the context of feminist controversies around the concept of woman and argue that it provides us with a comprehensive account of intersectionality as well as an explicative solution to the regress problem. (shrink)

This paper examines the Hegelian moment of the for-another in its negative relation to the other moment of particularity: the for-itself. I identify the dissolving, fluidifying action of the for-another by examining figures within the Philosophy of Nature, particularly comets and moons, but also Hegel’s physics of light and sound. The dissolution of the lunar for-itself at the hands of the cometary for-another illustrates how the dynamic relation between the two moments of particularity participates in the presentation of essence, (...) within the Hegelian syllogism, i.e. as mediating between the universal and the singular. The dynamic action of cometary negativity occurs throughout the Philosophy of Nature and therefore should be pivotal to how the work is read. (shrink)

L'objectivité dont s'occupe la science hégélienne n'est pas celle d'une réalité détachée, mue selon les lois dialectiques, et le discours scientifique n'est pas vrai et objectif parce qu'il serait la réflexion adéquate d'une telle réalité. L'objectivité scientifique chez Hegel doit être saisie comme le logos , c'est-à-dire le discours de la science elle-même dans son actualité existante. Il s'agit d'un discours qui est son objet et qui est l'objectivité véritable. Ce type de langage est seulement possible s'il est compris comme (...) le moyen terme entre l'être et le penser, c'est-à-dire en termes du syllogisme hégélien et non pas de la prédication. Concrètement, les objets de la science, ou les contenus de l' Encyclopédie , sont eux-mêmes des discours objectifs, c'est-à-dire existants et vrais. Ainsi, par exemple, c'est seulement comme discours que l'Etat peut constituer un objet/contenu de la science hégélienne.The objectivity that Hegelian science deals with is not a detached reality running on dialectical laws, and scientific discourse is not true and objective because it is the faithful reflection of such a reality. Scientific objectivity, in Hegel, should be understood as logos, as existing scientific discourse. It is a discourse that is its object and is true objectivity. This type of language is only possible if it is grasped as the middle term between mere being and thought, in terms of the Hegelian syllogism rather than predication. Concretely, the objects of science, or the contents of the Encyclopedia, are themselves existing, true discourses. Thus, for example, only as discourse does the State become an object/content of science. (shrink)

Supporters of G.W.F. Hegel's philosophy have largely shied away from relating his logic to modern symbolic or mathematical approaches. While it has predominantly been the non-Greek discipline of algebra that has informed modern mathematical logic, philosopher Paul Redding argues that the approaches of Plato and Aristotle to logic were deeply shaped by the arithmetic and geometry of classical Greek culture. And by ignoring the fact that Hegel's logic also has this deep mathematical dimension, conventional Hegelians have missed some of Hegel's (...) greatest insights. In Conceptual Harmonies, Redding develops an account of Hegel's logic against a classical and modern historical background that is rarely considered. He stresses Hegel's attention to the Platonic background of Aristotle's original syllogistic and beyond. He then links these Platonic elements to Leibniz's modern revitalization of the logical tradition and then to new forms of algebraic geometry emerging in Hegel's lifetime. Redding thereby reestablishes aspects of Hegel's philosophy that are essential if Hegel is to be taken as a thinker relevant not only to contemporary philosophy, but also to current philosophical conceptions of logic. (shrink)

RESUMENEste artículo trata del programa de la Enciclopedia de las ciencias filosóficas deHegel. Programa que Hegel delinea en la introducción a dicha obra, y que culmina en el silogismo de silogismos que concluye su obra. Desde su comienzo y término cabe advertir (1) la diferencia de la Enciclopedia hegeliana con las enciclopedias dieciochescas, y (2) la relación de la ciencia filosófica hegeliana con lo empírico.PALABRAS CLAVEHEGEL, ENCICLOPEDIA, FUNDACIONALISMO, CIENCIAABSTRACTThis paper focus on the Hegelian program of the Encyclopedia of the (...) Philosophical Sciences. Hegel explains his program in the ―Introduction‖, and in the last pages of the Encyclopedia, the syllogism of syllogisms. From its beginning and from its end, it is possible to notice (1) the difference between the Encyclopedia and other encyclopedias of the eighteenth-century, like the French one or the British one; and (2) the relationship of the Hegelian notion of philosophical science with the empirical.KEY WORDSHEGEL, ENCYCLOPEDIA, FOUNDATIONALISM, SCIENCE. (shrink)

There is a place in the Hegelian system where the notion of Objectivity is explicitly articulated. One entire section in the Science of Logic is entitled ‘Objectivity’, almost suggesting that that would be the appropriate place to unveil the question of the notion of objective thought. However, precisely that section, its content and its presence have always represented a problematic knot in the Hegelian logical system. The problem is two-fold: on the one hand, the categories of this section (...) (Mechanism, Chemism and Teleology) seem to refer to the realm of the Realphilosophie, and they seem not to respect the purity of logical thought. This makes the connection between such section and the remaining Hegelian system uncertain and questionable. On the other hand, the Objectivity section does not seem to present any character which determines its logical advancement over Subjectivity. This contribution focuses on the interpretation of the origin of this section and on the logical meaning that “objectivity” has in the Logic. This paper explores the roots of the problem attempting to undo a few of its theoretical knots via a particular interpretation of the Syllogism. The underlying thesis in this analysis is in fact that the section on Objectivity originates precisely in the internal dynamics of the Syllogism. The forms of self-reflection in the Syllogism justify the presence of the section Objectivity. In this context, Objectivity does not present itself with the characters of immediacy and externality that are normally attributed to it, but it expresses the result of the syllogistic process. The second fundamental aspect of this short analysis is that Mechanism, Chemism and Teleology as categories are not directly exported from the Realphilosophie. They are rather organizational forms in the relationships among concepts. Thus, they don’t figure as abstractions from the proper structures of nature and spirit. Rather, they express the relational structures reached by thought once the syllogism has been able to manifest a yet occult rational capacity of thought. (shrink)

ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental (...) rules which establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out. (shrink)

Comparative syllogism is a type of scientific reasoning widely used, explicitly or implicitly, for inferences from observations to conclusions about effectiveness, but its philosophical significance has not been fully elaborated or appreciated. In its simplest form, the comparative syllogism derives a conclusion about the effectiveness of a factor (e.g. a treatment or an exposure) on a certain property via an experiment design using a test (experimental) group and a comparison (control) group. Our objective is to show that the (...) comparative syllogism can be understood as encoding a simulation view of counterfactuals, in that counterfactual situations are conceptual constructs that can be correctly simulated by homogeneous comparison groups. In this simulation view, the empirical data from the comparison groups play an evidential role in the evaluation of counterfactuals and in obtaining counterfactual knowledge. We further indicate how successful experimental designs can help us to obtain correct simulations, and thus to bring us to scientifically-empirically based counterfactual knowledge. (shrink)

This book presents the first study of the development of the theory of modal syllogistic in the Middle Ages.

Extensions of traditional syllogistics have been increasingly researched in philosophy, linguistics, and areas such as artificial intelligence and computer science in recent decades. This is mainly due to the fact that syllogistics is seen as a logic that comes very close to natural language abilities. Various forms of extended syllogistics have become established. This paper deals with the question to what extent a syllogistic representation in CL diagrams can be seen as a form of extended syllogistics. It will be shown (...) that the ontology of CL enables numerically exact assertions and inferences. (shrink)

The debate over Hypothetical Syllogism is locked in stalemate. Although putative natural language counterexamples to Hypothetical Syllogism abound, many philosophers defend Hypothetical Syllogism, arguing that the alleged counterexamples involve an illicit shift in context. The proper lesson to draw from the putative counterexamples, they argue, is that natural language conditionals are context-sensitive conditionals which obey Hypothetical Syllogism. In order to make progress on the issue, I consider and improve upon Morreau’s proof of the invalidity of Hypothetical (...) Syllogism. The improved proof relies upon the semantic claim that conditionals with antecedents irrelevant to the obtaining of an already true consequent are themselves true. Moreover, this semantic insight allows us to provide compelling counterexamples to Hypothetical Syllogism that are resistant to the usual contextualist response. (shrink)

This article enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: Allxareyand Somexarey, There are at least as manyxasy, and There are morexthany. Herexandyrange over subsets (not elements) of a giveninfiniteset. Moreover,xandymay appear complemented (i.e., as$\bar{x}$and$\bar{y}$), with the natural meaning. We formulate a logic for our language that is based on the classical syllogistic. The main result is a soundness/completeness theorem. (...) There are efficient algorithms for proof search and model construction. (shrink)

In the first book of the Prior Analytics, Aristotle sets out, for the first time in Greek philosophy, a logical system. It consists of a deductive system (I.4-22), meta-logical results (I.23-26), and a method for finding and giving deductions (I.27-29) that can apply in “any art or science whatsoever” (I.30). After this, Aristotle compares this method with Plato’s method of division, a procedure designed to find essences of natural kinds through systematic classification. This critical comparison in APr I.31 raises an (...) interpretive puzzle: how can Aristotle reasonably juxtapose two methods that differ so much in their aims and approach? What can be gained by doing so? Previous interpreters have failed to show how this comparison is legitimate or what important point Aristotle is making. The goal of this paper is to resolve the puzzle. In resolving this puzzle we not only learn more about the relationship be- tween division and the syllogistic in Aristotle. We will also learn something about the motivation for the syllogistic itself, by seeing the role that it plays in his philosophy of science. (shrink)

This paper reviews current psychological theories of syllogistic inference and establishes that despite their various merits they all contain deficiencies as theories of performance. It presents the results of two experiments, one using syllogisms and the other using three-term series problems, designed to elucidate how the arrangement of terms within the premises affects performance. These data are used in the construction of a theory based on the hypothesis that reasoners construct mental models of the premises, formulate informative conclusions about the (...) relations in the model, and search for alternative models that are counterexamples to these conclusions. This theory, which has been implemented in several computer programs, predicts that two principal factors should affect performance: the figure of the premises, and the number of models that they call for. These predictions were confirmed by a third experiment. Dans cet article sont passées en revue les théories psychologiques du traitement des syllogismes. On établit qu'en dépit de leurs mérites variés toutes sont en défaut en tant que théories de la performance. On présente les résultats de deux expériences, une utilisant des syllogismes et l'autre des problémes avec des séries de trois termes conçues pour élucider comment l'arrangement des termes dans les premises affecte la performance. Ces données sont utilisées pour construire une théorie fondée sur l'hypothése que les gens construisent des modèles mentaux des premises, formulent des conclusions explicites sur les relations dans le modèle et cherchent des modèles qui seraient des contre-exemples pour leurs conclusions. Cette théorie, utilisée dans plusieurs programmes d'ordinateur, prédit que deux principaux facteurs affectent la performance: la figure des premises et le nombre de; modèles qu'ils mettent en jeu. Cette prédiction est confirmée dans les trois expériences. (shrink)

This book uncovers and examines the confusion in antiquity between Aristotle's hypothetical syllogistic and Stoic logic, and offers a fresh perspective on the ...

This article describes a specific pedagogical context for an advanced logic course and presents a strategy that might facilitate students’ transition from the object-theoretical to the metatheoretical perspective on logic. The pedagogical context consists of philosophy students who in general have had little training in logic, except for a thorough introduction to syllogistics. The teaching strategy tries to exploit this knowledge of syllogistics, by emphasizing the analogies between ideas from metalogic and ideas from syllogistics, such as existential import, the distinction (...) between contradictories and contraries, and the square of opposition. This strategy helps to improve students’ understanding of metalogic, because it allows the students to integrate these new ideas with their previously acquired knowledge of syllogistics. (shrink)

The relation between ex falso and disjunctive syllogism, or even the justification of ex falso based on disjunctive syllogism, is an old topic in the history of logic. This old topic reappears in contemporary logic since the introduction of minimal logic by Johansson. The disjunctive syllogism seems to be part of our general non-problematic inferential practices and superficially it does not seem to be related to or to depend on our acceptance of the frequently disputable ex falso (...) rule. We know that the acceptance of the ex falso is a sufficient condition for the acceptance of the disjunctive syllogism, but the interesting question is: is the ex falso a necessary condition for the acceptance of the disjunctive syllogism? The aim of the present paper is to discuss some possible ways to define systems that combines the preservation of the disjunctive syllogism with the rejection of the ex falso. (shrink)

European Journal of Philosophy, Volume 29, Issue 4, Page 864-886, December 2021.

This paper adds comparative adjectives to two systems of syllogistic logic. The comparatives are interpreted by transitive and irreflexive relations on the underlying domain. The main point is to obtain sound and complete axiomatizations of the valid formulas in the logics.

In 1883, while a student of C. S. Peirce at Johns Hopkins University, Christine Ladd-Franklin published a paper titled On the Algebra of Logic, in which she develops an elegant and powerful test for the validity of syllogisms that constitutes the most significant advance in syllogistic logic in two thousand years. Sadly, her work has been all but forgotten by logicians and historians of logic. Ladd-Franklin's achievement has been overlooked, partly because it has been overshadowed by the work of other (...) logicians of the nineteenth century renaissance in logic, but probably also because she was a woman. Though neglected, the significance of her contribution to the field of symbolic logic has not been diminished by subsequent achievements of others.In this paper, I bring to light the important work of Ladd-Franklin so that she is justly credited with having solved a problem over two millennia old. First, I give a brief survey of the history of syllogistic logic. In the second section, I discuss the logical systems called “algebras of logic”. I then outline Ladd-Franklin's algebra of logic, discussing how it differs from others, and explain her test for the validity of the syllogism, both in her symbolic language and the more familiar language of modern logic. Finally I present a rigorous proof of her theorem. Ladd-Franklin developed her algebra of logic before the methods necessary for a rigorous proof were available to her. Thus, I do now what she could not have done then. (shrink)

A formalism is developed that treats a robot as a subject that can interpret its own experience rather than an object that is interpreted within our experience. A regulative definition of a meaningful experience in robots is proposed in which the present sensible experience is considered meaningful to the agent, as the subject of the experience, if it can be related to the agent's temporal horizons. This definition is validated by demonstrating that such an experience in evolutionary autonomous agents is (...) embodied, contextual and normative, as is required for the maintenance of phenomenological accuracy. With this formalism it is shown how a dialectic similar to that described in Hegelian phenomenology can emerge in the robotic experience and why the presence of such a dialectic can serve as a constraint in the further development of cognitive agents. (shrink)

Chapter 1 presents BS, a basic syllogistic system based on Aristotle's logic, in natural deduction form. Chapters 2 and 3 treat the metatheory of BS: consitency, soundness, independence, and completeness. Chapter 4 and 5 deal with syllogistic and, in turn, propositional and predicate logic, chapter 6 is on existential import, chapter 7 on subject and predicate and chapter 8 on classes. Chapter 9 adds negative variables to BS, and proves its soundness and completeness.

The process of constructing a social reality where “difference” becomes a social asset rather than a monster that threatens peace and progress must commence with a phenomenological understanding of social interactions within and among human societies. In my opinion, Hegel, more than any other thinker, has constructed a phenomenological framework that adequately captures and represents the nature of group interactions within human societies. This paper explores the Hegelian phenomenon of social identity, and, especially, characterizes the interactions between and among (...) various social identities. It is a modest effort to contribute theoretically to the available discourse on the management of “difference” in multi-ethnic societies. (shrink)

Though Wilfrid Sellars portrayed himself as a latter-day Kantian, I argue here that he was at least as much a Hegelian. Several themes Sellars shares with Hegel are investigated: the sociality and normativity of the intentional, categorial change, the rejection of the given, and especially their denial of an unknowable thing-in-itself. They are also united by an emphasis on the unity of things—the belief that things do ‘‘hang together.’’ Hegel’s unity is idealist; Sellars’ is physicalist; the differences are substantial, (...) but so are the resonances. (shrink)

The question as to what makes a perfect Aristotelian syllogism a perfect one has long been discussed by Aristotelian scholars. G. Patzig was the first to point the way to a correct answer: it is the evidence of the logical necessity that is the special feature of perfect syllogisms. Patzig moreover claimed that the evidence of a perfect syllogism can be seen for Barbara in the transitivity of the a-relation. However, this explanation would give Barbara a different status (...) over the other three first figure syllogisms. I argue that, taking into account the role of the being-contained-as-in-a-whole formulation, transitivity can be seen to be present in all four first figure syllogisms. Using this wording will put the negation sign with the predicate, similar to the notation in modern predicate calculus. (shrink)

Does cognition sometimes literally extend into the extra-organismic environment (Clark, 2003), or is it always “merely” environmentally embedded (Rupert, 2004)? Underlying this current border dispute is the question about how to individuate cognitive processes on principled grounds. Based on recent evidence about the active role of representation selection and construction in learning how to reason (Stenning, 2002), I raise the question: what makes two distinct, modality-specific pen-and-paper manipulations of external representations – diagrams versus sentences – cognitive processes of the same (...) kind, e.g. episodes of syllogistic reasoning? In response, I defend a “division of labor” hypothesis, according to which external representations are dependent on perceptually grounded neural representations and mechanisms to guide our behavior; these internal mechanisms, however, are dependent on external representations to have their syllogistic content fixed. Only their joint contributions qualify the extended computational process as an episode of syllogistic reasoning in good standing. (shrink)

Syllogistics reduces to only two rules of inference: monotonicity and symmetry, plus a third if one wants to take existential import into account. We give an implementation that uses only the monotonicity and symmetry rules, with an addendum for the treatment of existential import. Soundness follows from the monotonicity properties and symmetry properties of the Aristotelean quantifiers, while completeness for syllogistic theory is proved by direct inspection of the valid syllogisms. Next, the valid syllogisms are decomposed in terms of the (...) rules they involve. The implementation uses Haskell [8], and is given in ‘literate programming’ style [9]. (shrink)

Traditionally, syllogisms are arguments with two premises and one conclusion which are constructed by propositions of the form “All… are…” and “At least one… is…” and their respective negated versions. Unfortunately, the practical use of traditional syllogisms is quite restricted. On the one hand, the “All…” propositions are too strict, since a single counterexample suffices for falsification. On the other hand, the “At least one …” propositions are too weak, since a single example suffices for verification. The present contribution studies (...) algebraic interpretations of syllogisms with comparative quantifiers (e.g., “Most… are…”) and quantitative quantifiers (e.g., “n/m… are…”, “all, except n… are…”). This modern version of syllogistics is intended to be a more adequate framework for argumentation theory than traditional syllogistics. (shrink)

The syllogism has often been criticized. Yet the theory of the syllogism cannot be omitted from logic. Even if it were not for its historical significance, its nature as a chapter of class logic assigns to it a place in any presentation of logic.The usual exposition of the theory of the syllogism, however, whether given by the use of the familiar rules of the syllogism, or by the help of diagrams, appears clumsy and lacks the lucidity (...) of modern chapters of logic. The reason seems to be given in the inefficient notation, taken over from ancient and medieval logic. In the following I should like to present an improved notation, which combines some of the traditional features with modern ones, and which is based on the criticism of the syllogistic theory which I have given elsewhere. It will be seen that in the revised form the theory of the syllogism is apt to meet the standards of modern logic. (shrink)

In elementary logic textbooks, Venn diagrams are used to analyze and evaluate the validity of syllogistic arguments. Although the method of Venn diagrams is shown to be a powerful analytical tool in these textbooks, it still has limitations. On the one hand, such method fails to represent singular statements of the form, “a is F.” On other hand, it also fails to represent identity statements of the form, “a is b.” Because of this, it also fails to give an account (...) of the validity of some obviously valid arguments that contain these types of statements as constituents. In this paper, owing to the developments in the literature on Venn diagrams, we offer a way of supplementing the rules of the Venn diagram found in textbooks, and show how this retooled Venn diagram technique could account for the problem cases. (shrink)

1. In both Benedetto Croce's and Hegel's own terminology, dialectics can be understood as dottrina degli opposti (the doctrine of the opposites – Lehre der Gegensätze).1 In the dialectical process,...

This paper presents a restructured set of axioms for categorical logic. In virtue of it, the syllogistic with indefinite terms is deduced and proved, within the categorical logic boundaries. As a result, the number of all the conclusive syllogisms is deduced through a simple and axiomatic methodology. Moreover, the distinction between immediate and mediate inferences disappears, which reinstitutes the unity of Aristotelian logic.

In this article, I present a schema for generating counterexamples to the argument form known as Hypothetical Syllogism with indicative conditionals. If my schema for generating counterexamples to HS works as I think it does, then HS is invalid for indicative conditionals.

Hegel’s practical philosophy has important insights for understanding the ethical role of the firm in modern society. From a broadly Hegelian perspective, the firm’s role in society is to facilitate freedom, that is, the concrete realization of rational agency. It does this by providing the institutional structures, norms, practices, and modes of discourse necessary for individuals to link their subjective aims with objectively valid societal aims, embodied in the firm’s purpose. Accordingly, I first present a Hegelian account of (...) the link between action and social structure, before arguing that the firm, when it functions properly, enables individuals to express their capacity for rational agency within concrete social contexts. I then draw upon Paul Adler’s Marxist analysis of the development of organizational structures and processes to support the viability of a broadly Hegelian account of the firm within a contemporary economic context. I conclude by outlining a number of practical implications of this account, before explaining its implications for two prominent theories of business ethics, the MacIntyrean perspective, and the Market Failures Approach. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

In the paper, a new syllogistic system is built up. This system simulates a massive-parallel behavior in the propagation of collectives of parasites. In particular, this system simulates the behavior of collectives of trematode larvae.

If we think of recognition as the practical relation consciously enacted by concerned individual subjects as social actors, which allows them to fulfil their intersubjectively valid social roles, this by no means exhausts the significance that recognition is accorded by Hegel. In fact the problem of recognition is central to the understanding and evaluation of Hegel’s metaphysical system. Thus a close scrutiny of the presentation of self-consciousness in Phenomenology of Spirit and the interpretative difficulties it poses leads on to the (...) question of the subject and Hegel’s distinction between finite, accidental individuals and the true subject in his system: the concept of Spirit, understood not as a separate entity but as a system of relations, objectified in the historical forms of the Absolute Spirit. But what is the price of Hegel’s metaphysics of subjectivity? Hegelian recognition signifies the recognition by individuals of recognition in its truth, that is, the self-recognition by finite individuals that they participate in Spirit as the true universal subject to the degree that they recognize their shared world of actions as the world of their own making. Modernity is therefore defined for Hegel as the recognition and realization of ‘conscious freedom’, whose telos lies in the actualization of universal reciprocal recognition that brings the unfreedom of history to an end. The idea of freedom and the thesis of the ‘end of history’ remain, however, the preserve of the thinking few. Hegelian recognition and with it Hegel’s whole metaphysical system founders on the rock of finitude, on the unfreedom of finite human beings. (shrink)

Does syllogistic logic have the resources to capture mathematical proof? This volume provides the first unified account of the history of attempts to answer this question, the reasoning behind the different positions taken, and their far-reaching implications. Aristotle had claimed that scientific knowledge, which includes mathematics, is provided by syllogisms of a special sort: 'scientific' ('demonstrative') syllogisms. In ancient Greece and in the Middle Ages, the claim that Euclid's theorems could be recast syllogistically was accepted without further scrutiny. Nevertheless, as (...) early as Galen, the importance of relational reasoning for mathematics had already been recognized. Further critical voices emerged in the Renaissance and the question of whether mathematical proofs could be recast syllogistically attracted more sustained attention over the following three centuries. Supported by more detailed analyses of Euclidean theorems, this led to attempts to extend logical theory to include relational reasoning, and to arguments purporting to reduce relational reasoning to a syllogistic form. Philosophical proposals to the effect that mathematical reasoning is heterogenous with respect to logical proofs were famously defended by Kant, and the implications of the debate about the adequacy of syllogistic logic for mathematics are at the very core of Kant's account of synthetic a priori judgments. While it is now widely accepted that syllogistic logic is not sufficient to account for the logic of mathematical proof, the history and the analysis of this debate, running from Aristotle to de Morgan and beyond, is a fascinating and crucial insight into the relationship between philosophy and mathematics. (shrink)