In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof? -/- The answer is an emphatic yes, as I explain in this article. I argue that non-deductive justification is in fact pervasive in mathematics, and (...) that it is in good epistemic standing. (shrink)

Logic, in one of the many sense of that term, is a theory about what follows from what and why. Arguably, the correct theory has to be determined by abduction. Over recent years, so called logical anti-exceptionalists have investigated this matter. Current discussions have been restricted to deductive logic. However, there are also, of course, various forms of non-deductive reasoning. Indeed, abduction itself is one of these. What is to be said about the way of choosing the best (...) theory of non-deductive inferences? It would seem clear that an anti-exceptionalist should hold that essentially the same method of choice should apply to non-deductive logic. A number of issues need to be faced in the process, not the least of which is the circularity involved in an abductive justification for a theory of abduction. This paper discusses matters. (shrink)

Le but principal de la communication est d’esquisser les traits essentiels de la méthode appliquée dans les sciences déductives.1. A quoi tend la méthode déductive? Termes primitifs et définis ; axiomes et théorèmes. Les sciences antérieures à une science donnée. La méthode déductive considérée comme propriété caractéristique des mathématiques.2. Liberté dans le choix des termes primitifs et des axiomes ; notion d’équivalence de deux systèmes de termes ou de propositions.Postulats d’indépendance des termes primitifs et des axiomes.3. Postulats de la formalisation (...) des définitions et des démonstrations. Règles de définition et de démonstration ; les démonstrations complètes ; les sciences déductives formalisées. Le rôle de la logique mathématique au point de vue des nouvelles exigences méthodologiques.4. Les notions de science non-contradictoire et complète. L’importance théorique de ces notions ; leur rapport avec le problème de la vérité des mathématiques. Pourquoi les notions en question n’exercent pas en pratique une grande influence sur la construction des sciences mathématiques.5. Certaines propriétés des sciences mathématiques découlant d’une application conséquente de la méthode déductive.ion du sens des termes primitifs dans la construction de la science. Modèles et interprétations du système d’axiomes. Concept de système formel. Valeur de la méthode déductive du point de vue de l’économie de la pensée. (shrink)

ABSTRACT We show that there is a polynomial over the rational number field corresponding to each propositional formula in a given many-valued logic. To decide whether a propositional formula can be deduced from a finite set of such formulas (deduction problem), we only need to decide whether a polynomial vanishes on an algebraic variety. By using Wu's method, an algorithm for this problem is presented.

The paraconsistent system CPQ-ZFC/F is defined. It is shown using strong non-finitary methods that the theorems of CPQ-ZFC/F are exactly the theorems of classical ZFC minus foundation. The proof presented in the paper uses the assumption that a strongly inaccessible cardinal exists. It is then shown using strictly finitary methods that CPQ-ZFC/F is non-trivial. CPQ-ZFC/F thus provides a formulation of set theory that has the same deductive power as the corresponding classical system but is more reliable in (...) that non-triviality is provable by strictly finitary methods. This result does not contradict Gödel's incompleteness theorem because the proof of the deductive equivalence of the paraconsistent and classical systemss use non-finitary methods. (shrink)

ABSTRACTThe new paradigm in the psychology of reasoning redirects the investigation of deduction conceptually and methodologically because the premises and the conclusion of the inferences are assumed to be uncertain. A probabilistic counterpart of the concept of logical validity and a method to assess whether individuals comply with it must be defined. Conceptually, we used de Finetti's coherence as a normative framework to assess individuals' performance. Methodologically, we presented inference schemas whose premises had various levels of probability that contained non-numerical (...) expressions and, as a control, sure levels. Depending on the inference schemas, from 60% to 80% of the participants produced coherent conclusions when the premises were uncertain. The data also show that except for schemas involving conjunction, performance was consistently lower with certain than uncertain premises, the rate of conjunction fallacy was consistently low (not exceeding 20%,.. (shrink)

A thought experiment involving an omniscient being and quantum mechanics is used to justify non-deductive methods in mathematics. The twin prime conjecture is used to illustrate what can be achieved.

The paper presents a compositional approach to the verification of CTL* properties over reactive systems. Both symbolic model-checking and deductive verification are considered. Both methods are based on two decomposition principles. A general state formula is decomposed into basic state formulas which are CTL* formulas with no embedded path quantifiers. To deal with arbitrary basic state formulas, we introduce another reduction principle which replaces each basic path formula, i.e., path formulas whose principal operator is temporal and which contain (...) no embedded temporal operators or path quantifiers, by a newly introduced boolean variable which is added to the system. Thus, both the algorithmic and the deductive methods are based on two statification transformations which successively replace temporal formulas by assertions which contain no path quantifiers or temporal operators. Performing these decompositions repeatedly, we remain with basic assertional formulas, i.e., formulas of the form Efp and Afp for some assertion p. In the model-checking method we present a single symbolic algorithm to verify both universal and existential basic assertional properties. In the deductive method we present a small set of proof rules and show that this set is sound and relatively complete for verifying universal and existential basic assertional properties over reactive systems. Together with two proof rules for the decompositions, we obtain a sound and relatively complete proof system for arbitrary CTL* properties. Interestingly, the deductive approach for CTL* presented here, offers a viable new approach to the deductive verification of arbitrary LTL formulas. The paper corrects a previous preliminary version of a deductive system for CTL*, in which some of the rules were unsound. The correction is based on the introduction of a new type of temporal testers which are guaranteed to be non blocking. That is, when composed with a deadlock-free system, which is a key operation in the verification process, the resulting composed system is guaranteed to remain deadlock free. (shrink)

In the Organon Aristotle describes some deductive schemata in which inconsistencies do not entail the trivialization of the logical theory involved. This thesis is corroborated by three different theoretical topics by him discussed, which are presented in this paper. We analyse inference schema used by Aristotle in the Protrepticus and the method of indirect demonstration for categorical syllogisms. Both methods exemplify as Aristotle employs classical reductio ad absurdum strategies. Following, we discuss valid syllogisms from opposite premises (contrary and (...) contradictory) studied by the Stagerian in the Analytica Priora (B15). According to him, the following syllogisms are valid from opposite premises, in which small Latin letters stand for terms such as subject and predicate, and capital Latin letters stand for the categorical propositions such as in the traditional notation: (i) in the second figure, Eba,Aba ` Eaa (Cesare), Aba, Eba ` Eaa (Camestres), Eba, I ba ` Oaa (Festino), and Aba,Oba ` Oaa (Baroco); (ii) in the third one, Eab,Aab ` Oaa (Felapton), Oab,Aab ` Oaa (Bocardo) and Eab, Iab ` Oaa (Ferison). Finally, we discuss the passage from the Analytica Posteriora (A11) in which Aristotle states that the Principle of Non-Contradiction is not generally presupposed in all demonstrations (scientific syllogisms), but only in those in which the conclusion must be proved from the Principle; the Stagerian states that if a syllogism of the first figure has the major term consistent, the other terms of the demonstration can be each one separately inconsistent. These results allow us to propose an interpretation of his deductive theory as a broad sense paraconsistent theory. Firstly, we proceed to a hermeneutical analysis, evaluating its logical significance and the interplay of the results with some other points of Aristotle’s philosophy. Secondly, we point to a logical interpretation of the Aristotelian syllogisms from opposite premises in the antilogisms method proposed by Christine Ladd-Franklin in 1883, and we also present a logical treatment of the Aristotelian demonstration with inconsistent material in the paraconsistent logics Cn, 1 _ n _!, introduced by da Costa in 1963. These two issues seem having not yet been analysed in detail in the literature. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of (...) cut admissibility. We then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91]. (shrink)

The expressive power of logic is believed to be able to model most of the fundamental aspects of information retrieval. However, it is also understood that classical logic is ineffective for handling partiality and uncertainty in IR. Applying non-classical logics such as the AGM belief revision logic and the possibilistic logic to adaptive information retrieval is appealing since they provide a powerful and rigorous framework to model partiality and uncertainty inherent in any IR processes. The maxi-adjustment method, which is an (...) effective computational apparatus of the AGM paradigm, is applied to develop the learning components of the adaptive information agents. Essentially, maxi-adjustment allows the partial representation K of a user's information needs N to be refined gradually based on the user's relevance feedback t. Generally speaking, learning in adaptive information agents is characterised by the AGM belief revision Kt*. On the other hand, possibilistic logic supports a gradated assessment of the uncertainty arising from matching K with the imperfect characterisation d of an information object D. Information matching in adaptive information agents is underpinned by K ƕ d, where ƕ is the possibilistic inference relation. This paper illustrates how maxi-adjustment and possibilistic deduction can be applied to develop the adaptive information agents. Their impact on the agents learning autonomy and explanatory power is also discussed. (shrink)

Exploring the early development of an area of medical literature can inform contemporary medical debates. Different methods of inference include deduction, induction, abduction, and inference to the best explanation. I argue that early shaken baby research is best understood as using abduction to tentatively suggest that infants with unexplained intracranial and ocular bleeding have been assaulted. However, this tentative conclusion was quickly interpreted, by some at least, as a general rule that infants with these pathological signs were certainly cases (...) of abuse. Rather than focusing on inductive arguments, researchers today may be better off focusing on making a compelling inference to the best explanation. (shrink)

This article demonstrates the fallacy of legal deduction as a method which is supposed to guarantee the certainty and predictability of the law. The Author asserts that legal deduction is in fact not of a logical nature. Their premises are of an uneven character or else one of them must be created in a non-mechanical way. This in turn makes legal deduction that is comprehended as a mode of inference that is infallible, provided its premises are true / valid nothing (...) more than misunderstanding and fallacy. In addition, in this paper, the outcomes of legal deduction are shown to be highly dependent on the direct and indirect context under the threat of their being utterly absurd. Since the lack of fixity and clarity, these contexts, however, may lead one to madness, i. e. at least unless she or he abandons the idea that only logical types of inferences govern legal deduction and are solely responsible for its outcomes. (shrink)

I argue for a novel, non-subjectivist interpretation of Kant’s transcendental idealism. Kant’s idealism is often interpreted as specifying how we must experience objects or how objects must appear to us. I argue to the contrary by appealing to Kant’s Transcendental Deduction. Kant’s Deduction is the proof that the categories are not merely subjectively necessary conditions we need for our cognition, but objectively valid conditions necessary for objects to be appearances. My interpretation centres on two claims. First, Kant’s method of self-knowledge (...) consists in his determining what makes our cognitive faculty finite in contrast to God’s infinite cognitive faculty. Second, Kant’s limitation of our knowledge to appearances consists in his developing an account according to which appearances and our finite cognitive faculty are conceived of in terms of each other and in contrast to noumena in the positive sense and God’s infinite cognitive faculty. (shrink)

This lecture will deal with the heuristic power of the deductive method and its contributions to the scientific task of finding new knowledge. I will argue for a new reading of the term 'deductive method.' It will be presented as an architectural scheme for the reconstruction of the processes of gaining reliable scientific knowledge. This scheme combines the activities of doing science with the activities of presenting scientific results. It combines the heuristic and the deductive side of (...) science. The heuristic side is represented, e.g., by the creative methods to find the 'best' hypotheses, to design experimental systems for empirical research in order to formulate general laws, or to create axiomatic systems. The other side consists of the production of deductive knowledge. This combination leads to a clear hierarchy: the heuristic side provides the basic presuppositions from which the deductive side takes off. The former is used to make deductions possible. The deductive method can be presented as an analysis-synthesis scheme as it can be found, e.g., in the tradition of Kant, Jakob Friedrich Fries, and Leonard Nelson. Nelson's critical philosophy can be seen as a key for understanding the philosophy behind David Hilbert's early axiomatic method. This axiomatic method is usually restricted to a non-philosophical approach to pure mathematics. But Hilbert was not an exclusive formalist; he proposed a mathesis universalis in the Cartesian-Leibnizian sense according to which mathematics is the syntactical tool for a general philosophy of science, applicable to all scientific disciplines. In this function, mathematics takes its problems from the sciences. Hilbert did not deny that mathematics should play a role in explaining the world. The analysis-synthesis scheme helps to provide a consistent interpretation of these two sides of Hilbert's attitude towards his working field. (shrink)

Le présent texte discute les méthodes utilisées par Philopon et Pseudo-Simplicius lorsqu’ils commentent la théorie de la perception sensible d’Aristote, et il marque leurs différences. Philopon fait un fréquent usage de théories médicales et de matériel empirique, empruntés principalement à Aristote, afin de mettre en lumière non pas seulement les activités des différents sens, mais aussi une certaine conscience ainsi que la façon dont nous expérimentons nos états intérieurs. Par contre, son contemporain Athénien, Pseudo-Simplicius, méconnaît complètement de tels aspects. Sa (...) méthode est déductive: il fait appel à une thèse générale, empruntée en partie à Iamblichus, dont il déduit des thèses relatives à la perception sensible. L’accent est mis ici sur la doctrine de Philopon, car son recours à des perspectives médicales donne lieu à un mélange intéressant de théories platoniciennes et de théories médicales empiriques. (shrink)

The claims of randomized controlled trials to be the gold standard rest on the fact that the ideal RCT is a deductive method: if the assumptions of the test are met, a positive result implies the appropriate causal conclusion. This is a feature that RCTs share with a variety of other methods, which thus have equal claim to being a gold standard. This article describes some of these other deductive methods and also some useful non-deductive (...) methods, including the hypothetico-deductive method. It argues that with all deductive methods, the benefit that the conclusions follow deductively in the ideal case comes with a great cost: narrowness of scope. This is an instance of the familiar trade-off between internal and external validity. RCTs have high internal validity but the formal methodology puts severe constraints on the assumptions a target population must meet to justify exporting a conclusion from the test population to the target. The article reviews one such set of assumptions to show the kind of knowledge required. The overall conclusion is that to draw causal inferences about a target population, which method is best depends case-by-case on what background knowledge we have or can come to obtain. There is no gold standard. (shrink)

Résumé — S’il fallait résumer les thèses défendues par Lessing dans le Laocoon, on arriverait à un résultat finalement malingre. L’idée que la peinture représente au moyen de « signes naturels » des corps coexistant dans l’espace, tandis que la poésie représente au moyen de « signes arbitraires » des actions se succédant dans le temps a été en effet maintes fois développée par d’autres auteurs avant l’essai de 1766. Ce constat n’ôte pourtant rien à l’intérêt du Laocoon. Car cet (...) intérêt réside ailleurs. Il se situe non pas dans l’exposé définitif de la thèse, mais dans la méthode qui y mène, une méthode que Lessing veut d’abord inductive, c’est-à-dire fondée sur l’exemple. Le présent article entend analyser la conception de la critique qui se dégage de l’essai : refus de la méthode déductive, rôle central dévolu à l’imagination, intronisation du public comme instance légitime de jugement – le tout sur fond de polémique antiwinckelmannienne.Zusammenfassung — Wollte man die in Lessings Laokoon verteidigten Thesen zusammenfassen, so würde man zu einem eher spärlichen Ergebnis kommen. Da die Malerei anhand « natürlicher Zeichen » nur koexistierende Körper im Raum und die Poesie anhand « willkürlicher Zeichen » nur sukzessive Handlungen in der Zeit darstellt, wurde in der Tat schon vor der Erscheinung dieses Traktats von anderen Autoren dargelegt. Dies gereicht aber dem Essay keineswegs zum Nachteil, denn seine Bedeutung liegt anderswo. Sie liegt weniger in dem Inhalt der Thesen selbst, als in der Methode, die zur Aufstellung dieser Thesen führt. Lessing sucht nach einer induktiven Methode, die auf der präzisen Analyse von zahlreichen Beispielen beruht. Ziel des vorliegenden Artikels ist es, den Lessingschen Kritik-Begriff zu untersuchen, der sich aus der Methodologie dieses Essays ergibt : Ablehnung der deduktiven Verfahrensweise, Erhebung der Einbildungskraft zum entscheidenden Vermögen sowohl für die Produktion als auch für die Rezeption der Kunstwerke, Inthronisation des Publikums als vollberechtigter Kunstrichter – lauter Aspekte, die vor dem Hintergrund einer scharfen Polemik gegen Winckelmann entwickelt werden. (shrink)

In this article, the focus of research attention is directed at understanding the principles of the deductive rationalist method of the French thinker of the New Age, R.Descartes, who formulated the rules of morality on its basis. The work involves a critical analysis of these principles, in particular morality, given the current conditions of existence, marked by social cataclysms that cause tragic changes and difficult life circumstances, the search for ways to minimize/overcome which should be directed by modern scientific (...) research, which is currently being developed in the field of philosophy and social work. The purpose of this study is to provide a philosophical critical analysis of the key principles of the rationalist method of cognition and the maxims built on its basis, proposed by R. Descartes for personal discretion. In addition, the paper focuses on those obstacles that, by obscuring the common sense of an individual, lead to the non-obviousness of the ocular, which results in the establishment and further adherence to a false conclusion instead of the true one, which will be facilitated by the use of the psychoanalytic method of research. The achievement of this goal will contribute to the formation of critical thinking in novice scientists, which, based on doubt, should ultimately be directed not at denial and removal, which is irrational destruction, but at achieving and comprehending the truth, which should be the basis for finding constructive solutions in such meaningful areas of human activity as philosophy and practical social work. The analysis made it possible to identify the following important points:1. In terms of ratio: special attention should be paid to understanding the essence of the first principle of R.Descartes’ rationalist method, which provides for a comprehensive scientific justification for the choice of simple and obvious truths as the basis for further development of scientific research, which will refute the non-obviousness of the obvious, as well as a careful approach to identifying keywords within it. 2. In terms of morality: a scientist should develop determination, firmness, and perseverance, which, despite doubts, will allow him or her to follow his or her own individual scientific trajectory. In the future, this work should focus on the conscious choice and use of methods of cognition, which will help to improve the efficiency of research activities and the quality of work of beginning scientists. (shrink)

Recent years have been blessed with an abundance of logical systems, arising from a multitude of applications. A logic can be characterised in many different ways. Traditionally, a logic is presented via the following three components: 1. an intuitive non-formal motivation, perhaps tie it in to some application area 2. a semantical interpretation 3. a proof theoretical formulation. There are several types of proof theoretical methodologies, Hilbert style, Gentzen style, goal directed style, labelled deductive system style, and so on. (...) The tableau methodology, invented in the 1950s by Beth and Hintikka and later per fected by Smullyan and Fitting, is today one of the most popular, since it appears to bring together the proof-theoretical and the semantical approaches to the pre of a logical system and is also very intuitive. In many universities it is sentation the style first taught to students. Recently interest in tableaux has become more widespread and a community crystallised around the subject. An annual tableaux conference is being held and proceedings are published. The present volume is a Handbook a/Tableaux pre senting to the community a wide coverage of tableaux systems for a variety of logics. It is written by active members of the community and brings the reader up to frontline research. It will be of interest to any formal logician from any area. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Meaning in Spinoza’s MethodAlan Nelson and Noa SheinAaron V. Garrett. Meaning in Spinoza’s Method. New York: Cambridge University Press, 2003. Pp. xii + 240. Cloth, $60.00.This is a book about some fundamental aspects of Spinoza's mature metaphysics. The principal focus is on Part I of the Ethics concerning infinite substance, and on Part V concerning the intuitive knowledge that is the goal of philosophy. Within this focus, Garrett (...) concentrates on Spinoza's method. Garrett explains how he has been influenced by interpretations that stress the structural parallel between the synthetic, geometrical presentation of the Ethics and the relations among the ideas of the well-ordered mind. The Ethics proceeds by deriving propositions concerning substance's modes from definitions of substance and related notions while the mind, insofar as it is free from emotions, intuitively knows things as following from God. Garrett's distinctive thesis, however, is that Spinozistic enlightenment depends on the process of emending the intellect as described in one of Spinoza's earliest works, the Treatise on the Emendation of the Intellect. While this work strongly foreshadows many of Spinoza's mature doctrines, its exposition much more closely resembles the analytic, non-geometric approach of Descartes's Meditations. Spinoza's therapy for the emendation of our "inadequate" ideas closely parallels Descartes's analytic therapy for making what he calls confused ideas "clear and distinct."Garrett is on firm ground in arguing that the process of enlightenment cannot be an unbroken chain of intuitive deductions. For one thing, the definitions and axioms printed in a copy of the Ethics are first sensed by the reader and are not likely to produce immediately the requisite adequate ideas (172). An example of one of Garrett's interesting conclusions is that the definition of God as consisting of an infinity of attributes, the sixth of Part I of the Ethics (1D6), is preliminary and replaced in the process of therapeutic emendation by a definition of God as self caused. It is only after our ideas have become adequate at the end of the process that we are able to fully appreciate 1D6 (171).One might think that Garrett does not go far enough here because his point can be naturally extended. Definitions might assist us in the process of bringing our adequate ideas to light, but our cognitive goal is making these ideas themselves adequate. What then are the ideas we can make most adequate and exactly what distinguishes them from highly refined ideas that are nevertheless more inadequate than these? On one interpretation it is divine essence, that is, the attributes of thought and extension that are individually most adequately conceived after successful therapy. It is tempting to suppose that a single idea of God as both thinking and extended would be more adequate still, but Spinoza holds that our conceptions of these attributes are "really distinct" in a sense analogous to Descartes's(1P10). Spinoza's 1D6 definition of God as a substance consisting of both these attributes (and perhaps other inconceivable ones) thus seems to indicate a subsumption of the attributes we can conceive under the image of some printed or spoken words. Such an idea would be more inadequate than our adequate conceptions of the individual attributes. One is here strongly reminded of Descartes's characterization in Article 63 of the first Part of his Principles of Philosophy, "... we more readily understand extended substance or thinking substance than substance alone."In his seventh and concluding chapter, Garrett tackles the grand problems of intuitive knowledge or knowledge of "the third kind" and the sense in which Spinoza thought a part of the human mind is eternal. The discussion is necessarily complex, but one great advantage of Garrett's treatment is that he realizes intuitive knowledge cannot consist in the explicit deduction of particular things from God. The detailed determination of particular "everyday objects" requires sensory knowledge which is, of course, never perfectly determinate and therefore inherently inadequate. But we can acquire the "second kind" of adequate knowledge by understanding what our minds have in common with other determinate, finite things. This finally enables us to ascend to an appreciation of these... (shrink)

Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the latter he saw evidence for the impossibility of capturing the intended interpretation by purely deductive methods. In the history of mathematics the concept of a nonstandard model is new. An analysis of some major innovations–the discovery of irrationals, the use of negative and complex numbers, the modern concept (...) of function, and non-Euclidean geometry–reveals them as essentially different from the introduction of non-standard models. Yet, non-Euclidean geometry, which is discussed at some length, is relevant to the present concern; for it raises the issue of intended interpretation. The standard model of natural numbers is the best candidate for an intended interpretation that cannot be captured by a deductive system. Next, I suggest, is the concept of a wellordered set, and then, perhaps, the concept of a constructible set. One may have doubts about a realistic conception of the standard natural numbers, but such doubts cannot gain support from non-standard models. Attempts to utilize non-standard models for an anti-realist position in mathematics, which appeal to meaning-as-use, or to arguments of the kind proposed by Putnam, fail through irrelevance, or lead to incoherence. Robinson’s skepticism, on the other hand, is a coherent position, though one that gives up on providing a detailed philosophical account. The last section enumerates various uses of non-standard models. (shrink)

There is a common understanding among logicians today that nonmonotonic types of reasoning, such as defeasible or presumptive, can clearly warrant a rational acceptance of its conclusion. Recognition of the significance and legitimacy of these forms of arguments, which were considered for long as fallacious, is believed to be very recent and many logicians tended to reject any discussions around it within the tradition of logic after Aristotle. In contrast, Islamic jurisprudence (fiqh), since medieval age, has recognised the validity and (...) importance of non-deductive forms of reasoning, analysed the structures of such arguments and demonstrated how the conventionally understood fallacies are not essentially fallacies. Qiyās – generally represents analogical reasoning – is taken here as a case of defeasible reasoning in Islamic argumentation. This paper will show through works of two emblematic medieval Muslim jurist-logicians, al-Juwainī and al-Ghazzālī, that understanding the legitimacy and the structure of non-deductive forms of arguments is at the core of Islamic legal theories (usūl al-fiqh), and that they recognised its legitimacy and proposed a dialectical method (jadal) of approaching it. Les logiciens comprennent généralement que les types de raisonnement non monotones peuvent justifier l’acceptation rationnelle des conclusions. L’importance et la légitimité de ces formes d’arguments, qui ont longtemps été considérées comme fallacieuses, ont été controversées parmi les logiciens de la tradition logique aristotélicienne. En revanche, la jurisprudence islamique (fiqh) a depuis longtemps reconnu l’importance des formes de raisonnement non déductives et a démontré que les sophismes conventionnellement compris ne sont pas essentiellement des sophismes. Le qiyās – qui représente généralement le raisonnement analogique – est un exemple de raisonnement réfutable dans l’argumentation islamique. Je montre à travers les travaux de deux juristes-logiciens musulmans médiévaux emblématiques, al-Juwainī et al-Ghazzālī, que la compréhension de la légitimité des formes d’arguments non déductives est au cœur des théories juridiques islamiques (usūl al-fiqh), qui ont proposé une méthode d’approche dialectique (jadal). (shrink)

There is widespread recognition at universities that a proper understanding of science is needed for all undergraduates. Good jobs are increasingly found in fields related to Science, Technology, Engineering, and Medicine, and science now enters almost all aspects of our daily lives. For these reasons, scientific literacy and an understanding of scientific methodology are a foundational part of any undergraduate education. Recipes for Science provides an accessible introduction to the main concepts and methods of scientific reasoning. With the help (...) of an array of contemporary and historical examples, definitions, visual aids, and exercises for active learning, the textbook helps to increase students’ scientific literacy. The first part of the book covers the definitive features of science: naturalism, experimentation, modeling, and the merits and shortcomings of both activities. The second part covers the main forms of inference in science: deductive, inductive, abductive, probabilistic, statistical, and causal. The book concludes with a discussion of explanation, theorizing and theory-change, and the relationship between science and society. The textbook is designed to be adaptable to a wide variety of different kinds of courses. In any of these different uses, the book helps students better navigate our scientific, 21st-century world, and it lays the foundation for more advanced undergraduate coursework in a wide variety of liberal arts and science courses. Selling Points Helps students develop scientific literacy—an essential aspect of _any_ undergraduate education in the 21 st century, including a broad understanding of scientific reasoning, methods, and concepts Written for all beginning college students: preparing science majors for more focused work in particular science; introducing the humanities’ investigations of science; and helping non-science majors become more sophisticated consumers of scientific information Provides an abundance of both contemporary and historical examples Covers reasoning strategies and norms applicable in all fields of physical, life, and social sciences, _as well as_ strategies and norms distinctive of specific sciences Includes visual aids to clarify and illustrate ideas Provides text boxes with related topics and helpful definitions of key terms, and includes a final Glossary with all key terms Includes Exercises for Active Learning at the end of each chapter, which will ensure full student engagement and mastery of the information include earlier in the chapter Provides annotated ‘For Further Reading’ sections at the end of each chapter, guiding students to the best primary and secondary sources available Offers a Companion Website, with: For Students: direct links to many of the primary sources discussed in the text, student self-check assessments, a bank of exam questions, and ideas for extended out-of-class projects For Instructors: a password-protected Teacher’s Manual, which provides student exam questions with answers, extensive lecture notes, classroom-ready Power Point presentations, and sample syllabi Extensive Curricular Development materials, helping any instructor who needs to create a Scientific Reasoning Course, ex nihilo. (shrink)

Since the publication of Husserl's Logische Untersuchungen at the outset of this century, the notion of phenomenology has had a long and important history on the European continent. Of the many claims made on its behalf perhaps the most interesting is that phenomenology is able to ground philosophical assertions in a manner which is neither purely formal nor purely empirical, i.e., that phenomenology as a method is capable of transcending this very distinction. For example, phenomenologists argue that their reduction of (...) essences provides a way of knowing which is neither analytic nor synthetic but both. Having reached such an eidetic intuition phenomenologists claim to have attained a non-trivial universality and necessity. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:“Though This Be Madness, Yet There Is Method in ’T”A Positive Account of MadnessValentina Cardella, PhD (bio)What does it mean to be mad? How can we define mental disorder? The question is still widely discussed among psychiatrists and philosophers, and what exactly distinguishes sanity and insanity remains unclear. Despite this lack of clarity, the common conceptualization of madness is that mental disorders are impairments in rationality: people with mental (...) disorders are generally believed to have impairments in reasoning and logic (Bermúdez, 2001; Edwards, 1981; Harvey, Watkins, Mansell, & Shafran, 2004).Yet, the result of this conception is that we fail to understand madness. We do not understand how, for example, schizophrenic patients can be more logical than healthy people in some reasoning tasks (Mellet et al., 2006; Owen, Cutting, & David, 2007), or delusional patients can be perfectly rational in defending their ideas (Cardella, 2020), or depressive people can be more realistic than non-clinical subjects (Moore & Fresco, 2012). In contrast, if we consider rationality as the core feature of sanity, we fail to understand sanity, too, because normal people exhibit many kinds of biases and heuristics, and thus irrationality is widespread in the general population (Bortolotti, 2020; Johnson-Laird, 2006; Evans, 2002; Kahneman, Tversky, & Slovic, 1982).In his paper, Garson focuses on Late Modern theorists of madness (Locke, Kant, Wigan, and Heinroth) to reframe the problem and provide a positive account of mental disorder. These authors contrast madness to idiocy, rather than sanity, and therefore their conception of madness is very different: “while idiocy is marked by the diminished, or even abolished, functioning of the reasoning power, madness is marked by its perverse functioning. The mad person “has” reason, but deploys it in a perverse, improper, or unseemly manner” (Garson, 2023, p. 286).In his reading of Late Modern authors, Garson provides a new ontology of mental disorder, that goes beyond an absence of reason, and has also political and social implications. Furthermore, he reminds us that every scientific discipline has its philosophical assumptions. In defining madness, [End Page 305] we cannot help but choose a contrasting class, and each choice will have very different consequences. Even if psychiatry is a medical discipline, which aims to endorse a neutral view of mental disorder, it is impossible for it to be unbiased and impartial. All the more so, for it has to deal with human suffering and fragility.But, in this article, there is one crucial question that remains unanswered. In his analysis of Late Modern theorization of madness, Garson comes to the conclusion that madness is marked by a perverse deployment of reason. Yet, who decides when reason is deployed in a perverse manner? While the lack of rationality is something that we can detect easily enough, the task of defining precisely what is an ‘improper’ or ‘perverse’ use of reason is much harder. In other words, it seems to me that, in replacing the ‘lack of rationality’ paradigm with ‘the perverse rationality paradigm,’ the question of what is madness still remains open, and, most important, some normative aspects remain necessary. To recognize when rationality is improperly used, we have to adopt an evaluative stance, which defines what a proper use of reason would be.Valentina Cardellavalentina.cardella@unime.it University of MessinaValentina Cardella Valentina Cardella is Associate Professor of Philosophy of Language at the University of Messina, Italy. Her research interests encompass cognitive sciences, philosophy of language, and psychopathology.The author reports no conflicts of interest.Commentary submitted on June 13, 2023Accepted on June 13, 2023ReferencesBermúdez, J. L. (2001). Normativity and rationality in delusional psychiatric disorders. Mind and Language 16, 457–93. Google ScholarBortolotti, L. (2020). The epistemic innocence of irrational beliefs. Oxford: Oxford University Press. Google ScholarCardella, V. (2020). Rationality in mental disorders: too little or too much? European Journal of Analytic Philosophy 16 (2), 13–36. Google ScholarEdwards, R. B. (1997). Mental health as rational autonomy. In Edwards, R. B. (ed.), Ethics of psychiatry: Insanity, rational autonomy, and mental health care. New York: Prometheus Books. Google ScholarEvans, J. (2002). Logic and human reasoning: an assessment of the deduction paradigm. Psychological Bulletin 128... (shrink)

In this account of the philosophical and scientific pursuits of Pierre Gassendi , I challenge a traditional view which says that the inspiration, motivation, and demonstrative grounds for his physical atomism consist not in his empiricism but in his historicist commitments. Indeed, Gassendi suggests that it's a consequence of our best theory of knowledge and sound scientific method that we get evidence which warrants his microphysical theory. ;The primary novelty of his theory of empirical knowledge is his proposal, against the (...) Stoics and Descartes, that it is not a necessary condition of our knowing some claim that we are certain of it. This move immediately broadens the scope of what we can know through the senses, as does his suggestion that we include among such knowledge claims those assertions about what is hidden to the senses which we legitimately infer from the evidence of the perceptually given. ;On the basis of these epistemological positions, Gassendi develops his views on scientific method according to which we attain and justify our best empirical claims by deduction yet, he believes, these claims are at root probabilistic, and we may maintain hypotheses as the basis of our scientific reasoning so long as there is some empirical evidence for them, however broadly construed. ;Hence Gassendi suggests that his atomism constitutes a 'most likely hypothesis' for which we have empirical evidence--which of necessity is found indirectly in 'indicative' signs, those surface level phenomena for which he takes the existence of atoms to be a sine qua non condition. Perhaps the paramount case of such evidence consists in those microscopic observations of crystalline formation and dissolution that he takes to demonstrate the molecular structure of matter--a key and innovative consequence of his atomism. ;The robust character of Gassendi's method emerges in his appeals to indirect evidence for claims about the unobservable and his willingness to count this as adequate empirical grounds for maintaining an atomist hypothesis. Remarkably, he stands at the dawn of the modern era with at least a proposal as to how to resolve one of empiricism's more vexing difficulties, still with us in some form today. (shrink)

In my thesis I explain why the common, pre-theoretical understanding of morality is an important part of Kant’s ethics, and I critically evaluate what the strengths and weaknesses are of doing ethics with the common perspective as a point of reference. In chapter 1, I discuss the significance of common rational capacities for the deduction in Groundwork III as well as for the Fact of Reason. Attention to the fundamental role of common rational capacities in the Second Critique reveals that (...) Kant intends to provide further warrant for the Fact than its introspective self-evidence. In chapter 2, I discuss what it means for a rational agent to be endowed with common rational capacities. The agent has everything she needs to reason on her own about what she ought to do and act from rational judgements. Furthermore, I critically evaluate Kant’s claim that his ethics spells out fundamental, pre-theoretical convictions. In chapter 3, I discuss Kant’s conception of rationalizing. I analyse rationalizing as a process of self-deception in which an agent tries to justify or excuse violations of the moral law. This can lead to loss of the reliable use of common rational capacities. I discuss what help critical practical philosophy and moral education can afford against rationalizing. In chapter 4, I argue that Kant saw dialogical engagement with ordinary agents as an important way of obtaining data concerning the correct starting point of practical philosophy. Kant demands that whatever we get from dialog and observation has to be isolated from its contingent elements. I conclude that the main problem for Kant’s method is how we can, on the one hand, exclude non-rational content, and, on the other hand, be open to what other agents actually have to say about morality. (shrink)

When an empirical prediction E of hypothesis H is observed to be true, such observation is said to confirm, i.e., support (although not prove) the truth of the hypothesis. But why? What justifies the claim that such evidence supports the hypothesis? The widely accepted answer is that it is justified by induction. More specifically, it is commonly held that the following argument, (1) If H then E; (2) E; (3) Therefore, (probably) H (here referred to as ‘hypothetico-deductive confirmation argument’), (...) is inductively strong. Yet this argument looks nothing like an inductive generalization, i.e., it does not seem inductive in the term’s traditional, enumerative sense. If anything, it has the form of the fallacy of affirming the consequent. This paper aims to solve this puzzle. True, in recent decades, ‘induction’ has been sometimes used more broadly to encompass any non-deductive, i.e., ampliative, argument. Applying Bayesian confirmation theory has famously demonstrated that hypothetico-deductive confirmation is indeed inductive in this broader, ampliative sense. Nonetheless, it will be argued here that, despite appearance, hypothetico-deductive confirmation can also be recast as enumerative induction. Hence, by being enumeratively inductive, the scientific method of hypothetico-deductive confirmation is justified through this traditional, more restrictive type of induction rather than merely by ampliative induction. (shrink)

Two theses are central to foundationalism. First, the foundationalist claims that there is a class of propositions, a class of empirical contingent beliefs, that are ‘immediately justified’. Alternatively, one can describe these beliefs as ‘self–evident’, ‘non–inferentially justified’, or ‘self–warranted’, though these are not always regarded as entailing one another. The justification or epistemic warrant for these beliefs is not derived from other justified beliefs through inductive evidential support or deductive methods of inference. These ‘basic beliefs’ constitute the foundations (...) of empirical knowledge. One can give a reason for the justification of a basic belief even though the justification for that belief is not based on other beliefs. Thus, according to Chisholm, if asked what one's justification was for thinking that one knew, presently, that one is thinking about a city one takes to be Albuquerque, one could simply say ‘what justifies me…is simply the fact that I am thinking about a city I take to be Albuquerque’. (shrink)

In the process of scientific discovery, knowledge ampliation is pursued by means of non-deductive inferences. When ampliative reasoning is performed, probabilities cannot be assigned objectively. One of the reasons is that we face the problem of the unconceived alternatives: we are unable to explore the space of all the possible alternatives to a given hypothesis, because we do not know how this space is shaped. So, if we want to adequately account for the process of knowledge ampliation, we need (...) to develop an account of the process of scientific discovery which is not exclusively based on probability calculus. We argue that the analytic view of the method of science advocated by Cellucci is interestingly suited to this goal, since it rests on the concept of plausibility. In this perspective, in order to account for how probabilities are in fact assigned in uncertain contexts and knowledge ampliation is really pursued, we have to take into account plausibility-based considerations. (shrink)

The question of whether or not to partition data for the purposes of inferring phylogenetic hypotheses remains controversial. Opinions have been especially divided since Kluge's (1989, Systematic Zoology 38, 7–25) claim that data partitioning violates the requirement of total evidence (RTE). Unfortunately, advocacy for or against the RTE has not been based on accurate portrayals of the requirement. The RTE is a basic maxim for non-deductive inference, stipulating that evidence must be considered if it has relevance to an inference. (...) Evidence is relevant if it has a positive or negative effect on a given conclusion. In the case of ℈partitioned’ phylogenetic inferences, the RTE is violated, and the basis for rational belief in any conclusion is compromised, unless it is shown that the partitions are evidentially irrelevant to one another. The goal of phylogenetic systematics is to hypothesize past causal conditions to account for observed shared similarities among two or more species. Such inferences are non-deductive, necessitating consideration of the RTE. Some phylogeneticists claim the parsimony criterion as justification for the RTE. There is no relation between the two – parsimony is a relation between a hypothesis and causal question(s). Parsimony does not dictate the content of premises prior to an inference. ℈Taxonomic congruence,’ ℈supertrees,’ and ℈conditional combination’ methods violate the RTE. Taxonomic congruence and supertree methods also fail to achieve the intended goal of phylogenetic inference, such that ℈consensus trees’ and ℈supertrees’ lack an empirical basis. ℈Conditional combination’ is problematic because hypotheses derived from partitioned data cannot be compared – a causal hypothesis inferred to account for a set of effects only has relevance to those effects, not any comparative relevance to other causal hypotheses. A similar problem arises in the comparisons of hypotheses derived from different causal theories. (shrink)

The subject of Labelled Non-Classical Logics is the development and investigation of a framework for the modular and uniform presentation and implementation of non-classical logics, in particular modal and relevance logics. Logics are presented as labelled deduction systems, which are proved to be sound and complete with respect to the corresponding Kripke-style semantics. We investigate the proof theory of our systems, and show them to possess structural properties such as normalization and the subformula property, which we exploit not only to (...) establish advantages and limitations of our approach with respect to related ones, but also to give, by means of a substructural analysis, a new proof-theoretic method for investigating decidability and complexity of (some of) the logics we consider. All of our deduction systems have been implemented in the generic theorem prover Isabelle, thus providing a simple and natural environment for interactive proof development. Labelled Non-Classical Logics is essential reading for researchers and practitioners interested in the theory and applications of non-classical logics. (shrink)

Purpose: finding out the features of logical methods for obtaining conclusions on humanitarian subjects. The task of this scientific research is to analyze the logical paradigmatic aspects of humanitarian knowledge, such as the differences in the formation of sets, logical forms and conclusions, and so on. Methods: a set of logical methods such as analysis, synthesis, abstraction and generalization, informal logic of understanding, etc. Scientific novelty: the logic of humanitarian knowledge is formed on the basis of the (...) logic of historical knowledge, the specificity of which is manifested in the special meaning of individual sets that receive direct and indirect meaning, form the corresponding logical connections in obtaining conclusions. When obtaining conclusions on humanitarian subjects, the formation of an exemplary individual is a plurality. Its ideal qualities are transferred to an unnamed set. Thus, the integrity of the humanitarian subject is created. In the transition from parts to the whole there is a “logical jump”, which is also the difference in the procedure for obtaining a conclusion. Widely used rational understanding by identifying the value of the whole. Understanding can be strong or weak. The difference in the logic of humanitarian knowledge is primarily due to the peculiarity of the work of human thinking, which uses specific tools in formatting various scientific disciplines and organizing the cognitive process. Among the many new approaches and methods of the humanities developed in phenomenology, existentialism, philosophy of life, structuralism, postmodernism, etc., hermeneutics should be noted. It is hermeneutics that occupies a special place in humanitarian knowledge due to its ontology and logic. Conclusions: the logic of humanitarian knowledge is formed on the basis of historical consciousness, relies on ontology and has signs of the logic of practical thinking. In its formation, it is advisable to use the theory of practical judgments and conclusions, taking into account subjective values of life experience, obtaining the logical values of truth and falsehood. The terms of deductive and non-deductive forms are based on judgments about values. The logic of humanitarian knowledge to a large extent is hermeneutical logic, which, in turn, is the logic of rational understanding. (shrink)

Reminiscences of Peter, by P. Oppenheim.--Natural kinds, by W. V. Quine.--Inductive independence and the paradoxes of confirmation, by J. Hintikka.--Partial entailment as a basis for inductive logic, by W. C. Salmon.--Are there non-deductive logics?, by W. Sellars.--Statistical explanation vs. statistical inference, by R. C. Jeffre--Newcomb's problem and two principles of choice, by R. Nozick.--The meaning of time, by A. Grünbaum.--Lawfulness as mind-dependent, by N. Rescher.--Events and their descriptions: some considerations, by J. Kim.--The individuation of events, by D. Davidson.--On properties, (...) by H. Putnam.--A method for avoiding the Curry paradox, by F. B. Fitch.--Publications (1934-1969) by Carl G. Hempel (p. [266]-270). (shrink)

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...) are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. (shrink)

SummaryThe central thesis advocated by the author is that the so‐called hermeneutic method is actually the same as the hypothetico‐deductive method applied to materials that are “meaningful” . Five different interpretations of the role of the stranger in Ibsens “Peer Gynt” are discussed and shown to be examples of how interpretation‐hypotheses can be judged by confronting them with the data . The conclusion drawn from the analysis is this: there is no fundamental methodological difference between natural sciences and humanities.

The non-Fregean logic SCI is obtained from the classical sentential calculus by adding a new identity connective = and axioms which say ?a = ß' means ?a is identical to ß'. We present complete and sound proof system for SCI in the style of Rasiowa-Sikorski. It provides a natural deduction-style method of reasoning for the non-Fregean sentential logic SCI.

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large (...) extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument. (shrink)

Drawing on the epistemology of logic literature on anti-exceptionalism about logic, we set out to investigate the following metaphilosophical questions empirically: Is philosophy special? Are its methods (dis)continuous with science? More specifically, we test the following metaphilosophical hypotheses empirically: philosophical deductivism, philosophical inductivism, and philosophical abductivism. Using indicator words to classify arguments by type (namely, deductive, inductive, and abductive arguments), we searched through a large corpus of philosophical texts mined from the JSTOR database (N = 435,703) to find (...) patterns of argumentation. The results of our quantitative, corpus-based study suggest that deductive arguments are significantly more common than abductive arguments and inductive arguments in philosophical texts overall, but they are gradually and steadily giving way to non-deductive (i.e. inductive and abductive) arguments in academic philosophy. (shrink)

Abductive conclusions are drawn in a special, co-hortative mood. Abductive conclusions are representative interpretants that represent abduction as a form of reasoning that can convey a general conception of the truth. The truth is not asserted; abduction merely delivers the idea of a matter of course, rendering that idea comparatively simple and natural, hence assuring us of its justified assertibility. Hence abductive reasoning is at home in addressing ‘How Possible’-questions in science. Abductive reasoning concerns the question of how things might, (...) could or would conceivably be such that they can be plausibly asserted. Peirce took all reasoning to be diagrammatic and representable using the graphical method of logic. Yet no examples have previously been found in his large manuscript corpus of what such non-deductive graphs might look like. This paper proposes a new interpretation of a sole exception, a sketch of two graphs from a rejected page from 1903, which might be the only surviving example of Peirce’s abductive graphs. The proposed interpretation takes them to be representative interpretants of this special inverse type of inference. (shrink)

Goldin thus begins the book by asking the basic question "What are we reading?" while also considering why it has been so rarely asked. Yet far from denigrating Chinese philosophy, he argues that liberating these texts from the mythic idea that they are the product of a single great mind only improves our understanding and appreciation. By no means does a text require single and undisputed authorship to be meaningful; nor is historicism the only legitimate interpretive stance. The first chapter (...) takes up a hallmark of Chinese philosophy that demands a Western reader's cognizance: its preference for non-deductive argumentation. Chinese philosophy is an art he demonstrates, more than it is a rigorous logical method. Then comes the core of the book, eight chapters devoted to the eight philosophical texts divided into three parts: Philosophy of Heaven, Philosophy of the Way, and Two Titans at the End of an Age.. (shrink)

Although there exist today a variety of non-deductive reliable processes able to determine the truth of certain mathematical propositions, proof remains the only form of justification accepted in mathematical practice. Some philosophers and mathematicians have contested this commonly accepted epistemic superiority of proof on the ground that mathematicians are fallible: when the deductive method is carried out by a fallible agent, then it comes with its own level of reliability, and so might happen to be equally or even (...) less reliable than existing non-deductive reliable processes—I will refer to this as the reliability argument. The aim of this article is to examine whether the reliability argument forces us to reconsider the commonly accepted epistemic superiority of the deductive method over non-deductive reliable processes. I will argue that the reliability argument is fundamentally correct, but that there is another epistemic property differentiating the deductive method from non-deductive reliable processes. This property is based on the observation that although mathematicians are fallible agents, they are also self-correcting agents. This means that when a proof is produced that only contains repairable mistakes, given enough time and energy, a mathematician or a group thereof should be able to converge towards a correct proof through a finite number of verification and correction rounds, thus providing a guarantee that the considered proposition is true, something that non-deductive reliable processes will never be able to produce. From this perspective, the standard of justification adopted in mathematical practice should be read in a diachronic way: the demand is not that any proof that is ever produced be correct—which would amount to requiring that mathematicians are infallible—but rather that, over time, proofs that contain repairable mistakes be corrected, and proofs that cannot be repaired be rejected. (shrink)