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 chapter. I argue that non-deductive justification is in fact pervasive in mathematics, and that it (...) is in good epistemic standing. (shrink)

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)

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)

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.

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.

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)

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)

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)

The nature of the scientific method has been a main concern of philosophy from Plato to Mill. In that period logic has been considered to be a part of the methodology of science. Since Mill, however, the situation has completely changed. Logic has ceased to be a part of the methodology of science, and no Discourse on method has been written. Both logic and the methodology of science have stopped dealing with the process of discovery, and generally with the actual (...) process of scientific research. As a result, several first-rate scientists, from Feynman and Weinberg to Dyson and Hawkins, have concluded that philosophy has become useless and totally irrelevant to science. The aim of this paper is to give some indications as to how to develop a logic concerned with the process of discovery and a methodology of science dealing with the actual process of scientific research. (shrink)

n this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods of Barbara and Datisi. By adding first-degree propositional negation to this system, we prove that the square of opposition holds without using many of the other rules of (...) classical logic (including double negation elimination). We then show that the Propositional Substructural Logic SLe is the best logic to study Aristotelian Syllogisms. Also, based on the IFLe square of opposition, the rules of conversation and the rules of negation are completely proved in Muzaffar's logic. For this purpose, we used the monadic first-order logic with the same standard deductive apparatus of quantifiers in classical logic, plus the axioms of "some a is an a" and "some not-a is a not-a". Finally, to show that there is no existential commitment to general terms in categorical logic, the Strong Four-Valued Relevant-classical Logic KR4 was used. With the same existential interpretation of the quantifiers and the standard translation of the quarter quantified. (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)

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)

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)

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)

The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic \({\mathbf{NC}}_{\mathbf{3}}\). As a result, for each binary extension of \({\mathbf{NC}}_{\mathbf{3}}\), we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining _n_-sequent proof systems for any _n_-valent logic (...) with deterministic or non-deterministic matrices is not applicable to \({\mathbf{NC}}_{\mathbf{3}}\) and its binary extensions. (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)

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)

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)

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)

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)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or (...) `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic. (shrink)

Weak Quantum Theory (WQT) and the Model of Pragmatic Information (MPI) are two psychophysical concepts developed on the basis of quantum physics. The present study contributes to their empirical examination. The issue of the study is whether WQT and MPI can not only explain ‘psi’-phenomena theoretically but also prove to be consistent with the empirical phenomenology of extrasensory perception (ESP). From the main statements of both models, 33 deductions for psychic readings are derived. Psychic readings are defined as settings, in (...) which psychics support or counsel clients by using information not mediated through the five senses. A qualitative approach is chosen to explore how the psychics experience extrasensory perceptions. Eight psychics are interviewed with a half-structured method. The reports are examined regarding deductive and inductive aspects, using a multi-level structured content analysis. The vast majority of deductions is clearly confirmed by the reports. Even though the study has to be seen as an explorative attempt with many aspects to be specified, WQT and MPI prove to be coherent and helpful concepts to explain ESP in psychic readings. (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)

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)

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.

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)

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)

Bruce Waller has defended a deductive reconstruction of the kinds of analogical arguments found in ethics, law, and metaphysics. This paper demonstrates the limits of such a reconstruction and argues for an alternative. non-deductive reconstruction. It will be shown that some analogical arguments do not fit Waller's deductive schema, and that such a schema does not allow for an adequate account of the strengths and weaknesses of an analogical argument. The similarities and differences between the account defended (...) herein and the Trudy Govier's account are discussed as well. (shrink)

Bruce Waller has defended a deductive reconstruction of the kinds of analogical arguments found in ethics, law, and metaphysics. This paper demonstrates the limits of such a reconstruction and argues for an alternative. non-deductive reconstruction. It will be shown that some analogical arguments do not fit Waller's deductive schema, and that such a schema does not allow for an adequate account of the strengths and weaknesses of an analogical argument. The similarities and differences between the account defended (...) herein and the Trudy Govier's account are discussed as well. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (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)

Dogmatism holds that an experience or seeming that p can provide prima facie immediate justification for believing p in virtue of its phenomenology. Dogmatism about perceptual justification has appealed primarily to proponents of representational theories of perceptual experience. Call dogmatism that takes perceptual experience to be representational "representational phenomenal dogmatism." As we show, phenomenal seemings play a crucial role in dogmatism of this kind. Despite its conventional appeal to representational theorists, dogmatism is not by definition committed to any particular view (...) of perceptual experience. Naive realists and disjunctivists who hold that perceptual experience is a perceptual relation of direct acquaintance can also endorse dogmatism. Indeed, we argue that they ought to do so. Otherwise, they cannot maintain that relationalism about perceptual experience has an epistemic advantage compared to the representational view. We then show that even if we grant that relationalism has this advantage, only dogmatism that takes perceptual experience to be representational can be extended to account for non-deductive inferential justification. As an account of the latter is required to avoid succumbing to skepticism, relationalism, we argue, does not have the epistemic advantage its defenders claim it has. (shrink)

There exist valuable methods for theorem proving in non classical logics based on translation from these logics into first-order classical logic (abbreviated henceforth FOL). The key notion in these approaches istranslation from aSource Logic (henceforth abbreviated SL) to aTarget Logic (henceforth abbreviated TL). These methods are concerned with the problem offinding a proof in TL by translating a formula in SL, but they do not address the very important problem ofpresenting proofs in SL via a backward translation. We (...) propose a framework for presenting proofs in SL based on a partial backward translation of proofs obtained in a familiar TL: Order-Sorted Predicate Logic. The proposed backward translation transfers some formulasF TL belonging to the proof in TL into formulasF SL , such that the formulasF SL either (a) belong to a corresponding deduction in SL (in the best case) or, (b) are semantically related in some precise way, to formulas in the corresponding deduction in SL (in the worst case). The formulasF TL andF SL can obviously be considered aslemmas of their respective proofs. Therefore the transfer of lemmas of TL gives at least a skeleton of the corresponding proof in SL. Since the formulas of a proof keep trace of the strategy used to obtain the proof, clearly the framework can also help in solving another fundamental and difficult problem:the transfer of strategies from classical to non classical logics. We show how to apply the proposed framework, at least to S5, S4(p), K, T, K4. Two conjectures are stated and we propose sufficient (and in general satisfactory) conditions in order to obtain formulas in the proof in SL. Two particular cases of the conjectures are proved to be theorems. Three examples are treated in full detail. The main lines of future research are given. (shrink)

This essay begins by summarizing the reasons why unregulated health-care markets are inefficient. The inefficiencies stem from the asymmetries of information among providers, patients and payers, which give rise to moral hazard and adverse selection. Attempts to ameliorate these inefficiencies by means of risk-adjusted insurance and monetary incentives such as co-pays and deductibles lessen the inefficiencies at the cost of increasing inequalities. Another possibility is to rely on non-monetary incentives, including ordeals. While not a magic bullet, these are feasible (...) class='Hi'>methods for addressing the inadequacies of market provision of health care, such as moral hazard. (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 [email protected] 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)

The current widespread belief that taxonomic methods used before Darwin were essentialist is ill-founded. The essentialist method developed by followers of Plato and Aristotle required definitions to state properties that are always present. Polythetic groups do not obey that requirement, whatever may have been the ontological beliefs of the taxonomist recognizing such groups. Two distinct methods of forming higher taxa, by chaining and by examplar, were widely used in the period between Linnaeus and Darwin, and both generated polythetic (...) groups. Philosopher William Whewell congratulated pre-Darwinian taxonomists for not adhering to the rigid ideal of definition used in the mathematical sciences. What he called the method of types is here called the method of exemplars because typology has been equated with essentialism, whereas the use of a type species as the reference point or prototype for a higher category was a practice inconsistent with essentialism. The story that the essentialism of philosophers dominated the development of systematics may prove to be a myth. (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 [email protected] 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)