The semantical structures called T x W frames were introduced in (Thomason, 1984) for the Ockhamist temporal-modal language, $[Unrepresented Character]_{o}$ , which consists of the usual propositional language augmented with the Priorean operators P and F and with a possibility operator ◇. However, these structures are also suitable for interpreting an extended language, $[Unrepresented Character]_{so}$ , containing a further possibility operator $\lozenge^{s}$ which expresses synchronism among possibly incompatible histories and which can thus be thought of as a cross-history 'simultaneity' operator. (...) In the present paper we provide an infinite set of axioms in $[Unrepresented Character]_{so}$ , which is shown to be strongly complete for T x W-validity. Von Kutschera (1997) contains a finite axiomatization of T x W-validity which however makes use of the Gabbay Irreflexivity Rule (Gabbay, 1981). In order to avoid using this rule, the proof presented here develops a new technique to deal with reflexive maximal consistent sets in Henkin-style constructions. (shrink)

The λ-continuum of inductive methods was derived from an assumption, called λ-condition, which says that the probability of finding an individual having property $x_{j}$ depends only on the number of observed individuals having property $x_{j}$ and on the total number of observed individuals. So, according to that assumption, all individuals with properties which are different from $x_{j}$ have equal weight with respect to that probability and, in particular, it does not matter whether any individual was observed having some property similar (...) to $x_{j}$ (the most complete proof of this result is presented in Carnap, 1980). The problem thus remained open to find some general condition, weaker than the λ-condition, which would allow for the derivation of probability functions which might be sensitive to similarity. Carnap himself suggested a weakening of the λ-condition which might allow for similarity sensitive probability functions (Carnap, 1980, p. 45) but he did not find the family of probability functions derivable from that principle. The aim of this paper is to present the family of probability functions derivable from Carnap's suggestion and to show how it is derived. In Section 1 the general problem of analogy by similarity in inductive logic is presented, Section 2 outlines the notation and the conceptual background involved in the proof, Section 3 gives the proof, Section 4 discusses Carnap's principle and the result, Section 5 is a brief review of the solutions which have previously been proposed. (shrink)

In this thesis I defend and pursue that line about the foundations of probability theory which has come to be known as "the logicist view about probability", and, in particular, the shape which it took in Carnap's Inductive Logic. ;Most philosophers who now deal with probability theory claim that Carnap's program of Inductive Logic has failed. The main aim of my thesis is to show that this judgment is based on a fundamental misunderstanding about the nature and the aim of (...) inductive logic. To that end, I follow, in chapter 1, the events in the history of probability, logic and mathematics which led to choosing certain requirements as the requirements which any theory about the foundations of probability must fulfill in order to be acceptable. ;In chapter 2 I explain how Carnap's inductive logic fulfills those requirements and how the method it uses to give foundations to probability theory can also be used to solve another major problem about probability, namely, the problem of statistical inference. ;Chapter 3 shows how the ideas explained in chapter 2 were developed, formally, by Carnap, and how Carnap's program can be continued in a particular case, that of reasoning by analogy. ;Finally, in chapter 4, I attempt to show that subjectivism about probability theory, the rival theory to Carnap's logicism, does not succeed in meeting the most basic requirement for the acceptability of any theory about the foundations of probability. (shrink)

This paper analyses and compares some of the automated reasoners that have been used in recent research for compliance checking. Although the list of the considered reasoners is not exhaustive, we believe that our analysis is representative enough to take stock of the current state of the art in the topic. We are interested here in formalizations at the _first-order_ level. Past literature on normative reasoning mostly focuses on the _propositional_ level. However, the propositional level is of little usefulness for (...) concrete LegalTech applications, in which compliance checking must be enforced on (large) sets of individuals. Furthermore, we are interested in technologies that are _freely available_ and that can be further investigated and compared by the scientific community. In other words, this paper does not consider technologies only employed in industry and/or whose source code is non-accessible. This paper formalizes a selected use case in the considered reasoners and compares the implementations, also in terms of simulations with respect to shared synthetic datasets. The comparison will highlight that lot of further research still needs to be done to integrate the benefits featured by the different reasoners into a single standardized first-order framework, suitable for LegalTech applications. All source codes are freely available at https://github.com/liviorobaldo/compliancecheckers, together with instructions to locally reproduce the simulations. (shrink)

Recent literature emphasizes an altered perception of the body in subjects practicing sport to pursue an ideal body image. Sometimes such bodily misperception is associated with the inability to re...

Infant-directed speech, the particular form of spontaneous language observed in interactions between parents and their infants, is a crucial aspect of the mother-infant interaction and an index of the attunement of maternal linguistic input to her infant communicative abilities and needs during dyadic interactions. The present study aimed to explore linguistic and pragmatic features of IDS during mother-infant interactions at 3-month of infant age. The effects of infant, maternal and dyadic factors on IDS were explored. Results evidenced few differences between (...) the groups on IDS linguistic characteristics. Moreover, observing the interaction of birth status and dyadic co-regulation, full-term mothers varied their IDS pragmatic features according to the quality of co-regulation while preterm mothers did not. Parenting stress was associated to specific linguistic IDS features independently from the birth status. Findings are discussed underling implications for the study of preterm dyads interactions and the importance to consider the interplay of several factors in affecting the quality of IDS. (shrink)

Applying value relevance analysis to a sample of European banks, we test the following: (i) the direct effects of the sustainability report on stock price; (ii) whether the report modifies the value relevance of financial accounting variables (indirect effects); and (iii) whether the value relevance of sustainability reports varies across countries. Results show that investors appreciate the additional and complementary disclosure provided by the sustainability report and that this disclosure produces a positive effect on stock prices. Estimates of the indirect (...) effects demonstrate that it has a negative influence on book value per share, whereas the effect on earnings per share is not significant. Cross-country analysis shows that the value relevance of the sustainability report varies across European countries, consistent with the hypothesis that the value relevance of the sustainability report is likely to be influenced by different institutional contexts. (shrink)

The negative impact of the COVID-19 pandemic on individuals’ psychosocial functioning was widely attested during the last year. However, the extent to which individual differences are associated with adaptive and maladaptive outcomes during quarantine in Italy remains largely unexplored. Using a person-oriented approach, the present study explored the association of personality profiles, based on three broad individual dispositions and two self-efficacy beliefs in the emotional area, with adaptive and maladaptive outcomes during the first Italian lockdown. In doing so, we focused (...) also on how different age groups differently faced the pandemic. The study was conducted through an online survey from May to June 2020 and included 1341 participants living in Italy, divided into two groups: 737 young adults aged 18–35 and 604 adults aged 36–60 years old. Latent Profile Analysis identified three personality profiles: resilient, vulnerable, and moderate. A subsequent path analysis model showed that the resilient profile was positively associated with prosocial behavior as an indicator of adaptive outcome, and negatively associated with three maladaptive outcomes: interpersonal aggression, depressive symptoms, and anxiety problems. Contrarily, the vulnerable profile resulted negatively associated with prosocial behavior and positively associated with the three maladaptive outcomes. Finally, regarding age group differences, young adults belonging to the vulnerable profile showed a greater association especially with interpersonal aggression, depression, and anxiety problems, as compared to adults belonging to the same profile. Overall, the results of the present study highlighted the importance to analyze individual functioning during an isolation period by using a person-oriented approach. Findings evidenced the existence of three different profiles and subsequent path analysis revealed, especially for the vulnerable profile and young adults, a greater maladaptive consequence of the quarantine. The practical implications will be discussed. (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)

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last (...) century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality. (shrink)

The view that the subject matter of epistemology is the concept of knowledge is faced with the problem that all attempts so far to define that concept are subject to counterexamples. As an alternative, this article argues that the subject matter of epistemology is knowledge itself rather than the concept of knowledge. Moreover, knowledge is not merely a state of mind but rather a certain kind of response to the environment that is essential for survival. In this perspective, the article (...) outlines an answer to four basic questions about knowledge: What is the role of knowledge in human life? What is the relation between knowledge and reality? How is knowledge acquired? Is there any a priori knowledge? (shrink)

In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by ‘enlightening’. This paper proposes an alternative approach, according to which a mathematical demonstration or theorem is beautiful when it provides understanding. Mathematical beauty thus considered can have a role in mathematical discovery because it can guide the mathematician in selecting which hypothesis to (...) consider and which to disregard. Thus aesthetic factors can have an epistemic role qua aesthetic factors in mathematical research. (shrink)

From antiquity several philosophers have claimed that the goal of natural science is truth. In particular, this is a basic tenet of contemporary scientific realism. However, all concepts of truth that have been put forward are inadequate to modern science because they do not provide a criterion of truth. This means that we will generally be unable to recognize a scientific truth when we reach it. As an alternative, this paper argues that the goal of natural science is plausibility and (...) considers some characters of plausibility. (shrink)

Concetta Pennuto, « Pulsations du corps en médecine », Histoire, médecine et santé, 11, été 2017, p. 55-76. Cet article propose une illustration de la manière dont la musique a fourni, par le biais de ses harmonies et son langage, un outil aux médecins pour comprendre et maîtriser les pulsations du coeur. Après une courte exploration de l'acte de prendre le pouls dans l'Antiquité et au Moyen Âge, l'étude présente des exemples de médecins modernes qui utilisent non seulement les (...) moyens de l'harmonie - Médecine – Nouvel article. (shrink)

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some (...) general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach. (shrink)

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the (...) effectiveness of mathematics in natural science. (shrink)

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...) role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

Although in the past three decades interest in mathematical explanation revived, recent literature on the subject seems to neglect the strict connection between explanation and discovery. In this paper I sketch an alternative approach that takes such connection into account. My approach is a revised version of one originally considered by Descartes. The main difference is that my approach is in terms of the analytic method, which is a method of discovery prior to axiomatized mathematics, whereas Descartes’s approach is in (...) terms of the analytic–synthetic method, which is a heuristic pattern in already axiomatized mathematics. (shrink)

The universal generalization problem is the question: What entitles one to conclude that a property established for an individual object holds for any individual object in the domain? This amounts to the question: Why is the rule of universal generalization justified? In the modern and contemporary age Descartes, Locke, Berkeley, Hume, Kant, Mill, Gentzen gave alternative solutions of the universal generalization problem. In this paper I consider Locke’s, Berkeley’s and Gentzen’s solutions and argue that they are problematic. Then I consider (...) an alternative formulation of universal generalization which depends on the view that mathematical objects are individual objects and are hypotheses introduced to solve mathematical problems, and that mathematical proofs are argument schemata. I argue that this alternative formulation allows one to overcome the problems of Locke’s, Berkeley’s and Gentzen’s solutions, and is related to the approach to generality in Greek mathematics. I also argue that there is a connection between the present formulation of universal generalization and a special form of the analogy rule which is implicit in Proclus’ approach to the universal generalization problem. (shrink)

This volume deals with the rapports among the extant Greek commentaries on Aristotle's _Metaphysics_: Alexander of Aphrodisias, Syrianus, Asclepius, Ps. Alexander. It traces the precise map of these texts and provides a starting point for any future research in the field.

Can philosophy still be fruitful, and what kind of philosophy can be such? In particular, what kind of philosophy can be legitimized in the face of sciences? The aim of this paper is to answer these questions, listing the characteristics philosophy should have to be fruitful and legitimized in the face of sciences. Since the characteristics in question demand that philosophy search for new knowledge and new rules of discovery, a philosophy with such characteristics may be called the ‘heuristic view’. (...) According to the heuristic view, philosophy is an inquiry into the world which is continuous with the sciences. It differs from them only because it deals with questions which are beyond the present sciences, and in order to deal with them must try unexplored routes. By so doing, when successful, it may even give birth to new sciences. In listing the characteristics that philosophy should have, the paper systematically compares them with classical analytic philosophy, because the latter has been the dominant philosophical tradition in the last century. (shrink)

This volume is a collection of papers on philosophy of mathematics which deal with a series of questions quite different from those which occupied the minds of the proponents of the three classic schools: logicism, formalism, and intuitionism. The questions of the volume are not to do with justification in the traditional sense, but with a variety of other topics. Some are concerned with discovery and the growth of mathematics. How does the semantics of mathematics change as the subject develops? (...) What heuristics are involved in mathematical discovery, and do such heuristics constitute a logic of mathematical discovery? What new problems have been introduced by the development of mathematics since the 1930s? Other questions are concerned with the applications of mathematics both to physics and to the new field of computer science. Then there is the new question of whether the axiomatic method is really so essential to mathematics as is often supposed, and the question, which goes back to Wittgenstein, of the sense in which mathematical proofs are compelling. Taking these questions together they give part of an emerging agenda which is likely to carry philosophy of mathematics forward into the twenty first century. (shrink)

Although there have never been so many professional philosophers as today, most of the questions discussed by today’s philosophers are of no interest to cultured people at large. Specifically, several scientists have maintained that philosophy has become an irrelevant subject. Thus philosophy is at a crossroads: either to continue on the present line, which relegates it into irrelevance, or to analyse the reasons of the irrelevance and seek an escape. This paper is an attempt to explore the second alternative.

The aim of this article is to show that intuition plays no role in mathematics. That intuition plays a role in mathematics is mainly associated to the view that the method of mathematics is the axiomatic method. It is assumed that axioms are directly (Gödel) or indirectly (Hilbert) justified by intuition. This article argues that all attempts to justify axioms in terms of intuition fail. As an alternative, the article supports the view that the method of mathematics is the analytic (...) method, a method originally used by the mathematician Hippocrates of Chios and the physician Hippocrates of Cos, and first explicitly formulated by Plato. The article examines the main features of the analytic method, and argues that, while intuition plays an essential role in the axiomatic method, it plays no role in the analytic method, either in the discovery or in the justification of hypotheses. That the method of mathematics is the analytic method involves that mathematical knowledge is not absolutely certain but only plausible, but the article argues that this is the best we can achieve. (shrink)

In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past” (Poincaré (...) 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks to the view that “science only comes to fruition in a system” because “only through a system can we achieve complete clarity and order” (Frege 1979, p. 242). He even goes so far as saying that “no science can be so enveloped in obscurity as mathematics, if it fails to construct a system” (Frege 1979, p. 242). By ‘system’ Frege means ‘axiomatic system’. In his view, in mathematics we cannot rest content with the fact that “we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction”, that is, to find “what the primitive truths are”, because “only in this way can a system be constructed” (Frege 1979, p. 205). The primitive truths are the principles of the axiomatic system. Frege’s stress on the role of systems also determines his views on the growth of mathematical knowledge.. (shrink)

Reuben Hersh is a champion of maverick philosophy of mathematics. He maintains that mathematics is a human activity, intelligible only in a social context; it is the subject where statements are capable in principle of being proved or disproved, and where proof or disproof bring unanimous agreement by all qualified experts; mathematicians' proof is deduction from established mathematics; mathematical objects exist only in the shared consciousness of human beings. In this paper I describe my several points of agreement and few (...) points of disagreement with Hersh's views. (shrink)

ellucci, C., Existential instantiation and normalization in sequent natural deduction, Annals of Pure and Applied Logic 58 111–148. A sequent conclusion natural deduction system is introduced in which classical logic is treated per se, not as a special case of intuitionistic logic. The system includes an existential instantiation rule and involves restrictions on the discharge rules. Contrary to the standard formula conclusion natural deduction systems for classical logic, its normal derivations satisfy both the subformula property and the separation property and (...) allow to establish a version of the midsequent theorem and Herbrand's theorem. (shrink)

While Gödel's (first) incompleteness theorem has been used to refute the main contentions of Hilbert's program, it does not seem to have been generally used to stress that a basic ingredient of that program, the concept of formal system as a closed system - as well as the underlying view, embodied in the axiomatic method, that mathematical theories are deductions from first principles must be abandoned. Indeed the logical community has generally failed to learn Gödel's lesson that Hilbert's concept of (...) formal system as a closed system is inadequate and continues to use it as if there were no incompleteness theorem. In this paper I will stress the role of Gödel's incompleteness theorem in showing the inadequacy of such a concept of formal system and the need for a more articulated view of mathematical theories. More generally I will argue that Gödel's result entails that, as an alternative to mathematical logic, a new concept of logic is required: logic as the theory of communicating inference processes. (shrink)

According to Bernard Williams, philosophy is a humanistic discipline essentially different from the sciences. While the sciences describe the world as it is in itself, independent of perspective, philosophy tries to make sense of ourselves and of our activities. Only the humanistic disciplines, in particular philosophy, can do this, the sciences have nothing to say about it. In this note I point out some limitations of Williams’ view and outline an alternative view.

Three decades ago Laudan posed the challenge: Why should the logic of discovery be revived? This paper tries to answer this question arguing that the logic of discovery should be revived, on the one hand, because, by Gödel’s second incompleteness theorem, mathematical logic fails to be the logic of justification, and only reviving the logic of discovery logic may continue to have an important role. On the other hand, scientists use heuristic tools in their work, and it may be useful (...) to study such tools systematically in order to improve current heuristic tools or to develop new ones. As a step towards reviving the logic of discovery, the paper follows Aristotle in asserting that logic must be a tool for the method of science, and outlines an approach to the logic of discovery based on the analytic method and on ampliative inference rules. (shrink)

onl y to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community and so seem worth discussing anyhow. Such claims will be used as reference to make some remarks about the present state of relations between mathematical logic and the philosophy of mathematics.

This article examines the current justifications of deductive inferences, and finds them wanting. It argues that this depends on the fact that all such justification take no account of the role deductive inferences play in knowledge. Alternatively, the article argues that a justification of deductive inferences may be given in terms of the fact that they are non-ampliative, in the sense that the content of the conclusion is merely a reformulation of the content of the premises. Some possible objections to (...) this view are discussed and found inadequate. (shrink)

According to a view going back to Plato, the aim of philosophy is to acquire knowledge and there is a method to acquire knowledge, namely a method of discovery. In the last century, however, this view has been completely abandoned, the attempt to give a rational account of discovery has been given up, and logic has been disconnected from discovery. This paper outlines a way of reconnecting logic with discovery.

It is generally agreed that for blame to be appropriate the wrongdoer must be blameworthy. However, blameworthiness is not sufficient for appropriate blame. It has been argued that for blame to be appropriate the blamer must have standing to blame. Philosophers writing on the topic have distinguished several considerations that might defeat someone’s standing to blame. This paper examines the underexplored consideration of how personal relationships can influence who has the standing to express blame. We seem to assume that if (...) we do not stand in the right relation to the wrongdoer, it is not our business to blame them. I identify three challenges to this view. First, we do not know what is wrong with meddling (whether it is morally or prudentially wrong). Second, there are cases where we have no close relation to the wrongdoer, but where we seem to have standing nonetheless – murder and other serious moral offences are examples of this. Third, we don’t seem to have a clear conception of what ‘standing’ means, and this makes it hard to see how discussion on standing can help us determine the propriety of blame. This paper defends the notion of standing as a propriety condition on blame in cases of meddling. I argue that personal relationships influence who has the standing to blame through constitution of relationship-specific norms; and that the role these norms play in developing and regulating our relationships can explain why meddlesome blame is morally wrong. (shrink)

The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. (...) One of such ideas is Boole’s view that logic is the study of the laws of thought. This is not to be meant in a psychologistic way. Frege himself states that the task of logic can be represented “as the investigation of the mind; [though] of the mind, not of minds” [17, p. 369]. Moreover Frege never charges Boole with being psychologistic and in a letter to Peano even distinguishes between the followers of Boole and “the psychological logicians” [16, p. 108]. In fact for Boole the laws of thought which are the object of logic belong “to the domain of what is termed necessary truth” [2, p. 404]. For him logic does not depend on psychology, on the contrary psychology depends on logic insofar as it is only through an investigation of logical operations that we could obtain “some probable intimations concerning the nature and constitution of the human mind” [2, p. 1]. Logic is normative, not descriptive. For, the laws of thought do not “manifest their presence otherwise than by merely prescribing the conditions of formal inference” [2, p. 419]. They are, “properly speaking, the laws of right reasoning only” [2, p. 408]. So they “form but a part of the system of laws by which the actual processes of reasoning, whether right or wrong, are governed” [2, p. 409]. Boole’s idea that logic is the study of the laws of thought was taken over by Hilbert. According to him logic is “a discipline which expresses the structure of all our thought” [31, p.. (shrink)

This article examines Quine's original proposal for a natural deduction calculus including an existential specification rule, it argues that it introduces a new paradigm of natural deduction alternative to Gentzen's but has some substantial defects. As an alternative the article puts forward a system of sequent natural deduction.

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, (...) notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways. (shrink)

Résumé Au cours de son histoire, la Commission Léonine a élaboré une méthode ecdotique de plus en plus complexe et raffinée. L’’édition de saint Thomas constitue, en effet, dans son ensemble, un cas unique dans l’histoire de la philologie moderne, car chaque étape a comporté l’acquisition de notions fondamentales pour tout éditeur de textes anciens et médiévaux. L’analyse minutieuse de chaque volume montre, contre toute idée reçue, que des principes de critique textuelle tout à fait corrects sont déjà à l’œuvre (...) dans les premières éditions (1882-1886) et que, dans la polémique qui opposa, à propos de l’’édition de la I a Pars, Cl. Baeumker et les éditeurs de la Léonine, ce sont ces derniers qui défendent le point de vue philologiquement le plus rigoureux. La découverte du phénomène de la pecia à l’occasion de l’’édition du Supplementum (1906) marque un tournant décisif dans l’’élaboration de la technique d’édition en donnant naissance à la philologie des textes universitaires, qui, portée à sa perfection dans les derniers volumes, constitue un des résultats les plus importants dans l’histoire de la philologie. (shrink)

Résumé L’œuvre philologique du Père Louis Jacques Bataillon (1914-2009) a profondément marqué les études médiévales en général et la philologie en particulier. L’élaboration de notions-clefs telles que celle d’original, la grande variété de traditions textuelles étudiées, la maîtrise hors pair des techniques d’édition élaborées dans le cadre de l’Édition Léonine de Saint Thomas d’Aquin, ainsi que la connaissance exceptionnelle des fonds manuscrits les plus importants d’Europe, font de l’œuvre du Père Bataillon une référence incontournable pour tout philologue.