Denis Bonnay

Philosophy of mathematics deals both with ontological issues (what is it that mathematics studies?) and epistemological issues (how is mathematical knowledge possible?). This chapter reviews the main answers given to these two sets of issues, stressing how interrelated they are. It starts from the classical opposition between empiricist, rational, and critical approaches to set the sage and poses the question of mathematics’ relationship with experience as well as the one of the respective roles of intuition and logical principles. A detailed account of two anti-realist programs (finitism and intuitionism) is provided. Arguments in favor of realism are presented, and distinct realist views are distinguished. Having confronted the epistemological difficulties of various realist views, the last part of the chapter deals with naturalist perspectives and mathematical structuralism.

In this chapter, I argue in favor of a new approach to collective beliefs in unorganized groups, in terms of doxastic clustering. When a group does not have dedicated mechanisms for production of collective beliefs, and when individual beliefs of members of the group are diverse, it does not make much sense to attribute to the group some average beliefs or any other kind of collective beliefs produced by aggregating individual beliefs. Rather, beliefs are meaningfully attributed to coherent subgroups of individuals who share similar opinions. In this case, attribution of collective beliefs involves both clustering, that is partitioning the group into coherent doxastic units, and aggregation, that is aggregating individual opinions within coherent clusters. Adapting standard judgment aggregation theory, I propose a formal framework for doxastic clustering and provide an axiomatic characterization of majoritarian intra-cluster aggregation.

One of the major aims of science, it is commonly held, is to provide explanations. Philosophers of science have tried to understand what it is to provide a scientific explanation, what distinguishes good from bad explanations, and why explanations are valuable. This chapter goes through the main answers that have been elaborated in the last decades. It starts with a detailed discussion of the famous deductive-nomological (DN) model of explanation proposed by Hempel and Oppenheim. Then, the two main rivals to the DN model, the causal theory and the unificationist theory, are introduced and discussed. Some other contemporary approaches are sketched out in the closing section.

Philosophy of science studies the methods, theories, and concepts used by scientists. It mainly developed as a field in its own right during the twentieth century and is now a diversified and lively research area. This book surveys the current state of the discipline by focusing on central themes such as confirmation of scientific hypotheses, scientific explanation, causality, the relationship between science and metaphysics, scientific change, the relationship between philosophy of science and science studies, the role of theories and models, and unity of science. These themes define general philosophy of science. The book also presents subdisciplines in the philosophy of science dealing with the main sciences: logic, mathematics, physics, biology, medicine, cognitive science, linguistics, social sciences, and economics. Although it is common to address the specific philosophical problems raised by physics and biology in such a book, the place assigned to the philosophy of special sciences is much more unusual. Most authors collaborate on a regular basis in their research or teaching and share a common vision of philosophy of science and its place within philosophy and academia in general. The chapters have been written in close accordance with the three editors, thus achieving strong unity of style and tone.

We discuss two interpretations of two-dimensional semantics (2DMS) due to D. Chalmers and R. Stalnaker. The main problem with both interpretations of the formal framework is the relinquishng of rigidity for terms. They are in a sense unfaithful to an agent's beliefs. We present alternative principles to capture what we take to be agents's beliefs, namely: the principles of hyper-rigidity and backward reference to actuality. We propose then to go back to a one-dimensional semantics which affords a satisfactory model of beliefs reports. Our proposal, like 2DMS, addresses typical problems of representation for beliefs and epistemological difficulties related to modal knowledge.

Le Précis de philosophie des sciences vise à présenter, de manière pédagogique, l'état actuel des grandes questions et des grands domaines de la philosophie des sciences. C'est un ouvrage de niveau "intermédiaire", entre les ouvrages d'initiation et les ouvrages de recherche. Il peut être utilisé comme manuel pour des cours de philosophie des sciences au niveau Master, ainsi que dans le cadre de la préparation aux nouvelles épreuves d'épistémologie des CAPES scientifiques. Il a notamment pour vocation de servir de support de cours pour des enseignements en épistémologie et philosophie des sciences de niveau L3, M1 et M2.

Computer vision is one of AI’s most successful fields. In the last twenty years, machines have become increasingly good at extracting information from images and at identifying objects. But does this mean that machines really can see, or is computer vision just a fancy metaphor for object detection? This paper aims to provide a reasoned answer to the question. First, three criteria for vision attribution are reviewed and it is argued that a functionalist criterion, in terms of exploitable internal representations based on visual stimuli, fares better than behaviourist or phenomenological ones. The functionalist criterion is then applied to vision algorithms based on neural networks and it is argued that such machines can indeed see, insofar as neural networks are precisely trained to generate representations of the visual data they are fed with. Those representations present the specific traits associated with successful training: They are hierarchical, robust, and versatile. We argue that these properties may be used as further constraints on representational devices in general, helping to solve some of the classical issues faced by teleosemantic theories of representation such as that expounded by Dretske.

We take Carnap’s problem to be to what extent standard consequence relations in various formal languages fix the meaning of their logical vocabulary, alone or together with additional constraints on the form of the semantics. This paper studies Carnap’s problem for basic modal logic. Setting the stage, we show that neighborhood semantics is the most general form of compositional possible worlds semantics, and proceed to ask which standard modal logics (if any) constrain the box operator to be interpreted as in relational Kripke semantics. Except when restricted to finite domains, no modal logic characterizes all the Kripkean interpretation of P. Moreover, we show that, in contrast with the case of first-order logic, the obvious requirement of permutation invariance is not adequate in the modal case. After pointing out some known facts about modal logics that nevertheless force the Kripkean interpretation, we focus on another feature often taken to embody the gist of modal logic: locality. We show that invariance under point-generated subframes (properly defined) does single out the Kripkean interpretations, but only among topological interpretations, not in general. Finally, we define a notion of bisimulation invariance — another aspect of locality — that, together with a reasonable closure condition, gives the desired general result. Along the way, we propose a new perspective on normal neighborhood frames as filter frames, consisting of a set of worlds equipped with an accessibility relation, and a free filter at every world.

The thesis that truth is a logical notion has been stated repeatedly by deflationists in philosophical discussions on the nature of truth. However, to prove the point, one would need to show that the truth predicate does classify as logical according to some reasonable criterion of logicality. Following Tarski, invariance criteria have been considered to provide an adequate rendering of the generality and formality of logic. In this article, we show how the deflationist can use invariance criteria in support of her claim that deflationary truth is a logical notion.

Jeffrey conditioning tells an agent how to update her priors so as to grant a given probability to a particular event. Weighted averaging tells an agent how to update her priors on the basis of testimonial evidence, by changing to a weighted arithmetic mean of her priors and another agent’s priors. We show that, in their respective settings, these two seemingly so different updating rules are axiomatized by essentially the same invariance condition. As a by-product, this sheds new light on the question how weighted averaging should be extended to deal with cases when the other agent reveals only parts of her probability distribution. The combination of weighted averaging and Jeffrey conditioning is a comprehensive updating rule to deal with such cases, which is again axiomatized by invariance under embedding. We conclude that, even though one may dislike Jeffrey conditioning or weighted averaging, the two make a natural pair when a policy for partial testimonial evidence is needed.

The dual character of invariance under transformations and definability by some operations has been used in classical works by, for example, Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this article, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in L∞∞ so as to cover the cases that are of interest in the logicality debates, getting McGee’s theorem about quantifiers invariant under all permutations and definability in pure L∞∞ as a particular case. We also prove some optimality results along the way, regarding the kinds of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms.

What is a logical constant? In which terms should we characterize the meaning of logical words like “and”, “or”, “implies”? An attractive answer is: in terms of their inferential roles, i.e. in terms of the role they play in building inferences. More precisely, we favor an approach, going back to Dosen and Sambin, in which the inferential role of a logical constant is captured by a double line rule which introduces it as reflecting structural links (for example, multiplicative conjunction reflects comma on the right of the turnstyle). Rule-based characterizations of logical constants are subject to the well known objection of Prior’s fake connective, tonk. We show that some double line rules also give rise to such pseudo logical constants. But then, we are able to find a property of a double line rules which guarantee that it defines a genuine logical constant. Thus we provide an alternative answer to Belnap’s requirement of conservatity in terms of a local requirement on double line rules.

Since the ground-breaking contributions of M. Dummett (Dummett 1978), it is widely recognized that anti-realist principles have a critical impact on the choice of logic. Dummett argued that classical logic does not satisfy the requirements of such principles but that intuitionistic logic does. Some philosophers have adopted a more radical stance and argued for a more important departure from classical logic on the basis of similar intuitions. In particular, J. Dubucs and M. Marion (?) and (Dubucs 2002) have recently argued that a proper understanding of anti-realism should lead us to the so-called substructural logics (see (Restall 2000)) and especially linear logic. The aim of this paper is to scrutinize this proposal. We will raise two kinds of issues for the radical anti-realist. First, we will stress the fact that it is hard to live without structural rules. Second, we will argue that, from an anti- realist perspective, there is currently no satisfactory justi cation to the shift to substructural logics.

The problem of logical constants consists in finding a principled way to draw the line between those expressions of a language that are logical and those that are not. The criterion of invariance under permutation, attributed to Tarski, is probably the most common answer to this problem, at least within the semantic tradition. However, as the received view on the matter, it has recently come under heavy attack. Does this mean that the criterion should be amended, or maybe even that it should be abandoned? I shall review the different types of objections that have been made against invariance as a logicality criterion and distinguish between three kinds of objections, skeptical worries against the very relevance of such a demarcation, intensional warnings against the level at which the criterion operates, and extensional quarrels against the results that are obtained. I shall argue that the first two kinds of objections are at least partly misguided and that the third kind of objection calls for amendment rather than abandonment.

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation ⇒x. We investigate a function going in the other direction, thus extracting the constants of a given consequence relation, and we show that this function (a) retrieves the usual logical constants from the usual logical consequence relations, and (b) is an inverse to—more precisely, forms a Galois connection with—the Bolzano-Tarski function.

Consider any logical system, what is its natural repertoire of logical operations? This question has been raised in particular for first-order logic and its extensions with generalized quantifiers, and various characterizations in terms of semantic invariance have been proposed. In this paper, our main concern is with modal and dynamic logics. Drawing on previous work on invariance for first-order operations, we find an abstract connection between the kind of logical operations a system uses and the kind of invariance conditions the system respects. This analysis yields (a) a characterization of invariance and safety under bisimulation as natural conditions for logical operations in modal and dynamic logics, and (b) some new transfer results between first-order logic and modal logic.

Hintikka makes a distinction between two kinds of games: truthconstituting games and truth-seeking games. His well-known game-theoretical semantics for first-order classical logic and its independence-friendly extension belongs to the first class of games. In order to ground Hintikka’s claim that truth-constituting games are genuine verification and falsification games that make explicit the language games underlying the use of logical constants, it would be desirable to establish a substantial link between these two kinds of games. Adapting a result from Thierry Coquand, we propose such a link, based on a slight modification of Hintikka’s games, in which we allow backward playing for ∃loïse. In this new setting, it can be proven that sequent rules for first-order logic, including the cut rule, are admissible, in the sense that for each rule, there exists an algorithm which turns winning strategies for the premisses into a winning strategy for the conclusion. Thus, proofs, as results of truth-seeking games, can be seen

Hintikka and Sandu have developed IF logic as a genuine alternative to classical first-order logic : liberalizing dependence schemas between quantifiers, IF would carry out all the ideas already underlying classical logic. But they are alternatives to Hintikka’s game-theoretic approach; one could use instead Henkin quantifiers. We will present here some arguments of both technical and philosophical nature in favor of IF. We will show that its notion of independence, once extended to connectives, can indeed claim to be fully general, and that IF logic provides an analysis of independence patterns. This last point will be argued for thanks to an explanation of the epistemic content of IF, through a partial translation into modal logic.

La logique est un compagnon naturel de la philosophie. Qu'est-ce qu'un raisonnement correct? Qu'est-ce qu'une preuve? Peut-on definir le concept de verite? Que faire face aux paradoxes? Ces questions sont debattues par les philosophes depuis l'Antiquite; et la logique moderne, usant de langages formels, developpe une analyse rigoureuse de ces concepts les plus fondamentaux. Les onze textes classiques reunis ici proposent un retour reflexif sur cette discipline et sur la signification philosophique de ses achevements. Ils s'adressent a quiconque souhaite prendre la mesure des enjeux conceptuels de la logique, et a tous les etudiants desireux de completer leur apprentissage de la discipline par une reflexion epistemologique sur ses fondements.

Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidean. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson’s luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. We formulate a generalized version of the semantics, called token semantics, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resource-sensitive logic with implications for the representation of common knowledge in situations of bounded rationality.