This article puts forward the notion of “evolving diagram” as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter “diagrams*”) in the context of “sketch theory,” a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, (...) that it helps to overcome the rivalry between set theory and category theory as a general semantical framework, and that it suggests a more flexible understanding of the opposition between formal proofs and diagrammatic reasoning. Thus, the aim of the paper is twofold. First, it claims that diagrams* provide a clear example of evolving diagrams, and shed light on them as a general phenomenon. Second, in return, it uses sketches, understood as evolving diagrams, to show how diagrams* in general should be re-evaluated positively. (shrink)

The purpose of this paper is to generalize Kripke semantics for propositional modal logic by geometrizing it, that is, by considering the space underlying the collection of all possible worlds as an important semantic feature in its own right, so as to take the idea of accessibility seriously. The resulting new modal semantics is worked out in a setting coming from Riemannian geometry, where Kripke semantics is shown to correspond to a particular case, namely, the discrete one. Several correspondence results, (...) established between variants of well-known modal systems and corresponding geometric properties, illustrate the import of this new framework. (shrink)

In the Tractatus, Wittgenstein describes the general validity of logical truths as being “essential,” as opposed to merely “accidental” general truths. He does not say much more, and little has been said about it by commentators. How to make sense of the essential general validity by which Wittgenstein characterizes logic? This chapter aims to clarify this crucial concept.

English summary: Is universal truth fully necessary? Are all necessary truths universal? From Kant to Tarski, the philosophical tradition as a whole has accepted the existence of a correlation between these two properties. The purpose of this book is to question this assumption and challenge the belief that universality and necessity are basic components of truth. French text. French description: Toute verite universelle est-elle necessaire? Toute verite necessaire est-elle universelle? De Kant a Tarski, la tradition philosophique a dans son ensemble (...) admis, sans la discuter, l'existence d'une correlation entre ces deux proprietes. L'objet de ce livre est de remettre en question ce presuppose et de contester que l'universalite et la necessite soient des proprietes equivalentes de verites, et meme qu'elles soient des proprietes de verites. Correles, l'universel et le necessaire renvoient chacun a une totalite de possibles dont l'origine est nexpliquee. Penses separement, ils peuvent au contraire etre compris dans leur genese. L'universel s'eclaire a la lumiere de l'idee de genericite, qui prevaut sur celle de totalite; le necessaire se definit, non plus par le fait d'etre vrai dans tous les cas possibles, mais en reference a un possible qui apparait n'etre ni totalisable, ni reductible a un seul et meme plan. (shrink)

L’accointance d’un attribut est-elle de même nature que celle d’un individu? Bien entendu, l’assimilation des attributs à des universaux, et par suite à des objets, conduit immédiatement à une réponse positive. Mais que dire des attributs en position de prédicat? Le présent article vise à soutenir l’univocité de l’accointance en montrant que les prédicats sont bien, malgré d’importantes différences, les objets possibles d’une accointance comparable à l’accointance d’individus. Pour cela, on envisagera les choses négativement, en examinant la façon dont, dans (...) l’un et l’autre cas, une absence d’accointance se trouve suppléée. Dans le cadre d’une interprétation épistémique de la théorie des types des Principia Mathematica justifiée par des raisons indépendantes, cette perspective amènera à comprendre l’axiome de réductibilité comme un principe d’accointance par procuration pour les prédicats, principe dont l’analyse russellienne des descriptions définies d’individu partage le schéma. (shrink)

Abstract:My aim is to examine logical types in Principia Mathematica from two (partly independent) perspectives. The first one pertains to the ambiguity of the notion of logical type as introduced in the Introduction (to the first edition). I claim that a distinction has to be made between types as called for in the context of paradoxes, and types as logical prototypes. The second perspective bears on typical ambiguity as described in Russell and Whitehead’s “Prefatory Statement of Symbolic Conventions”, inasmuch as (...) it lends itself to a comparison with specific systems of modern typed lambda-calculus. In particular, a recent paper shows that the theory of logical types can be formalized in the way of a lambda-calculus. This opens the way to an interesting reconciliation between type theories in the Russellian sense of the word, and type theories in the modern sense. But typical ambiguity is not taken into account in the paper. I would like to take up that question of typical ambiguity, by extending the typed lambda-calculus to be used. (shrink)

Drawing on the analogy between any unary first-order quantifier and a "face operator," this paper establishes several connections between model theory and homotopy theory. The concept of simplicial set is brought into play to describe the formulae of any first-order language L, the definable subsets of any L-structure, as well as the type spaces of any theory expressed in L. An adjunction result is then proved between the category of o-minimal structures and a subcategory of the category of linearly ordered (...) simplicial sets with distinguished vertices. (shrink)

Logical universalism, a label that has been pinned on to Frege, involves the conflation of two features commonly ascribed to logic: universality and radicality. Logical universality consists in logic being about absolutely everything. Logical radicality, on the other hand, corresponds to there being the one and the same logic that any reasoning must comply with. The first part of this paper quickly remarks that Frege’s conception of logic makes logical universality prevail and does not preclude the admission of different contexts (...) of discourse. The paper then aims to make it clear how Frege’s universalism can make sense of contextuality. Drawing on a suggestion made by Frege in his discussion of Hilbert, it shows that a properly Fregean notion of model can be devised. Taking up a suggestion from Wilfrid Hodges and William Demopoulos that the non-logical constants of a formal language can be compared to indexicals, this paper shows,paceHodges and Demopoulos, that such an understanding of non-logical constants is not beyond Frege’s horizon. A formal framework, based on the modern tool of fibrations, is set out to explain and justify this point. This framework allows one to compare Frege and Tarski, by formalizing Frege’s suggestion and by presenting Tarski’s semantics in a generalized setting. (shrink)

Une théorie de la science qui permette une « étude systématique des formes » est l’ἐπιστήμη ζητοῦμένη de Cavaillès : ce dernier ne la trouve ni dans l’analytique kantienne, ni dans la théorie de la science de Bolzano ; pas plus que chez Frege, Carnap ou Tarski ; et ni dans la théorie de la démonstration, ni dans la mathesis husserlienne. Cet article défend l’idée qu’il aurait pu la trouver, ou du moins s’en approcher, s’il avait accordé davantage d’attention à (...) une figure de la logique qu’il a délaissée : celle qu’incarnent Russell et son héritier Wittgenstein. L’analyse de la notion même de forme, que celle-ci désigne un complexe symbolique, une structure mathématique ou un schéma démonstratif, y aurait également gagné. (shrink)

Kreisel’s set-theoretic problem is the problem as to whether any logical consequence of ZFC is ensured to be true. Kreisel and Boolos both proposed an answer, taking truth to mean truth in the background set-theoretic universe. This article advocates another answer, which lies at the level of models of set theory, so that truth remains the usual semantic notion. The article is divided into three parts. It first analyzes Kreisel’s set-theoretic problem and proposes one way in which any model of (...) set theory can be compared to a background universe and shown to contain internal models. It then defines logical consequence with respect to a model of ZFC, solves the model-scaled version of Kreisel’s set-theoretic problem, and presents various further results bearing on internal models. Finally, internal models are presented as accessible worlds, leading to an internal modal logic in which internal reflection corresponds to modal reflexivity, and resplendency corresponds to modal axiom 4. (shrink)

Ce texte est tiré d’un exposé élémentaire, destiné à des étudiants agrégatifs, d’introduction aux rapports entre la logique (entendue au sens large) et la question du monde. Son objet est de décrire la place variable que cette question occupe en logique.

Algebraic Set Theory, a reconsideration of Zermelo-Fraenkel set theory in category-theoretic terms, has been built up in the mid-nineties by André Joyal and Ieke Moerdijk. Since then, it has developed into a whole research program. This paper gets back to the original formulation by Joyal and Moerdijk, and more specifically to its first three axioms. It explains in detail that these axioms set up a framework directly linked to descent theory, a theory having to do with the shift from local (...) data to a global item in modern algebraic geometry. Fibered categories, introduced by Grothendieck, provide a powerful framework for descent theory: They formalize in a very general way the consideration of local data of different kinds over the objects of some base category. The paper shows that fibered categories fit Joyal and Moerdijk’s axiomatization in a natural way, since the latter actually aims to secure a descent condition. As a result, Algebraic Set Theory is shown to accomplish, not only an original and fruitful combination of set theory with category theory, but the genuine graft of a deeply geometric idea onto the usual setting of zfc. (shrink)

This paper pursues two goals. Its first goal is to clear up the “identity problem” faced by the structuralist interpretation of mathematics. Its second goal, through the consideration of examples coming in particular from the theory of permutations, is to examine cases of misunderstandings in mathematics fit to cast some light on mathematical understanding in general. The common thread shared by these two goals is the notion of setting. The study of a mathematical object almost always goes together with the (...) choice of a particular setting, and the understanding of the workings of mathematical settings is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings, as features distinct from both mathematical structures and the systems which instantiate those structures, allows one to classify most of understandable misunderstandings in mathematics, and also to solve the identity problem. (shrink)

Intervertir a et b, c’est mettre b à la place de a et a à la place de b. Dans le cas où a occupait la première place et b la deuxième, c’est mettre b à la première place et a à la deuxième. Dans le cas où, de plus, a est le chiffre ‘1’ et b le chiffre ‘2’, c’est mettre 2 à la place n˚ 1 et 1 à la place n˚ 2. La théorie des substitutions combine ainsi (...) naturellement chiffres et numéros de place. Cet article montre que la confusion de ces deux éléments constitue un cas exemplaire de mécompréhension en mathématiques. Positivement, reconnaître à leur juste valeur toutes les numérotations ou paramétrisations qui interviennent en mathématiques, c’est reconnaître que les mathématiques ont rarement affaire à des structures sans l’intermédiaire essentiel de repères permettant de « fixer les idées ». L’article montre en particulier qu’une description structuraliste satisfaisante de l’objectivité mathématique ne peut faire l’économie de tels repères.To swap a and b is to put b in a’s place and a in b’s place. In case a was at the first place and b at the second, it is to put b at the first place and a at the second. In case, moreover, a is the numeral ‘1’ and b is the numeral ‘2’, it is to put 2 at the place no. 1 and 1 at the place no. 2. Substitution theory is thus led to combine numerals and numbers. This paper shows that many cases of misunderstanding in mathematics can be compared to a confusion between numerals and numbers. On a positive side, it shows how important are all the numberings or parametrizations that turn out to be pervasive throughout mathematics, as devices to « set ideas ». A mathematical structure cannot in general be reached without the medium of such points of reference. The paper shows in particular that any consistent structuralist explanation of mathematical objectivity cannot be sustained without giving them full attention. (shrink)

Drawing on Russell's substitutional theory, this paper examines the notion of ‘structured variable’, in order to compare Russell's and Tarski's conceptions of variables. The framework of syntactic fibrations, coming from categorical logic, is used as a common setting. The main objective of this paper is to make sense of the notion of structured variable beyond the context of Russell's theory, to question the Tarskian way of understanding what it is to be a possible value for a variable, and to bring (...) out semantically structured variables as a renewed and fruitful way of conceiving of logical structure. (shrink)

In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ‘no loss of generality’ problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53–54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. (...) The purpose of this article is to clarify Russell's claim and to solve the ‘no loss of generality’ problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (‘ramif-types’). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other hand, makes it possible to give us a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the ‘no loss of generality’ problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed. (shrink)

Dire d’une proposition que, nécessairement, elle est nécessairement vraie, c’est affirmer incomparablement plus que ce que l’on affirme en disant simplement qu’elle est nécessairement vraie. C’est en effet, intuitivement, affirmer qu’elle est nécessaire quelle que puisse être la donnée de tous les mondes possibles à l’aune de laquelle sa nécessité est établie. C’est faire de cette don­née elle-même un possible parmi d’autres, et faire ainsi référence à des mondes possibles d’ordre supérieur. Cet article vise à formaliser la notion de monde (...) possible d’ordre supérieur au moyen d’outils empruntés à la géométrie riemannienne. Le cadre sémantique proposé repose sur une collection ouverte de mondes possibles de niveaux croissants, par opposition à l’héritage leibnizien d’une totalité close de mondes possibles. (shrink)

Dire d’une proposition que, nécessairement, elle est nécessairement vraie, c’est affirmer incomparablement plus que ce que l’on affirme en disant simplement qu’elle est nécessairement vraie. C’est en effet, intuitivement, affirmer qu’elle est nécessaire quelle que puisse être la donnée de tous les mondes possibles à l’aune de laquelle sa nécessité est établie. C’est faire de cette don­née elle-même un possible parmi d’autres, et faire ainsi référence à des mondes possibles d’ordre supérieur. Cet article vise à formaliser la notion de monde (...) possible d’ordre supérieur au moyen d’outils empruntés à la géométrie riemannienne. Le cadre sémantique proposé repose sur une collection ouverte de mondes possibles de niveaux croissants, par opposition à l’héritage leibnizien d’une totalité close de mondes possibles. (shrink)

Dire d’une proposition que, nécessairement, elle est nécessairement vraie, c’est affirmer incomparablement plus que ce que l’on affirme en disant simplement qu’elle est nécessairement vraie. C’est en effet, intuitivement, affirmer qu’elle est nécessaire quelle que puisse être la donnée de tous les mondes possibles à l’aune de laquelle sa nécessité est établie. C’est faire de cette don­née elle-même un possible parmi d’autres, et faire ainsi référence à des mondes possibles d’ordre supérieur. Cet article vise à formaliser la notion de monde (...) possible d’ordre supérieur au moyen d’outils empruntés à la géométrie riemannienne. Le cadre sémantique proposé repose sur une collection ouverte de mondes possibles de niveaux croissants, par opposition à l’héritage leibnizien d’une totalité close de mondes possibles. (shrink)

Kant, dans l’Introduction de la seconde édition de la Critique de la raison pure, reconnaît l’universalité et la nécessité comme deux critères extensionnellement équivalents de l’a priori :Nécessité et rigoureuse universalité sont […] des critères sûrs d’une connaissance a priori et renvoient en outre, inséparablement, l’une à l’autre.Le terme de « critère » signifie dans ce passage qu’universalité...