It has been known for six years that the restriction of Girard's polymorphic system $\text{\bfseries\upshape F}$ to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each $\beta$-reduction step of the full intuitionistic propositional calculus translates into one or more $\beta\eta$-reduction steps in the restricted Girard system. As a consequence, we obtain (...) a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to $\eta$-conversions. (shrink)

We introduce a new typed combinatory calculus with a type constructor that, to each type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, associates the star type σ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*$$\end{document} of the nonempty finite subsets of elements of type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory (...) calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements. (shrink)

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex (...) and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions. (shrink)

It is known that there is a sound and faithful translation of the full intuitionistic propositional calculus into the atomic polymorphic system F at, a predicative calculus with only two connectives: the conditional and the second-order universal quantifier. The faithfulness of the embedding was established quite recently via a model-theoretic argument based in Kripke structures. In this paper we present a purely proof-theoretic proof of faithfulness. As an application, we give a purely proof-theoretic proof of the disjunction property of the (...) intuitionistic propositional logic in which commuting conversions are not needed. (shrink)

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is animpassable barrierin the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted inbut also what cannot be so (...) interpreted. We explain some of these results and rely on them to discuss Nelson's position. (shrink)

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but (...) also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson's position. (shrink)

We give an elementary proof of the strong normalization of the atomic polymorphic calculus Fat.

Suppose that it is possible to integrate real functions over a weak base theory related to polynomial time computability. Does it follow that we can count? The answer seems to be: obviously yes! We try to convince the reader that the severe restrictions on induction in feasible theories preclude a straightforward answer. Nevertheless, a more sophisticated reflection does indeed show that the answer is affirmative.

Leo Harrington showed that the second-order theory of arithmetic WKL 0 is ${\Pi^1_1}$ -conservative over the theory RCA 0. Harrington’s proof is model-theoretic, making use of a forcing argument. A purely proof-theoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cut-elimination argument.

The aim of this article is to offer a view of the current status of women in medical physics and biomedical engineering, while focusing on solutions towards gender balance and providing examples of current activities carried out at national and international levels. The International Union of Physical and Engineering Scientists in Medicine is committed to advancing women in science and health and has several initiatives overseen by the Women in Medical Physics and Biomedical Engineering Task Group. Some of the main (...) strategies proposed by the Task Group to attain gender balance are: (a) identify and promote female role models that achieve successful work-life balance, (b) establish programs to develop female leaders, (c) create opportunities for females to increase the international visibility within the scientific community, and (d) establish archives and databases of women in STEM. (shrink)

The first half of this article follows Mario Bunge’s early realist moves, his efforts to articulate the achievements of theoretical physics as gains in the quest for objective truth and understanding, particularly in the context of the fights against the idealist and subjectivist interpretations of quantum mechanics that, at least until the mid-1970s, prevailed in physics. Bunge’s answers to the problems of quantum mechanics provide a good angle for understanding how his realist positions grew on the “battlefield.” The second half (...) discusses Bunge’s general conception of the scientific realist stance and confronts it with some current approaches to realism in the mainstream literature. (shrink)

Les multiples histoires des idées qui retracent la constitution, relativement récente, de la discipline appelée «esthétique», mentionnent souvent le nom d�Alexander Gottlieb Baumgarten (1714-1762) en lui attribuant l�«invention» du néologisme «æsthetica». Très rapidement, ce terme latin, traduit en allemand, entre dans l�usage courant. De nos jours, il désigne un ensemble de discours souvent très hétérogènes sur les beaux-arts. Mais le sens originaire d�æsthetica, c�est-à-dire le contenu de l�ouvrage éponyme de 1750, demeure largement méconnu. Le présent article souhaite contribuer à souligner (...) l�importance du geste inaugural de Baumgarten, dont l�intérêt n�est pas seulement historiographique. Car ce philosophe instaure de manière durable un nouveau rapport à l��uvre d�art. Nous insisterons tout particulièrement sur l�inscription métaphysique de son projet. (shrink)

Le débat qui s’est tenu entre Sartre et Lévi-Strauss suite à la publication de la Critique de la raison dialectique est resté célèbre comme l’une des disputes intellectuelles majeures du xxe siècle. En première approche, les termes en sont bien connus : le structuralisme montant contre l’existentialisme caractéristique du moment de l’après-guerre, les sciences humaines contre la philosophie, la diversité...