The intention here is that of giving a formal underpinning to the idea of 'meaning-is-use' which, even if based on proofs, it is rather different from proof-theoretic semantics as in the Dummett-Prawitz tradition. Instead, it is based on the idea that the meaning of logical constants are given by the explanation of immediate consequences, which in formalistic terms means the effect of elimination rules on the result of introduction rules, i. e. the so-called reduction rules. For that we suggest an (...) extension to the Curry-Howard interpretation which draws on the idea of labelled deduction, and brings back Frege's device of variable-abstraction to operate on the labels (i. e., proof-terms) alongside formulas of predicate logic. (shrink)

SummaryLooking at proof theory as an attempt to ‘code’ the general pattern of the logical steps of a mathematical proof, the question of what kind of rules can make the meaning of a logical connective completely explicit does not seem to have been answered satisfactorily. The lambda calculus seems to have been more coherent simply because the use of ‘λ’ together with its projection 'apply' is specified by what can be called a 'reduction' rule: β‐conversion. We attempt to analyse the (...) role of proof rules, making use of a set of formal rules designed to capture both the notions of proof theory and those of the lambda‐calculus: Martin‐Löf's Intuitionistic Type Theory.RésuméSi on considère la théorie de la démonstration comme une tentative de ‘codifier’ les pas logiques d'une démonstration mathématique, on se rendra compte qu'on n'a pas répondu de façon satisfaisante à la question suivante: quelles sont les règies qui rendent complètement explicite le sens d'un connecteur logique? Le lambda‐calcul a été apparemment plus cohérent, tout simplement parce que l'utilisation du‘λ’ avec sa projection 'apply' est spécifiée par une regie de 'réduction': β‐conversion. Nous essayons d'analyser le rôle des règies de démonstration, en utilisant un système formel de règies conçu pour englober à la fois les notions de la théorie de la démonstration et celles du lambda‐calcul: la Théorie Intuitionniste des Types de Martin‐Löf.ZusammenfassungWenn man Beweistheorie als einen Versuch, ein allgemeines Muster der logischen Schritte in einem mathematischen Beweis zu kodifizieren, betrachtet, so scheint die Frage nach der Art von Regeln , die die Bedeutung von logischen Operatoren vollständig beschreiben, nicht zufriedenstellend beantwortet. Der Lambda‐Kalkül erscheint, kohärenter, einfach deshalb weil der Gebrauch von ‘λ’ zusammen mit dessen Projektion 'apply' durch Regeln bestimmt wird, die man 'Reduktions'‐Regeln nennen kann: β‐Konversion. Wir versuchen, die Rolle von Beweisregeln zu analysieren, indem wir ein Regelsystem verwenden, das sowohl die Begriffe der Beweistheorie als auch diejenigen des Lambda‐Kalküls erfasst, nämlich die Martin‐Löfsche Typentheorie. (shrink)

The intention here is that of giving a formal underpinning to the idea of ‘meaning-is-use’ which, even if based on proofs, it is rather different from proof-theoretic semantics as in the Dummett–Prawitz tradition. Instead, it is based on the idea that the meaning of logical constants are given by the explanation of immediate consequences, which in formalistic terms means the effect of elimination rules on the result of introduction rules, i.e. the so-called reduction rules. For that we suggest an extension (...) to the Curry– Howard interpretation which draws on the idea of labelled deduction, and brings back Frege’s device of variable-abstraction to operate on the labels (i.e., proof-terms) alongside formulas of predicate logic. (shrink)

We are concerned with showing how ‘labelled’ Natural Deduction presentation systems based on an extension of the so-called Curry-Howard functional interpretation can help us understand and generalise most of the deduction calculi designed to deal with the logical notion of existential quantification. We present the labelling mechanism for ‘’ using what we call ‘ɛ-terms’, which have the form of ‘a’) in a dual form to the ‘Ax.f’ terms of in the sense that the ‘witness’ is chosen at the time of (...) assertion/introduction.With the intention of demonstrating the generality of our framework, we provide a brief comparison with some other calculi of existentials, including the model-theoretic interpretations of Skolem and Herbrand, indicating the way in which some of the classical results on prenex normal forms can be extended to a wide range of logics. The essential ingredient is to look at the conceptual framework of labelled natural deduction as an attempted reconstruction of Frege's functional calculus where the devices of the Begriffsschrift work separately from, yet in harmony with, those of the Grundgesetze 1. (shrink)

Further to the connections between meaning and use, it seems useful to consider the consequences one is allowed to draw from a proposition as something directly related to its meaning/use. And indeed, Wittgenstein's references to the connections between meaning and the consequences, as well as between use and consequences are sometimes as explicit as his celebrated ‘definition’ of meaning as use given in the Investigations. Here we attempt to collect some of these references, discussing how an intuitive basis for the (...) construction of a more convincing proof-theoretic semantics for the mathematical language can arise out of this connection meaning/use/ consequences. (shrink)

As a celebration of theTractatus100th anniversary it might be worth revisiting its relation to the later writings. From the former to the latter, David Pears recalls that “everyone is aware of the holistic character of Wittgenstein’s later philosophy, but it is not so well known that it was already beginning to establish itself in theTractatus” (The False Prison, 1987). From the latter to the former, Stephen Hilmy’s (The Later Wittgenstein, 1987) extensive study of theNachlasshas helped removing classical misconceptions such as (...) Hintikka’s claim that “Wittgenstein in thePhilosophical Investigationsalmost completely gave up the calculus analogy.” Hilmy points out that even in theInvestigationsone finds the use of the calculus/game paradigm to the understanding of language, such as “in operating with the word” (Part I, §559) and “it plays a different part in the calculus”. Hilmy also quotes from a late (1946) unpublished manuscript (MS 130) “this sentence has use in the calculus of language”, which seems to be compatible with “asking whether and how a proposition can be verified is only a particular way of asking ‘How do you mean?’” Central in this back and forth there is an aspect which seems to deserve attention in the discussion of a semantics for the language of mathematics which might be based on (normalisation of) proofs and/or Hintikka/Lorenzen game-dialogue: the explication of consequences. Such a discussion is substantially supported by the use of the open and searchableThe Wittgenstein Archives at the University of Bergen. These findings are framed within the discussion of the meaning of logical constants in the context of natural deduction style rules of inference. (shrink)

Este artículo aborda la contextualización de la ética de Butler en un marco teórico que da por supuesta la perspectiva deconstructiva y que, por su parte, es reconstruido de acuerdo a dos criterios, las nociones de “matriz disciplinal” y de “paradigma”. Mediante ellas se trata de proporcionar un sistema de conceptos o categorías usuales en la teoría ética en general, pero redefinidas en términos butlerianos. Estas categorías a las que nos referimos son la acción, los sujetos, los juicios y los (...) criterios morales y, que, aplicadas a la ética de Butler, dan lugar a una ética precaria en la que la humanidad normativa se revela como imposible universalmente y, por tanto, no puede ser utilizada como “fundamento” de un juicio moral universal. Por ejemplo, la que se sugiere mediante las manifestaciones de duelo por la humanidad de “los nuestros”, víctimas de la violencia, de las que queda excluida estructuralmente la humanidad de “los otros”. (shrink)

This paper aims of this to describe, from Hannah Arendt’s point of view the theoretical and historical route that provide the emergence of one at the three faculties of the spirit: the faculty of the will or want. Drawing initially on ancient Greece, the paper also deals with contributions of classic, mediavel and modern thinkers. In a retrospective approach Arendt examines the perception and installation process of the faculty of the will, by correlating it with the advent and consolidation of (...) the linear conception of time, essencially from christian origins, against the background of cyclical conception of time, from Greek origins. It also reflects about the clash between the faculties of will and thinking, by following the controversy between Saint Tomas Aquino and Duns Scotus about the subject. Finally, the paper links the faculty of the will to the idea of freedom which is rationale of human condition. (shrink)

In recent years, animal welfare has become an important element of sustainable production that has evolved along with the transformation of animal production systems. Consumer attitudes towards farm animal welfare are changing around the world, especially at emerging markets of Asia, Africa and Latin America. Survey-based research on consumer attitudes towards farm animal welfare has increased. However, the geographical coverage of studies on consumer attitudes and perceptions about farm animal welfare has mostly been limited to Europe, and North America. Until (...) now, Latin American consumers’ attitudes towards animal welfare have not been well studied. Despite the fact that Mexico, Chile and Brazil belong to the same region, there are marked differences between these countries in terms of their economical and geographical characteristics among other factors. Those differences potentially have an impact on consumer attitudes towards animal welfare and livestock production systems in general. Given the evidence from the literature review, it seems advisable that Latin American producers and food industry who engage in animal welfare-enhancing practices should clearly label their products with information on the type of husbandry system to reach those consumers who want to make an informed choice. Therefore, there are some aspects that need to be studied and cannot be worked separately in order to promote and understand consumer attitudes towards dairy and beef systems, such as geography, economic development, and politics. (shrink)

The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the application of a term/function to any other term/function, including itself. The calculus can be seen as a formal theory with certain pre-established axioms and inference rules, which can be interpreted by models. Dana Scott proposed the first non-trivial model of the extensional lambda calculus, (...) known as $ D_{\infty }$, to represent the $\lambda $-terms as the typical functions of set theory, where it is not allowed to apply a function to itself. Here we propose a construction of an $\infty $-groupoid from any lambda model endowed with a topology. We apply this construction for the particular case $D_{\infty }$, and we see that the Scott topology does not provide enough information about the relationship between higher homotopies. This motivates a new line of research focused on the exploration of $\lambda $-models with the structure of a non-trivial $\infty $-groupoid to generalize the proofs of term conversion to higher-proofs in $\lambda $-calculus. (shrink)