Theabstract variable binding calculus (VB-calculus) provides a formal frame-work encompassing such diverse variable-binding phenomena as lambda abstraction, Riemann integration, existential and universal quantification (in both classical and nonclassical logic), and various notions of generalized quantification that have been studied in abstract model theory. All axioms of the VB-calculus are in the form of equations, but like the lambda calculus it is not a true equational theory since substitution of terms for variables is restricted. A similar problem with the standard formalism (...) of the first-order predicate logic led to the development of the theory of cylindric and polyadic Boolean algebras. We take the same course here and introduce the variety of polyadic VB-algebras as a pure equational form of the VB-calculus. In one of the main results of the paper we show that every locally finite polyadic VB-algebra of infinite dimension is isomorphic to a functional polyadic VB-algebra that is obtained from a model of the VB-calculus by a natural coordinatization process. This theorem is a generalization of the functional representation theorem for polyadic Boolean algebras given by P. Halmos. As an application of this theorem we present a strong completeness theorem for the VB-calculus. More precisely, we prove that, for every VB-theory T that is obtained by adjoining new equations to the axioms of the VB-calculus, there exists a model D such that T s=t iff D s=t. This result specializes to a completeness theorem for a number of familiar systems that can be formalized as VB-calculi. For example, the lambda calculus, the classical first-order predicate calculus, the theory of the generalized quantifierexists uncountably many and a fragment of Riemann integration. (shrink)

We continue the investigation, initiated in Salibra et al. (Found Sci, 2020), of Boolean-like algebras of dimension n (nBA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document}s), algebras having n constants e1,⋯,en\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf e_1,\dots,\mathsf e_n$$\end{document}, and an (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBA\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. Putting to good use the concept of a central element, we extend the Boolean power construction to that of a semiring power and we prove two representation theorems: (i) Any pure nBA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\textrm{BA}$$\end{document} is isomorphic to the algebra of n-central elements of a Boolean vector space; (ii) Any member of a variety of nBAs with one generator is isomorphic to a Boolean power of this generator. This yields a new proof of Foster’s theorem on primal varieties. (shrink)

A new hypothesis on the basic features characterising the Foundations of Mathematics is suggested. By means of them the entire historical development of Mathematics before the 20th Century is summarised through a table. Also the several programs, launched around the year 1900, on the Foundations of Mathematics are characterised by a corresponding table. The major difficulty that these programs met was to recognize an alternative to the basic feature of the deductive organization of a theory - more precisely, to Hilbert’s (...) main tenet. Ironically, already half a century before the births of these programs the alternative organization has been substantially represented by Lobachevsky's theory on parallel lines. Moreover, although each program’s founder recognised the basic features in a partial way only, all together these programs represented just the four possible foundational approaches. (shrink)

Over the years, research in the field of narratological studies has converged towards a unitary vision of narrativity, which is based on a shared interpretation of Aristotle's Poetics. The object of this article is to describe the main characteristics of an alternative model of narrativity, which I propose to call “Nietzschean narratology”. In particular, I emphasize the role that the notion of melancholy plays within the model of aesthetic experience of The Birth of Tragedy. I wish to show that, in (...) this work, Nietzsche criticizes the artificiality of the Aristotelian catharsis, by opposing to it the naturalness of the “Dionysiac melancholy” which, according to him, vivifies the Attic drama. (shrink)

We introduce Boolean-like algebras of dimension n ($$n{\mathrm {BA}}$$ n BA s) having n constants $${{{\mathsf {e}}}}_1,\ldots,{{{\mathsf {e}}}}_n$$ e 1, …, e n, and an $$(n+1)$$ ( n + 1 ) -ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of $$n{\mathrm {BA}}$$ n BA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The $$n{\mathrm {BA}}$$ n BA s provide the (...) algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, $$n{\mathrm {CL}}$$ n CL, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in $$n{\mathrm {CL}}$$ n CL, and, via the embeddings, in all the finite tabular logics. (shrink)

This book proposes a radically evolutionary approach to biolinguistics that consists in considering human language as a form of species-specific intelligence entirely embodied in the corporeal structures of Homo sapiens. The book starts with a historical reconstruction of two opposing biolinguistic models: the Chomskian Biolinguistic Model (CBM) and the Darwinian Biolinguistic Model (DBM). The second part compares the two models and develops into a complete reconsideration of the traditional biolinguistic issues in an evolutionary perspective, highlighting their potential influence on the (...) paradigm of biologically oriented cognitive science. The third part formulates the philosophical, evolutionary and experimental basis of an extended theory of linguistic performativity within a naturalistic perspective of pragmatics of verbal language. The book proposes a model in which the continuity between human and non-human primates is linked to the gradual development of the articulatory and neurocerebral structures, and to a kind of prelinguistic pragmatics which characterizes the common nature of social learning. In contrast, grammatical, semantic and pragmatic skills that mark the learning of historical-natural languages are seen as a rapid acceleration of cultural evolution. The book makes clear that this acceleration will not necessarily favour the long-term adaptations for Homo sapiens. (shrink)

Die Erforschung des historischen und systematischen Kontextes von Kants Philosophie weist bis auf den heutigen Tag signifikante Lücken auf. Dies gilt umso mehr für unser Verständnis der Bedeutung der gegenüber Kants Philosophie erhobenen Einwände und Kritikpunkte sowohl für die verschiedenen Entwicklungsphasen seiner Philosophie wie für ihre Rezeption und Darstellung in späteren Jahrhunderten. Die im vorliegenden Band versammelten Beiträge wollen einen Beitrag zur Schließung dieser Lücken leisten. Sie gehen auf Vorträge zurück, die auf dem VII. Multilateralen Kant-Kolloquium zum Thema „Kant und (...) seine Kritiker – Kant and his critics – Kant et ses critiques“ gehalten worden sind, das vom 28.-30. April 2017 an der Martin-Luther-Universität Halle-Wittenberg stattfand. There are still significant gaps in the study of the historical and systematic contexts of Kant’s philosophy. This is true even more of our understanding of the significance of the objections and criticisms levelled at Kant’s philosophy and of the development of his philosophy and its reception and depiction in later centuries. The essays in this volume seek to close these gaps. They are based on the papers given at the 7th multilateral Kant Colloquium with the theme “Kant und seine Kritiker – Kant and his critics – Kant et ses critiques” which took place from 28th-30th April 2017 at the Martin-Luther-University, Halle-Wittenberg. (shrink)

The article analyses some aspects of Schmitt’s theories on international law, in particular the notion of Gro ß raum. The assumption is that with this notion Schmitt tries to re-think politics and international relations beyond the classical categories of the State. From this point of view, there is an essential affinity between the concept of politics explained in the homonymous essay published in 1927 and the concept of Gro ß raum. After the Second World War, Schmitt distances himself from the (...) notion of Gro ß raum, too close to the nazi theories, and focuses on that of nomos, explained in Der Nomos der Erde, and on the crisis of the Jus publicum europaeum. However, Schmitt fails to define which system of international relations should follow the Jus publicum europaeum and the same notion of nomos remains rather undefined. In the last part of the paper the author compares Schmitt’s theories about international law and some theories of Kelsen, with particular reference to the theory of bellum justum. (shrink)

The paper deals with a method of hierarchization of alternatives in a multicriteria environment by means of statistical nonparametric procedures. For each criterion, alternatives are disposed on an ordinal scale. After that a procedure similar to ANOVA is activated on the data. The differences relating to the average ranks of each action are used to build the hierarchical algorithm. The concept of outranking, in a probabilistic meaning, is reached in this way. Thereafter, we arrive at the concept of dominance using (...) pairwise comparisons. A case study is presented which implements the methodological suggestion. (shrink)

According to Gilead and colleagues, to be efficient abstraction requires a hierarchical organization of information into long-term memory. But, how and when are abstract representations consolidated into long-term memory and how are they integrated with pre-existing abstracta are questions not discussed by Gilead and colleagues. Here, we propose that these processes occur preferentially during offline periods such as sleep.

This article presents the results of the longitudinal study of Addiopizzo, a successful anti-bribery organization founded in Sicily in 2004. It analyzes how this organization has used information disclosure as a strategy to fight adverse environmental conditions and the immoral activities of the Sicilian Mafia. This article extends the business ethics and corporate social responsibility literature by showing how multi-level strategic information disclosure processes can help gain organizational legitimacy in adverse social environments and successfully fight against social resistance to change, (...) low levels of moral imagination and attacks from criminal organizations. This article provides an additional contribution to the literature by linking the three research streams on corporate transparency, the fight against corruption, and organizational legitimacy. The results of this research also contribute to the special issue of the EBEN AC 2010, “Which values for which organizations”, since it provides a unique example of an organization capable of spreading the values of social justice and honesty in a difficult social environment plagued by Mafia. (shrink)