Dipartimento di Lettere e Filosofia

We try to recast in modern terms a choice principle conceived by Beppo Levi, who called it the Approximation Principle (AP). Up to now, there was almost no discussion about Levi’s contribution, due to the quite obscure formulation of AP the author has chosen. After briefly reviewing the historical and philosophical surroundings of Levi’s proposal, we undertake our own attempt at interpreting AP. The idea underlying the principle, as well as the supposed faithfulness of our version to Levi’s original intention, are then discussed. Finally, an application of AP to a property of metric spaces is presented, with the aim of showing how AP may work in contexts where other forms of choice are commonly at use.(Received March 07 2013)

The paper introduces Hilbert– and Gentzen-style calculi which correspond to systems \({\mathsf{C}_{n}}\) from Gupta and Belnap [3]. Systems \({\mathsf{C}_{n}}\) were shown to be sound and complete with respect to the semantics of finite revision. Here, it is shown that Gentzen-style systems \({\mathsf{GC}_{n}}\) admit a syntactic proof of cut elimination. As a consequence, it follows that they are consistent.

This paper presents and comments the content of a note by Beppo Levi on logical paradoxes. Though the existence of this contribution is known, very little analysis of it is available in the literature. I put the emphasis on Levi’s usage of “elementation procedures” for solving the set-theoretical paradoxes, which is the most original part of Levi’s approach to the topic.

This paper introduces theories for arithmetical quasi-inductive definitions (Burgess, 1986) as it has been done for first-order monotone and nonmonotone inductive ones. After displaying the basic axiomatic framework, we provide some initial result in the proof theoretic bounds line of research (the upper one being given in terms of a theory of sets extending Kripke–Platek set theory).(Received May 04 2009)

Paradoxes have represented an area of research where the interplay between mathematical methods of investigation and conceptual analysis proved to be particularly fruitful. As is well known, the discovery of paradoxes at the end of the nineteenth century occasioned the debate on the foundations of mathematics, which in turn was pivotal for the development of contemporary logic. However, rather than remaining a phenomenon confined in this area, paradoxes also arose in other contexts and became a matter of dispute which was also important for domains such as philosophy of logic and language, epistemology, decision theory, computer science.The book by Roy T. Cook under review aims at illustrating the richness of the debate on paradoxes, as well as suggesting a unifying view of them. This is done by taking into account a representative number and variety of paradoxical arguments which the author explains and discusses in detail. Cook’s initial remarks are intended as a guide for the inexpe