Bi-Modal Naive Set Theory

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding 'forward looking' and 'backward looking' modal operators. We show that this 'bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects

"A Smack of Irrelevance" in Inconsistent Mathematics?

Recently, some proponents and practitioners of inconsistent mathe- matics have argued that the subject requires a conditional with ir- relevant features, i.e. where antecedent and consequent in a valid conditional do not behave as expected in relevance logics —by shar- ing propositional variables, for example. Here we argue that more fine-grained notions of content and content-sharing are needed to ex- amine the language of (inconsistent) arithmetic and set theory, and that the conditionals needed in inconsistent mathematics are not as irrelevant as it is suggested in the current literature

"A Smack of Irrelevance" in Inconsistent Mathematics?

Recently, some proponents and practitioners of inconsistent mathe- matics have argued that the subject requires a conditional with ir- relevant features, i.e. where antecedent and consequent in a valid conditional do not behave as expected in relevance logics —by shar- ing propositional variables, for example. Here we argue that more fine-grained notions of content and content-sharing are needed to ex- amine the language of (inconsistent) arithmetic and set theory, and that the conditionals needed in inconsistent mathematics are not as irrelevant as it is suggested in the current literature

Leibniz's law and its paraconsistent models

This paper aims at discussing the importance of Leibniz Law to getting models for Paraconsistent Set Theories.Comment: No comment

Bi-Modal Naive Set Theory

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding 'forward looking' and 'backward looking' modal operators. We show that this 'bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects

Revisiting Semilattice Semantics

The operational semantics of Urquhart is a deep and important part of the development of relevant logics. In this paper, I present an overview of work on Urquhart’s operational semantics. I then present the basics of collection frames. Finally, I show how one kind of collection frame, namely, functional set frames, is equivalent to Urquhart’s semilattice semantics

Teoria degli Insiemi in una logica paraconsistente

Utilizziamo una particolare logica rilevante (paraconsistente) per ricostruire la teoria ingenua degli insiemi senza che questa scada nella banalità. L'obiettivo del testo è mostrare come una logica più debole possa produrre una teoria soddisfacente gestendo le contraddizioni. Metteremo in luce alcune difficoltà formali mentre mostreremo alcuni teoremi insiemistici basilari. Infine cercheremo di definire i numeri ordinali

Non-Classical Circular Definitions

Circular denitions have primarily been studied in revision theory in the classical scheme. I present systems of circular denitions in the Strong Kleene and supervaluation schemes and provide complete proof systems for them. One class of denitions, the intrinsic denitions, naturally arises in both schemes. I survey some of the features of this class of denitions