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Trial and error mathematics: Dialectical systems and completions of theories

Amidei, Jacopo; Andrews, Uri; Pianigiani, Duccio; San Mauro, Luca and Sorbi, Andrea (2019). Trial and error mathematics: Dialectical systems and completions of theories. Journal of Logic and Computation, 29(1) pp. 157–184.

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DOI (Digital Object Identifier) Link: https://doi.org/10.1093/logcom/exy033
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Abstract

This paper is part of a project that is based on the notion of a dialectical system, introduced by Magari as a way of capturing trial and error mathematics. In Amidei et al. (2016, Rev. Symb. Logic, 9, 1–26) and Amidei et al. (2016, Rev. Symb. Logic, 9, 299–324), we investigated the expressive and computational power of dialectical systems, and we compared them to a new class of systems, that of quasi-dialectical systems, that enrich Magari’s systems with a natural mechanism of revision. In the present paper we consider a third class of systems, that of p-dialectical systems, that naturally combine features coming from the two other cases. We prove several results about p-dialectical systems and the sets that they represent. Then we focus on the completions of first order theories. In doing so, we consider systems with connectives, i.e. systems that encode the rules of classical logic. We show that any consistent system with connectives represents the completion of a given theory. We prove that dialectical and q-dialectical systems coincide with respect to the completions that they can represent. Yet, p-dialectical systems are more powerful; we exhibit a p-dialectical system representing a completion of Peano Arithmetic that is neither dialectical nor q-dialectical.

Item Type: Journal Item
Copyright Holders: 2018 The Authors
ISSN: 1465-363X
Keywords: dialectical system; q-dialectical system; p-dialectical system; completion
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Computing and Communications
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 58983
SWORD Depositor: Jisc Publications-Router
Depositing User: Jisc Publications-Router
Date Deposited: 05 Feb 2019 08:56
Last Modified: 30 Mar 2019 06:22
URI: http://oro.open.ac.uk/id/eprint/58983
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