Inconsistent models of arithmetic. I: Finite models
type: Item
instance of: scholarly article
title: Inconsistent models of arithmetic. I: Finite models
zbMATH Open document ID: 0878.03017
DOI: 10.1023/A:1004251506208
author: Graham Priest
published in: Journal of Philosophical Logic
publication date: 1997-08-05T00:00:00Z
review text: This paper demonstrates the existence of interpretations of the paraconsistent logic LP which model theories properly containing all the sentences of first-order arithmetic. The metatheorem about LP used, the Collapsing Lemma, takes us from any interpretation to a new one, using an equivalence relation on the domain of interpretation which is also a congruence relation on the interpretations of the function symbols of the language. This means that in a process of collapse from one interpretation to another, truth values are not lost: anything true (false) in the original interpretation is true (false) in the collapsed interpretation. One group of models use identity modulo \(n\) for some \(n\) as the consequence relation, and the successor function becomes cyclic. Another group use a relation which identifies all numbers equal to or greater than some \(n\), giving a successor relation described as a heap. The general structure of a finite model is a tail, plus a cycle of regular numbers, plus a collection of cycles of irregular numbers, such that the period of each is a divisor of the cycle of regular numbers.
Mathematics Subject Classification ID: 03B53
zbMATH DE Number: 1044162
zbMATH Keywords: inconsistent models
zbMATH Keywords: paraconsistent logic
zbMATH Keywords: first-order arithmetic
zbMATH Keywords: Collapsing Lemma
zbMATH Keywords: interpretations
zbMATH Keywords: heap
zbMATH Keywords: finite model
zbMATH Keywords: tail
reviewed by: Jim Mackenzie
Wikidata QID: Q56852437
MaRDI profile type: Publication
cites work: Q4052072
cites work: A theorem in 3-valued model theory with connections to number theory, type theory, and relevant logic
cites work: Q4001935
cites work: Inconsistent models for relevant arithmetics
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cites work: Minimally inconsistent LP
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description: scientific article; zbMATH DE number 1044162