Constructive predicate logic with strong negation and model theory
type: Item
instance of: scholarly article
title: Constructive predicate logic with strong negation and model theory
zbMATH Open document ID: 0647.03005
author: Seiki Akama
published in: Notre Dame Journal of Formal Logic
publication date: 1988-01-01T00:00:00Z
review text: Strong negation differs from intuitionistic (Heyting) negation in that, by the former but not the latter, \(\sim (A\wedge B)\) is equivalent to the derivability of either \(\sim A\) or \(\sim B\), and \(\sim \forall xA(x)\) is equivalent to \(\sim A(t)\). Intuitionistic negation, \(\neg\), may, however, be defined in terms of strong negation, as \(\neg A\equiv A\supset \sim A\). In this paper a first-order constructive logic with strong negation is defined. It is presented first axiomatically, and then a Kripke-type many-worlds semantics is defined for it (the key feature of which is that formulas are evaluated in two ways, both positively and negatively). Using Henkin-style techniques, the axiomatic system is proved consistent and complete with respect to the semantics, likewise theorems for compactness and the Löwenheim property, that A is valid iff A is valid in all models with denumerable parameters. In addition, it is shown to embed intuitionistic logic, H, within the present system, S, with strong negation, and also how to embed the present system, S, within classical logic, C.
Mathematics Subject Classification ID: 03B20
zbMATH DE Number: 4055565
zbMATH Keywords: strong negation
zbMATH Keywords: first-order constructive logic
zbMATH Keywords: Kripke-type many-worlds semantics
zbMATH Keywords: intuitionistic logic
zbMATH Keywords: classical logic
reviewed by: Louis F. Goble
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093637767
OpenAlex ID: W2060687784
DOI: 10.1305/NDJFL/1093637767
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description: scientific article; zbMATH DE number 4055565