Nonstandard connectives of intuitionistic propositional logic
type: Item
instance of: scholarly article
title: Nonstandard connectives of intuitionistic propositional logic
zbMATH Open document ID: 0658.03005
author: Michael Kaminski
published in: Notre Dame Journal of Formal Logic
publication date: 1988-01-01T00:00:00Z
review text: This article is organized as follows: In Section 1 the author proposes the question: ``What is an intuitionistic propositional connective?'', and defines a new n-ary connective ``{\#}'' which is said to be extensional (or weakly extensional). In Section 2 he introduces a new unary connective ``\(\circ ''\) and its related axioms. He extends also the ordinary intuitionistic propositional calculus I to the logical system \(I^{\circ}\) with connective \(\circ\). The extended system has a Gentzen system with the subformula property and cut-free deductions. In Section 3 he explains a possibility of introducing new connectives by their interpretation in Kripke models. He proves that for any n-ary connective {\#} there exists a formula \(\psi (A_ 1,...,A_ n)\) with ordinary connectives only, such that \(I\vdash \psi (A_ 1,...,A_ n)\) implies \(I^{\#}\vdash \#(A_ 1,...,A_ n)\), and \(I^{\#}\vdash \#(A_ 1,...,A_ n)\) implies \(I\vdash \neg \neg \psi (A_ 1,...,A_ n).\) In Section 4 he considers some extension \(I^{\#}\) of I such that {\#} can be described by rules of inference in a Gentzen system. Under natural and not strong restrictions on rules of inference he proves that \(I^{\#}\) can be interpreted either in \(I^{\circ}\) or in propositional intuitionistic modal logic S4.
Mathematics Subject Classification ID: 03B20
zbMATH DE Number: 4075017
zbMATH Keywords: intuitionistic propositional connective
zbMATH Keywords: Gentzen system
zbMATH Keywords: Kripke models
reviewed by: Satoshi Miura
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093637931
OpenAlex ID: W1964474114
DOI: 10.1305/NDJFL/1093637931
description: scientific article; zbMATH DE number 4075017