A deduction theorem schema for deductive systems of propositional logics
type: Item
instance of: scholarly article
title: A deduction theorem schema for deductive systems of propositional logics
zbMATH Open document ID: 0755.03014
DOI: 10.1007/BF00370679
author: Janusz Czelakowski
author: W. Dziobiak
published in: Studia Logica
publication date: 1992-09-27T00:00:00Z
review text: The authors propose a new schema for a deduction theorem for propositional logics. They prove that the logic \(L\) fulfills the schema if and only if there exists a finite set \(A(p,q)\) of propositional formulas involving only \(p\), \(q\) such that \(\vdash A(p,p)\) and \(p,A(p,q)\vdash q\) for each \(A(p,q)\). The new schema is defined in terms of a family \(F\) of finite sets of propositional formulas involving \(p\), \(q\) and perhaps other propositional letters, a set of formulas \(X\), single formulas \(A\), \(B\), and a sequence of propositional formulas \(s\). The schema is \(X,A\vdash B\) iff there exists \(P\) in \(F\) and a sequence \(s\) of appropriate length such that \(X\vdash P(p/A,q/B,s),P(p/A,q/B,s)\) is the result of simultaneously substituting in all formulas of \(P\), \(A\) for \(p\), \(B\) for \(q\) and the elements of \(s\) for propositional letters of \(P\) other than \(p\), \(q\). In addition to proving their theorem they illustrate application of the schema to familiar deduction theorems.
Mathematics Subject Classification ID: 03B99
zbMATH DE Number: 58407
zbMATH Keywords: deduction theorem
zbMATH Keywords: propositional logics
reviewed by: Charles F. Kielkopf
MaRDI profile type: Publication
cites work: Algebraizable logics
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cites work: Algebraic aspects of deduction theorems
cites work: Local deductions theorems
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full work available at URL: https://doi.org/10.1007/bf00370679
OpenAlex ID: W2011556534
description: scientific article; zbMATH DE number 58407