An alternative generalisation of the concept of duality
type: Item
instance of: scholarly article
title: An alternative generalisation of the concept of duality
zbMATH Open document ID: 0105.24702
DOI: 10.1007/BF01440954
author: Alan Rose
published in: Mathematische Annalen
publication date: 1962-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/160974
review text: The symbol \([Y, X_1, X_2, \ldots, X_m, Y]\) was introduced in an earlier paper of the author [Math. Ann. 123, 76--78 (1951; Zbl 0042.00702)]. As the generalization of the dual of a formula in the two-valued propositional calculus, there is defined a relation between two formulae \(\Lambda\), \(\Phi\) of the \(m\)-valued propositional calculus, namely that \(\Lambda\) is an \(m\)-al of type \(i\) of \(\Phi\) (the definition is inductive according to \(i)\). There are proved four theorems each of which asserts or denies the existence of a functor of the following nature: the \(m\)-valued propositional calculus with the functor (as the only primitive) is functionally complete and the functor satisfies various conditions expressed in terms of \([\ldots]\) and \(m\)-ality. The last theorem is a generalization of a result of \textit{A. Church} [Port. Math. 7, 87--90 (1948; Zbl 0034.29101)]; it states that certain \(m-1\) functors form a complete set of independent primitives for the \(m\)-valued propositional calculus.
Mathematics Subject Classification ID: 03-XX
zbMATH DE Number: 3171970
zbMATH Keywords: mathematical logic
MaRDI profile type: Publication
cites work: Conditioned disjunction as a primitive connective for the \(m\)-valued propositional calculus
cites work: Some self-dual primitive functions for propositional calculi
cites work: Self-dual binary and ternary connectives for \(m\)-valued propositional calculi
description: scientific article; zbMATH DE number 3171970