# On Pairs of Dual Consequence Operations

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Logica Universalis

, Volume 5, Issue 2, pp 177–203

First Online: 15 June 2011Received: 23 June 2010Accepted: 07 June 2011

Abstract

In the paper, the authors discuss two kinds of consequence operations characterized axiomatically. The first one are consequence operations of the type Cn that, in the intuitive sense, are infallible operations, always leading from accepted true sentences of a deductive system to accepted true sentences of the deductive system see Tarski in Monatshefte für Mathematik und Physik 37:361–404, 1930, Comptes Rendus des Séances De la Société des Sciences et des Lettres de Varsovie 23:22–29, 1930; Pogorzelski and Słupecki in Stud Logic 9:163–176, 1960, Stud Logic 10:77–95, 1960. The second kind are dual consequence operations of the type Cn that can be regarded as anti-infallible operations leading from non-accepted rejected, false sentences of a deductive system to non-accepted rejected, false sentences of the system see Słupecki in Funkcja Łukasiewicza, 33–40, 1959; Wybraniec-Skardowska in Teoria zdań odrzuconych, 5–131, Zeszyty Naukowe Wyższej Szkoły Inżynierskiej w Opolu, Seria Matematyka 481:35–61, 1983, Ann Pure Appl Logic 127:243–266, 2004, in On the notion and function of rejected propositions, 179–202, 2005. The operations of the types Cn and Cn can be ordinary finitistic consequence operations or unit consequence operations. A deductive system can be characterized in two ways by the following triple:$$\begin{array}{ll}{ m by\,the\,triple}:\hspace{1.4cm}+ -\hspace{0,6cm} \\ { m or\,by\,the\,triple}:\hspace{1.0cm}-, +\hspace{0,6cm} .\end{array}$$We compare axiom systems for operations of the types Cn and Cn, give some methodological properties of deductive systems defined by means of these operations e.g. consistency, completeness, decidability in Łukasiewicz’s sense, as well as formulate different metatheorems concerning them.Mathematics Subject Classification 2010Primary 03B22 Secondary 01A60 03B99 Communicated by Jean-Yves Beziau.

KeywordsAxiom systems of theories of deductive systems consequence operations unit consequence operations a rejection consequence operation a dual consequence operation asserted system refutation system  Download to read the full article text

Autor: Urszula Wybraniec-Skardowska - Jacek Waldmajer