Logica Universalis

, Volume 5, Issue 2, pp 177–203 | Cite as

On Pairs of Dual Consequence Operations

Open Access


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 4(81):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:
$$\begin{array}{ll}{\rm by\,the\,triple}:\hspace{1.4cm} (+ , -)\hspace{0,6cm}<S, Cn^{+},Cn^{-}> \\ {\rm or\,by\,the\,triple}:\hspace{1.0cm} (-, +)\hspace{0,6cm} <S, Cn^{-}, Cn^{+}>.\end{array}$$
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 (2010)

Primary 03B22 Secondary 01A60 03B99 


Axiom systems of theories of deductive systems consequence operations unit consequence operations a rejection consequence operation a dual consequence operation asserted system refutation system 



We would like to thank the Referees for their useful remarks and comments which have contributed to improvement of our paper.

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2011

Authors and Affiliations

  1. 1.Group of Logic, Language and InformationUniversity of OpoleOpolePoland
  2. 2.Department of PhilosophyUniversity of OpoleOpolePoland

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