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 fur Mathematik und Physik 37:361–404, 1930, Comptes Rendus des Seances De la Societe des Sciences et des Lettres de Varsovie 23:22–29, 1930; Pogorzelski and Slupecki 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 Slupecki in Funkcja Łukasiewicza, 33–40, 1959; Wybraniec-Skardowska in Teoria zdan odrzuconych, 5–131, Zeszyty Naukowe Wyzszej Szkoly Inzynierskiej 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: #R##N##R##N#$$\begin{array}{ll}{\rm by\,the\,triple}:\hspace{1.4cm} (+ , -)\hspace{0,6cm}  \\ {\rm or\,by\,the\,triple}:\hspace{1.0cm} (-, +)\hspace{0,6cm}  .\end{array}$$#R##N##R##N#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.
['Urszula Wybraniec-Skardowska', 'Jacek Waldmajer']
On Pairs of Dual Consequence Operations