On Pairs of Dual Consequence Operations

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 S lupecki 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 antiinfallible operations leading from non-accepted (rejected, false) sentences of a deductive system to non-accepted (rejected, false) sentences of the system (see S lupecki in Funkcja Lukasiewicza, 33–40, 1959; Wybraniec-Skardowska in Teoria zdań odrzuconych, 5–131, Zeszyty Naukowe Wyższej Szko ly 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: by the triple: (+,−) < S,Cn, Cn− > or by the triple: (−,+) < S,Cn−, Cn >. 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 Lukasiewicz’s sense), as well as formulate different metatheorems concerning them. Mathematics Subject Classification (2010). Primary 03B22; Secondary 01A60, 03B99.