On Pairs of Dual Consequence Operations

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 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.Keywords. Axiom systems of theories of deductive systems, consequence operations, unit consequence operations, a rejection consequence operation, a dual consequence operation, asserted system, refutation system. 178U. Wybraniec-Skardowska and J. Waldmajer Log. Univers.

On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency

“…It is given, e.g., in [7], below Theorem 3.3. If we accept that the consequence operation Cn + is infallible: from true (or accepted as true) sentences of the set X it leads to true (or accepted as true) consequences of the set Cn + X, then in accordance with T1 the operation Cn´is an anti-infallible consequence operation: from false (or not accepted as true, rejected) sentences of set SzX it leads to false (or not accepted as true, rejected) sentences of set Cn´(SzX).…”

“…In Sections 2-4, we present the axiom systems of these three theories: T + , T´and T 1 (The theories T + and T´were presented or sketched earlier in [19][20][21]; see also axiom systems in [7,22]. The axioms of the theory T 1 were discussed in [23].).…”

Logical Squares for Classical Logic Sentences

“…The third connective is also called the dual implication: d(p → q) = df ∼ (q → p) (see [16,17]) and the last one is known as binegation .…”

A meta-logic of inference rules: Syntax

“…The complementary systems contain, essentially, two subsystems: one for deriving the asserted propositions, and another -for deriving the rejected propositions. For instance, in [49,50,59] the authors consider two closure operators: the regular one Cn that gives the theorems, and the complementary one Cn − , that gives anti-theorems (see also [45,Section 5.2]).…”