Accepting/rejecting propositions from accepted/rejected propositions: A unifying overview

Abstract

Looking at inference as a way of transforming information so as to make it more easily usable (or interpretable) allows to consider accepted and rejected propositions as equally relevant and naturally gives a bipolar view of reasoning. The four possibilities of transforming information from accepted or rejected propositions into accepted or rejected ones are analyzed and examples illustrating them are given. This analysis is not only interesting per se but can also be useful in increasing capabilities of existing theorem provers. A unified framework based on former work by the authors is extended by incorporating the idea of theory-anti-subsumption related to Plotkin's generalization. Working on some technical details of this framework should allow automated reasoning tools to deal with different ways of connecting accepted and rejected propositions. C 2008 Wiley Periodicals, Inc.

Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology

“…This domain has recently motivated work in several directions, for instance for applications in knowledge representation, preference modeling, argumentation, multi-criteria decision analysis, cooperative games, among others [1,9,28,31,47,49,52,54,63,64,66]. In particular, fuzzy and possibilistic formalisms for bipolar information have been proposed [43,46,8,9].…”

A meta-logic of inference rules: Syntax

“…It is the most common type of the refutation systems. The complete (mixed) systems were constructed for various of logics (see, for instance [9,3,57,11,1,44,52,2]). R. Carnap suggested that we can use the regular notion of derivation (that allows to derive a formula (proposition) from a set of formulas (propositions), and that we reject a formula A if we can derive from A a formula that we know is rejected, is an anti-axiom.…”

Bipolar Fuzzy Spatial Information: Geometry, Morphology, Spatial Reasoning

“…This domain has recently motivated work in several directions, for instance for applications in knowledge representation, preference modeling, argumentation, multi-criteria decision analysis, cooperative games, among others [1,5,21,23,37,38,42,43,49,50,51]. In particular, fuzzy and possibilistic formalisms for Another example, for the pair of relations close/far, is illustrated in Figure 2.…”