On inference from inconsistent premisses

Abstract

The main object of this paper is to provide the logical machinery needed for a viable basis for talking of the 'consequences', the 'content', or of 'equivalences' between inconsistent sets of premisses.With reference to its maximal consistent subsets (m.c.s.), two kinds of'consequences' of a propositional set S are defined. A proposition P is a weak consequence (W-consequence) of S if it is a logical consequence of at least one m.c.s, of S, and P is an inevitable consequence (/-consequence) of S if it is a logical consequence of all the m.c.s. of S. The set of W-consequences of a set S it determines (up to logical equivalence) its m.c.s. (This enables us to define a normal form for every set such that any two sets having the same W-consequences have the same normal form.) The W-consequences and/-consequences will not do to define the 'content' of a set S. The first is too broad, may include propositions mutually inconsistent, the second is too narrow. A via media between these concepts is accordingly defined: P is a P-consequence of S, where P is some preference criterion yielding some of the m.c.s, of S as preferred to others, and P is a consequence of all of the P-preferred m.c.s, of S. The bulk of the paper is devoted to discussion of various preference criteria, and also surveys the application of this machinery in diverse contexts -for example, in connection with the processing of mutually inconsistent reports.

Formal Representations of Uncertainty

“…The second is called disjunctive fusion. It corresponds to a classical mode for dealing with inconsistency in logic (Rescher and Manor [100]): If the propositions of the form x ∈ A i are contradictory, then one looks for maximal consistent subsets of propositions, assuming that reality corresponds to one of these subsets (here reduced to {A 1 } and {A 2 }). The third mode is of another nature: the hypothesis of independence of the sources allows for a counting process.…”

Which style of reasoning to choose in the face of conflicting information?

“…Although the original RM-consequence relations were defined in [18] at the propositional level, we shall extend them to the predicative level. Let MCS(Γ) stand for the set of maximal consistent subsets of Γ and let F ree(Γ) stand for {A | A ∈ ∆, for each ∆ ∈ MCS(Γ)}.…”

Toward a General Framework for Information Fusion

“…, n}) is the set of maximal consistent subsets of sources. It was first proposed by [15]. It satisfies all core properties.…”