Restricted access logics for inconsistent information
For practical reasoning with classically inconsistent information, desiderata for an appropriate logic L could include (1) it is an extension of classical logic — in the sense that all classical tautologies are theorems of L, and (2) contradictions do not trivialize L — in the sense that ex falso quodlibet does not hold. Two ways of realizing the second desideratum, for any database that may be inconsistent, include (A) take weaker than classical proof rules, but use all the data, or (B) take all the classical proof rules, but restrict the access of the data to the proof rules. The problem with adopting option (A) is that desideratum (1) is then not realizable. In this paper, we pursue option (B) by adding extra conditions on the proof rules to stop certain subsets of the data using the classical proof rules. To facilitate the presentation, we use the approach of Labelled Deductive Systems — formulae are labelled, and proof rules defined to manipulate both the formulae and the labels. The extra conditions on the proof rules are then defined in terms of the labels. This gives us a class of logics, called restricted access logics, that meet the desiderata above.
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