Realizability semantics for error-tolerant logics: preliminary version

• Authors: John C. Mitchell;Michael J. O'Donnell
• Venue: TARK '86 Proceedings of the 1986 conference on Theoretical aspects of reasoning about knowledge
• Year: 1986

Classical and constructive logics have shortcomings as foundations for sophisticated automated reasoning from large amounts of data because a single error in the data could produce a contradiction, logically implying all possible conclusions. Relevance logics have the potential to support sensible reasoning from data that contains a few errors, limiting the impact of those errors to assertions that are naturally related to the erroneous information. There are a number of competing formal systems for relevance logic in the literature, with different sets of theorems. Applications of relevance logics, and particularly choices between formalisms, are hampered by the lack of clear intuitive semantic treatment of relevance. This paper proposes plausible semantic treatments of relevance logic based on intuitive restrictions on the behavior of realizability functions. We examine two versions of realizability semantics. The first uses models which consist entirely of realizability functions that preserve independence of evidence, while the second semantics requires functions to be strictly monotone with respect to strength of evidence. We show soundness for the first semantics, and soundness and completeness theorems over a "nonstandard" set of models for the second. The second approach also yields completeness over "nonstandard" models for intuitionistic implication.