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Nonmonotonicity in the Framework of Parametric Logic

Abstract

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base T, a hierarchy of sentences that are generalised logical consequences of T is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies.

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Acknowledgements

The author would like to thank the anonymous reviewers for extremely detailed and most valuable comments that greatly helped improve the paper, and Kevin Kelly for having organised the stimulating workshop on the Logic of simplicity that led to this special issue of Studia Logica.

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Correspondence to Éric Martin.

Additional information

Presented by Jacek Malinowski

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Martin, É. Nonmonotonicity in the Framework of Parametric Logic. Stud Logica 107, 1025–1077 (2019). https://doi.org/10.1007/s11225-018-9831-7

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Keywords

  • Generalised logical consequence
  • Hierarchies
  • Closed world assumption
  • Preferential entailment
  • Nonmonotonicity