On the ‘in many cases’ Modality: Tableaux, Decidability, Complexity, Variants

Abstract

The modality ‘true in many cases’ is used to handle non-classical patterns of reasoning, like ‘probably φ is the case’ or ‘normally φ holds’. It is of interest in Knowledge Representation as it has found interesting applications in Epistemic Logic, ‘Typicality’ logics, and it also provides a foundation for defining ‘normality’ conditionals in Non-Monotonic Reasoning. In this paper we contribute to the study of this modality, providing results on the ‘majority logic’ Θ of V. Jauregui. The logic Θ captures a simple notion of ‘a large number of cases’, which has been independently introduced by K. Schlechta and appeared implicitly in earlier attempts to axiomatize the modality ‘probably φ’. We provide a tableaux proof procedure for the logic Θ and prove its soundness and completeness with respect to the class of neighborhood semantics modelling ‘large’ sets of alternative situations. The tableaux-based decision procedure allows us to prove that the satisfiability problem for Θ is NP-complete. We discuss a more natural notion of ‘large’ sets which accurately captures ‘clear majority’ and we prove that it can be also used, at the high cost however of destroying the finite model property for the resulting logic. Then, we show how to extend our results in the logic of complete majority spaces, suited for applications where either a proposition or its negation (but not both) are to be considered ‘true in many cases’, a notion useful in epistemic logic.