Logical Connectives for Nonmonotonicity: A Choice Function-Based Approach

摘要

Several semantics for logics that model defeasible inference are based on the idea that not all models of a set F of classical formulas should be considered, but only some of them, the preferred ones. Recently, Daniel Lehmann proved that a very general family of nonmonotonic inference relations can be obtained by using choice functions, that pick some of the models of a given set of logical formulas. However, in this setting the choice function is fixed. This paper describes a semantics where the choice function is defined by formulas: instead of associating a set of models with each formula of the language, we associate a choice function which picks some models. The choice functions are defined for atomic formulas first, and then inductively for every formula, using for each connective a corresponding operator for combining choice functions. We show that this approach generalises classical logic: the choice function associated to a classical formula φ is the function that picks, from a set of models M, the elements of M that satisfy φ in the classical sense. We then describe operations on choice functions that correspond to connectives meaning for example: "p if it is consistent" or "p prior to q".