Nonmonotonic Logics and Semantics

TL;DR

This paper explores a generalized semantics for nonmonotonic logic based on choice functions and importance measures, showing classical propositional connectives can be justified without monotonicity.

Contribution

It introduces a new semantics for nonmonotonic reasoning using choice functions and importance measures, extending Tarski's framework.

Findings

Equivalent semantics based on importance measures and choice functions.

Classical propositional connectives are justified without monotonicity.

Weakening of Tarski's properties characterizes nonmonotonic consequence operations.

Abstract

Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas iff a holds in all models in which each of the elements of A holds. A more liberal semantics has been considered: a formula a may be deduced from a set A of formulas iff a holds in all of the "preferred" models in which all the elements of A hold. Shoham proposed that the notion of "preferred" models be defined by a partial ordering on the models of the underlying language. A more general semantics is described in this paper, based on a set of natural properties of choice functions. This semantics is here shown to be equivalent to a semantics based on comparing the relative "importance" of sets of models, by what amounts to a qualitative probability measure. The consequence operations defined by the equivalent semantics are then characterized by a weakening of Tarski's properties in which the monotonicity requirement is replaced by three weaker conditions. Classical propositional connectives are characterized by natural introduction-elimination rules in a nonmonotonic setting. Even in the nonmonotonic setting, one obtains classical propositional logic, thus showing that monotonicity is not required to justify classical propositional connectives.