On Rational Entailment for Propositional Typicality Logic

TL;DR

This paper explores propositional typicality logic (PTL), introducing multiple entailment notions based on rational closure, and demonstrates the impossibility of satisfying all desirable postulates simultaneously, advocating for diverse entailment approaches.

Contribution

It presents an impossibility theorem for PTL entailment postulates and investigates three semantic entailment variants based on rational closure and minimality.

Findings

Impossibility theorem for PTL entailment postulates

Development of three different PTL entailment notions

Analysis of minimality-based rational closure in PTL

Abstract

Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator capturing the most typical (alias normal or conventional) situations in which a given sentence holds. The semantics of PTL is in terms of ranked models as studied in the well-known KLM approach to preferential reasoning and therefore KLM-style rational consequence relations can be embedded in PTL. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for PTL remains monotonic and is therefore not appropriate in many contexts. Our first important result is an impossibility theorem showing that a set of proposed postulates that at first all seem appropriate for a notion of entailment with regard to typicality cannot be satisfied simultaneously. Closer inspection reveals that this result is best interpreted as an argument for advocating the development of more than one type of PTL entailment. In the spirit of this interpretation, we investigate three different (semantic) versions of entailment for PTL, each one based on the definition of rational closure as introduced by Lehmann and Magidor for KLM-style conditionals, and constructed using different notions of minimality.