Logics for Approximate Entailment in ordered universes of discourse

• We consider a variant of the Logic of Approximate Entailment in the sense of Ruspini.
• Propositions are interpreted by subsets of a chain, or a product of chains.
• We present the calculi LAEC and LAEPC and show their completeness.
• In these calculi, it is possible to combine conclusions in a conjunctive way.

The Logic of Approximate Entailment (LAELAE) is a graded counterpart of classical propositional calculus, where conclusions that are only approximately correct can be drawn. This is achieved by equipping the underlying set of possible worlds with a similarity relation. When using this logic in applications, however, a disadvantage must be accepted; namely, in LAELAE it is not possible to combine conclusions in a conjunctive way. In order to overcome this drawback, we propose in this paper a modification of LAELAE where, at the semantic level, the underlying set of worlds is moreover endowed with an order structure. The chosen framework is designed in view of possible applications.