Logic Methods for Many-valued Logics: Higher-order Autoepistemic Language Concepts

The higher-order types of Herbrand interpretations for databases arise often in practice when we have to deal with uncertain or imprecise information, or in constrained databases. They are a consequence of the introduction of a kind of the observational equivalence for the hidden information in the ordinary Herbrand interpretations. It means that the logics for such databases are particular functional many-valued logics, where the set of algebraic truth values is a complete distributive lattice of functions from the set of values of hidden attributes to the 2-valued lattice of classic truth values. In this paper we present a general definition for higher-order Herbrand interpretation types and their standard flattening into the ordinary 2-valued Herbrand interpretations. Unfortunately such a flattening is not able to support the higher-order language concepts as those used in the original databases. Because of that we introduce a modal flattening into the quotient, modulo observation equivalence, Herbrand interpretations. Then we show how we are able to define the higherorder autoepistemic language concepts, but in a multi-modal logics framework where the original many-valued ground formulae are transformed into modal 2valued autoepistemic concepts. We define a multi-modal Kripke model semantics for them based only on theoretical considerations for autoepistemic semantics of concepts. We explain why, when we are dealing with uncertainty or with predicate compression, that is, with higher-order Herbrand interpretation types for such many-valued logics, we are dealing fundamentally with the logics with a possible worlds semantics. After that we show that each universal multi-modal operator can be canonically replaced by a unique existential modal operator in order to obtain a normal (mono)modal logic for higher-order language concepts useful for practical applications.