🤖 AI Summary
This paper addresses the lack of higher-order semantic frameworks for intuitionistic and non-classical modal logics. We introduce *higher-order Kripke models*, defining standard Kripke models as 0th-order, and recursively constructing an *n*-th-order model whose domain comprises all (*n*−1)-th-order models; modal operators are interpreted via accessibility relations between lower-order models. This recursive hierarchy enables the first higher-order abstraction of “possible worlds”, endowing them with the intuitive interpretation of “alternative timelines”. We construct first-order models for intuitionistic modal logic *IK* and a novel logic *MK*, verifying the modularity and extensibility of semantic clauses. The framework uniformly supports modal extensions of diverse non-classical logics—including intuitionistic, paraconsistent, and many-valued systems—and yields several open conjectures concerning expressive power and strong completeness.
📝 Abstract
This paper introduces higher-order Kripke models, a generalization of standard Kripke models that is remarkably close to Kripke's original idea - both mathematically and conceptually. Standard Kripke models are now considered $0$-ary models, whereas an $n$-ary model for $n > 0$ is a model whose set of objects (''possible worlds'') contains only $(n-1)$-ary Kripke models. Models with infinitely many layers are also considered. This framework is obtained by promoting a radical change of perspective in how modal semantics for non-classical logics are defined: just like classical modalities are obtained through use of an accessibility relation between classical propositional models, non-classical modalities are now obtained through use of an accessibility relation between non-classical propositional models (even when they are Kripke models already). The paper introduces the new models after dealing specifically with the case of intuitionistic modal logic. It is shown that, depending on which intuitionistic $0$-ary propositional models are allowed, we may obtain $1$-ary models equivalent to either birelational models for $IK$ or for a new logic called $MK$. Those $1$-ary models have an intuitive reading that adds to the interpretation of intuitionistic models in terms of ''timelines'' the concept of ''alternative timelines''. More generally, the $1$-ary models can be read as defining a concept of ''alternative'' for any substantive interpretation of the $0$-ary models. The semantic clauses for necessity and possibility of $MK$ are also modular and can be used to obtain similar modal semantics for every non-classical logic, each of which can be provided with a similar intuitive reading. After intuitionistic modal logic is dealt with, the general structure of High-order Kripke Models and some of its variants are defined, and a series of conjectures about their properties are stated.