Novel three-valued non-deterministic level semantics for modal logics $\textbf {T}$ and $\textbf {S4}$ are presented. A criterion for partial level valuations is given, making it possible to create truth tables. Additionally, semantics and truth tables for $\textbf {0}$ (defined as $\textbf {PC}$ plus rule of necessitation) and $\textbf {0T}$ with only two values are based on Ivlev’s work. We need Kearns’ notion of level valuations: a generalization of Dugundji’s theorem shows that there is no non-deterministic semantics for modal logics up to $\textbf {S5}$, containing the rule of necessitation.

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Motivated by the colloquial language term of a “glass gummy bear”, an additional type of concept composition for description logics is suggested. This composition type is then axiomatically formalized and called concept generalization. Consistency of the formalization is checked. By proving axiom K and Go ̈del rule, it is shown that this logic is in fact a multi-modal logic. Concepts could be both modal operators and predicate symbols. A Kripke semantics is presented (the adequacy is future work). In this semantics, the TBox axioms hold for any view, assertions in the ABox hold for the natural view (a selected world in the Kripke structure) only. The relationship to other formalisms is outlined. Further examples are discussed at the end.

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