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Internal definability and completeness in modal logic. (English)
Berlin: FU Berlin, Fachbereich Mathematik, 97 p. (1991).
The semantical understanding of modal logic has developed along two courses, via a theory of completeness (e.g. Kripke) and via a theory of correspondence (e.g. van Benthem). This thesis unites these two approaches by formalizing a general theory of internal definability. (Given a generalized Kripke-frame, $X$, a concept $C$ applying to $X$ is internally described in $X$ by an $n$-tuple $\vert P\vert$ of modal formulas if $C$ holds of an $n$-tuple $\vert w\vert$ of worlds in $G\in X$ iff for some valuation $γ$ into $G$, $w\sb i\in γ(P\sb i)$ for all $i\in n$. A concept is internally definable in $X$ iff its complement or negation is internally describable in $X$.) The theory of internally definable elementary relations is the theory of correspondence; moreover, completeness proofs can generally be reduced to internally defining specific properties of worlds within specified classes of generalized Kripke-frames. This theory of internal definability generates simple, constructive completeness proofs for appropriate systems and allows one to determine such characteristics of systems as the finite model property. The thesis contains two parts. The first deals with generalized correspondence and internal definability and explores what concepts are internally definable under what conditions; the second part explores completeness questions and the finite model property, especially for extensions of K4.
Reviewer: L.F.Goble (Salem)