Global definability in basic modal logic. (English) Zbl 0998.03014

de Rijke, Maarten; Sturm, Holger

Wansing, Heinrich (ed.), Essays on non-classical logic. Including papers from the workshop, Konstanz, Germany, October 5-6, 1999. Singapore: World Scientific Publishing. Adv. Log. 1, 111-135 (2001).

A class \(K\) of models is called globally definable iff it is the model class of a set \(\Phi\) of modal formulas with respect to validity in models. In this way the usual notion of (local) definability is ‘globalized’, which is defined with respect to pointed models. The authors present, in particular, global versions of the Goldblatt-Thomason Theorem and van Benthem’s Bisimulation Theorem, respectively. I.e., both a characterization theorem for globally definable classes and a determination of the first-order formulas globally equivalent to some modal formula is provided [see, e.g., P. Blackburn, M. de Rijke and Y. Venema: Modal logic, Cambridge: Cambridge Univ. Press (2001; Zbl 0988.03006)]. Moreover, corresponding characterization results and some corollaries on preservation of modal formulas are presented for model classes globally definable by universal and positive formulas, respectively.
For the entire collection see [Zbl 0982.00038].

Reviewer: Bernhard Heinemann (Hagen)

MSC:

03B45	Modal logic (including the logic of norms)
03C40	Interpolation, preservation, definability

Keywords:

modal model theory; definability; preservation