A New Modal Lindström Theorem

We prove new Lindström theorems for the basic modal propositional language, and for the guarded fragment of first-order logic. We find difficulties with such results for modal languages without a finite-depth property, high-lighting the difference between abstract model theory for fragments and for extensions of first-order logic. 1 What is modal logic? I would broadly endorse the 'minimal design view' in van Benthem 1996, Andréka, van Benthem & Németi 1998, Blackburn, de Rijke & Venema 2000, which says that modal languages are well-balanced fragments of classical ones combining good expressive power with reasonable computational complexity for model checking and satisfiability. But one can also try to understand what makes modal logic tick in other ways. One obvious alternative format is that of a Lindström theorem. Indeed, de Rijke 1993 contains one such result – and we sketch its modern proof as our starting point. 2 A first modal Lindström theorem Blackburn, de Rijke & Venema 2000 define an 'abstract modal language' as a formalism satisfying the usual base constraints from abstract model theory (Barwise & Feferman, eds., 1985), plus the modal characteristic of bisimulation invariance for all formulas. They then high-light the following semantic property, saying that modal formulas only look at models up to some finite depth: Finite Depth Property For any formula , there is a natural number k such that, for all models, (M, w) |= iff (M|k, w) |= , where M|k is the model M with its domain restricted to just those points that can be reached from w in k or fewer successive R-steps. Then we have the following 'maximality version' of a modal Lindström result, valid for abstract modal languages L with a finite vocabulary: Theorem 1 Any abstract modal language extending the basic modal one which has the Finite Depth Property is the basic modal language itself. The proof of this result revolves around the following fact.