We generalize two well-known model-theoretic characterization theorems from propositional modal logic to first-order modal logic. We first study FML-definable frames and give a version of the Goldblatt–Thomason theorem for this logic. The advantage of this result, compared with the original Goldblatt–Thomason theorem, is that it does not need the condition of ultrafilter reflection and uses only closure under bounded morphic images, generated subframes and disjoint unions. We then investigate Lindström type theorems for first-order modal logic. We show that FML has (...) the maximal expressive power among the logics extending FML which satisfy compactness, bisimulation invariance and the Tarski union property. (shrink)

For a subset K ⊆ [0, 1], the notion of K-satisfiability is a generalization of the usual satisfiability in first order fuzzy logics. A set Γ of closed formulas in a first order language τ is K-satisfiable, if there exists a τ-structure such that ∥ σ ∥ ∈ K, for any σ ∈ Γ. As a consequence, the usual compactness property can be replaced by the K-compactness property. In this paper, the K-compactness property for Łukasiewicz first order logic is investigated. (...) Using the ultraproduct construction, it is proved that for any closed subset K and set Γ of closed formulas, Γ is K-satisfiable if and only if it is finitely K-satisfiable. (shrink)