First-order modal logic

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Commutativity of quantifiers in varying-domain Kripke models

Robert Goldblatt and Ian Hodkinson
Proceedings of Studia Logica International Conference 'Towards Mathematical Philosophy' (Trends in Logic IV).
D. Makinson, J. Malinowski, H. Wansing (eds), Toruń, Poland, Sept 2006.
Trends in Logic vol. 28, Springer, 2008, pp. 9-30.
A possible-worlds semantics is defined that validates the main axioms of Kripke's original system for first-order modal logic over varying-domain structures.  The novelty of this semantics is that it does not validate the commutative quantification schema ∀x∀yφ → ∀y∀xφ, as we show by constructing a counter-model.