A Note on Axiomatisations of Two-Dimensional Modal Logics

Abstract

Abstract. We analyse the role of the modal axiom corresponding to the first-order formula "∃y (x = y)" in axiomatisations of two-dimensional propositional modal logics.One of the several possible connections between propositional multi-modal logics and classical first-order logic is to consider finite variable fragments of the latter as 'multi-dimensional' modal formalisms: First-order variable-assignment tuples are regarded as possible worlds in Kripke frames, and each first-order quantification ∃v i and ∀v i as 'coordinate-wise' modal operators 3 i and 2 i in these frames. This view is implicit in the algebraisation of finite variable fragments using finite dimensional cylindric algebras [6], and is made explicit in [15,12].Here we look at axiomatisation questions for the two-dimensional case from this modal perspective. (For basic notions in modal logic and its Kripke semantics, consult e.g. [2,3].) We consider the propositional multi-modal language ML δ 2 having the usual Boolean operators, unary modalities 3 0 and 3 1 (and their duals 2 0 , 2 1 ), and a constant δ:Formulas of this language can be embedded into the two-variable fragment of first-order logic by mapping propositional variables to binary atoms P (v 0 , v 1 ) (with this fixed order of the two available variables), diamonds 3 i to quantification ∃v i , and the 'diagonal' constant δ to the equality atom v 0 = v 1 . Semantically, we look at first-order models as multimodal Kripke frames (fitting to the above language) of the formWe call frames of this kind square frames. The above embedding is validitypreserving in the sense that a modal ML δ 2 -formula ϕ is valid in all square frames iff its translation ϕ † is a first-order validity.