A Note on Axiomatisations of Two-Dimensional Modal Logics

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 ∃vi and ∀vi as ‘coordinate-wise’ modal operators 3i and 2i 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 ML2 having the usual Boolean operators, unary modalities 30 and 31 (and their duals 20, 21), and a constant δ: ML2 : p | ¬φ | φ ∨ ψ | 30φ | 31φ | 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 (v0, v1) (with this fixed order of the two available variables), diamonds 3i to quantification ∃vi, and the ‘diagonal’ constant δ to the equality atom v0 = v1. Semantically, we look at first-order models as multimodal Kripke frames (fitting to the above language) of the form 〈U × U,≡0,≡1, Id〉, where, for all u0, u1, v0, v1 ∈ U, 〈u0, u1〉≡0〈v0, v1〉 iff u1 = v1, 〈u0, u1〉≡1〈v0, v1〉 iff u0 = v0, and Id = {〈u, u〉 : u ∈ U}. We call frames of this kind square frames. The above embedding is validitypreserving in the sense that a modalML2-formula φ is valid in all square frames iff its translation φ† is a first-order validity.