On the Complexity of Modal Axiomatisations over Many-dimensional Structures.
Agi Kurucz · 2010
We show that all the complexities of a possible axiomatisation of S5, the n-modal logic of products of n equivalence frames, are already present in any axiomatisation of Kn. Then we show that if 3 ≤ n < ω then, for any set L of n-modal formulas between Kn and S5, the class of all frames for L is not closed under ultraproducts and is therefore not elementary. So any modal axiomatisation for a Kripke complete logic in the interval between Kn and S5 must contain modal formulas with no first-order correspondents. The proof is based on a construction of Hirsch and Hodkinson [15] showing that the class of strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts. We show that this construction can be carried through in a diagonal-free setting.