It is shown that the many-dimensional modal logic K n , determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any $n > 2$ . On the other hand, K n is determined by a class of frames satisfying a single first-order sentence.

It is shown that the many-dimensional modal logic K$^n$, determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any $n > 2$. On the other hand, K$^n$ is determined by a class of frames satisfying a single first-order sentence.