A note on the tense logic of dominoes (Q1187979)

The tense logic of a semantics of a two-dimensional nature, the so-called domino relation, is the subject of this paper. More precisely, a domino- relation is a frame with base set of kind \(U\times U\) and accessibility relation \(R\) such that \((x,y)R(x',y')\iff y=x'\). The author notes that an axiomatization of modal logic for domino-relation semantics is given by S. Kuhn. This axiomatization has infinitely many axioms and a derivation rule of the kind of the Gabbay irreflexivity rule. This paper offers some finite axiomatization for the tense logic of a domino relation which has no rules besides modus ponens and necessitation (right and left type). The main step is the invention of the ``matrix method'', which allows to reach the completeness theorem considerably easier.