 The strategy of defining a  product  of modal logics (introduced in Segerberg 1973 and Šehtman 1978) shares the idea of using a language that is freely generated by the union of the modalities of the original languages. But, on the semantic side, the approach is quite different: instead of working on a one-dimension domain, it works on a multi-dimensional domain that has one dimension for each one of the involved aspects (i.e., modalities). More precisely, if the semantic models of the to-be-combined systems are \(M_1 = {\langle W_1, R_1, V_1 \rangle}\) and \(M_2 = {\langle W_2, R_2, V_2 \rangle}\), the models where formulas of the resulting language are evaluated are now of the form \(M' = {\langle W_1 \times W_2, R'_1, R'_2, V_1 \times V_2 \rangle}\). The domain \(W_1 \times W_2\) is, then, the standard Cartesian product of the original domains, and the valuation \(V_1 \times V_2\) is such than an atom p is true in a world \((w_1, w_2)\) if and only if p was true at \(w_1\) in \(M_1\) and also true at \(w_2\) in \(M_2\) (i.e., \((V_1 \times V_2)(p) := V_1(p) \times V_2(p))\). For the relations, each one of them is given as in their original models, restricted now to their respective dimensions: 