Semantics for Some Intermediate Logics

Milan Božić

We give semantics for intermediate logics of the form $H+\vee S$, where $\vee S$ is the schema $$ \underset{(i,j)\in S}\to\vee(A_i\to A_j) $$ and $S$ is a nonempty subset of $\{1,\ldots,n\}^2$. It is proved that such a logic is complete with respect to the class of Kripke frames $(X,R)$ which satisfy the universal closure of the formula $$\underset{(i,j),(k,i)\in S}\to\vee x_{ij}Rx_{ki} $$