 One of the simplest intermediate propositional logics is the Gödel-Dummett logic \(\mathbf{LC},\) obtained by adding to \(\mathbf{IPC}\) the schema \((A \rightarrow B) \vee (B \rightarrow A)\) which is valid on all and only those Kripke frames in which the partial order of the nodes is linear. Gödel [1932] used an infinite sequence of successively stronger intermediate logics to show that \(\mathbf{IPC}\) has no finite truth-table interpretation. For each positive integer \(n,\) let \(\mathbf{G_n}\) be \(\mathbf{LC}\) plus the schema \((A_1 \rightarrow A_2) \vee \ldots \vee (A_1 \oldand \ldots \oldand A_n \rightarrow A_{n + 1}).\) Then \(\mathbf{G_n}\) is valid on all and only those linearly ordered Kripke frames with no more than \(n\) nodes. 