G\"odel-Dummett and $\mathsf{BD_2}$: Linearity and Depth-Two Branching in Kripke Semantics

We study the semantic relationship between G\"odel-Dummett logic $\mathsf{GL}$ and bounded-depth-2 logic $\mathsf{BD_2}$, two well-known intermediate logics. While $\mathsf{GL}$ imposes linearity on Kripke frames, $\mathsf{BD_2}$ bounds their depth to two. We prove these logics are incomparable (neither contains the other) through minimal frame conditions. Notably, their combination $\mathsf{GL+BD_2}$ collapses to the logic of one or two world frames, bringing it remarkably close to classical logic. This illustrates how controlling breadth and depth in intuitionistic semantics leads to mutually exclusive structural constraints. Finally, we give a conceptual and philosophical interpretation of the previous results. This is an extended abstract of work in progress. Comments and suggestions welcome at: [email protected]