The Case for Non-Classical Logic

In this introductory level talk, I lay out the main motivations for adopting non-classical logics, from the 'naive' theory of truth: such logics promise to let us keep Tarski's unrestricted truth schema. Pros and cons are considered, for example weighing the gain in expressive power against the (apparent) loss of inferential power. Symmetries between theories with truth value gaps and truth value gluts are considered. I then survey some recent developments in substructural logic, and point to ongoing challenges and prospects for these projects.