A Simple Logic of Concepts

Thomas F. Icard III, Lawrence S. Moss

Abstract: In Pietroski (2018) a simple representation language called SMPL is introduced, construed as a hypothesis about core conceptual structure. The present work is a study of this system from a logical perspective. In addition to establishing a completeness result and a complexity characterization for reasoning in the system, we also pinpoint its expressive limits, in particular showing that the fourth corner in the square of opposition (“Some_not”) eludes expression. We then study a seemingly small extension, called SMPL+, which allows for a minimal predicate-binding operator. Perhaps surprisingly, the resulting system is shown to encode precisely the concepts expressible in first-order logic. However, unlike the latter class, the class of SMPL+ expressions admits a simple procedural (context-free) characterization. Our contribution brings together research strands in logic—including natural logic, modal logic, description logic, and hybrid logic—with recent advances in semantics and philosophy of language.