Non-Fregean Propositional Logic with Quantifiers

Abstract

We study the non-Fregean propositional logic with propositional quantifiers, denoted by ${\mathsf{SCI}}_{\mathsf{Q}}$. We prove that ${\mathsf{SCI}}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in ${\mathsf{SCI}}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of ${\mathsf{SCI}}_{\mathsf{Q}}$-sentences. Finally, we present a translation of ${\mathsf{SCI}}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove some model-theoretic properties of ${\mathsf{SCI}}_{\mathsf{Q}}$.