Non-Fregean propositional logic with quantifiers

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