I present the principles of a logic I have developed, in which quantified arguments occur in the argument position of predicates. That is, while the natural language sentence 'Alice is polite' is formalised P(a), the sentence 'Some students are polite' is formalised P(∃S). In several ways, this logic is closer to Natural Language more than is any verson of Frege's Predicate Calculus (PC). I proceed to discuss further features of this new logic, the Quantified Argument Calculus (Quarc). For instance, Quarc incorporates, like Natural Language and unlike PC, both sentential negation and predication negation, as well as converse relation-terms. It also sheds light on the necessity for expressive completeness of Natural Language of these devices. The use of anaphors vis-à-vis variables is also discussed. I next describe the system's power -- it is no less powerful than first-order PC -- and say a few words on its meta-logical properties. I then extend Quarc to modal logic and show how its version of the Barcan Formulas and of their converses come out straightforwardly invalid, which is arguably an advantage of modal Quarc over modal PC. Finally, I mention forthcoming papers by other authors, which extend Quarc in various ways, and also directions for further work, some currently pursued.