Modal QUARC and Barcan

Abstract I develop a modal extension of the QUantified ARgument Calculus ()—a novel logical system introduced by Hanoch Ben-Yami. is meant to better capture the logic of natural language. The purpose of this paper is to evaluate this claim by considering how modal () handles the Barcan and Converse Barcan Formulas and how this correlates to surrounding debates. To do so, I develop a variable domain semantics for and show that even if the usual domain conditions are imposed on models with variable domains, simple - analogues of the Barcan and Converse Barcan Formulas are not valid. I introduce new conditions and show that they validate the formulas. I go on to extend the language of to simulate unrestricted quantification and show that in this setting the domain conditions validate the Barcan and Converse Barcan Formulas. Based on these results, I evaluate the relationship of the formal systems and standard quantified modal logic with respect to one another and to natural language. I argue that if does capture the logic of natural language, then natural language is capable to express neither the Barcan and Converse Barcan Formulas nor counterexamples to them. Given that, I raise doubts that captures quantification as it occurs in natural language.