My dissertation is composed of three papers: In "Proof-Theoretic Justification of Logic" I develop a theory for justifying introduction rules on the basis of elimination rules and show that intuitionistic logic is the strongest logic which can be justified in this way. In "The Barcan Formulae for Determinacy" I argue that there is a logical problem with denying the Barcan Formula for a determinacy operator and I develop a novel intuitionistic quantified modal logic to get around this problem. In "Is the Vagueness Argument Valid?" I argue that Sider’s "Vagueness Argument" isn’t valid and that this has important consequences for the notions of supervenience and metaphysical dependence.