The tautologies and admissible rules of a formal system may exceed those of its underlying logic. For example, the Diaconescu–Goodman–Myhill-theorem shows that the axiom of choice entails the law of excluded middle with respect to many intuitionistic set theories.The goal of this dissertation is, roughly speaking, to study situations where this is not the case. We show that many intuitionistic and constructive set theories are loyal to their underlying logic: We say that a formal system is (propositional/first-order) tautology loyal if its (propositional/first-order) tautologies are exactly those of its underlying logic. We call a formal system (propositional/first-order) rule loyal if its (propositional/first-order) admissible rules are exactly those of its underlying logic.Using Kripke models with classical domains, we show that intuitionistic Kripke--Platek set theory (IKP) is first-order loyal. Moreover, we introduce a realisability notion based on Ordinal Turing Machines that allows us to prove that IKP is propositional rule loyal, as well. This notion of realisability also lends itself to realising infinitary set theories.We introduce blended models for intuitionistic Zermelo--Fraenkel set theory (IZF) to show that IZF is propositional tautology loyal. A variation of this technique allows to prove that various constructive set theories are propositional rule loyal.Finally, we also prove that constructive Zermelo--Fraenkel set theory (CZF) is both first-order tautology loyal and propositional rule loyal. To this end, we introduce a new notion of transfinite computability, the so-called Set Register Machines and combine the resulting notion of realisability with Beth models.