This thesis is based on the following paper Goudsmit, J.P.: “Admissibility and refutation: Some characterisations of intermediate logics” Utrecht University, Janskerkhof 13.3512 BL, Utrecht, The Netherlands. Springer–Verla, Berlin Heidelberg (2014) According to ?ukasiewicz, we assert true propositions, and reject false ones. He remarked that rejection had been neglected in the study of formal logic, and introduced a formal system to inductively derive rejections of false propositions. We call such systems refutation systems, following Scott and Skura. The general theory of such systems has been studied extensively by S?upecki et al. A refutation system can be thought of as a proof system for rejection. Instead of deriving that one can correctly assert a statement through a series of truth-preserving inferences from given axioms, as one does in a proof system of assertion, one derives the refutability of a propositional statement through a series of non-truth preserving inferences from given anti-axioms. Proofs in a refutation system will be called refutations, and a formula will be called refutable whenever a refutation exists ending in this formula. Let us, by way of example, present a reformulation of the original refutation system for the stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" o:preferrelative="t" o:spt="75" coordsize="21600,21600" denotes a propositional variable, ? and both denote propositional formulae, and denotes a substitution. This refutation system is both sound (all refutable formulae are not derivable in CPC) and complete (all formulae that are not derivable in CPC are refutable). Refutation systems are formal systems for inferring the falsity of formulae. These systems can, in particular, be used to syntactically characterise logics. In this thesis, we focus on refutation systems for intermediate logics. we explore the close connection between refutation systems and admissible rules. In particular, we provide a refutation system for the logics of bounded branching , also known as the Gabbay–de Jongh logics, making use of admissible rules similar to those given by Goudsmit and Iemhoff. We develop technical machinery to construct refutation systems, employing techniques from the study of admissible rules. Concretely, we provide a refutation system for the intermediate logics of bounded branching, known as the Gabbay–de Jongh logics. We show that this gives a characterisation of these logics in terms of their admissible rules. To illustrate the technique, we also provide a refutation system for Medvedev’s logic.