The hypothetical notion of consequence is normally understood as the transmission of a categorical notion from premisses to conclusion. In model-theoretic semantics this categorical notion is ‘truth’, in standard proof-theoretic semantics it is ‘canonical provability’. Three underlying dogmas, (I) the priority of the categorical over the hypothetical, (II) the transmission view of consequence, and (III) the identification of consequence and correctness of inference are criticized from an alternative view of proof-theoretic semantics. It is argued that consequence is a basic semantical concept which is directly governed by elementary reasoning principles such as definitional closure and definitional reflection, and not reduced to a categorical concept. This understanding of consequence allows in particular to deal with non-wellfounded phenomena as they arise from circular definitions.