We propose a reading of Kreisel's informal rigour as a dialectic interplay between a formal and an intuitive level in the study of logically relevant notions. To substantiate our reading, we provide two case-studies: Prawitz semantics and Walton's analysis of argument schemes. Prawitz's semantics suggests an interaction between informal presuppositions, confirmed and refined by results about formal setups, and formal setups, which can be improved by suitably formalising informal presuppositions. In Walton's Argumentation Theory, argument schemes are introduced to account for arguments that support their conclusion in a presumptive way. The possibility of developing such informal notion through the formal means provided by research in AI shows the importance of the interaction between these two levels in the analysis of defeasibility.