The thesis underlying this paper is that informal rigour – a term famously coined by Kreisel in [63] – corresponds to a more definite method than has previously been recognized. We will illustrate this by first presenting a scheme for what we will refer to as an informally rigorous argument and then showing in detail how the three central examples considered by Kreisel [64] conform to this. These respectively pertain to the analysis of first-order validity, the status Markov’s Principle in intuitionistic analysis, and the definiteness of the Continuum Hypothesis. Upon comparing these arguments systematically, we will suggest that the latter is the most tendentious in virtue of the status which Kreisel himself came to assign to the notion of second-order validity. We conclude by providing some brief observations about the scope and continued significance of Kreisel’s method.
