There are alternatives to the Neyman–Pearson formulation of the problem of testing statistical hypotheses. Although it is important that we recognize and understand the differences between the various formulations, there is no standard terminology to help us. Many authors have distinguished between what we are calling Neyman–Pearson tests and tests that have a different form and purpose, often calling the latter significance tests and usually citing R.A. Fisher as a particularly influential developer or proponent. Although Fisher was not the originator of significance tests, we call them ‘Fisherian’ because of his consistent emphasis on the distinction between the problems addressed by the Neyman–Pearson theory of hypothesis testing and problems of evidential interpretation of scientific data, for which significance tests are intended. We will draw a further distinction, describing two varieties of significance test, both of which seem to have been advocated by Fisher. The first we will call p-value procedures, and will consider in sections 3.2–3.4. These are prominent in the statistical analyses used in science. The second variety, also influential in scientific applications, we call rejection trials. These are particularly interesting because they link statistical hypothesis testing directly to formal logic and to the philosophy of science; they will be discussed in section 3.5.