The Phenomenology of Second-Level Inference: Perfumes in The Deductive Garden - Makinson, David (Bulletin of the Section of Logic, 2020)
We comment on certain features that second-level inference rules commonly used in mathematical proof sometimes have, sometimes lack: suppositions, indirectness, goal-simplification, goal-preservation and premise-preservation. The emphasis is on the roles of these features, which we call 'perfumes', in mathematical practice rather than on the space of all formal possibilities, deployment in proof-theory, or conventions for display in systems of natural deduction.