Proper definitions for Risk and Uncertainty (July update with (a) better notation, (b) relative risk (c) an application to insurance)

The commonly adopted definitions of risk and uncertainty generate conceptual problems and inconsistencies, and they are a source of confusion in general. However, alternative and proper definitions are: (1) First there is the distinction between certainty and uncertainty. (2) Uncertainty forks into known (assumed) and unknown probabilities. (3) Unknown probabilities forks into known categories and unknown categories. (4) Known categories forks into 'including the uncertainties in the probabilities by explicitly assuming a uniform distribution' (Laplace) or neglect (or use other non-probabilistic techniques). Note that the term 'risk' has not been used in the 4 points above, so that an independent definition is possible. 'Risk' can be defined as the absolute value of probable loss, i.e. as rho = -E[x < 0]. Also, relative risk is the probable loss with respect to a target t, giving rho(t) = t - E[x < t]. The definitions provided here are directly in line with the Oxford English dictionary. It turns out that economics textbooks generally can keep their mathematics but will best rewrite their texts to these definitions. Not only the students and the general public will benefit from this sudden clarity, but eventually also economic theory itself. Note: This update on the February 1999 version has: (a) better notation, (b) relative risk (c) an application to insurance