It is well-known that Sobol indices, which count among the most popular sensitivity indices, are based on the Sobol decomposition. Here we challenge this construction by redefining Sobol indices without the Sobol decomposition. In fact, we show that Sobol indices are a particular instance of a more general concept which we call sensitivity measures. A sensitivity measure of a system taking inputs and returning outputs is a set function that is null at a subset of inputs if and only if, with probability one, the output actually does not depend on those inputs. A sensitivity measure evaluated at the whole set of inputs represents the uncertainty about the output. We show that measuring sensitivity to a particular subset is akin to measuring the expected output's uncertainty conditionally on the fact that the inputs belonging to that subset have been fixed to random values. By considering all of the possible combinations of inputs, sensitivity measures induce an implicit symmetric factorial experiment with two levels, the factorial effects of which can be calculated. This new paradigm generalizes many known sensitivity indices, can create new ones, and defines interaction effects independently of the choice of the sensitivity measure. No assumption about the distribution of the inputs is required.