Fifty years ago Birdsall, Tanner, and colleagues made rapid progress in developing signal detection theory into a powerful psychophysical tool. One of their major insights was the utility of adding external noise to the signals of interest. These methods have been enhanced in recent years by the addition of multipass and classification-image methods for opening up the black box. There remain a number of as yet unresolved issues. In particular, Birdsall developed a theorem that large amounts of external input noise can linearize nonlinear systems, and Tanner conjectured, with mathematical backup, that what had been previously thought of as a nonlinear system could actually be a linear system with uncertainty. Recent findings, both experimental and theoretical, have validated Birdsall's theorem and Tanner's conjecture. However, there have also been experimental and theoretical findings with the opposite outcome. In this paper we present new data and simulations in an attempt to sort out these issues. Our simulations and experiments plus data from others show that Birdsall's theorem is quite robust. We argue that uncertainty can serve as an explanation for violations of Birdsall's linearization by noise and also for reports of stochastic resonance. In addition, we modify present models to better handle detection of signals with both noise and pedestal backgrounds.