Generalizing the sigmoid function using continuous-valued logic

Abstract

In this study, we present a continuous-valued logical approach to generalize the sigmoid function. Our starting point is the kappa function, which is known as a unary operator in continuous-valued logic. First we extend the kappa function to the (a,b) interval, and then we interpret the generalized sigmoid function as the limit of the extended kappa function when a and b tend to the negative and positive infinity, respectively. Since the extended kappa function is induced by an additive generator of a strict t-norm or strict t-conorm, the generalized sigmoid function is operator dependent. Based on the properties of this new function, we show that it can be viewed as the generalization of the classical sigmoid function. Also, we demonstrate that the classical sigmoid function is a special case of the generalized sigmoid function. Next, we provide a sufficient condition for the equality of two generalized sigmoid functions. It is well known that the classical sigmoid function can be utilized in logistic regression and in preference modeling. Here, we demonstrate how the logistic regression can be generalized using the generator function-based sigmoid function. Also, we show that the generalized sigmoid function can be viewed as a preference measure.