In universal logic, S-norm is the mathematical model of “OR” operation. S-norm and S-generator were defined on interval [0, 1] in previous work. In the related work, authors put forward a kind of logic based on generalized interval [a, b]. This paper studied the S-norm and S-generator on any interval [a, b], discussed the two kinds of generalized S-generators: “Automorphic increase S-generator” and “Infinite decrease S-generator”, and proved the important generating theorem of generalized S-norm. Based on the conceptions of integrated cluster of generalized S-norm and S-generator, authors put forward a new uniform “OR” operation model. The simulation shows that it is not only flexible but controllable. It enlarged the study domain of universal logic, and offered important theory for uncertainties reasoning of complex system.