Product probability property, known in the literature as statistical independence, is examined first. Then generalized entropies are introduced, all of which give generalizations to Shannon entropy. It is shown that the nature of the recursivity postulate automatically determines the logarithmic functional form for Shannon entropy. Due to the logarithmic nature, Shannon entropy naturally gives rise to additivity, when applied to situations having product probability property. It is argued that the natural process is non-additivity, important, for example, in statistical mechanics [C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1988) 479-487; E.G.D. Cohen, Boltzmann and Einstein: statistics and dynamics-an unsolved problem, Pramana 64 (2005) 635-643.], even in product probability property situations and additivity can hold due to the involvement of a recursivity postulate leading to a logarithmic function. Generalized entropies are introduced and some of their properties are examined. Situations are examined where a generalized entropy of order a leads to pathway models, exponential and power law behavior and related differential equations. Connection of this entropy to Kerridge's measure of ''inaccuracy'' is also explored.