  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, even in
product probability property situations and additivity can hold due to the
involvement of a recursivity postulate leading to a logarithmic function.
Generalizations, including Mathai's generalized entropy are introduced and some
of the properties are examined. Situations are examined where Mathai's entropy
leads to pathway models, exponential and power law behavior and related
differential equations. Connection of Mathai's entropy to Kerridge's measure of
"inaccuracy" is also explored.
