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Product probability property, known in the literature as statistical independence, is examined first. Then various generalized entropies are introduced, all of which give gen-eralizations to Shannon entropy. It is shown that the na-ture 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 prod-uct probability property. It is argued that the natural process is non-additivity leading to non-extensive statisti-cal mechanics, even in product probability property situ-ations and additivity can hold due to the involvement of a recursivity postulate leading to a logarithmic function. Various generalizations, including Mathai’s generalized en-tropy are introduced and some of the properties are ex-amined. Various situations are examined where Mathai’s entropy leads to pathway models, Tsallis statistics, su-perstatistics, power law and related differential equations. Connection of Mathai’s entropy to Kerridge’s measure of “inaccuracy ” is also explored. Key words:Generalized entropies, Mathai’s entropy, characterizations, additivity, non-extensivity, Tsallis statistics, superstatistics, power law