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We propose a unified framework for both Shannon-Khinchin and Shore-Johnsonaxiomatic systems. We do it by rephrasing Shannon-Khinchine axioms in terms ofgeneralized arithmetics of Kolmogorov and Nagumo. We prove that the twoaxiomatic schemes yield identical classes of entropic functionals -- Uffinkclass of entropies. This allows to re-establish the entropic parallelismbetween information theory and statistical inference that has seemed to be"broken" by the use of non-Shannonian entropies.

When Shannon and Khinchin meet Shore and Johnson: equivalence of information theory and statistical inference axiomatics