Thermodynamic laws are limiting behavior of the statistics of repeated measurements of an arbitrary system with a priori probability distribution. A duality symmetry arises, between Massieu-Guggenheim entropy and Gibbs entropy, in the limit of large number of measurements. This yields the fundamental thermodynamic relation and Hill-Gibbs-Duhem (HGD) equation as a dual pair. We show if the system itself has a second macroscopic limit that satisfies Callen’s postulate that entropy being an Eulerian homogeneous function of all extensive variables, the symmetry is lost: the HGD equation reduces to the Gibbs-Duhem equation. This theory provides better logic to textbook thermodynamics, a clarification on nanothermodynamics, as well as novel ideas for a thermodynamic-like framework for single-cell biology.

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