Abstract <p>The issue of non-equilibrium states and proof of the Clausius inequality in classical thermodynamics is examined in the context of the works of famous theoretical physicist and mathematician Truesdell, as well as Uffink’s paper criticizing classical thermodynamics. The paper discusses aspects of the history of thermodynamics of the 19th century, the development of non-equilibrium thermodynamics, as well as the question of terminology—how thermodynamics should be called, classical or phenomenological. The Clausius inequality is interpreted as a research program that connects classical thermodynamics, continuum mechanics, and non-equilibrium thermodynamics.</p>