  Since the 1909 work of Carath\'eodory, formulations of thermodynamics have
gained ground which highlight the role of the the binary relation of adiabatic
accessibility between equilibrium states. A feature of Carath\'eodory's system
is that the version therein of the second law contains an ambiguity about the
nature of irreversible adiabatic processes, making it weaker than the
traditional Kelvin-Planck statement of the law. This paper attempts first to
clarify the nature of this ambiguity, by defining the arrow of time in
thermodynamics by way of the Equilibrium Principle ("Minus First Law"). It then
argues that the ambiguity reappears in the important 1999 axiomatisation due to
Lieb and Yngvason.
