You asked, about logical inconsistencies and logical ambiguities in classical thermodynamics and classical statistical mechanics. As far as classical thermodynamics is concerned, there are no inconsistencies or ambiguities. Classical thermodynamics is based on several axioms (known as laws), and a bunch of definitions. All the rest is deduction. If this makes it seem like Euclidian plane geometry, it is.
Classical statistical mechanics (SM), on the other hand, has big problems. SM attempts to compute thermodynamics properties, i.e. macroscopic variables, from microscopic properties. Classical SM assumes that atoms and molecules obey classical mechanics. They don't. The result is that one cannot reproduce macroscopic reality that way without a number of additional assumptions. The problem of course is that on a microscopic level matter inherently obeys quantum mechanics.
For example classical mechanics assumes that identical particles are, in principle, distinguishable because you can mark them with a sufficiently small pen. In quantum mechanics, identical properties are really identical. You can't mark them, and if you interchange the positions and velocities of two of them, NOTHING changes -- so you can't even tell that the interchange occurred.
----- Paul J. Gans