Learning the ABC of statistical mechanics

This is a great moment. I’ve collected all the three puzzle pieces of classical mechanics, quantum mechanics and statistical mechanics. Now, I finally get started to learn physics.

 

A relatively rigorous statement of the 2nd Law of Thermodynamics:

It is almost impossible for the sum of the entropies of the microcanonical ensembles of all the macroscopic subsystems of an isolated system which has not reached statistical equilibrium to decrease.

a priori

a priori judgement: proposition to be justified independent of all experience beyond what is needed to grasp the concepts involved

a posterior judgement: proposition to be justified dependent on some experience beyond what is needed to grasp the concepts involved

analytic judgement: proposition in which all concepts are related solely by their definitions

synthetic judgement: proposition in which some concepts are not related solely by their definitions

How is a synthetic proposition a priori possible?