Thanks for noting our work in this area. This has been worked out in even more detail here:
Lattice duality: The origin of probability and entropy. Neurocomputing. 67C: 245-274. DOI: 10.1016/j.neucom.2004.11.039 http://knuthlab.rit.albany.edu/papers/knuth-neurocomp-05-published.pdf
Its fundamental application to a wide array of problems is first noted and described here arXiv:physics/0403031v1
which was, in part, the inspiration for the quantum mechanics paper you kindly note above: arXiv:0907.0909
Last, I have a more recent derivation of Bayesian probability theory from lattices with simple graphical proofs: arXiv:0909.3684
This work is literally a derivation of measure theory from much the more basic symmetry of associativity. It is immediately applicable to Boolean lattices of statements, which results in Bayesian probability theory. And, as we have shown, results in information theory and the complex Feynman rules of Quantum Mechanics.