Expressivity of probabilistic modal logic, the easy way

Probabilistic modal logic is a very basic modal logic (propositional logic plus a diamond-like modality) interpreted on probabilistic transition systems. It is expressive, i.e., its logical equivalence coincides with bisimilarity, and the proof of this is easy.

Conjunctive and disjunctive fragments of PML turn out to be expressive as well. Contrasting with the proof for the full logic, original proofs of these results span several pages, and invoke measure theory and/or coalgebra theory.

I will present short and completely elementary expressivity proofs for both fragments, and more!