On the closure ordinals of first-order logics of probability with distribution on possible worlds

We are going to discuss why, in terms of closure ordinals, a great many of infinitary calculi for first-order logics of probability of ‘type 2’, which utilize distributions on possible worlds (like various probability logics studied by J. Y. Halpern, Z. Ognjanović and others), are as hard as possible: the corresponding closure ordinals coincide with $\omega_1^{\mathrm{CK}}$, i.e. the least non-constructive ordinal.