Completeness theorems for σ-additive probabilistic semantics

We study propositional probabilistic logics (LPP-logics) with probability operators of the form P->= r ("the probability is at least r") with sigma-additive semantics. For regular infinite cardinals kappa and lambda, the probabilistic logic LPP kappa,lambda has lambda propositional variables, allows conjunctions of < kappa, formulas, and allows iterations of probability operators. LPP kappa,lambda,2 denotes the fragment of LPP kappa,lambda where iterations of probability operators is not allowed. Besides the well known non-compactness of LPP-logics, we show that LPP kappa,lambda,2-logics are not countably compact for any lambda >= omega(1) and any kappa and that are not 2(aleph 0+)-compact for kappa >= omega(1) and any lambda. We prove the equivalence of our adaptation of the Hoover's continuity rule (Rule (5) in [13]) and Goldblat's Countable Additivity Rule [9] and show their necessity for complete axiomatization with respect to the class of all sigma-additive models. The main result is the strong completeness theorem for countable fragments LPPA and LPPA,2 of LPP omega 1,omega. (C) 2019 Published by Elsevier B.V.