Completeness theorems for σ–additive probabilistic semantics

Abstract

We study propositional probabilistic logics (LPP–logics) with probability operators of the form P≥r (“the probability is at least r”) with σ–additive semantics. For regular infinite cardinals κ and λ, the probabilistic logic LPPκ,λ has λ propositional variables, allows conjunctions of <κ formulas, and allows iterations of probability operators. LPPκ,λ,2 denotes the fragment of LPPκ,λ where iterations of probability operators is not allowed. Besides the well known non-compactness of LPP–logics, we show that LPPκ,λ,2–logics are not countably compact for any λ≥ω1 and any κ, and that are not 2ℵ0+–compact for κ≥ω1 and any λ. 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 σ–additive models. The main result is the strong completeness theorem for countable fragments LPPA and LPPA,2 of LPPω1,ω.