This paper provides some model-theoretic analysis for probability (modal) logic (⁠|$PL$|⁠). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of |$PL$|⁠, namely basic probability logic (⁠|$BPL$|⁠), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of |$PL$|⁠, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending |$PL$| and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models (⁠|$\mathcal{F}\mathcal{P}\mathcal{M}$|⁠) and introducing positive sublogic of |$PL$| including |$BPL$|⁠, it is proved that this sublogic possesses the compactness property with respect to |$\mathcal{F}\mathcal{P}\mathcal{M}$|⁠.

中文翻译：

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概率逻辑：模型理论的观点

本文提供了概率（模态）逻辑（⁠| $ PL $ |⁠）的一些模型理论分析。已知该逻辑不具有紧凑性。但是，通过传入| $ PL $ |⁠子逻辑（即基本概率逻辑（⁠| $ BPL $ |⁠）），可以证明该逻辑满足紧凑性。此外，通过特别关注| $ PL $ |⁠的某些基本模型理论性质，研究了Lindström表征定理的一个版本。实际上，已证明概率逻辑在扩展| $ PL $ |的那些抽象逻辑中具有最大的表达能力。并同时满足过滤和脱节联合属性。最后，通过将语义替换为有限加性概率模型（⁠| $ \ mathcal {F} \ mathcal {P} \ mathcal {M} $ |⁠），并引入| $ PL $ |的正子逻辑。包括| $ BPL $ |⁠，已证明该子逻辑相对于| $ \ mathcal {F} \ mathcal {P} \ mathcal {M} $ |⁠具有紧凑性。