 Without compactness, a logic might be weakly complete (every valid formula is provable in the axiomatic system), but not strongly complete (for every set \(\Gamma\) of formulas, every logical consequence of \(\Gamma\) is provable from \(\Gamma\) in the axiomatic system). In Fagin et al. (1990), a proof system involving linear combinations was given and the logic was shown to be both sound and weakly complete. In Ognjanović and Rašković (1999), a sound and strongly complete proof system is given for propositional probability logic without linear combinations. In Heifetz and Mongin (2001), a proof system for a variation of the logic without linear combinations that uses a system of types to allow for iteration of probability formulas (we will see in   Section 4  how such iteration can be achieved using possible worlds) was given and the logic was shown to be sound and weakly complete. They also observe that no finitary proof system for such a logic can be strongly complete. Ognjanović et al. (2008) present some qualitative probabilistic logics with infinitary derivation rules (which require a countably infinite number of premises), and prove strong completeness. Goldblatt (2010) presents a strongly complete proof system for a related coalgebraic logic. Perović et al. (2008) give a proof system and proof of strong completeness for propositional probability logic with polynomial weight formulas. Finally, another strategy for obtaining strong completeness involves restricting the range of the probability functions to a fixed, finite set of numbers; for example, Ognjanović et al. (2008) discuss a qualitative probabilistic logic in which the range of the probability functions is not the full real unit interval \([0,1]\), but rather the ‘discretized’ version \(\{0,\frac{1}{n},\frac{2}{n},\dots,\frac{n-1}{n},1\}\) (for some fixed number \(n\in\mathbb{N}\)). See Chapter 7 of Ognjanović et al. (2016) for an overview of completeness results.