In this paper we present a probability logic which allows Boolean combinations of formulas of the form of $a_1w(\varphi_1)+\cdots+a_kw(\varphi_k)\geq c$, where $\varphi_1,\ldots,\varphi_k$ are propositional formulas, $a_1,\ldots,a_k,c$ are rational numbers. The logic with such syntax was introduced in [1]. We present axiomatic system along with the ideas from [2, 3] and prove extended completeness, instead of simple completeness shown in [1].