We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal {M}$ , or an axiomatization S thereof, we find a modal logic M such that a modal sentence $\varphi $ is a theorem of M if and only if the sentence $\varphi ^*$ obtained by translating the modal operator with the truth predicate is true in $\mathcal {M}$ or a theorem of S under all such translations. To this end, we introduce a novel version of possible worlds semantics featuring both classical and nonclassical worlds and establish the completeness of a family of noncongruent modal logics whose internal logic is nonclassical with respect to this semantics.

中文翻译：

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克里普克-费尔曼真理的模态逻辑

我们通过 Solovay 风格的完整性结果确定了 Solomon Feferman 的定点真值模型及其公理化的模态逻辑。给定一个定点模型$\数学{M}$, 或公理化小号其中，我们找到了一个模态逻辑米这样一个情态句$\varphi $是一个定理米当且仅当句子$\varphi ^*$通过用真值谓词翻译模态算子获得$\数学{M}$或一个定理小号在所有此类翻译下。为此，我们引入了具有经典和非经典世界的可能世界语义的新版本，并建立了一系列非全等模态逻辑的完整性，其内部逻辑就该语义而言是非经典的。