Justification logics are a family of modal logics whose non-normal modalities are parametrised by a type-theoretic calculus of terms. The first justification logic was developed by Sergei Artemov to provide an explicit modal logic for arithmetical provability in which these terms were taken to pick out proofs. But, justification logics have been given various other interpretations as well. In this paper, we will rely on an interpretation in which the modality $$\llbracket \tau \rrbracket \varphi $$ is read ‘S accepts $$\tau $$ as justification for $$\varphi $$ ’. Since it is often important to specify just how much confidence agents have in propositions on the basis of justifications, the logic will need to be extended if it is to provide a sufficiently general account. The primary purpose of this paper is to extend justification logic with the expressive resources to needed to do so. Thus, we will construct the justification logic with confidence ( $${\mathsf {JC}}$$ ). While $${\mathsf {JC}}$$ will be extremely general, capable of accommodating a wide range of interpretations, we provide motivation in terms of the notion of confidence deriving recent work by Paul and Quiggin (Episteme 15(3):363–382, 2018). Under this understanding, confidence must only correspond to a partial ordering. We axiomatise $${\mathsf {JC}}$$ and provide a sound and complete semantics.

中文翻译：

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有信心的证明逻辑

证明逻辑是一系列模态逻辑，其非正规模态由项的类型理论演算参数化。第一个证明逻辑是由 Sergei Artemov 开发的，它为算术可证明性提供了一个明确的模态逻辑，其中这些项被用来挑选证明。但是，证明逻辑也被赋予了各种其他解释。在本文中，我们将依赖一种解释，其中模态 $$\llbracket \tau \rrbracket \varphi $$ 读作'S 接受 $$\tau $$ 作为 $$\varphi $$ 的理由'。由于在论证的基础上指定代理人对命题的信心程度通常很重要，因此如果要提供足够概括的说明，则需要扩展逻辑。本文的主要目的是将证明逻辑扩展到所需的表达资源。因此，我们将有信心地构建证明逻辑（ $${\mathsf {JC}}$$ ）。虽然 $${\mathsf {JC}}$$ 将非常通用，能够适应广泛的解释，但我们提供了源自 Paul 和 Quiggin 最近工作的信心概念的动机（Episteme 15(3)： 363–382, 2018)。在这种理解下，置信度只能对应于偏序。我们公理化 $${\mathsf {JC}}$$ 并提供健全和完整的语义。我们根据源自 Paul 和 Quiggin 最近工作的信心概念提供动机（Episteme 15(3):363–382, 2018）。在这种理解下，置信度只能对应于偏序。我们公理化 $${\mathsf {JC}}$$ 并提供健全和完整的语义。我们根据源自 Paul 和 Quiggin 最近工作的信心概念提供动机（Episteme 15(3):363–382, 2018）。在这种理解下，置信度只能对应于偏序。我们公理化了 $${\mathsf {JC}}$$ 并提供了健全和完整的语义。