We present the first-order logic of change, which is an extension of the propositional logic of change $\textsf {LC}\Box $ developed and axiomatized by Świętorzecka and Czermak (2015, Log. et Anal., 232, 511–527). $\textsf {LC}\Box $ has two primitive operators: ${\mathcal {C}}$ to be read it changes whether and $\Box $ for constant unchangeability. It implements the philosophically grounded idea that with the help of the primary concept of change it is possible to define the concept of time. One of the characteristic axioms for ${\mathcal {C}}$ is ${\mathcal {C}} A \to {\mathcal {C}}\neg A$ and one of the primitive rules is $\omega $-rule introducing $\Box $. It turns out that the next operator is definable in $\textsf {LC}\Box $. Logic $\textsf {LC}\Box $ with next is equivalent to certain fragment of $\textsf {LTL}$ extended by the appropriate definition of ${\mathcal {C}}$. Recently, $\textsf {LC}\Box $ has also been modified to the propositional logic of ‘branching’ changes $\textsf {BTC}$ by Łyczak (2022, Log. J. IGPL) and the logic of ‘parallel’ changes $\textsf {LC}\Box $ ↳ by Świętorzecka and Łyczak (2022, Log. Log. Philios.). Extended in an appropriate manner, $\textsf {BTC}$ is equivalent to a certain fragment of logic $\textsf {CTL}$ to which definitions of two kinds of changes have been added. Here we propose an extension of $\textsf {LC}\Box $ to first-order logic in which, again, the only primitive modal operators are ${\mathcal {C}}$ and $\Box $. We interpret our logic in the semantics of histories of changes. We give an axiomatic system $\textsf {LC}\Box Q$ for the considered logic, and we show selected theses about some relationships between ${\mathcal {C}}$, $\Box $ and $\forall $ (in particular, we prove two versions of the Barcan formula). Then we prove the completeness of $\textsf {LC}\Box Q$. Finally, we compare our logic with $\textsf {FOLT}$ and show the relation between $\textsf {LC}\Box Q$ and a certain fragment of the Kröger system $\varSigma $ to which the definition of ${\mathcal {C}}$ was added.

中文翻译：

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变化的一阶逻辑

我们提出了变化的一阶逻辑，它是由 Świętorzecka 和 Czermak (2015, Log. et Anal., 232, 511–527) 开发和公理化的变化的命题逻辑 $\textsf {LC}\Box $ 的扩展）。$\textsf {LC}\Box $ 有两个原始运算符：${\mathcal {C}}$ 用于读取它是否改变和 $\Box $ 用于恒定不变性。它实现了基于哲学的理念，即借助变化的主要概念，可以定义时间的概念。${\mathcal {C}}$ 的特征公理之一是 ${\mathcal {C}} A \to {\mathcal {C}}\neg A$，原始规则之一是 $\omega $-引入 $\Box $ 的规则。事实证明，下一个运算符可以在 $\textsf {LC}\Box $ 中定义。逻辑 $\textsf {LC}\Box $ 等价于 $\textsf {LTL}$ 的某个片段，该片段由 ${\mathcal {C}}$ 的适当定义扩展。最近，$\textsf {LC}\Box $ 也被 Łyczak (2022, Log. J. IGPL) 修改为“分支”变化 $\textsf {BTC}$ 的命题逻辑和“并行”变化的逻辑$\textsf {LC}\Box $ ↳，作者 Świętorzecka 和 Łyczak（2022，Log.Log.Philios.）。以适当的方式扩展，$\textsf {BTC}$ 相当于添加了两种变化定义的逻辑$\textsf {CTL}$ 的某个片段。在这里，我们建议将 $\textsf {LC}\Box $ 扩展到一阶逻辑，其中唯一的原始模态运算符是 ${\mathcal {C}}$ 和 $\Box $。我们用变化历史的语义来解释我们的逻辑。我们为所考虑的逻辑给出了一个公理系统 $\textsf {LC}\Box Q$，并展示了关于 ${\mathcal {C}}$、$\Box $ 和 $\forall $ 之间的一些关系的选定论文（在特别是，我们证明了 Barcan 公式的两个版本）。然后我们证明$\textsf {LC}\Box Q$ 的完备性。最后，我们将我们的逻辑与 $\textsf {FOLT}$ 进行比较，并显示 $\textsf {LC}\Box Q$ 与 Kröger 系统 $\varSigma $ 的某个片段之间的关系，其中 ${\mathcal 的定义{C}}$ 已添加。