David Kaplan observed in Kaplan ( 1995 ) that the principle ∀ p ♢ ∀ q ( Q q ⇔ q = p ) $\forall p \Diamond \forall q (Qq \leftrightarrow q = p)$ cannot be verified at a world in a standard possible worlds model for a quantified bimodal propositional language. This raises a puzzle for certain interpretations of the operator Q : it seems that some proposition p is such that is not possible to query p , and p alone. On the other hand, Arthur Prior had observed in Prior ( 1961 ) that on pain of contradiction, ∀ p ( Q p →¬ p ) is Q only if one true proposition is Q and one false proposition is Q . The two observations are related: ∀ p ( Q p →¬ p ) is elusive in that it is not possible for the proposition to be uniquely Q . Kaplan based his model-theoretic observation on Cantor’s theorem, but there is a less well-known link between this simple set-theoretic observation and Prior’s remark. We generalize the link to develop a heuristic designed to move from Cantor’s theorem to the observation that a variety of sentences of the bimodal language express propositions that cannot be Q uniquely. We highlight the analogy between some of these results and some set-theoretic antinomies and suggest that the phenomenon is richer than one may have anticipated.

中文翻译：

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难以捉摸的命题

David Kaplan 在 Kaplan ( 1995 ) 中观察到原则 ∀ p ♢ ∀ q ( Q q ⇔ q = p ) $\forall p \Diamond \forall q (Qq \leftrightarrow q = p)$ 不能在 a 的世界中得到验证量化双峰命题语言的标准可能世界模型。这对运算符 Q 的某些解释提出了一个难题：似乎某些命题 p 是不可能查询 p 和 p 单独的。另一方面，Arthur Prior 在Prior (1961) 中观察到，在矛盾的痛苦中，仅当一个真命题为 Q 且一个假命题为 Q 时，∀ p ( Q p →¬ p ) 才是 Q 。这两个观察是相关的： ∀ p ( Q p →¬ p ) 是难以捉摸的，因为该命题不可能是唯一的 Q 。卡普兰基于康托定理的模型理论观察，但是在这个简单的集合论观察和Prior 的评论之间有一个鲜为人知的联系。我们将链接概括为开发一种启发式方法，旨在从康托尔定理转移到双峰语言的各种句子表达不能唯一 Q 的命题的观察。我们强调其中一些结果与一些集合论二律背反之间的类比，并表明这种现象比人们预期的要丰富。