We treat the problem of reasoning with ambiguous propositions. Even though ambiguity is obviously problematic for reasoning, it is no less obvious that ambiguous propositions entail other propositions (both ambiguous and unambiguous), and are entailed by other propositions. This article gives a formal analysis of the underlying mechanisms, both from an algebraic and a logical point of view. The main result can be summarized as follows: sound (and complete) reasoning with ambiguity requires a distinction between equivalence on the one and congruence on the other side: the fact that \(\alpha \) entails \(\beta \) does not imply \(\beta \) can be substituted for \(\alpha \) in all contexts preserving truth. Without this distinction, we will always run into paradoxical results. We present the (cut-free) sequent calculus \(\mathsf {AL}^{\textit{cf}}\), which we conjecture implements sound and complete propositional reasoning with ambiguity, and provide it with a language-theoretic semantics, where letters represent unambiguous meanings and concatenation represents ambiguity.

我们用模棱两可的命题来处理推理问题。尽管歧义显然在推理上是有问题的，但歧义命题必然包含其他命题（模棱两可和模棱两可），而其他命题也必然存在歧义。本文从代数和逻辑的角度对底层机制进行了正式分析。主要结果可归纳如下：声音（和完整）与模糊推理需要在在另一侧的一个和同余等价区别的事实：\（\阿尔法\）嗣继承\（\测试\）并没有暗示可以用\（\ beta \）代替\（\ alpha \）在所有情况下都保持真理。没有这种区别，我们将总是遇到矛盾的结果。我们介绍了（免割）后续演算\（\ mathsf {AL} ^ {\ textit {cf}} \），我们猜想该演算可实现含糊的合理命题推理，并提供语言理论上的语义，字母表示明确的含义，而串联表示模棱两可。