The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency.

多年来，所谓的实质蕴含悖论推动了许多非经典逻辑的发展，例如相关逻辑，超一致逻辑，模糊逻辑等。在本说明中，我们以最小的逻辑研究了其中一些悖论并将其分类。我们在适当的地方提供了等效的证明和语义模型，可将悖论分开。出现了许多等价的基团，所有这些基团由于无限制地使用双重否定消除而崩溃。有趣的是，前另类原则和一些较弱的原则被证明是可区分的，这或许为在可能存在不一致的情况下采用最小逻辑作为推理的环境逻辑提供了支持动力。