Statistical Hypothesis Testing (SHT) is a class of inference methods whereby one makes use of empirical data to test a hypothesis and often emit a judgment about whether to reject it or not. In this paper, we focus on the logical aspect of this strategy, which is largely independent of the adopted school of thought, at least within the various frequentist approaches. We identify SHT as taking the form of an unsound argument from Modus Tollens in classical logic, and, in order to rescue SHT from this difficulty, we propose that it can instead be grounded in t-norm based fuzzy logics. We reformulate the frequentists’ SHT logic by making use of a fuzzy extension of Modus Tollens to develop a model of truth valuation for its premises. Importantly, we show that it is possible to preserve the soundness of Modus Tollens by exploring the various conventions involved with constructing fuzzy negations and fuzzy implications (namely, the S and R conventions). We find that under the S convention, it is possible to conduct the Modus Tollens inference argument using Zadeh’s compositional extension and any possible t-norm. Under the R convention we find that this is not necessarily the case, but that by mixing R-implication with S-negation we can salvage the product t-norm, for example. In conclusion, we have shown that fuzzy logic is a legitimate framework to discuss and address the difficulties plaguing frequentist interpretations of SHT.

中文翻译：

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统计假设检验逻辑问题的模糊理解

统计假设检验（SHT）是一类推论方法，通过该方法，人们可以利用经验数据来检验假设，并经常做出是否拒绝该假设的判断。在本文中，我们将重点放在该策略的逻辑方面，该策略在很大程度上独立于所采用的思想流派，至少在各种频率论者方法中是这样。我们将SHT识别为经典逻辑中取自Modus Tollens的不合理论证的形式，并且为了从此困难中解脱出SHT，我们建议可以将其替代基于t范数的模糊逻辑。我们通过使用Modus Tollens的模糊扩展来重新设计常客的SHT逻辑，以开发用于其前提的真值评估模型。重要的，我们表明，通过探索与构建模糊否定和模糊含义有关的各种约定（即S和R约定），可以保留Modus Tollens的健全性。我们发现，在S约定下，可以使用Zadeh的成分扩展和任何可能的t范数来进行Modus Tollens推论。在R约定下，我们发现并不一定是这种情况，但是例如，通过将R蕴涵与S负混合在一起，我们可以挽救乘积t范数。总而言之，我们已经表明模糊逻辑是讨论和解决困扰SHT频繁使用者解释的难题的合法框架。可以使用Zadeh的成分扩展和任何可能的t范数进行Modus Tollens推理论证。在R约定下，我们发现并不一定是这种情况，但是例如，通过将R蕴涵与S负混合在一起，我们可以挽救乘积t范数。总而言之，我们已经表明模糊逻辑是讨论和解决困扰SHT频繁使用者解释的难题的合法框架。可以使用Zadeh的成分扩展和任何可能的t范数进行Modus Tollens推理论证。在R约定下，我们发现并不一定是这种情况，但是例如，通过将R蕴涵与S负混合在一起，我们可以挽救乘积t范数。总而言之，我们已经表明模糊逻辑是讨论和解决困扰SHT频繁使用者解释的难题的合法框架。