A fuzzy take on the logical issues of statistical hypothesis testing
This addresses foundational logical issues in frequentist statistical hypothesis testing, which is a core method in data analysis across many scientific fields, though it is incremental as it builds on existing fuzzy logic concepts.
The paper tackles the logical unsoundness of statistical hypothesis testing (SHT) by identifying it as an unsound Modus Tollens argument in classical logic and proposes grounding SHT in t-norm based fuzzy logics to preserve soundness, showing that under the S convention, Modus Tollens inference works with any t-norm, and under the R convention, it can be salvaged for specific cases like the product t-norm.
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.