A fuzzy take on the logical issues of statistical hypothesis testing

TL;DR

This paper explores the logical foundations of statistical hypothesis testing, proposing fuzzy logic as a means to address its classical logical shortcomings and improve its theoretical robustness.

Contribution

It introduces a fuzzy logic framework to reformulate and analyze the logical structure of SHT, demonstrating how to preserve soundness through specific fuzzy negation and implication conventions.

Findings

Fuzzy logic can address logical issues in SHT.

Soundness of Modus Tollens can be preserved with appropriate fuzzy conventions.

Different fuzzy conventions impact the validity of inference methods.

Abstract

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.