A Fuzzy Take on the Logical Issues of Statistical Hypothesis Testing

Booth, Matthew and Paillusson, Fabien (2021) A Fuzzy Take on the Logical Issues of Statistical Hypothesis Testing. Philosophies, 6 (1). p. 21. ISSN 2409-9287

Full content URL: https://doi.org/10.3390/philosophies6010021

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A Fuzzy Take on the Logical Issues of Statistical Hypothesis Testing
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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.

Keywords:Fuzzy Logic, Statistics
Subjects:G Mathematical and Computer Sciences > G300 Statistics
V Historical and Philosophical studies > V550 Philosophy of Science
G Mathematical and Computer Sciences > G120 Applied Mathematics
G Mathematical and Computer Sciences > G320 Probability
Divisions:College of Science > School of Mathematics and Physics
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ID Code:50076
Deposited On:07 Jul 2022 10:49

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