  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.
