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.
Highlights
According to Popper [1], a theory can be deemed scientific if it implies the impossibility of certain events that are otherwise perfectly conceivable
Based on the above discussion we agree with Sober [17] that “there is no probabilistic analog of modus tollens” but we have shown that a) this fundamental state of affairs could not be salvaged by comparing hypotheses and b) the logical problem has been shown to occur when propositions need to be exactly true or false
To look for a logic on which to ground the intuition behind P value based hypothesis rejection, we have explored t-norm based fuzzy logics under the S- and Rconventions for three of the most commonly employed t-norms
Summary
According to Popper [1], a theory can be deemed scientific if it implies the impossibility of certain events that are otherwise perfectly conceivable (or even expected). Upon a little consideration, that the mere fact that a particular sample may be expected to occur very rarely in sampling from Π [i.e., the hypothetical population] would not in itself justify the rejection of the hypothesis that it had be so drawn, if there were no more probable hypotheses conceivable This line of argument in the philosophy of SHT [10,12,17] considers that the MT reasoning developed in Equations (4)–(6) is basically fallacious but can be rectified by considering H as being rejected in favour of an alternative, more likely hypothesis, rather than being rejected as a standalone hypothesis.
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