A Note on Statistical Hypothesis Testing: Probabilifying Modus Tollens Invalidates Its Force? Not True!

Abstract

The import or force of the result of a statistical test has long been portrayed as consistent with deductive reasoning. The simplest form of deductive argument has a first premise with conditional form, such as p→q, which means that “if p is true, then q must be true.” Given the first premise, one can either affirm or deny the antecedent clause (p) or affirm or deny the consequent claim (q). This leads to four forms of deductive argument, two of which are valid forms of reasoning and two of which are invalid. The typical conclusion is that only a single form of argument—denying the consequent, also known as modus tollens—is a reasonable analog of decisions based on statistical hypothesis testing. Now, statistical evidence is never certain, but is associated with a probability (i.e., a p-level). Some have argued that modus tollens, when probabilified, loses its force and leads to ridiculous, nonsensical conclusions. Their argument is based on specious problem setup. This note is intended to correct this error and restore the position of modus tollens as a valid form of deductive inference in statistical matters, even when it is probabilified.