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A 'logical fallacy' is an error in reasoning that undermines the logic of an argument. In Sir Ronald A. Fisher's quotation, he mentions a specific logical fallacy. This fallacy occurs when someone believes a hypothesis is true simply because no evidence has contradicted it. Imagine you claim, 'It has not rained today because no one has told me they saw rain.' This is faulty reasoning because the absence of evidence against your claim doesn't necessarily validate it. Fisher warns against this kind of thinking in statistical reasoning. Just because a hypothesis isn’t contradicted by current data doesn't mean it’s true. This fallacy is important to understand because it can mislead researchers into accepting unproven hypotheses.

In the context of hypothesis testing, 'tests of significance' are tools used by statisticians to determine if their data supports or contradicts a given hypothesis. These tests help assess whether observed outcomes are consistent with what would be expected under the null hypothesis. Common tests include the t-test, chi-square test, and ANOVA. These tests produce a p-value, which tells us whether the observed results could have happened by random chance alone. According to Fisher, while these tests can indicate when data contradicts a hypothesis, they cannot prove the hypothesis is true. This is crucial because it sets a limit on what statistical tests can achieve. They help us invalidate or reject hypotheses that don't align with the data but don’t confirm the absolute truth of a hypothesis.

When conducting a statistical test, 'hypothesis rejection' occurs if the data significantly contradicts the null hypothesis. The null hypothesis generally states there is no effect or no difference. If a test returns a p-value below a predetermined significance level (commonly 0.05), the null hypothesis can be rejected. Rejection implies that the observed data is unlikely to have occurred under the null hypothesis. Fisher's point is that while rejection of the hypothesis suggests it is not true, not being able to reject it does not confirm it is true. There could be many reasons for not rejecting a hypothesis, such as insufficient data or sample size. This understanding helps prevent the logical fallacy of believing a hypothesis is true just because it hasn’t been disproven.