If you claim that “a good and proper life” is evidence that a woman is a witch, then an evil and improper life must be evidence that she is not a witch.
To clarify for myself and/or others: my understanding is that the specific issue with the witch-hunter reasoning isn’t that they treated both of the opposing observations as confirmation (which might be implied to some readers); the issue is that there were no alternatives to either of those two opposing observations, and as a result, their hypothesis was not falsifiable.
It's not always the case that opposing observations comprise the complete possibility set: water can be not only hot or cold, but also room temperature.
Two seemingly opposite observations can, in some cases, both be valid evidence for a hypothesis. An example is bipolar disorder, where both manic and depressive episodes — though opposite in presentation — are more probable under the bipolar model than under a neurotypical or unipolar depression model. So both increase the posterior.
In general, a hypothesis can be supported by contrasting data — as long as there are still possible observations that would disconfirm it.
To clarify for myself and/or others: my understanding is that the specific issue with the witch-hunter reasoning isn’t that they treated both of the opposing observations as confirmation (which might be implied to some readers); the issue is that there were no alternatives to either of those two opposing observations, and as a result, their hypothesis was not falsifiable.
It's not always the case that opposing observations comprise the complete possibility set: water can be not only hot or cold, but also room temperature.
Two seemingly opposite observations can, in some cases, both be valid evidence for a hypothesis. An example is bipolar disorder, where both manic and depressive episodes — though opposite in presentation — are more probable under the bipolar model than under a neurotypical or unipolar depression model. So both increase the posterior.
In general, a hypothesis can be supported by contrasting data — as long as there are still possible observations that would disconfirm it.