Euclidean Precision: A swap is undeniable and follows from a “poisoned premise”

The poisoned premise: what is claimed to be a defining characteristic actually has an equally pervasive variety in two populations. More specifically to our topic, scores based on defining characteristics have an equally pervasive variety in both a target and an off-target population.  Of course, this is a contradiction: it’s not really “defining” if the characteristic is shared with an off-target population. With this premise, the argument has already swallowed a poison-pill and cannot be saved. The swap must follow by mathematical necessity. There are two critical points:

Point 1, Left: The two words, “defining characteristic,” is a misunderstanding in this context. We make distinctions with differences and cannot make them with similarities.  Despite the connotation of “differences” in the usual usage of the word, “variety,” a similar variety in two populations is still a similarity and not a difference. This misunderstanding, unchallenged, effectively re-labels the off-target as the target population: due to similarity of variety, the cutoff point involves no true distinguishing characteristic. 

Point 2, Right: Any cutoff point at the absolute minimum is meaningless; it is synonymous with the entire population. Any cutoff point anywhere above the minimum necessarily excludes all below itself.

Below, review of the mathematical necessity of the swap:

• The threshold of any scoring system cannot make the intended distinction if the “defining characteristics” of its intended target are also similarly present in the off-target population. Imposing the cutoff point does not “identify” the target. It simply confuses the higher scores with the intended target. It inflates the results with errors.
• With an equally pervasive variety, a scoring system cannot overcome the existence of those below the cutoff point. Imposing the cutoff point not only fails to “identify” the target accurately; it actually abandons a portion of them.

If the “defining characteristic,” or scores based on characteristics, has an equally pervasive variety in both the target and the off-target, then the swap is undeniable and can be demonstrated with Euclidean precision.