[PeterisP]

The argument for general cognitive ability is because the correlation is observed among all the various measures related to cognitive ability that we have. If you test people on the ability to play the piano, organ, and harpsichord, and assert that this is a reflection of general cognitive ability, then this is prediction is testable by later correlating this ability with other measures, and gets falsified by observing the metrics with which it does not correlate. So far, the concept of "G" survives because it actually does succeed in various tests of alternative metrics of cognitive ability - e.g. "emotional intelligence" metrics, biological metrics such as reaction speed, and various 'outcomes' e.g. job performance and future income controlled for socioeconomic background.

In essence, the answer to "why positive correlation between any particular set of tests leads to the conclusion that this is truly a general cognitive ability" is that many respectable people have tried to find measures of aspects of cognitive ability that are separate and uncorrelated, however, as far as I know, all such attempts have failed so far and their experiments on alternative metrics did correlate with everything else and just became more data supporting the concept.