I know the interest in Veriphysics is limited here, hence the separate site devoted to the philosophy, but since this question has popped up in several places, I thought I should at least mention that it has been answered in substantive detail over there.
I don’t understand why it is necessary for there to be three different elements of the Triveritas. Aren’t L and M basically the same thing, because math is logic?
Here is the abridged version of the complete answer to it.
Each of the three dimensions of the Triveritas has characteristic failure modes that the other two dimensions cannot detect from within their own domain. That is why relying on any one, or even any two, leaves a structural blind spot that historically produces false confidence…
The critical insight from the historical record is that false claims survive by trading on their strong dimensions to deflect scrutiny from their weak one. The defenders of phlogiston pointed to its empirical success and quantitative accounting to avoid the question of logical coherence. The defenders of caloric theory pointed to Fourier’s mathematics and the theory’s logical elegance to deflect Rumford’s empirical challenge. The defenders of Ptolemy pointed to centuries of accurate predictions to deflect the question of explanatory unity.
And in every resolved historical case, the refutation arrived from the specific dimension that was missing. Not from a random direction, but from the precise blind spot the theory’s defenders were trying to hide. Newtonian mechanics, steady-state cosmology, and caloric theory all satisfied L and M but failed E, and all three were killed by empirical observation. Continental drift and the plum pudding model satisfied L and E but failed M, and both were killed by mathematical incoherence. Ptolemaic epicycles, phlogiston, and miasma theory satisfied M and E but failed L, and all three were killed by the arrival of logically coherent replacements.
Also, for those who are interested in applying the Triveritas, the reference scales for L, M, and E are all now complete.