Catuṣkoṭi or tretralemma is mostly made up by intellectuals who commented on what they thought was the dhamma, and they tried to push their sloppy reasoning as the dhamma too, like Nagarjuna, so it is mostly in Mahayana. None of that stuff is strongly buddhist. The intellectual works like the Upanishads use it too, with their neti neti, and the jains also, to support exactly the same claim like Nagarjuna, ie ''you can't talk about reality'', ie the absolute truth cannot be said with conventional language. Buddhist intellectuals are not really rigorous logicians, even more so when compared to today logicians.
Plenty of people have tried to formalize the tretralemma like here https://arxiv.org/pdf/1405.7744.pdf
Hence the interpretation of this modified catus.kot.i is: If somebody denies all these alternatives, then he/she intends to mean by abbreviation using the representatives that he/she declares that the ultimate truth or the reality can- not be described by any formula of any propositional calculus in which A is a generic formula. Particularly any propositional calculus for which A is a sentence letter doesn’t work to describe the reality.
https://plato.stanford.edu/entries/contradiction/#LNCBudTet
In the sutras, the tetralemma is just rejecting a claim and also rejecting the negation of the claim. This is from the claim being ill-defined, completely out of topic. But intellectuals prefer to view it has a deep thing about reality.