 One might expect that Boole was building toward claiming an axiomatic foundation for his algebra of logic that, as he had (erroneously) claimed in MAL, justified using all the processes of common algebra. Indeed he did discuss the rules of inference, that adding or subtracting equals from equals gives equals, and multiplying equals by equals gives equals. But then the development of an axiomatic approach came to an abrupt halt. There was no discussion as to whether the stated axioms (which he called laws) and rules of inference (which he called axioms) were sufficient for  his algebra of logic. (They were not.) Instead he simply and briefly, with remarkably little fanfare, presented a radically new foundation for his algebra of logic (LT pp. 37,38):