Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the “perfect syllogisms”) must be supplemented by rules for equation solving.
The task of developing the modern account of Boolean algebra fell to Boole's successors in the tradition of algebraic logic (Jevons 1869, Peirce 1880, Jevons 1890, Schröder 1890, Huntington 1904).
In Boole's account of his algebra, terms are reasoned about equationally, without a systematic interpretation being assigned to them.
The coherences of the whole enterprise is justified by Boole in what Stanley Burris has later called the "rule of 0s and 1s", which justifies the claim that uninterpretable terms cannot be the ultimate result of equational manipulations from meaningful starting formulae (Burris 2000).
In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined.