 Boole’s aspiration to capture logic in terms of an algebraic system has inspired many mathematicians and logicians who are interested in connecting the two disciplines, logic and mathematics. As seen in the previous section, Peirce is clearly one of them. He pushed the Boolean idea of an algebraic system further in two ways: One is to improve Boole’s representation of particular proposition (i.e., “Some A is B”) so that traditional Aristotelian syllogisms may fit in an algebraic system. The other is to represent not only qualities but also relations so that a new algebraic system may reach beyond traditional syllogisms. In that process, new notations for quantifiers and variables were invented.