 Let’s focus on hidden but assumed presence of quantifiers in the case of multiplication and exponentiation: \(\binary{l}\unary{w}\) is interpreted as “a lover of some woman” and \(l^{\unary{w}}\) as “a lover of every woman”. Interestingly, in the process of introducing polyadic predicates, Peirce ends up bringing in quantifiers, some and every. On the other hand, considering that Aristotle’s syllogisms have two quantifiers, Peirce’s algebraic notation for quantifiers—some and every—should not surprise us. However, a crucial aspect of this development is that Boole’s unsatisfactory representation of existential propositions (as opposed to universal propositions) pushed Peirce and his student O. H. Mitchell to go beyond Boole’s  logic.[13]  The way Peirce interprets a relative product—“\(\binary{l}\unary{w}\)” meaning “lover of some woman”—allows an existential quantifier to be expressed implicitly in terms of multiplication.