Resumen: Christine Ladd was a promising mathematician who converted to logic and philosophy under the guidance of Charles S. Peirce. Her contribution to the algebra of logic is known among historians of logic, and her famous ¿antilogism¿ received acknowledgement from the algebraic logicians of the time. But the interest of Ladd¿s logical system is not restricted to the development of Boole¿s project. Concentrating on the application of Ladd¿s algebra to the analysis of syllogisms, I highlight Ladd¿s innovations with respect to the Aristotelian-Scholastic tradition: I show how Ladd abandons the old point of view (predication), in favour of the new algebraic one (combination of terms), with the interesting feature of giving preference to symmetrical ways of combination. In Ladd¿s logical system, both the traditional figures and moods, and the traditional validity principles are translated into one single formula for every valid syllogism, which exhibits the inconsistency of affirming the premises while negating the conclusion. This result not only involves Peirce¿s ideal of iconicity, but is also the basis for an innovative view of logic in which discussing, rather than concluding, captures the true form of reasoning.