Form, Content, Extensions

In contrast to Kant’s metatheoretical (“transcendental”) approach grounded in the operations of an abstract logical subject, Herbart and Bolzano aimed to avoid subject-dependent foundations of logic. They instead proposed formal logics founded, respectively, on the form of conceptual content (Herbart) and on the extensions of logical forms (Bolzano). Among the traditions originating from Herbart and Bolzano, we consider the role of Bolzano’s informal disciple Robert Zimmermann, who subsequently adopted Herbartianism, and Zimmermann’s own disciple, the mathematical logician and philosopher Albino Nagy. Special attention is given to Nagy’s analysis of expressive completeness and his treatment of the Jevons-Clifford problem: how many and which types and representatives of types there are of Boolean propositional functions. We transform this latter problem into the question of types of sequent refutations and examine the corresponding admissibility of the cut rule.