Sequent Calculi and Bidirectional Natural Deduction: On the Proper Basis of Proof-theoretic Semantics

Philosophical theories of logical reasoning are intrinsically related to formal models. This holds in particular of Dummett–Prawitz-style proof-theoretic semantics and calculi of natural deduction. Basic philosophical ideas of this semantic approach have a counterpart in the theory of natural deduction. For example, the “fundamental assumption” in Dummett’s theory of meaning (Dummett, 1991, p. 254 and Ch. 12) corresponds to Prawitz’s formal result that every closed derivation can be transformed into introduction form (Prawitz, 1965, p. 53). Examples from other areas in the philosophy of logic support this claim. If conceptual considerations are genetically dependent on formal ones, we may ask whether the formal model chosen is appropriate to the intended conceptual application, and, if this is not the case, whether an inappropriate choice of a formal model motivated the wrong conceptual conclusions. We will pose this question with respect to the paradigm of natural deduction and proof-theoretic semantics, and plead for Gentzen’s sequent calculus as a more adequate formal model of hypothetical reasoning. Our main argument is that the sequent calculus, when philosophically re-interpreted, does more justice to the notion of assumption than does natural deduction. This is particularly important when it is extended to a wider field of reasoning than just that based on logical constants. To avoid confusion, a terminological caveat must be put in place: When we talk of the sequent calculus and the reasoning paradigm it represents, we mean, as its characteristic feature, its symmetry or bidirectionality, i.e.,