Commentary and Illocutionary Expressions in Linear Calculi of Natural Deduction

Abstract

Abstract. We argue that the need for commentary in commonly used linear calculi of natural deduction is connected to the "deletion" of illocutionary expressions that express the role of propositions as reasons, assumptions, or inferred propositions. We first analyze the formalization of an informal proof in some common calculi which do not formalize natural language illocutionary expressions, and show that in these calculi the formalizations of the example proof rely on commentary devices that have no counterpart in the original proof. We then present a linear natural deduction calculus that makes use of formal illocutionary expressions in such a way that unique readability for derivations is guaranteed  thus showing that formalizing illocutionary expressions can eliminate the need for commentary.

Holistic Inferential Criteria of Adequate Formalization

“…Note that this choice is motivated by the relative ease with which informal derivations are formalized and formal derivations instantiated while the view on calculi sketched above can be applied to other types of calculi as well Cordes and Reinmuth (2017). discuss the formalization of informal derivations in different types of linear calculi of natural deduction.Dialectica vol.…”

The constituents of an explication

“…Here is one example rule of introduction: If φ is a closed formula consistent with any axioms and definitions already set, then one may set φ as an axiom. For a full treatment see Reinmuth (2013).…”