Investigations on a Pedagogical Calculus of Constructions

TL;DR

This paper explores pedagogical propositional natural deduction systems, introduces a restricted calculus called CCr, analyzes its logical and computational properties, and proposes a formal definition for pedagogical calculi of constructions.

Contribution

It constructs a pedagogical restriction of the calculus of constructions, analyzes its properties, and formalizes what constitutes a pedagogical calculus of constructions.

Findings

CCr lacks negation at the logical level.

CCr's computational expressiveness is comparable to Godel system T.

A formal definition for pedagogical calculi of constructions is proposed.

Abstract

In the last few years appeared pedagogical propositional natural deduction systems. In these systems, one must satisfy the pedagogical constraint: the user must give an example of any introduced notion. First we expose the reasons of such a constraint and properties of these "pedagogical" calculi: the absence of negation at logical side, and the "usefulness" feature of terms at computational side (through the Curry-Howard correspondence). Then we construct a simple pedagogical restriction of the calculus of constructions (CC) called CCr. We establish logical limitations of this system, and compare its computational expressiveness to Godel system T. Finally, guided by the logical limitations of CCr, we propose a formal and general definition of what a pedagogical calculus of constructions should be.