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Journal of Logic and Computation Advance Access published online on April 22, 2008
Journal of Logic and Computation, doi:10.1093/logcom/exn001
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© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

Pedagogical Second-order Propositional Calculi

Loïc Colson and David Michel

University of Metz, Metz, France E-mail: colson{at}univ-metz.fr; david.michel{at}univ-metz.fr

Received 18 June 2007.


  Abstract

The present work introduces the notion of pedagogical natural deduction systems, which are natural deduction systems with the following additional constraint: all hypotheses made in a proof must be motivated by an example. Technically speaking, we replace the rule (Hyp):

Formula

with the rule (PHyp):

Formula

with {sigma} denoting a substitution replacing all variables of {Gamma} with an example. This substitution is called the motivation of {Gamma}. These systems are in essence negationless. In the present article, we study the second-order propositional calculus, since it is the simplest non-trivial natural deduction system in which the negation is definable. Some pedagogical versions of the second-order propositional calculus are proposed. We argue that these pedagogical calculi are negationless and we study their expressive power.

Keywords: Constructive mathematics; mathematical logic; natural deduction; negationless mathematics; typed {lambda}-calculus


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