 
Summary: Re
ective calculus
Jesse Alt and Sergei Artemov ?
Cornell University,
Ithaca, NY 14853, U.S.A.
jma35@cornell.edu, artemov@cs.cornell.edu
Abstract. We introduce a general purpose typed calculus 1 which
contains intuitionistic logic, is capable of internalizing its own derivations
as terms and yet enjoys strong normalization with respect to a natural
reduction system. In particular, 1 subsumes the typed calculus. The
CurryHoward isomorphism converting intuitionistic proofs into terms
is a simple instance of the internalization property of 1 . The standard
semantics of 1 is given by a proof system with proof checking capaci
ties. The system 1 is a theoretical prototype of re
ective extensions of
a broad class of typebased systems in programming languages, provers,
AI and knowledge representation, etc.
1 Introduction
According to the CurryHoward isomorphism, the calculus of intuitionistic pro
positions (types) and the calculus of typed terms (proof terms) constitute a
pair of isomorphic though distinct structures. Combining those logical and com
putational universes has been considered a major direction in theoretical logic
