 
Summary: DAVID ALBRECHT CurryHoward terms for
FRANK A. B ¨
AUERLE Linear Logic \Lambda
JOHN N. CROSSLEY
JOHN S. JEAVONS
Abstract. In this paper we 1. provide a natural deduction system for
full firstorder linear logic, 2. introduce CurryHowardstyle terms for this
version of linear logic, 3. extend the notion of substitution of CurryHoward
terms for term variables, 4. define the reduction rules for the CurryHoward
terms and 5. outline a proof of the strong normalization for the full system
of linear logic using a development of Girard's candidates for reducibility,
thereby providing an alternative to Girard's proof using proofnets.
Key words: Linear Logic, Curry Howard Terms, and Strong Normal
ization.
1 Introduction
The logical system is a further development of those of Troelstra [10] and
Benton et al. [1] and the rules follow fairly traditional patterns.
On the other hand the new CurryHoward terms involve substantial
modifications to the lambda calculus, in particular the introduction of new
operators ', Ÿ, ø and ! with their associated reduction rules. More impor
