One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations

Authors:
Murdoch J. Gabbay;Dominic P. Mulligan
Affiliations:
No Affiliations,;No Affiliations,
Venue:
WoLLIC '08 Proceedings of the 15th international workshop on Logic, Language, Information and Computation
Year:
2008

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Abstract

The Curry-Howard correspondence connects Natural Deduction derivation with the lambda-calculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason `backwards' from the desired conclusion towards the assumptions. At intermediate stages we may have an `incomplete derivation', with `holes'.This is natural in informal practice; the challenge is to formalise it. To this end we use a one-and-a-halfth order technique based on nominal terms, with twolevels of variable. Predicates are types, derivations are terms -- and the two levels of variable are respectively the assumptions and the `holes' of an incomplete derivation.