An untyped higher order logic with Y combinator

We define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and...

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Bibliographic Details
Published in:The Journal of symbolic logic 2007-12, Vol.72 (4), p.1385-1404
Main Author: Andrews, James H.
Format: Article
Language:English
Subjects:
Children
Entailment
Higher order logics
Induction assumption
Inductive reasoning
Logical proofs
Logical theorems
Mathematical induction
Model theory
Sequents
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Summary:We define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.
ISSN:0022-4812
1943-5886
DOI:10.2178/jsl/1203350794