Source: J. Symbolic Logic
Volume 72, Issue 4
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.
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