Review of Symbolic Logic 13 (4):829-844 (2020)

Authors
Michael Caie
University of Toronto, St. George Campus
Abstract
Systems of illative logic are logical calculi formulated in the untyped λ-calculus supplemented with certain logical constants.1 In this short paper, I consider a paradox that arises in illative logic. I note two prima facie attractive ways of resolving the paradox. The first is well known to be consistent, and I briefly outline a now standard construction used by Scott and Aczel that establishes this. The second, however, has been thought to be inconsistent. I show that this isn’t so, by providing a nonempty class of models that establishes its consistency. I then provide an illative logic which is sound and complete for this class of models. I close by briefly noting some attractive features of the second resolution of this paradox.
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DOI 10.1017/s1755020319000054
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References found in this work BETA

Outline of a Theory of Truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
The Inconsistency of Certain Formal Logic.Haskell B. Curry - 1942 - Journal of Symbolic Logic 7 (3):115-117.
The Combinatory Foundations of Mathematical Logic.Haskell B. Curry - 1942 - Journal of Symbolic Logic 7 (2):49-64.
A Fixed Point Theorem for the Weak Kleene Valuation Scheme.Anil Gupta & Robert L. Martin - 1984 - Journal of Philosophical Logic 13 (2):131 - 135.
A Paradox in Illative Combinatory Logic.M. W. Bunder - 1970 - Notre Dame Journal of Formal Logic 11 (4):467-470.

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