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Satoru Niki, Peter Schuster, On Scott’s semantics for many-valued logic, Journal of Logic and Computation, Volume 30, Issue 6, September 2020, Pages 1291–1302, https://doi.org/10.1093/logcom/exaa036
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Abstract
The semantics in ordered abelian groups Scott proposed for Łukasiewicz’s many-valued logic fails to be sound for one direction of one of the rules Scott gave for implication. We show this by a counterexample Urquhart has used to justify that in his own semantics, every formula has to have a least point at which it is valid. While this condition would make Scott’s semantics sound, it would cause a problem with its completeness. The question arises whether one can still amend Scott’s semantics so as to make it both sound and complete or better stick to Urquhart’s semantics anyway.