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.
On Scott’s semantics for many-valued logic
Schuster, Peter
2020-01-01
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.
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