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.

中文翻译：

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论斯科特的多值逻辑语义

斯科特为sukasiewicz的多值逻辑提出的有序阿贝尔组中的语义对于斯科特给出的暗示规则之一的一个方向来说是不合理的。我们通过一个反例来说明这一点，厄克特（Urquhart）曾经用他自己的语义来证明，每个公式都必须有一个最小的有效点。虽然这种情况会使Scott的语义听起来很合理，但会引起完整性方面的问题。问题是，是否仍然可以修改Scott的语义以使其听起来既完整又完整，还是更好地坚持Urquhart的语义。