In this article, I present a semantically natural conservative extension of Urquhart’s positive semilattice logic with a sort of constructive negation. A subscripted sequent calculus is given for this logic and proofs of its soundness and completeness are sketched. It is shown that the logic lacks the finite model property. I discuss certain questions Urquhart has raised concerning the decision problem for the positive semilattice logic in the context of this logic and pose some problems for further research.

中文翻译：

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无有限模型性质的正半格逻辑的保守否定扩展

在这篇文章中，我提出了厄克特的正半格逻辑的语义自然保守扩展，并带有一种建设性的否定。给出了这个逻辑的下标序列演算，并勾勒出其健全性和完整性的证明。结果表明，该逻辑缺乏有限模型特性。我在这个逻辑的背景下讨论了厄克特提出的关于正半格逻辑的判定问题的某些问题，并提出了一些问题以供进一步研究。