The preferential conditional logic $ \mathbb{PCL} $, introduced by Burgess, and its extensions are studied. First, a natural semantics based on neighbourhood models, which generalizes Lewis’ sphere models for counterfactual logics, is proposed. Soundness and completeness of $ \mathbb{PCL} $ and its extensions with respect to this class of models are proved directly. Labelled sequent calculi for all logics of the family are then introduced. The calculi are modular and have standard proof-theoretical properties, the most important of which is admissibility of cut that entails a syntactic proof of completeness of the calculi. By adopting a general strategy, root-first proof search terminates, thereby providing a decision procedure for $ \mathbb{PCL} $ and its extensions. Finally, semantic completeness of the calculi is established: from a finite branch in a failed proof attempt it is possible to extract a finite countermodel of the root sequent. The latter result gives a constructive proof of the finite model property of all the logics considered.

中文翻译：

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基于邻域语义的优先条件逻辑的统一标记演算

研究了 Burgess 引入的优先条件逻辑 $ \mathbb{PCL} $ 及其扩展。首先，提出了一种基于邻域模型的自然语义，该模型将 Lewis 的球面模型推广到反事实逻辑。直接证明了$ \mathbb{PCL} $ 及其对此类模型的扩展的健全性和完整性。然后介绍了该族所有逻辑的标记顺序演算。演算是模块化的，并具有标准的证明理论属性，其中最重要的是允许切分，这需要对演算完整性的句法证明。通过采用一般策略，根优先证明搜索终止，从而为 $ \mathbb{PCL} $ 及其扩展提供决策过程。最后，建立演算的语义完整性：从失败的证明尝试中的有限分支中，可以提取根序列的有限反模型。后一个结果为所考虑的所有逻辑的有限模型属性提供了建设性的证明。