Intuitionistic Non-normal Modal Logics: A General Framework
Intuitionistic Non-normal Modal Logics: A General Framework
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直觉非正态模态逻辑：通用框架
DOI：
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发表时间：
2019
影响因子：
1.5
通讯作者：
N. Olivetti
中科院分区：
文献类型：
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作者：
Tiziano Dalmonte;Charles Grellois;N. Olivetti
我们定义了一系列直觉非正规模态逻辑；它们可以被视为经典的直觉对应物。我们首先考虑单模态逻辑，它只包含必然性或可能性。然后我们考虑双模态逻辑的更重要的情况，其中包含两个模态运算符。在这种情况下，我们定义了增强强度的必要性和可能性之间的几种相互作用，尽管比二元性弱。由此我们获得了 24 个不同双模逻辑的格。对于所有逻辑，我们提供希尔伯特公理化和无割序列演算，在此基础上我们还证明了它们的可判定性。然后，我们根据邻域模型定义逻辑的语义特征，邻域模型包含与两种模态相对应的两个不同的邻域函数。我们的语义框架不仅以模块化方式捕获我们的系统，而且还捕获已知的直觉非正规模态逻辑，例如构造性 K (CK) 和 Wijesekera 的构造性并发动态逻辑的命题片段。
We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic.