论可解释性的逻辑与语义

Luka Mikec
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引用次数: 3

摘要

摘要研究形式化相对可解释性的各种性质。本文的中心部分研究了不同可解释性逻辑的模态语义完备性、可判决性和算法复杂度。特别地,我们研究了可解释性逻辑的两种基本类型的关系语义。一种是维尔特曼语义学,我们称之为规则语义学或普通语义学;另一种叫做广义维特曼语义。近年来,特别是在撰写本文期间,广义Veltman语义被证明特别适合作为可解释性逻辑的关系语义。特别是,在某些情况下,模态完备性结果更容易得到;可决性可以通过过滤在所有已知的情况下证明。我们证明了一些关于广义语义的新的完备性结果,并对一些旧的完备性结果进行了修正。我们用过滤的方法得到了各种逻辑的有限模型性质。除了关于语义本身的结果外,我们还应用语义的方法来确定某些可解释性逻辑的可判决性(由有限模型性质隐含)和可证明性(和一致性)问题的复杂性。从算术的角度出发,我们探讨了三种不同的可解释性原则。对于其中两个已知算术和模态稳健性的矩阵,给出了一个新的算术稳健性证明。第三个系列是由模态考虑得出的。我们证明了它在算术上是合理的,并利用普通的Veltman语义刻画了框架条件。我们还证明了关于新级数和广义Veltman语义的结果。摘要由Luka Mikec准备。电子邮件:luka.mikec1@gmail.com URL: http://hdl.handle.net/2445/177373
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On Logics and Semantics for Interpretability
Abstract We study various properties of formalised relativised interpretability. In the central part of this thesis we study for different interpretability logics the following aspects: completeness for modal semantics, decidability and algorithmic complexity. In particular, we study two basic types of relational semantics for interpretability logics. One is the Veltman semantics, which we shall refer to as the regular or ordinary semantics; the other is called generalised Veltman semantics. In the recent years and especially during the writing of this thesis, generalised Veltman semantics was shown to be particularly well-suited as a relational semantics for interpretability logics. In particular, modal completeness results are easier to obtain in some cases; and decidability can be proven via filtration in all known cases. We prove various new and reprove some old completeness results with respect to the generalised semantics. We use the method of filtration to obtain the finite model property for various logics. Apart from results concerning semantics in its own right, we also apply methods from semantics to determine decidability (implied by the finite model property) and complexity of provability (and consistency) problems for certain interpretability logics. From the arithmetical standpoint, we explore three different series of interpretability principles. For two of them, for which arithmetical and modal soundness was already known, we give a new proof of arithmetical soundness. The third series results from our modal considerations. We prove it arithmetically sound and also characterise frame conditions w.r.t. ordinary Veltman semantics. We also prove results concerning the new series and generalised Veltman semantics. Abstract prepared by Luka Mikec. E-mail: luka.mikec1@gmail.com URL: http://hdl.handle.net/2445/177373
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POUR-EL’S LANDSCAPE CATEGORICAL QUANTIFICATION POINCARÉ-WEYL’S PREDICATIVITY: GOING BEYOND A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp.
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