Provability and Interpretability Logics with Restricted Realizations

IF 0.6 3区 数学 Q2 LOGIC
Thomas F. Icard, J. Joosten
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引用次数: 6

Abstract

The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set $\Gamma$, where each sentence in $\Gamma$ has a well understood (meta)-mathematical content in T, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and I$\Sigma_1$. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) $\subset$ ILM.
具有受限实现的可证明性和可解释性逻辑
理论T的可证性逻辑是模态公式的集合,这些模态公式在任何算术实现下在T中都是可证的。我们稍微修改了这个概念,要求算术实现来自一个指定的集合$\Gamma$。我们对可解释性逻辑做了类似的修改。这是2012年的一篇论文。我们首先研究了具有限制实现的可证明性逻辑,并表明对于理论T和限制集$\Gamma$的各种自然候选者,其中$\Gamma$中的每个句子在T中都有一个很好的理解(元)数学内容,结果是线性框架的逻辑。然而,对于理论原始递归算法(PRA),我们定义了一个片段,通过充分研究PRA和I之间的关系$\Sigma_1$,产生了一个更有趣的可证明性逻辑。然后,我们研究了可解释性逻辑,得到了IL(PRA)的一些上界,其表征仍然是可解释性逻辑中的一个主要开放问题。这个上界与线性坐标系密切相关。该技术还应用于IL(PRA) $\subset$ ILM的非平凡结果。
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来源期刊
CiteScore
1.00
自引率
14.30%
发文量
14
审稿时长
>12 weeks
期刊介绍: The Notre Dame Journal of Formal Logic, founded in 1960, aims to publish high quality and original research papers in philosophical logic, mathematical logic, and related areas, including papers of compelling historical interest. The Journal is also willing to selectively publish expository articles on important current topics of interest as well as book reviews.
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