The Calculus of Higher-Level Rules, Propositional Quantification, and the Foundational Approach to Proof-Theoretic Harmony
The Calculus of Higher-Level Rules, Propositional Quantification, and the Foundational Approach to Proof-Theoretic Harmony
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高级规则的演算、命题量化以及证明理论和谐的基本方法
DOI：
10.1007/s11225-014-9562-3
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发表时间：
2014
期刊：
影响因子：
0.7
通讯作者：
Renkawitz
中科院分区：
文献类型：
--
作者：
Herold;Panzer;Demmers;Kruger;Scharfe;Bartkuhn;Renkawitz
我们提出了更高级别规则的演算，并在规则内扩展了命题量化。这使得有可能目前的一般模式的引进和淘汰规则的任意命题运营商和定义它意味着什么，引进和淘汰是相互协调。这一定义并不以任何逻辑系统为前提，而是根据规则本身来制定的。因此，我们说的是一个基础（而不是还原）的证明理论的和谐帐户。对于每一组引入规则，一个规范消除规则，并且对于每一组消除规则，一个规范引入规则以这样的方式相关联，即该规范规则与它所关联的规则集相协调。本文以哈岑和佩尔蒂埃给出的一个例子来说明，存在着一些有意义的连接词，这些连接词的特征在于它们的排除规则，而它们的引入规则是与这些排除规则相关联的规范引入规则。由于更高层次的规则和命题量化的可用性，所开发的框架的表达手段足以确保规范的消除或引入规则的构造总是可能的，并且不会超出这个框架。
We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational (rather than reductive) account of proof-theoretic harmony. With every set of introduction rules a canonical elimination rule, and with every set of elimination rules a canonical introduction rule is associated in such a way that the canonical rule is in harmony with the set of rules it is associated with. An example given by Hazen and Pelletier is used to demonstrate that there are significant connectives, which are characterized by their elimination rules, and whose introduction rule is the canonical introduction rule associated with these elimination rules. Due to the availabiliy of higher-level rules and propositional quantification, the means of expression of the framework developed are sufficient to ensure that the construction of canonical elimination or introduction rules is always possible and does not lead out of this framework.
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