On the algebraization of Henkin-type second-order logic

Pub Date : 2022-02-06 DOI:10.1002/malq.202100057
Miklós Ferenczi
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Abstract

There is an extensive literature related to the algebraization of first-order logic. But the algebraization of full second-order logic, or Henkin-type second-order logic, has hardly been researched. The question arises: what kind of set algebra is the algebraic version of a Henkin-type model of second-order logic? The question is investigated within the framework of the theory of cylindric algebras. The answer is: a kind of cylindric-relativized diagonal restricted set algebra. And the class of the subdirect products of these set algebras is the algebraization of Henkin-type second-order logic. It is proved that the algebraization of a complete calculus of the Henkin-type second-order logic is a class of a kind of diagonal restricted cylindric algebras. Furthermore, the connection with the non-standard enlargements of standard complete second-order structures is investigated.

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关于henkin型二阶逻辑的代数化
关于一阶逻辑的代数化有大量的文献。但是关于全二阶逻辑的代数化,即henkin型二阶逻辑的代数化研究却很少。问题来了:二阶逻辑的henkin型模型的代数版本是什么样的集合代数?这个问题是在圆柱代数理论的框架内研究的。答案是:一种圆柱相对对角限制集代数。而这些集合代数的子直积的类就是henkin型二阶逻辑的代数化。证明了henkin型二阶逻辑的完全演算的代数化是一类对角限制柱代数。进一步研究了标准完全二阶结构与非标准扩展的关系。
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