论文标题
通过翻译通过：HIZ的谓词逻辑的警告
Axiomatization via translation: Hiz's warning for predicate logic
论文作者
论文摘要
多年来，公理体逻辑翻译和原始操作员的选择的问题已经浮出水面多次。 H. Hi {\提出了早期问题。 Z}在1950年代，关于翻译的计算的不完整。 W. Frank和S. Shapiro在1970年代以及随后的几十年中其他其他工作进行了进一步的相关工作，其中一些涉及到这里。正如我们将看到的那样，忽略这种可能性导致了不正确的完整性主张（例如J. L. Bell和A. B. Slomson以及J. N. Crossley以及J. N. Crossley）对经典谓语逻辑的公理化，这是通过从适用于适用于不同选择的逻辑原始原始原始原始词的转换获得的经典谓语逻辑的。在本说明中，我们首先讨论了W. Frank的早期文章的一些有问题的方面，讨论了获得翻译计算的完整定理的困难。夏皮罗（Shapiro）通过表现出不可证明的命题重言式研究来确定Crossley的公理化的不完整。相反，要处理贝尔和斯洛姆森的系统，该系统已完成命题重言术，我们继续表明，使用原始$ \的经典谓词微积分的形式系统，设置$ \ forall x ϕ（x）\ stackrel \ stackRel用$ \ forall $而不是$ \的$ rest $作为原始$的$ \ forall $的规则不保证所得系统的完整性。 In particular, instances of the valid schema $\exists x ϕ(x) \rightarrow \exists x \neg \negϕ(x)$ are not provable, which is analogous to what occurs in modal logic with $\Box$ and $\Diamond$.
The problems of logical translation of axiomatizations and the choice of primitive operators have surfaced several times over the years. An early issue was raised by H. Hi{\. z} in the 1950s on the incompleteness of translated calculi. Further pertinent work, some of it touched on here, was done in the 1970s by W. Frank and S. Shapiro, as well as by others in subsequent decades. As we shall see, overlooking such possibilities has led to incorrect claims of completeness being made (e.g. by J. L. Bell and A. B. Slomson as well as J. N. Crossley) for axiomatizations of classical predicate logic obtained by translation from axiomatizations suited to differently chosen logical primitives. In this note we begin by discussing some problematic aspects of an early article by W. Frank on the difficulties of obtaining completeness theorems for translated calculi. Shapiro had established the incompleteness of Crossley's axiomatization by exhibiting a propositional tautology that was not provable. In contrast, to deal with Bell and Slomson's system which is complete for propositional tautologies, we go on to show that taking a formal system for classical predicate calculus with the primitive $ \exists$, setting $\forall x ϕ(x) \stackrel{\text{def}}{=}\neg \exists x \neg ϕ(x)$, and writing down a set of axioms and rules complete for the calculus with $\forall $ instead of $ \exists$ as primitive, does not guarantee completeness of the resulting system. In particular, instances of the valid schema $\exists x ϕ(x) \rightarrow \exists x \neg \negϕ(x)$ are not provable, which is analogous to what occurs in modal logic with $\Box$ and $\Diamond$.
