What we call the Hilbert‐Bernays (HB) Theorem establishes that for any satisfiable first‐order quantificational schema S, there are expressions of elementary arithmetic that yield a true sentence of arithmetic when they are substituted for the predicate letters in S. Our goals here are, first, to explain and defend W. V. Quine's claim that the HB theorem licenses us to define the first‐order logical validity of a schema in terms of predicate substitution; second, to clarify the theorem by sketching an accessible and illuminating new proof of it; and, third, to explain how Quine's substitutional definition of logical notions can be modified and extended in ways that make it more attractive to contemporary logicians.

中文翻译：

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一阶逻辑有效性和希尔伯特-贝纳耶斯定理

我们所谓的希尔伯特-贝纳斯（HB）定理确定，对于任何可满足的一阶量化模式S，都有基本算术表达式可以在替换S中的谓词字母时产生真正的算术句子。首先，我们的目标是解释和捍卫奎因（WV Quine）的主张，即HB定理允许我们定义基于谓词替换的模式的一阶逻辑有效性。其次，通过画出一个易于理解的例子并阐明它的新证明来阐明该定理；第三，解释奎因对逻辑概念的替代定义如何以使之对当代逻辑学​​家更具吸引力的方式进行修改和扩展。