The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism.

中文翻译：

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新逻辑主义及其逻辑

批判性地审查了在新逻辑主义纲领中相当不受限制地使用二阶逻辑。有人详细论证了它带来了真正的集合论存在假设，并且休谟原理在推导弗雷格定理时似乎提供的数学力量主要来自假定的“逻辑”而不是休谟原理。结果表明，休谟原理实际上并不比非常弱的罗宾逊算术 Q 强。因此，只有少数算术的基本事实可以从休谟原理逻辑上推导出来。这几乎不能算是对逻辑主义的辩护。