Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn’t. In fact, 2FA is not conservative over n-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic.

新弗雷格逻辑学家声称，休谟原理 (HP) 可以被视为基数的隐含定义，仅凭命令即可成立。新弗雷格逻辑主义的一个长期存在的问题是HP对于纯公理二阶逻辑来说并不演绎保守。这似乎阻止了 HP 成为现实。在本文中，我们研究了 Richard Kimberly Heck 的二排序弗雷格算术 (2FA)，它是 HP< 的变体/span> 在二阶或高阶逻辑上不是演绎场保守的。HP。由此可见，在通常的一排序设置中，$n \geq 2$ 阶逻辑并不保守，对于所有n 对于 2FA 被认为比二阶逻辑演绎保守。我们证明事实并非如此。事实上，