  Before receiving the famous letter from Bertrand Russell informing him of the inconsistency in his system, Frege thought that he had shown that arithmetic is reducible to the truths of logic. It is recognized today, however, that at best Frege showed that arithmetic is reducible to second-order logic extended only by Hume’s Principle. Some philosophers think Hume’s Principle is analytically true (i.e., true in virtue of the very meanings of its words), while others resist the claim, and there is an interesting debate on this issue in the literature (see, e.g., Boolos 1997, Wright 1999).