 Notice that Hume’s Principle bears an obvious formal resemblance to Basic Law V. Both are biconditionals asserting the equivalence of an identity among singular terms (the left-side condition) with an equivalence relation on concepts (the right-side condition). Indeed, both correlate concepts with certain objects. In the case of Hume’s Principle, each concept \(F\) is correlated with \(\#F\). However, whereas Basic Law V problematically requires that the correlation between concepts and extensions be one-to-one, Hume’s Principle only requires that the correlation between concepts and numbers be many-to-one. Hume’s Principle often correlates distinct concepts with the same number. For example, the distinct concepts author of Principia Mathematica (‘\([\lambda x \, Axp]\)’) and positive integer between 1 and 4 (‘\([\lambda x \, 1\lt x\lt 4]\)’) are equinumerous (both have two objects falling under them). So \(\#[\lambda x\, Axp]\) = \(\#[\lambda x \, 1\lt x\lt 4]\). Thus, Hume’s Principle, unlike Basic Law V, does not require that the domain of numbers be as large as the domain of concepts. Indeed, several authors have developed models that show Hume’s Principle can be consistently added to second-order logic. See the independent work of Geach (1976, 446–7), Hodes (1984, 138), Burgess (1984) and Hazen (1985).