 I was motivated to write the present entry after reading an early draft of an essay by William Demopoulos. (The draft was eventually published as Demopoulos and Clark 2005.) Demopoulos kindly allowed me to quote certain passages from that early draft in the footnotes to the present entry. I am also indebted to Roberto Torretti, who carefully read this piece and identified numerous infelicities; to Franz Fritsche, who noticed a quantifier transposition error in Fact 2 about the strong ancestral; to Seyed N. Mousavian, who noticed some typographical errors in some formulas; to Xu Mingming, who noticed that Fact 8 about the Weak Ancestral (Section 4, subsection “The Weak Ancestral of \(R\)”) was missing an important condition (namely, that \(R\) must be 1–1); to Evgeni Latinov, who noted that the discussion in Section 2.7 (of how the Russell paradox is engendered in Frege’s system) also requires that the material equivalence of \(F\) and \(G\) be a sufficient for the identify of \(F\) and \(G\); and to Paul Pietroski, who noticed an infelicity in the first statement of the principle of induction in Section 4. I am indebted to Kai Wehmeier, who (a) reminded me that, strictly speaking, the result of replacing Basic Law V by Hume’s Principle in Frege’s system does not result in a subsystem of the original until we replace the primitive notion “the course of values of the function \(f\)” with the primitive notion “the number of \(F\)s”, and (b) refereed the July 2013 update to this entry and developed numerous, insightful suggestions for improvement. Finally, I am indebted to Jerzy Hanusek for pointing out that Existence of Extensions principle can be derived more simply in Frege’s system directly from the classical logic of identity, and to Rowena Conway for noticing some unnoticed corruption  caused by our new MathJax rendering of the formulas.