From the introduction:
No tax policy analysis stands complete without examination of equity implications. But despite its role as a traditional pillar of tax policy analysis, equity itself remains a controversial concept. What is meant by the term equity? How should it be measured? Is there more than one type of equity? What is the relationship of different types of equity to each other? For decades, scholars and policy makers have explored the possibility that equity is best understood as two distinct concepts — vertical equity and horizontal equity — both of which must be evaluated. Horizontal equity (HE) is defined to mean that equals should be treated alike. Vertical equity (VE) is defined to mean that an appropriate distinction should be made in the treatment of people who are not alike. Although disagreement exists, HE in our tax system has generally been thought to require that individuals with the same income should pay the same tax. VE has generally been thought to require a progressive rate structure that imposes progressively higher rates on individuals with higher incomes. Despite frequent reliance on both HE and VE in tax policy analysis over the years, scholars have engaged in an active and vibrant debate about whether HE has any significance independent of VE in designing a tax system. This dispute has been best captured by the debate between two economists, Richard Musgrave and Louis Kaplow.
James R. Repetti and Diane M. Ring. "Horizontal Equity Revisited." Florida Tax Review 13, no.3 (2013): 135-155.