In 1999, Liping Ma's study of US and Chinese elementary teachers’ mathematical knowledge presented readers with a puzzle. Why did the US teachers, all of whom had tertiary degrees and were enrolled in intensive post-graduate programs, show less knowledge of school mathematics than the Chinese teachers, who had so much less education? Why were the Chinese teachers able to connect procedural topics such as algorithms with conceptual topics such as place value and the distributive law? We believe that an important part of the explanation is that the school arithmetic in the two countries was profoundly different: Unlike the school arithmetic of their US counterparts, the solid substance of the school arithmetic learned and taught by the Chinese teachers afforded mathematical reasoning. We illustrate this claim and the meaning of “solid substance” by presenting a logical system distilled from 19th-century US and 20th-century Chinese elementary textbooks. This system extends the system described in our 2018 article “The Theory of School Arithmetic: Whole Numbers.”