Arithmetic algorithms include two types of rules: conventional rules that may be changed by authority, and may legitimately vary from one classroom or country to another (e.g. putting the sum below, rather than above, the numbers added) and logical rules that involve the logic of the algorithm. Changes in the logical rules produce incorrect answers. Hence these rules are not legitimately alterable by authority. Second-order logical rules depend on the particular conventions of the symbol system used (e.g. the rule for carrying in place-value addition). Given the symbol system used, these rules are not legitimately alterable by authority. However, as a result of their dependence on the symbol system, children may have difficulty distinguishing second-order logical rules from conventional rules. Ninety-eight children in grades 2 through 5 were interviewed about the correctness of answers obtained using alternatives to standard conventional and second-order logical rules, and about the legitimacy of authorities to change the rules. Half the children across this age range treated second-order logical rules as somewhat like conventions, judging that an answer resulting from an alternative to a second-order logical rule is correct if sanctioned by authority. With increasing age children increasingly limited the jurisdiction of authority over second-order logical rules.