 After the early 1940s, proof theory did not broaden and diversify so as to address a potentially fertile agenda: an investigation of the various forms that introduction and elimination rules might take, as it examines rule-governed expressions whose rules are not quite so neatly classifiable as introduction and elimination rules. This is the case, for example, with families of ‘coeval’ and interdependent concepts of a nevertheless logico-mathematical kind. An example of such a family is that of the ordered pair of any two things; the first member of any ordered pair; and the second member of the same. An important feature of this example, and of other examples that could be given, is that the operators in question are term-forming operators. Gentzen had confined his study to sentence-forming operators. Perhaps it was Tarski’s theory of truth for formalized languages (see Tarski 1956 [1933]) that deflected interest away from further development of this essentially inferentialist approach to the meanings of logical and mathematical operators. 