The Stoic “indemonstrables” were inference rules; a rule about rules was the synthetic theorem: if from certain premisses a conclusion follows and from that conclusion and certain further premisses a second conclusion follows, then the second conclusion follows from all the premisses together. Similar things occur as medieval “rules of consequence”, although not usually on a metametalevel; and (with the same proviso) the following might be deemed a contemporary avatar of that Stoic theorem.

If every formula which occurs once or more often in the list A1, A2, …, An, B1, B2, …, Bm occurs also at least once in the list C1, C2, …, Cr then:

This rule [Church: Introduction to Mathematical Logic, 1956, pp. 94, 165], which may be called the rule of modus ponens under hypotheses (MPH), is worthy of attention for the following reasons:

A. MPH and the axioms A ⊃ A yield precisely the positive implicative calculus (and very easily, too).

B. MPH and the axioms A ⊃ f ⊃ f ⊃ A yield a new formulation of the full classical propositional calculus (in terms of f and ⊃).

C. MPH and the axioms ∼A ⊃ A ⊃ A and A ⊃. ∼A ⊃ B yield the classical calculus in terms of ∼ and ⊃.