obligationes - medieval EF games

From Logic and Games (Stanford Encyclopedia of Philosophy):

Several medieval texts describe a form of debate called obligationes. There were two disputants, Opponens and Respondens. At the beginning of a session, the disputants would agree on a ‘positum’, typically a false statement. The job of Respondens was to give rational answers to questions from Opponens, assuming the truth of the positum; above all he had to avoid contradicting himself unnecessarily. The job of Opponens was to try to force Respondens into contradictions. So we broadly know the answer to the Dawkins question, but we don’t know the game rules! The medieval textbooks do describe several rules that the disputants should follow. But these rules are not stipulated rules of the game; they are guidelines which the textbooks derive from principles of sound reasoning with the aid of examples. (Paul of Venice justifies one rule by the practice of ‘great logicians, philosophers, geometers and theologians’.) In particular it was open to a teacher of obligationes to discover new rules. This open-endedness implies that obligationes are not logical games in our sense.

Imagine ∃ taking an oral examination in proof theory. The examiner gives her a sentence and invites her to start proving it. If the sentence has the form

ϕ∨ψ then she is entitled to choose one of the sentences and say ‘OK, I’ll prove this one’. (In fact if the examiner is an intuitionist, he may insist that she choose one of the sentences to prove.) On the other hand if the sentence is

ϕ∧ψ then the examiner, being an examiner, might well choose one of the conjuncts himself and invite her to prove that one. If she knows how to prove the conjunction then she certainly knows how to prove the conjunct.

The case of ϕ→ψ is a little subtler. She will probably want to start by assuming ϕ in order to deduce ψ; but there is some risk of confusion because the sentences that she has written down so far are all of them things to be proved, and ϕ is not a thing to be proved. The examiner can help her by saying ‘I’ll assume ϕ, and let’s see if you can get to ψ from there’. At this point there is a chance that she sees a way of getting to ψ by deducing a contradiction from ϕ; so she may turn the tables on the examiner and invite him to show that his assumption is consistent, with a view to proving that it isn’t. The symmetry is not perfect: he was asking her to show that a sentence is true everywhere, while she is inviting him to show that a sentence is true somewhere. Nevertheless we can see a sort of duality.