Handbook of the History of Logic: - Fordham University Faculty

Logic in the 14 th Century after Ockham 489
If R(φn) = 0, then Γn =Γn−1 ∪{¬φn}
If R(φn) =?, then Γn =Γn−1
In particular, if R(φ0) = 1, then Γ0= {φ0}. If R(φ0) =0orR(φ0) = ?, then
there is no disputation.
These rules mirror closely the clauses of Lindenbaüm’s lemma, the main idea
being that propositions are gradually added to a set of propositions (which starts
with one single element, the positum), while consistency is also maintained. There
is a significant difference, though, in that, in the construction of a maximal consistent
set according to this lemma, if the set Γn =Γn−1∪{φn} formed is inconsistent,
then the construction simply continues with Γn−1, i.e. the so far largest consistent
set built. In the obligationes framework, however, if an inconsistent set is constructed,
the procedure comes to a halt, as respondent has responded badly and
thus lost the game.
Outcome. O wins the game if it is recognized that Γn ⊥ ;thatis,ifR has conceded
a contradictory set of propositions. R wins the game if, when the disputation
is declared to be over, it is recognized that Γn ⊥ . The clause about the stipulated
time concerns the feasibility of the game: the construction of maximal-consistent
sets of propositions is not feasible within human time, therefore respondent is
expected to keep consistency only during a certain time.
4.2.2 Can respondent always win?
The rules of the obligational game as defined guarantee that there always be a
winning strategy for R. This is due to two facts: one is a stipulated rule of the
game and the other is a general logical fact. 64 The relevant rule of the game
is: a paradoxical positum should not be accepted. As stated by Burley himself,
the point of this clause is exactly to guarantee that R stands a chance to win.
Therefore, R always starts out with a consistent set of propositions.
It is a general principle of logic (and also the backbone of Lindenbaüm’s lemma)
that any consistent set of propositions can always be consistently expanded with
one of the propositions of a contradictory pair φnand ¬φn. 65 R starts with a consistent
set of propositions (the set composed of the positum); so at each move, there
is in theory at least one ‘correct’ way of answering, i.e. either accepting or denying
φn, which maintains the set of accepted and denied propositions consistent.
4.2.3 Why does R not always win?
But why does the game remain hard? It is a fact that R often makes wrong
choices and is not able to keep consistency, even though to keep consistency is
64 This fact has already been noticed by J. Ashworth: ‘a certain kind of consistency was
guaranteed for any correctly-handled disputation’ (Ashworth 1981, 177).
65 Proof: Assume that Γ is consistent. Assume that Γ ∪{φ} is inconsistent. Thus Γ ¬φ (1).
Moreover, assume that Γ ∪{¬φ} is inconsistent. Thus Γ φ (2). From (1) and (2), it follows
that Γ φ&¬φ, that is, that Γ is inconsistent, which contradicts the original assumption. The
principle to be proven follows by contraposition.