Handbook of the History of Logic: - Fordham University Faculty

488 Catarina Dutilh Novaes
begins. Then O puts forward a further proposition, R responds to it according to
R(φn), and this procedure is repeated until the end of the game.
The logical rules of the game are defined by R(φn), in the following way:
DEFINITION 2 (Rules for positum).
R(φ0) =0iffφ0 62 ⊥
R(φ0) =1iffφ0 ⊥
The rule defining the response that R should give to φ0 (the positum — [Burley,
1988, 378]) has interesting consequences for the idea that obligationes are games of
consistency maintenance. If R is obliged to accept at the beginning a proposition
that entails a contradiction — for example, any paradoxical proposition such as
Liar sentences and the like — then there is no possible winning strategy for R.
He cannot maintain the consistency of a set of propositions that, from the outset,
contains a contradictory paradoxical proposition. So the rules of the game stipulate
that there always be a winning strategy for R, starting from this restriction
upon the positum. Burley expresses this clause by saying that it must be in the
Respondent’s power to satisfy the requirement (of not falling in contradiction) (cf.
[Burley, 1988, 376]).
DEFINITION 3 (Rules for proposita).
Γn−1 φn, or
R(ϕn) =1iff
Γn−1 φn, Γn−1 ¬φn and Kc 63 φn
Γn−1 ¬φn, or
R(φn) =0iff
Γn−1 φn, Γn−1 ¬φn and KC ¬φn
R(φ)n) =?iffΓn−1 φn, Γn−1 ¬φn,KC φn,KC ¬φn
That is, if Respondent fails to recognize inferential relations and if he does not
respond to a proposition according to its truth-value within common knowledge,
then he responds badly (cf. [Burley, 1988, 381]).
Formation of Γn. The different sets of propositions accepted by R (i.e., the
propositions to which R has committed himself in the disputation) are formed in
the following way:
DEFINITION 4 (The sets Γn).
If R(φn) = 1, then Γn =Γn−1 ∪{φn}
62 I use the forcing turnstyle throughout to express the relation of semantic implication
between propositions. That is, within obligationes the relation of ‘following’ is not defined
syntactically or proof-theoretically, but rather semantically.
63 In case of KC, it is not so much that KC semantically implies a proposition φn, but rather
that φn is contained in KC (therefore a fortiori KC also implies φn). For the sake of simplicity,
I use only the forcing turnstyle.