Handbook of the History of Logic: - Fordham University Faculty

Logic in the 14 th Century after Ockham 491
This means that, during a disputation, it may occur that (1) φ0,φ1 φ2 but
φ0,φ2 φ1, or else that (2) φ0,φ1 φ2 but φ0 φ2. (1) is related to the
obvious asymmetric character of implication, and (2) to the dynamic nature of the
game, what I shall call the ‘expansion of the informational base Γn’. This can be
best seen if we examine what happens in terms of models during an obligational
disputation. For that, here are some definitions:
DEFINITION 5. Γn= Informational base, i.e. a set of propositions.
DEFINITION 6. UM n = The class of models that satisfy informational base Γn.
DEFINITION 7. UM φn = The class of models that satisfy φn.
DEFINITION 8. UM n Γn iff UM n P for all P in Γn.
A model that satisfies a set of propositions satisfies each of them (i.e. they are all
true in this model). It is clear that, if Γk= {φn}∪ {φm}, then UM k= UM φn ∩
UM φm. So, the set of models that satisfy Γk is the intersection of all the models
that satisfy each of the elements of Γk. Similarly, if Γn+1 =Γn∪{φn+1}, then
UM n+1 = UM n ∩ UM φn+1.
THEOREM 9. If Γn φn+1 and φn+1 is accepted, then UMn = UMn+1.
Assume that, at a given state of the game, Γn φn+1. AccordingtoR(φn),φn+1
must be accepted, forming Γn+1 =Γn ∪{φn+1}. Now take UM n, that is, all the
models that satisfy Γn. According to the model-theoretic definition of implication
(i.e. P Q iff Q is true in all models where P is true, that is, if UM P Q),
if UM n Γn and Γn φn+1, then UM n φn+1. Since Γn+1 =Γn ∪{φn+1},
UM n Γn and UM n φn+1, then UM n Γn+1. It is defined that UM n+1
Γn+1, soUM n = UM n+1.
Thus, all the models that satisfy Γn also satisfy Γn+1.
THEOREM 10. If Γn φn+1 and φn+1 is accepted, then UMn+1 ⊂ UMn.
Assume that, at a given state of the game, Γn φn+1 and KC φn+1. According
to R(φn), φn+1 must be accepted, forming Γn+1 =Γn∪ {φn+1}. UM n+1 is
the intersection of UM n and UM φn+1 (UM n+1 = UM n
∩ UM φn+1). But
because Γn φn+1, UM n φn+1. So not all models that satisfy Γnalso satisfy
φn+1. Since Γn+1 =Γn∪ {φn+1}, not all models that satisfy Γn also satisfy Γn+1.
Thus UM n+1 = UM n. But Γn is contained in Γn+1, so all models that satisfy
Γn+1 also satisfy Γn – UM n+1 Γn. SoUM n+1 ⊂ UM n.
Thus, all the models that satisfy Γn+1 are contained in the set of models that
satisfy Γn.
Summing up; in an obligational game, UM n+1 ⊆ UM n. If Γn φn+1, Γn
¬φn+1 or R(φn+1) = ?, then UM n = UM n+1, otherwise UM n+1 ⊂ UM n.
That is, the larger the informational base, the fewer models will satisfy it, and
greater the constraints on the choice between ¬φn and φn will be (a model-theoretic
way to see why a larger base implies that more propositions will have inferential
relations with Γn). Clearly, the base is expanded (and therefore the range of
models that satisfy it is reduced) only by inclusion of ‘irrelevant’ propositions.