in which each element incorporates all its predecessors. Second, it is a time-
dated event (a subset of the state space) which may be denoted Et, consisting
of all elements beginning with the Nt terms having values υn.
Et ≡ {ω : ωn = 2-*n+1) (un + 1) for all n ∈ NtJ
Third, it is a compound proposition pt, which may be defined using the Λ
operator, corresponding to logical AND. For pairs of elementary propositions,
let Λ be defined in terms of Ω as
pn Λ pn' = {ω : ωn = ωn< = 1}
= Epn ∩ Epn
More generally, for any collection of elementary propositions indexed by
N Ç N, we define
Λ pn = ∩ Epn
n } n }
The proposition representing the event Et associated with history h (t) is
then given by
pt = Λ pn Λ v∙
{n∈Nf.vn=1} {n∈Nt'.vn=-1}
More generally, a compound proposition is derived by assigning truth
values of 1 or —1 to all pn where n is a member of some (possibly empty)
subset N (p) Ç N, leaving all pn , n ∈L N (p) unconsidered. The set N (p)
is referred to as the scope of p, and is the disjoint union of N- (p) , the set
of elementary propositions false under p, and N+ (p), the set of elementary
propositions true under p. The simple proposition pn has scope N (pn) = {n} .
We define the null proposition p® such that pn = 0, Vn and do not assign a
truth value to p®.
The OR operator is
pn V pn = {ω : ωn = 1} U {ω : ωn' = 1} .
The class of all propositions in the model is denoted by P = { — 1, 0,1}N.
It is useful to consider more general classes of propositions P Ç P. To any
class of propositions P, given state ω, we assign the truth value
t (P; ω) = sup {t (p; ω)} .
p∈F
That is, P is true if any p ∈ P is true, and false if all p ∈ P are false. In
terms of the logical operations defined below, the set P has the truth value