The name is absent

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 E_t, consisting
of all elements beginning with the N_t terms having values υⁿ.

E_t ≡ {ω : ω_n = 2^-*ⁿ⁺¹) (uⁿ + 1) for all n ∈ N_tJ

Third, it is a compound proposition p_t, 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 = ω_n< = 1}

= E_pn ∩ E_pn

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 E_t 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 : ω_n = 1} U {ω : ω_n' = 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

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