The name is absent

derived by applying to its members the OR operator defined for elementary
propositions by

pⁿ V pⁿ = {ω : ω_n = 1} U {ω : ω_n∣ = 1}

and, more generally, by

V p = ∪ '■ ∙

{pEP }     {ptP }

The relationship between events, propositions and histories may now be
developed further. First, there is a 1-1 correspondence between complete
histories, having a value h (t) for each t E T. This generates a natural
correspondence between events (subsets of the state space) and collections of
histories.

Any (non-null) compound proposition p corresponds, from the external
viewpoint, to an event

E (p) = {ω ∈ [0,1] : ω_n = 0, Vn ∈ N_ (p) ; ω_n = 1, Vn ∈ N₊ (p)} C Ω.

Since distinct compound propositions may be logically equivalent, this cor-
respondence is not 1-1.

4 The decisionmaker’s viewpoint

We consider a decision maker who is not aware of all the propositions in P.
The class of all propositions considered by individual i at time t is denoted P^.
This class must include all the decisions that are to made by decisionmaker
i at time t + 1 ³and will also include a variety of propositions about the state
of the world. As in the Grant and Quiggin (2004), we can define logical
operations with respect to this subclass of P.

4.1 Full rationality in a bounded domain

An important special case is one that may be described as ‘full rationality in
a bounded domain’. In this case, the individual has access to a fixed set of
propositions P^г Ç P closed under V and Λ∙ Without loss of generality, it may
be assumed that the set P^г is generated by a set of elementary propositions
Nⁱ Ç N, that is, Pⁱ = 2ⁿ. The individual is unaware of elementary proposi-
tions in N ^г = N-N^г. Let P ^г = 2ⁿ ’ be the set of propositions generated
by elements of N '.

³We do not allow for ‘unconscious decisions’.

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