All the a-objective formulas true at a model M, essentially the objective formulas true *wrt.* w and the b-subjective formulas true wrt. e j b
, are given by obj+
a
(M). Note that these formulas do not strictly correspond to a's possibilities. Rather, we define Obj+
a on her epistemic state e ka
, and this gives us all the a-objectives formulas that a considers possible. We shall now argue that the intuition of all of Levesque's properties is maintained.8 Property 1. In the single agent case, this property ensured that an agent's epistemic possibilities are not affected on evaluating N. This is immediately the case here. Given a model, say (e ka
, e j b
, w),
a's epistemic possibilities are determined by Obj+
a(e ka). To evaluate Naα, we consider all models
(e ka
, ek−1 b, w′) such that hw
′
, ek−1 b**i 6∈** e ka
. Again, a's possibilities are given by Obj+
a(e ka) for all these models, and does not change.

Property 2. In the single agent case, this property ensured that evaluating Lα and Nα is always wrt. the set of all possibilities, and completely independent of e. As discussed, in the many agent case, possibilities mean i-objective formulas and analogously, if α is a possibility in a's view, say an a-objective formula of maximal b-depth of k, then we should interpret Laα and Naα *wrt.* all aobjective possibilities of max. depth k: the set of (k + 1)-structures. Clearly then, the result is fixed and independent of the corresponding e k+1. The following lemma is a direct consequence of the definition of the semantics.

Lemma 3.2: Let α be a i-objective formula of j-depth k, for j 6= i. Then, the set of k+1*-structures* that evaluate Liα and Niα is E
k+1.

Property 3. The third property ensures that one can characterize epistemic states from any set of iobjective formulas. Intuitively, given such a set, we must have a model where *precisely* this set is the epistemic state. Earlier attempts at clarifying this property involved constructing a set of maximally K45n-consistent sets of basic i-objective formulas, and showing that there exist an epistemic state that precisely corresponds to this set. But, defining possibilities via K45n proof-theoretic machinery inevitably leads to some limitations, as we shall see. We instead proceed semantically, and go beyond basic formulas. Let Ω be a satisfiable set of i-objective formulas, say of maximal j-depth k, for j 6= i. Let Ω
′ be a set obtained by adding a i-objective formula γ of maximal j-depth k such that Ω
′
is also satisfiable. By considering all i-objective formulas of maximal j-depth k, let us construct Ω
′
, Ω
′′
, **. . .** by adding formulas iff the resultant set remains satisfiable. When we are done, the resulting Ω
∗
is what we shall call a maximally satisfiable i-objective set.9 Naturally, there may be many such sets corresponding to Ω. We show that given a set of maximally satisfiable i-objective sets, there is a model where precisely this set characterizes the epistemic state.

Theorem 3.3: Let Si be a set of maximally satisfiable sets of i-objective formulas, and σ *a satisfiable* objective formula. Suppose Sa is of max. b-depth k − 1 and Sb is of max. a-depth j − 1*. Then there is a* model M∗ = he
∗k a
, e∗jb
, w∗i *such that* M∗|= σ, Sa = Obj+
a
(e
∗k a
) and Sb = Obj+
b
(e
∗j b
).

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