Affirming an individual's belief in the state does not affirm the existence of the state except under your recursive definition.
The definition that "the state is people believing in the state" can be shown to be semantically meaningless by defining it using a Chomsky context-free grammar. (See:
http://en.wikipedia.org/wiki/Context-free_grammar )
Grammar:
(State) ::-> One or more people believing in a (State)
Statement:
People believe in a State therefore a State exists.
Expansion:
People believe in a (State) therefore a (State) exists.
People believe in a (One or more people believing in a(One or more people believing in a(One or more people believing in a(... to infinity ...)))) therefore the (One or more people believing in a(One or more people believing in a(One or more people believing in a(... to infinity ...)))) exists.
You'll have to find a non-recursive definition, for the reasons above. Namely, that a recursive definition imparts no meaning, and therefore does not satisfy the requirements of a definition.