I've been considering the possible-worlds semantics for simple forms of modal logic, such as Kripke modal logic. This reading of modal logic seems to be a reduction to restricted truth-tables, where each row of the truth-table corresponds to the truth-assignment to propositions in (one or more) conceivable worlds; and the worlds which we deem possible are some subset of these rows.
Consider a proposition A, and some selection W of possible worlds which corresponds to some particular rows of a truth-table. We also introduce a (trivial) modal notion of a conceivable world, which is more general than a "possible" world in that all rows of the truth table are considered to correspond to conceivable worlds; and let U correspond to the set of all concievable worlds, containing the actually possible worlds as a subset. The usual notion of the truth of A in some world w ∈ W (or w ∈ U) is then written v(A,w). In this reading of modal logic, □A means that in all admissible rows, A holds, i.e.
∀w∈W: v(A,w)=T ;
and ◊A means that in some one or more admissible rows, A holds, i.e.
∃w∈W: v(A,w)=T .
This makes immediately clear why A⇒□A is valid in Kripke modal logic, while □A⇒A is not; if A is a theorem (and our logical system is sound), this means that
∀w∈U: v(A,w)=T ,
that is A holds in all conceivable worlds, without restriction merely to the possible worlds, so that it clearly implies that
∀w∈W: v(A,w)=T. Thus clearly A⇒□A; is valid. On the other hand, □A⇒A is not valid, as the truth of A in all conceivable worlds (all worlds in U) is not implied by its truth in only the possible worlds (all worlds in W ⊆ U).
Question. How is the restriction to possible worlds, in this case, not simply equivalent to adopting a supplementary propositional premise which is true in precisely the set of possible worlds, so that □A is synonymous with W⇒A where W is true in all possible worlds and only the possible worlds?