  Clauses \(({\sim}), (\rightarrow)\), and (5) allow us to calculate the truth value of any sentence at any world on a given valuation. A definition of validity is now just around the corner. An argument is 5-valid for a given set W (of possible worlds) if and only if every valuation of the atomic sentences that assigns the premises \(T\) at a world in \(W\) also assigns the conclusion \(T\) at the same world. An argument is said to be 5-valid iff it is valid for every non empty set \(W\) of possible worlds.