The plain meaning of an expression α is represented as [[α]] and the implicature is ¬S(α ALT ), where S(α ALT ) is the weakest alternative of α that entails α.
the strengthened meaning of α is the conjunction of the two meanings: [[α]] ∧ ¬S(α ALT ), which entails other strengthened meanings from the stronger alternatives of α. Chierchia introduces the negation operator to get a stronger meaning.