Additive presuppositions are derived through activating focus alternatives

The additive presupposition of particles like too/even is uncontested, but usually stipulated.This paper proposes to derive it based on two properties. (i) too/even is crosslinguisticallyfocus-sensitive, and (ii) in many languages, too/even builds negative polarityitems and free-choice items as well, often in concert with other particles. (i) is the sourceof its existential presupposition, and (ii) offers clues regarding how additivity comes about.(i)-(ii) together demand a sparse semantics for too/even, one that can work with differentkinds of alternatives (focus, subdomain, scalar) and invoke suitably different furtheroperators.