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 cross-linguistically focus-sensitive, and (ii) in many languages, too/even builds negative polarity items and free-choice items as well, often in concert with other particles. (i) is the source of 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 different kinds of alternatives (focus, subdomain, scalar) and invoke suitably different further operators.