The particles "too" and "either" are classically recognized as hard triggers, and "even" as a soft(er) trigger of additive presuppositions. Cross-linguistic counterparts have the same properties.
1. (It's not the case that) BILL yawned too.
hard presupposition: someone other than Bill yawned
2. (It's not the case that) BILL didn't yawn either.
hard presupposition: someone other than Bill didn't yawn
3. (It's not the case that) even BILL yawned.
hard presupposition: Bill was very unlikely to yawn
soft presupposition: someone other than Bill yawned
The additive presuppositions are typically stipulated, not compositionally derived. Recent literature has even ignored them (Chierchia 2013:148) or treated them as part of the assertion (Ahn 2014:29, Gajic' 2016). While following that literature in various other respects, this talk aims to account for the source, shape and presuppositional nature of the additive component. It builds on focus alternatives, presuppositional in the sense of Abusch 2010. The details of the analysis are motivated with reference to the fact that the same particles form free choice items and negative polarity items in many languages, from Hindi to Hungarian.