A licenser is the operator a construction places over the position of a polarity item, given
as a family of operators indexed by what the construction leaves open (the model, a restrictor, a
bound or an accessibility relation). It is classical (a map between bounded lattices), scalar (one
with stronger scale-mates), Strawsonian (von Fintel's map into trivalent propositions), modal, or
a wh-question. A licenser has a plain meaning below the exhaustivity operator and, after
Gajewski, a meaning above it that adds its direct implicature, the exhaustification against its
stronger scale-mates; only a scalar licenser has a direct implicature.
PolarityItem.Licenser.Carries: a strength of negation carried. Every negative polarity item
needs the plain meaning downward entailing modulo presupposition (von Fintel) and not upward
entailing as well (Bylinina and Nouwen); a stronger strength must hold outright of the
exhaustified meaning.
Gajewski asks only for downward entailment of the exhaustified meaning; Zwarts's strengths are
kept here, and agree with Gajewski's condition at the endpoints of the scale.
The unit conditions of ProjectivitySignature.HoldsFor do not exclude constant maps, which are
downward entailing and, if constantly false, anti-additive; requiring weak strength not to be
upward entailing as well excludes them.
The question route states the weak form the theory of question value proves, that widening
never lowers the question's value; van Rooy's strict condition for rhetorical questions is not
formalized.
A family of operators on a scale with stronger mates, each paired with its meaning enriched
by its direct implicature, the exhaustification against those mates (Gajewski 2011's O with
ALT-1).
The operator that a construction places over the position of a polarity item is classical,
scalar (with its enrichment), into trivalent propositions, modal, or a question.
A context licenses polarity items by strengthening, as a generic context licensing free choice
items (Kadmon & Landman 1993, Dayal 1996), or by the relevance of a question
(van Rooy 2003).
A licenser holds a signature outright when every operator in its family does; an operator into
trivalent propositions holds it on its truth set, and a scalar operator on its plain meaning.
A licenser is Strawson upward entailing when every operator in its family is upward entailing
modulo its presupposition; a constant operator is both this and downward entailing.
A licenser carries weak strength when it is downward and not upward entailing modulo
presupposition. This is the condition every negative polarity item puts on the meaning below the
exhaustivity operator (von Fintel 1999, Gajewski 2011's Condition 1,
Bylinina & Nouwen 2018's (53)). It carries a stronger strength when, in addition, its exhaustified
meaning holds that strength outright.
A licenser licenses by strengthening when, under each of its operators, widening the domain of
an existential strengthens the result: the existential over the wider domain entails, modulo
presupposition, the existential over the narrower one (Kadmon & Landman 1993).
A modal licenser licenses free choice when, under one of its operators and over a domain of at
least two possible witnesses, the antiexhaustified existential is consistent
(Chierchia 2006).
A question licenses a domain widener by relevance, since widening the domain of a wh-question
never lowers its value in any decision problem (van Rooy 2003).
A non-trivial licenser that is downward entailing modulo presupposition, and not outright once
its direct implicature is factored in, carries exactly weak strength (Gajewski 2011).