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Domains of Polarity Items

Homer, V. (2021). Journal of Semantics, 38(1), 1–48

Source Bibliography Paper

This file formalizes the localist, environment-based theory of polarity items: an item is evaluated in its domains, the constituents containing it and the polarity head of their own clause, and it is acceptable only if it is acceptable in some domain, one carrying the strength of negation that licenses it and lacking the strength that blocks it. For quoi que ce soit, any and, as the paper concludes, some, that strength is downward entailment. A second downward-entailing expression in a domain undoes the first, so quoi que ce soit and some are anti-licensed and licensed again as such expressions stack up, and some survives three of them in a sentence downward entailing with respect to it, since a smaller domain is not. A scalar implicature can make a domain non-monotone: any is anti-licensed and some licensed under not … and, but not under not … or.

Occurrences of some and any are entangled: an item is marked as licensed in a domain only if every other item there counts as acceptable there, being acceptable or marked as licensed in a smaller constituent, so licensing proceeds bottom-up. Below impossible, anyone … something is licensed and someone … anything is not, something being marked as licensed in its own clause first. Entanglement gives the paper's argument that some is blocked by downward entailment rather than anti-additivity: below not sure, downward entailing without being anti-additive, an anti-additive classification would license someone together with anything. The same holds, without entanglement, below not impossible … not, a derivation of this file.

Implementation notes

  • The paper states its conditions as necessary ones. For sentences with one polarity item they are read here as biconditionals, as the paper does when it predicts which of them are acceptable.
  • A world records a fact independent of the polarity item and which of two things the subject did. The environment of a domain is a function of the meaning of the indefinite, a quantifier over things, evaluated at the things the subject did: the abstraction over the position of the item.
  • Each clause contributes its polarity phrase, whose head is the layer of the minimal domain, and its TP and, when embedded, CP, which are identity layers. A superordinate polarity phrase composes its polarity head with the predicate below it.
  • The modals quantify over a fixed modal base, the worlds in which the subject did something, so that it is possible that the subject did not do the first thing, and possible that the subject did not do the second, but not that the subject did neither: not impossible … not is not anti-additive, as it would be over a base of one world. Sure and claim are both necessity, only their upward monotonicity mattering.
  • A conditional is the material conditional with its consequent fixed, so the CP of the conditional clause denotes the whole conditional. This is downward entailing in the antecedent, which is all the sign needs, where a natural-language conditional is only Strawson downward entailing.
  • The environments of not … and and not … or are Chierchia's strong values, negation being his doubt. The paper works the strengthened meaning through for not every only, so its application to the connectives is this file's. Tense adds no scalar alternatives, so the strengthened TP has the meaning of the strengthened polarity phrase.
  • Quoi que ce soit and some are this file's entries, licensed and blocked by downward entailment; any and someone are the Fragment's, someone blocked by anti-additivity.
  • The when-clause of the sentence with clausemate negation is left out, as in its logical form.
  • In the sentences with two indefinites, a world records which of two people did which of two things, every two items are entangled, and the constituents are numbered from the bottom of the clausal spine. A constituent below the position of the subject denotes a proposition for each value of its trace, and the modal base is the worlds in which someone did something.

Errata

  • The strengthened meaning conjoins the negation of the weakest alternative asymmetrically entailing the constituent, which is undefined when there is none: under not … or and in every constituent without a scalar term. Such a constituent is left as it is, as by Krifka's rule. The alternatives of a constituent without a scalar term are given as its meaning rather than the singleton of it.

TODO

  • The entanglement of some with itself and of any with itself, and at most five people sold someone anything, which needs a model of counting.
  • The disruption by not every, which needs a model of individuals, and the rescuing of some by a quantifier in the specifier of TP or by inferences that are not downward entailing.
  • Strong negative polarity items such as a single.

References

Declarations

L108–110
L112–114

Layers

Paths

L154–159

The items

L209–211

Flip-flop

Quoi que ce soit is anti-licensed under pas impossible, two clausemate downward-entailing expressions, and licensed again when its own clause is negated; some is anti-licensed under clausemate negation, rescued by a superordinate impossible, and anti-licensed again under a superordinate not impossible.

theorem flipFlopL293–302
:
(¬notImpossible.Licenses quoiQueCeSoit ∧ notImpossibleNot.Licenses quoiQueCeSoit) ∧ ¬clausemateNot.Licenses someItem ∧ impossibleNot.Licenses someItem ∧ ¬notImpossibleNot.Licenses someItem

Built from , , , , ,

Downward entailment or anti-additivity

Disruption by scalar implicatures

L343
L354–358

John didn't drink wine and … is downward entailing with respect to the indefinite, but not once strengthened by its implicature that John drank one of them, so any is anti-licensed and some licensed in it; John didn't drink wine or … is its own strengthened meaning, and the reverse holds.

theorem disruptionL435–445
:
((Antitone fun (Q : Quantifier (Fin 2)) => (notConn Chierchia2004.conj Q).ordinary) ∧ ¬notAnd.Licenses English.PolarityItems.any ∧ notAnd.Licenses someItem) ∧ (∀ (Q : Quantifier (Fin 2)), (notConn Chierchia2004.disj Q).strong = (notConn Chierchia2004.disj Q).ordinary) ∧ notOr.Licenses English.PolarityItems.any ∧ ¬notOr.Licenses someItem

Built from , , ,

Entanglement

L452–453
L473–481
L487–490
L497–501
L503–507
L511–512
L522–531
L554–555
L557–558

Monotonicity of the constituents

L564
L575

The logical forms

Acceptability in the logical forms

L643
L659–660
L662–663

With some and any entangled, it is impossible that someone stole anything is anti-licensed and it is impossible that anyone stole something licensed, something being marked as licensed in its own clause before anyone is evaluated; evaluated together rather than cyclically, the second would be anti-licensed as well.

theorem cyclicityL777–785
:
¬impossibleSomeoneAnything.Licensed ∧ impossibleAnyoneSomething.Licensed ∧ ¬impossibleAnyoneSomething.LicensedSimultaneously

Built from , , ,

Below not impossible … not and below not sure, environments are downward entailing without being anti-additive. Blocked by downward entailment, some is anti-licensed in both, with narrowest scope in the first and entangled with anything in the second, as judged; blocked only by anti-additivity, as the Fragment classifies someone, it would be licensed in both.

theorem downwardEntailmentL916–926
:
(¬notImpossibleNot.Licenses someItem ∧ notImpossibleNot.Licenses English.PolarityItems.someone) ∧ ¬notSureSomeoneAnything.Licensed ∧ (notSureSubjObj English.PolarityItems.any English.PolarityItems.someone).Licensed

Built from , , , ,

The judgments

L995–1010