The modal horizon

The standard Ladusaw-Fauconnier analysis of Negative Polarity Items (Fauconnier, 1975, 1978; Ladusaw, 1980) proposes that NPIs are licensed in downward-entailing contexts. This explains much of their distribution;

however, it famously leaves their acceptability in the antecedents of coun-terfactual conditionals unexplained. In contrast to the strict conditional analysis, the Lewis-Stalnaker analysis (Lewis, 1973b; Stalnaker, 1968) does not verify strengthening of the antecedent and therefore does not ensure that antecedents are downward-entailing. For this reason, von Fintel (1999,

2001) proposes an adaptation of both analyses in two steps: First, adjusting the definition of downward-entailingness used in the Ladusaw-Fauconnier analysis, and second, moving from the variably strict Lewis-Stalnaker se-mantics to a version of the strict conditional sese-mantics that preserves the advantages of Lewis-Stalnaker sketched in Section 2.2 while at the same time being downward-entailing in the adjusted sense.

Standardly, downward-entailingness (DE) is defined as follows (von Fintel, 1999):

(101) A function f of typehσ, τiis downward-entailing (DE) ifffor all x, yof typeσsuch thatx⇒ y: f(y)⇒ f(x)

Von Fintel calls his adjusted definition Strawson-downward-entailing-ness (SDE), and defines it as follows:

(102) A function fof typehσ, τiis Strawson-downward-entailing (SDE) iff for allx,yof typeσsuch thatx⇒ yand f(x) is defined: f(y)⇒ f(x) The difference between (101) and (102) is in the additional condition that f(x) be defined. This weakens the notion of downward-entailingness, as cases in which this additional condition does not hold do not count against a function being SDE, that is, a larger class of environments will be SDE than DE. One application for this weakened definition is in explaining the licensing of NPIs underonly, as in (88). Consider the following sentence:

(103) Only John reads a book.

Under the classical definition of downward-entailingness in (101), this is not downward-entailing, as it does not allow to infer (104):

(104) Only John readsGrisella.

Why can we not infer (104) from (103)? Because it might well be true that no one but John reads a book – verifying (103) –, but not that John reads Grisella. However, under von Fintel’s adapted notion of Strawson-DE, we are allowed to draw the inference by adding the additional premise that the presuppositionπof (104) be defined⁶:

(105) π=John reads Grisella.

(106) Only John reads a book∧π⇒Only John reads Grisella

The second step is to ensure that we have an analysis of the counter-factual that makes antecedents SDE. The crucial difference between a strict semantics (which is DEsimpliciter) and a variably strict semantics is in the fixedness of the domain of quantification. Von Fintel proposes a middle position between the two: In a dynamic strict semantics, the domain is quasi-fixed, but can dynamically be updated through a process of presup-position accommodation. Since SDE only requires monotonicity under the assumption of fulfilled presuppositions, this accommodation is invisible to it, rendering the antecedents of counterfactuals SDE.

In terms of semantics for the counterfactual, we return to the strict picture:

(107) ~if p would q^w=1 iff∀w⁰ ∈Dw:w⁰ ∈ ~p →w⁰ ∈~q

However, there is a twist to the strict account: We now take Dw to be a dynamic object, the modal horizon of w, which evolves throughout discourse. It is primarily updated through a presupposition that is attached to the definition in (107):

(108) ~if p would q^wis defined only if~φ∩Dw,∅

6 As with the Kadmon and Landman (1993) analysis, there are independent arguments against this analysis that we will set aside for the remainder of this discussion. Note that the analysis of the prejacent as a presupposition ofonlyemployed in this argument is not uncontroversial, see e.g. Ippolito (2006, 2008) for an overview.

That is, a counterfactual conditional is only defined if there are an-tecedent-worlds in the modal horizon. If the modal horizon does not con-tain antecedent worlds, i.e. the presupposition fails, it is accommodated by expanding the horizon minimally to include at least one antecedent world. This expansion follows the rule in (109)⁷:

(109) φ-expandedDw=Dw∪ {w⁰ | ∀v∈~φ :w⁰ ≤_w v}

That is, the modal horizon is expanded to include any world which is closer or equally close to the actual world w than all antecedent-worlds.

This includes all worlds up to (and including) the closest antecedent-worlds. In effect, this is entirely equivalent to the variably strict read-ing in those cases where the modal horizon did not previously include antecedent-worlds⁸.

However, in contrast to the variably strict semantics, this semantics is SDE, providing an explanation for the licensing of NPI-any in the an-tecedent: For any sentencesφ,ψandτ, and any domainDwfor which both

~φ€ψand~(φ∧τ)€ψare defined, the former will entail the latter.

This is guaranteed because a domainDwthat is defined for the antecedent (φ∧τ) will ensure that there are τ-worlds amongst the φ-worlds; conse-quently, if ψ holds throughout all φ-worlds, it will also hold throughout those (φ∧τ)-worlds. The variably strict semantics does not achieve this, because it cannot guarantee that the (φ∧τ)-worlds of the second counter-factual are included in theφ-worlds the first counterfactual is quantifying over.

7 Note that the modal horizon is defined relative to a world, as in von Fintel (1999, 2001), a detail that is somewhat obscured in Walker and Romero (2015), where the subscript w is not shown, while our update procedure in that paper still clearly assumes a dependency on world-assignment pairs.

8 In cases where the modal horizon already includes antecedent-worlds, there is no guarantee that these are the closest ones. This leads to the desired behaviour in so-called reverse Sobel-sequences, an additional motivation for von Fintel’s (1999) account that is, however, orthogonal to the issues here.