Number in NPI licensing

Abstract

The acceptability of any-DPs in existential modal sentences presents a challenge for theories of NPI licensing: existential modal sentences appear to differ substantially from other environments in which any-DPs are acceptable (in particular, they lack a downward-entailing operator). One approach to this challenge has been to, first, take any-DPs to be subject to an environment-based downward-entailingness condition—they have to occur in an environment that is Strawson downward-entailing with respect to their domain (cf. Kadmon and Landman 1993)—and, second, to derive such an environment in existential modal sentences by means of exhaustification (e.g., Fox 2007). This note presents new evidence for such a two-layered approach (cf. Crnič 2017, 2019). The evidence comes from a striking contrast in the behavior of singular vs. plural any-DPs in existential modal sentences. The paper concludes by charting some relations between any-DPs and other polarity items.

Appendices

Appendix A: Singular any-DPs in existential modal sentences, in detail

This section elaborates on the presentation in Sect. 2.1 (see Crnič 2017, 2019 for further details).

In the following, we take the resource domain D of any to consist of three books, {a, b, c}, and we represent alternatives with their translations into propositional logic formulas. Moreover, existential quantification is represented with disjunction, while universal quantification (and distributive predication over plurals) is represented with conjunction in the standard manner.

The sentence in (111-a) has the LF in (111-b).

• (111)

  

Formal alternatives

The alternatives to the sister of exh in (111-b) are provided in (112) (following Krifka 1995 and, esp., Chierchia 2013): they are derived by replacing the domain of any with its subdomains, and any with every. In the following, we rely on the characterization in the bottom row of (112), where ‘◊(a∨b)’ stands for the meaning corresponding to Gal being allowed to read a book in {a, b}, etc. (The sentence has other alternatives, say, those in which Gal is replaced with John, but these are irrelevant for the purpose at hand.)

• (112)

  

Excludable alternatives

Recall the definition of excludable alternatives:

• (113)

  

What are the excludable alternatives in (112)? One maximal set of alternatives in (113) that can be jointly negated with the sister of exh being true is provided in (114). This is witnessed by (115) being consistent, and by the fact that if the set of the alternatives that get negated is expanded by any of the subtracted alternatives, we obtain a contradiction.

• (114)

  

• (115)

  

All other maximal sets of alternatives in (112) that can be jointly negated with the sister of exh being true have a similar form and are provided in (116)–(117).

• (116)

  

• (117)

  

The intersection of these maximal sets—that is, the set of excludable alternatives in (112)—is provided in (118): it consists of all the alternatives that have a (non-trivial) conjunctive meaning (if a proposition is conjoined with itself, we obtain a trivial conjunction).

• (118)

  

More generally, the sets of the alternatives that can be jointly negated with the sister of exh being true have the form in (119), while their instersection has the form in (120).

• (119)

  

• (120)

  

Includable alternatives

Recall Bar-Lev and Fox’s (2020) definition of includable alternatives:

• (121)

  

What are the includable alternatives in (112)? They are all the disjunction alternatives (incl. the trivial disjunction alternatives), as given in (122). This is witnessed by the consistency of (123), which is a conjunction of all the alternatives in (122) with the negations of all the excludable alternatives; moreover, asserting a non-trivial conjunction alternative (which is the only type of alternative that is not in (122)) would obviously contradict the negation of one of the excludable conjunctive alternatives, which shows that the set in (122) is indeed maximal.

• (122)

  

• (123)

  

Exhaustification and obligatory pruning

Recall the definition of exh:

• (124)

  

What is the meaning of the structure (111-b)? This depends on what alternatives are relevant, that is, on the resolution of R. If all the formal alternatives to the sister of exh are relevant, ALT([◊ [any$_{\text{D}}$ book$_{\text{x}}$ [Gal read x]]]) ⊆ 〚R〛, the meaning we obtain is the one provided in (125) (the same result is obtained for any R that denotes a superset of the set consisting of all the excludable alternatives, Excl([◊ [any$_{\text{D}}$ book$_{\text{x}}$ [Gal read x]]]) ⊆ 〚R〛). On the other hand, if none of the alternatives are relevant, the meaning that we obtain is the one provided in (126). There are further conceivable resolutions of R (but see Katzir 2014; Crnič et al. 2015; Bar-Lev 2018 on some constraints on the pruning of alternatives).

• (125)

  

• (126)

  

How do we choose between the potential resolutions of R? It turns out that the choice is not completely free—it is partly dictated by the Any Condition. In order to obtain a Strawson downward-entailing environment with respect to the domain of any, all the disjunctive (incl. the single disjunct alternatives) must count as irrelevant (unless they are equivalent to the sister of exh). In other words, from the set in (112), only the excludable alternatives in (118) may be in 〚R〛 (together with the alternatives that are equivalent to the sister of exh) (cf. Buccola and Haida 2019 on obligatorily irrelevant alternatives):

• (127)

  

Let us first show that we indeed get Strawson downward-entailingness with respect to the domain of any if (127) obtains. The interpretation of the structure in (111-b) is provided in (128) (the parentheses indicate that the exclusion of the alternatives depends on R).

• (128)

  

If we replace D with a proper (non-empty) subdomain of D (and keep R fixed), we obtain a Strawson weaker meaning. Two such replacements are provided in (129), where each of the meanings is entailed by the meaning in (128): there are fewer asserted includable alternatives, while the negated excludable alternatives are constant across all the substitutions of D.

• (129)

  

What happens, however, if (127) does not obtain? In this case, at least one replacement of domain D will not result in a Strawson weaker meaning. For illustration, assume that ◊(a∨b) ∈ 〚R〛 (falsifying (127), since ◊(a∨b) is not an excludable alternative). Now, the meaning of the sentence with domain D is provided in (130), where the content of ϕ depends on what alternatives are in 〚R〛 besides ◊(a∨b). The meaning of the corresponding structure in which D is replaced by a stronger domain $D'$, namely 〚D′〛={c}, is provided in (131), where ψ depends on what alternatives are in 〚R〛 besides ◊(a∨b).

• (130)

  

• (131)

  

The meaning in (130) clearly does not entail the meaning in (131), no matter what the content of ϕ and ψ is—in fact, (130) entails the negation of (131). The same conclusion is reached for every other R that fails to satisfy the condition in (127). This is summarized in (132) , where the alternatives equivalent to the sister of exh are omitted for brevity.

• (132)

  

Summary

Exhaustification over the alternatives introduced by singular any-DPs in existential modal sentences, as in (111-b), creates a Strawson downward-entailing environment with respect to the domain of any—which results in the any-DP satisfying the Any Condition—if none of the includable alternatives are in the resource domain of exh, R (besides the sister of exh and any alternatives equivalent to it).

Appendix B: Plural any-DPs in existential modal sentences, in detail

This section elaborates on the presentation in Sect. 2.2.

We assume in the following that the domain D corresponds to the closure of the set of three books under sum formation, ∗{a,b,c}, that is, {a, b, c, a+b, a+c, b+c, a+b+c}. We adopt the notation of the preceding section, with existential quantification (distributive predication) over {a+b} being represented as a∧b, etc.

The sentence in (133-a) may have the LF in (133-b).

• (133)

  

Formal alternatives

The alternatives to the sister of exh in (133-b) are in (134): they are the alternatives derived by replacing the domain of any with its subdomains, and any with all. (An element is in the disjunctive closure of a set P iff it is in P or if it is a disjunction of two elements each of which is in the disjunctive closure of P.)

• (134)

  

Excludable alternatives

What are the excludable alternatives in (134)? One maximal set of alternatives in (134) that can be jointly negated with the sister of exh being true is provided in (135). This is witnessed by (136) being consistent, and by the fact that if the set of alternatives that get negated is expanded by any of the subtracted alternatives, we obtain a contradiction.

• (135)

  

• (136)

  

All other maximal sets of alternatives in (134) that can be jointly negated with the sister of exh being true have a similar form and are provided in (137)–(138).

• (137)

  

• (138)

  

The intersection of these maximal sets—that is, the set of excludable alternatives in (134)—is provided in (139): it consists of all the conjunctive alternatives and all the disjunctive alternatives in which all the disjuncts are (non-trivial) conjunctions.

• (139)

  

Excludable alternatives more generally

More generally, the maximal sets of alternatives that can be jointly negated with the sentence being true are those that have the form in (140): all the alternatives in (140) can be negated with the sentence being true (namely, it can hold that Gal is allowed to read book or books x).

• (140)

  

The set of excludable alternatives is, then, the set of alternatives based on universal quantifiers (and their equivalents based on any) whose meaning asymmetrically entails one of the alternatives (or their disjunctions) that have a singleton domain containing a plurality of books from D none of whose proper parts are also in the domain D of any.

• (141)

  

Includable alternatives

What are the includable alternatives in (134)? They are all the disjunction alternatives in which at least one disjunct is not a (non-trivial) conjunction, as given in (142). This is witnessed by the consistency of (143), which is a conjunction of all the alternatives in (142) with the negations of all the excludable alternatives (it is verified in any situation in which ◊a∧◊b∧◊c is true and all excludable alternatives are false); moreover, asserting a conjunction or a disjunction of conjunctions (which is the only type of formal alternative that is not in (142)) would obviously contradict a negation of one of the excludable conjunctive alternatives, which shows that the set in (142) is indeed maximal.

• (142)

  

• (143)

  

Includable alternatives more generally

On the assumption of the more general formulation of excludable alternatives in (141), the includable alternatives are those based on any (and their equivalents based on every) such that the domain of any in them contains at least one plurality none of whose proper parts are in the domain D.

• (144)

  

Exhaustification

What is the meaning of the structure in (133-b)? This depends on which alternatives are relevant, that is, on the resolution of R. If we assume that all the excludable alternatives are relevant, we obtain the meaning in (145).

• (145)

  

Given this resolution of R, the structure in (133-b) is not Strawson downward-entailing with respect to the domain of any. Consider the counterpart of (133-b) with a stronger domain D^∗ that consists of all proper pluralities, that is, D^∗ = {a+b, a+c, b+c, a+b+c}. The sets of excludable and includable alternatives in ALT([◊ [$\text{any}_{\text{D}^{*}}$ books_x [Gal read x]]]) are provided in (146)–(147).

• (146)

  

• (147)

  

Consequently, this structure has the meaning in (148).

• (148)

  

This meaning is obviously not entailed by the meaning in (145), which demonstrates that the structure (133-b) on the proposed resolution of R is not Strawson downward-entailing with respect to the domain D. Now, adding to R alternatives that are not excludable cannot improve the situation (see the discussion in Appendix A) nor can pruning of excludable alternatives: the includable alternatives that are asserted when the domain of any is D^∗ are not entailed by the includable alternatives that are asserted when the domain of any is D.

Exhaustification more generally

Combining the more general formulations of the excludable and includable alternatives, (141) and (144), and the same assumptions about R as above, we obtain the meaning in (149). This corresponds to every plurality of books that does not have proper parts in D being such that Gal is allowed to read it, and every plurality of books that does have proper parts in D being such that Gal is not allowed to read it.

• (149)

  

• (150)

  

The meaning in (149)–(150) need not entail one of the sentence in which the domain D is replaced with a subdomain, in particular, if the subdomain has distinct minimal elements than D. This means that the Strawson downward-entailingness required by the Any Condition does not obtain in (133-b). The generalization about free choice inferences pertaining to the minimal elements, provided in (33) in the main text, can be read off from the representations in (149)–(150).

Strong meanings of plural NPs

The state of the affairs does not change substantively if the plural NPs do not have number-neutral interpretations. For example, assume that D = {a+b, a+c, b+c, a+b+c} and that D^∗ = {a+b+c}. The meanings of the pertinent structures with domains D and D^∗ are provided in (151) and (152), respectively (again, the exclusion depends on the choice of R, thus the parentheses). Given that (151) fails to entail (152), the requisite Strawson downward-entailingness does not obtain here either.

• (151)

  

• (152)

  

Summary

Exhaustification over the alternatives introduced by plural any-DPs in existential modal sentences does not create a Strawson downward-entailing environment with respect to the domain of any—resulting in the any-DP not being able to satisfy the Any Condition.

Appendix C: The even approach to any-DPs

We describe how the results obtained in the main text can be derived in the approach of Crnič (2017, 2019) (who builds on Kadmon and Landman 1993; Lahiri 1998). The approach has as a consequence a condition on the distribution of any-DPs that differs from the Any Condition when it comes to non-monotone environments (as detailed in Crnič 2019).

The setup

On this approach, any-DPs are accompanied by a covert even operator that quantifies over the alternatives built on the alternatives to the domain of any (see Lahiri 1998 for a derivation in which even associates with the determiner). We follow Krifka (1995) and Chierchia (2013) in assuming that the relevant alternatives to a (focused) domain are its subdomains.

• (153)

  

Even primarily triggers the ordering presupposition in (154), where the relevant ordering relation (<) is filled in by the context (it could be logical entailment, Strawson entailment, contextual entailment, lower-likelihood, or some other relation that respects additivity).

• (154)

  

The precise predictions of the theory depend on the resolution of < in the context. If this is resolved to (contextual) Strawson entailment relation, we obtain straightforwardly that any-DPs are acceptable in Strawson downward-entailing environments that are not also Strawson upward-entailing, that singular any-DPs are acceptable in existential modal sentences, and that plural any-DPs are unacceptable in existential modal sentences (see Crnič 2019 for a detailed discussion of the first two cases).

Singular any-DPs

The sentence in (155-a) has the LF in (155-b).

• (155)

  

The presupposition of (155-b)) is provided in (156). Since the exhaustified constituent is Strawson downward-entailing with respect to the domain of any, as seen in Sect. 2.1 and Appendix A, the presupposition is trivially satisfied, and thus the any-DP is predicted to be acceptable.

• (156)

  

Plural any-DPs

In the case of plural any-DPs, the scalar presupposition triggered by even is inconsistent (at least on the assumption that < picks out the contextual Strawson entailment relation). The sentence in (157-a)) has the LF in (157-b).

• (157)

  

Let us focus on the following subdomain of D: D^∗ = {a+b+c} (recall our assumption from the preceding appendix section that D = ∗{a, b, c}). The presupposition of even in (157) depends on how we resolve the resource domain of exh, R. There are several options available to us, but it suffices to consider just two. First: if 〚R〛 contains no excludable alternatives, the meanings of the exhaustified sentences with domains D and D^∗ are the following:

• (158)

  

• (159)

  

Since the meaning in (159) is not equivalent to that in (158), even presupposes that it is (contextually) Strawson entailed by (158). This is not the case since the logical entailment goes the other way around. On this resolution of R, the any-DP is correctly predicted to be unacceptable. Second: if R contains an excludable alternatives, say, just ◊(a∧b), the meanings of the exhaustified sentences with domains D and D^∗ are the following:

• (160)

  

• (161)

  

Since the meaning in (161) is not equivalent to that in (160), even presupposes that it is (contextually) Strawson entailed by (160). This is not the case since (160) actually entails the negation of (161). On this resolution of R, the any-DP is also correctly predicted to be unacceptable.

Other ordering relations

If the ordering used by even is resolved to a weaker relation than (contextual) Strawson entailment, say, lower-likelihood in the context (cf., e.g., Karttunen and Peters 1979; Lahiri 1998; Crnič 2014b), nothing changes in the analysis of the singular any-DP case (see Crnič 2019 for details). In the case of plural any-DPs, on the other hand, we obtain contradictory presuppositions on some resolutions of R, and contingent (potentially implausible) presuppositions on other resolutions. We discuss two cases here. First: if no alternatives to the sister of exh are in R, we obtain a contradictory presupposition, namely, the meaning of the structure in which D^∗ replaces D is stronger than the meaning of the structure with D, and thus at most as likely. On this resolution of R, the any-DP is correctly predicted to be unacceptable.

• (162)

  

Second: if the excludable alternatives to the sister of exh are relevant, we obtain a contingent presupposition. Namely, while the meaning of the structure in which D^∗ replaces D contradicts the meaning of the structure with D, it may nonetheless be more likely than it.

• (163)

  

Consequently, on this resolution of < and R, the prediction of the approach described above is not that we have an unacceptable occurrence of an any-DP, but rather one whose felicity depends on the plausibility of the scalar presupposition in the context (if this is implausible, the sentence is marked). However, another property of even affects the acceptability of the sentence on this resolution: even tends to be infelicitous when the alternatives it quantifies over are incompatible with its sister (see Guerzoni 2003; Francis 2018 for a discussion). Since this is necessarily the case if 〚R〛 contains excludable alternatives, the unacceptability of the plural any-DPs may be attributed to this property of even (see Crnič 2019 for some related discussion).