Featural variables

Abstract

A long line of research takes some sign language loci to be the overt realization of variables. But as argued in Kuhn (2015), this analysis fails in ASL in two cases. (i) First, loci sometimes appear to be inherited through agreement rather than directly interpreted, in particular in those environments in which phi-features are known to remain uninterpreted (= ‘Kuhn’s Generalization’). (ii) Second, there are cases in which one and the same locus can refer to different individuals, in contradiction with the predictions of the standard theory. Kuhn concludes that sign language loci are an open class of features rather than of variables; and he provides a variable-free treatment of them, although without accounting for their deictic uses. While granting the correctness of Kuhn’s Generalization, we offer an alternative in which ASL loci are both features and variables: some loci (in particular deictic ones) obtain their value from an assignment function, and introduce presuppositions on the value of other (covert) variables; but loci are also subject to the same rules of agreement as phi-features, and they can thus remain uninterpreted in some other environments. We discuss their behavior both from the perspective of morpho-syntactic and of semantic theories of (apparent) feature agreement. Finally, we argue that in the tense domain spoken languages also have expressions that are featural while containing a variable element.

1 Loci as variables or loci as features?

1.1 Loci as variables

Lillo-Martin and Klima (1990) argued that logical variables or ‘indices’, which are usually covert in spoken languages, can be overtly realized in sign language by positions in signing space or ‘loci’. In case a pronoun is used deictically or indexically, its locus usually corresponds to the actual position of its denotation, be it the speaker, the addressee, or some third person (e.g. Meier 2012). If the pronoun is used anaphorically, the antecedent typically establishes a locus, which is then ‘indexed’ (= pointed at) by the pronoun. In (1a) (American Sign Language, henceforth ASL), the sign names Bush and Obama establish loci by being signed in different positions; in (1b), the antecedent DPs are accompanied with pointing signs that establish the relevant loci. In quantificational examples, indexing disambiguates among readings, as in (2) (French Sign Language, henceforth LSF).¹

• (1)

  

• (2) deputy _b senator _a cl _b-cl _a ix-b a-tell-b ix-a / ix-b win election

  ‘An MP_b told a senator_a that he_a / he_b (= the deputy) would win the election.’ (LSF; 4, 233)

In addition, it was argued in recent research that if loci are indeed the realization of indices, they can shed new light on some foundational issues in semantics, for instance on the necessity of a dynamic logic to handle anaphora (Schlenker 2011), on the existence of a uniform anaphoric system in the nominal, temporal and modal domains (Schlenker 2013), or even on the very nature of variables (Schlenker et al. 2013).

1.2 Kuhn’s objections

Against this tradition, Kuhn (2015) argues that loci are features rather than variables. He gives two arguments: first, there are uninterpreted loci under only; second, multiple occurrences of the same locus may refer to different individuals.

• Variable capture: Uninterpreted loci under ‘only’

Kuhn shows that under only the loci-as-variables view undergenerates, as in (3). (Both Kuhn’s and our ASL examples are assessed on a 7-point scale, with 7 = best; numerical averages appear in lieu of the standard ∗, ?, ??, etc., right before the examples.)

• (3) 7 ix-a jessica told-me ix-b billy only-one _(b) ² finish-tell mother poss -b favorite color.

  ‘Jessica told me that only Billy told his mother his favorite color.’ Can be read as: bound-bound, bound-free, free-bound, or free-free. (Kuhn 2015, based on the judgments of two consultants)

Let us unpack this example. JESSICA is associated with locus a by way of the initial pointing sign IX-a. Then BILLY is associated with locus b by way of the pointing sign IX-b. Finally, the possessive pronouns and POSS-b both index that same locus b. Now let us focus on the (available) ‘bound-free’ reading, on which the boxed possessive is read as bound by ONLY-ONE while the underlined possessive refers to Billy.³ For the boxed possessive to be bound, ONLY-ONE must somehow bind this variable, say by way of a Logical Form akin to (4), with the assumption that IX-b BILLY comes with a requirement that b denotes Billy, and that there is an empty copula preceding ONLY-ONE to yield a meaning such as: ‘Billy is the only person who…’. (Unless otherwise noted, our Logical Forms are based on the kind of syntax/semantics interface discussed in Heim and Kratzer 1998, among others.)

• (4) ix-b billy only-one $\lambda \mathrm{b}\ \mathrm{t}_{\mathrm{b}}$ finish-tell mother poss -b favorite color

But if the boxed possessive is bound by λb, the underlined pronoun POSS-b, which is lower in the structure, shouldn’t be able to get a deictic reading on which it denotes Billy (nor could it denote further salient individuals—a possibility which is not tested in Kuhn’s paper).⁴

On the view that loci may be interpreted, these data suggest that there are some environments in which they can be disregarded as well. Precisely this view is standard for phi-features, which are believed to be interpreted on free pronouns but to remain uninterpreted on bound variables under only (see Partee 1989, fn. 3 for an initial observation); this similarity between loci and phi-features is what we call ‘Kuhn’s Generalization’. There are two general directions to analyze the behavior of phi-feature under only.

(i) According to semantic analyses, a feature F on a pronoun pro remains uninterpreted due to its semantics in the focus dimension (Heim 2005; Spathas 2007; Jacobson 2012). An implementation for feminine features is sketched in (5). For ease of comparison with the rest of our discussion, we adopt a framework compatible with a variable-full treatment, as in Rooth (1996); # is used to encode presupposition failure.

• (5)

  

To illustrate, we assume that the ordinary value in (6a) is computed as usual, hence a feminine presupposition; while the focus value in (6b) ignores the features on her:

• (6)

  

What is notable is that the feminine features have no semantic contribution in the focus dimension specified in (6b). Thus if only works on top of (6a,b), the feminine feature f will play no role in the focus dimension, as is desired.

(ii) According to morpho-syntactic analyses, a feature F on a pronoun pro can remain uninterpreted if pro is bound by an element with feature F—henceforth ‘deletion under agreement’ (see Heim 1991, 2008; Kratzer 2009; Schlenker 1999, 2003; von Stechow 2004). A simple-minded rule is displayed in (7)–(8), where the gender/person features her _i and my _i remain uninterpreted.⁵

• (7)

  

• (8)

  

We add for future reference that Schlenker (1999) and Stechow (2004) posit that expressions like now can trigger the deletion of present tense features as well (this point will matter in the analysis of (10) below); while their analysis is morpho-syntactic, it could be re-cast within a semantic framework as well. Their argument is based on examples such as (9):

• (9)

  

The reasoning in Schlenker (1999) is as follows. Interpretable features are maximally used to constrain the denotation of expressions of referential type, hence the present tense feature is added to now, as is represented in (9b). Importantly, in this Logical Form $i_{k}$ is a time variable and it carries present tense features, but despite this it can range over past moments as well: it excludes the possibility that at earlier times the Concord was in Paris. The proposal is that tense features are deleted by agreeing with unpronounced features on now.⁶

Kuhn’s Generalization may be further strengthened by considering the behavior of feature-like elements within sign language itself. Schlenker (2014) argues that height specifications of loci behave like phi-features in that (i) they normally have a presuppositional semantics (e.g. high loci normally denote tall, important or powerful individuals), while (ii) under only (and ellipsis), they can remain uninterpreted. The data about only, illustrated in (10), suggest that Kuhn is exactly right: not just loci but other feature-like elements can be disregarded under only. As was the case for now in (9), we must posit in (10) that the subject HEIGHT _a (signed neutrally, and meaning something like ‘the tall one’) is given a ‘high’ feature (compatible with its semantics), which in turn triggers feature deletion on the boxed object pronoun, as in (10b); as a result, we obtain a reading which excludes the possibility that the short German person likes people who support him (despite the fact that he is short and thus that the high features specifications shouldn’t be appropriate to refer to him).⁷

• (10)

  

• Locus re-use

Kuhn (2015) offers a second argument against the loci-as-variables approach. In (11) a single locus is assigned to John and Mary, and another locus is assigned to Bill and Suzy. As a result, the boxed pronouns and refer to John and Bill respectively, while the underlined pronouns IX-a and IX-b refer to Mary and Suzy.

• (11) 6 every-day, john _a tell mary _a love ix -a. bill _b never tell suzy _b love ix -b.

  ‘Every day, John_i tells Mary_j that he_i loves her_j. Bill_k never tells Suzy_l that he_k loves her_l.’ (ASL, Kuhn 2015)

As Kuhn observes, this example is problematic for the variable-based view. The initial association of the proper name JOHN with variable a should force a to refer to John; but then how can a also refer in the same clause, and without any intervening binder, to Mary? By contrast, these data are unproblematic for the feature-based analysis of loci: just like two DPs may bear the same feminine gender features while denoting different individuals, so it is with loci-as-features. (There might be pragmatic constraints—such as clarity—that explain why this pattern isn’t more prevalent in sign languages.)

• Kuhn’s theory

Kuhn (2015) solves these problems by treating loci as features which are not interpreted (so that neither the problem of variable capture nor of variable re-use can arise in the first place), but are inherited by a mechanisms of morpho-syntactic agreement; this allows him to provide a variable-free treatment of loci. He accepts the consequence that features need not be part of a closed inventory, since there is no natural upper bound on the number of loci that can appear in a sentence (though there are clear performance limitations). On the other hand, he does not provide an account of deictic loci; we come back to their treatment below.

Besides positing that features are not part of a closed inventory, Kuhn’s analysis must also accept a puzzling consequence: setting aside deictic and indexical loci, if in a given sentence a noun phrase N is associated with locus L and a noun phrase $\mathrm{N}'$ is associated with locus $\mathrm{L}'$, it is also the case that, in another sentence, N can be associated with $\mathrm{L}'$ and $\mathrm{N}'$ can be associated with L. The reason is that outside of the deictic/indexical realm, locus assignment is not determined by intrinsic properties of noun phrases, but by other considerations—including the order in which they appear in the sentence (for instance, our ASL consultant usually introduces loci from right to left). We do not know of cases in spoken language in which feature choice is arbitrary in this way; on the other hand, this arbitrariness is expected on the loci-as-variables view.

2 Loci as variables and as features

We now suggest that loci are both variables and features: deictic loci are interpreted as free variables; while bound loci may remain uninterpreted in some environments.

• Deictic readings

As mentioned, when individuals are present in the discourse situation, the signer normally points towards them to realize deixis. Kuhn’s system could be extended to capture these uses, by analogy with the treatment of gender and person features given in Jacobson (1999, 2012). Jacobson’s proposal is that she and you behave as other pronouns in denoting an identity function, but that they come with a domain restriction to female individuals for she and to addressees for you. On this view, then, the pronoun you evaluated in a context c denotes the partial identity function $\lambda x_{e}$: x is an addressee of c. x. In words, this is the function which is defined on an individual x just in case x is an addressee of c, and which outputs x itself if this condition is met. Given the rest of Jacobson’s system, the sentence You left ends up denoting a partial function of type <e,t>, restricted to a domain of addressees⁸; and a truth value is finally obtained when that function is applied to an addressee. The same analysis could be extended to a deictic locus a corresponding to the position of an individual s(a): the meaning of a could be seen as an identity function restricted to individual s(a). Importantly, however, this takes Kuhn’s and Jacobson’s ‘features’ one step closer to variables: not only is there a potentially infinite number of them (as there is no natural upper bound on the number of loci), but in addition all the deictic ones must be associated with potentially distinct individuals—a relation that comes close to the reference relation established by assignment functions. (As was noted above, non-deictic Kuhnian features also resemble variables in being arbitrarily assigned to noun phrases irrespective of their intrinsic properties.)

Without refuting the variable-free analysis, we can offer an alternative within a variable-full system. Specifically, the loci-as-variables view can simply posit that deictic loci are free variables whose value is given by an assignment function (provided by the context), and come with a presupposition that their realization in signing space must spatially correspond to the actual position of their denotations. Importantly, deictic loci are no different from other loci in their ability to give rise to bound readings, as shown in (12); it is thus desirable to develop an analysis that handles all loci on a par.⁹

• (12)

  

• Phi-features

To develop our theory, let us start with a standard treatment of gender, extended to person (e.g. Cooper 1983; Schlenker 2003a,b; Heim 2008; Sauerland 2008).¹⁰ Assuming that the semantics is relativized to a context, a world and an assignment function, we posit the interpretive rules in (13), where as before # encodes presupposition failure. An example is given in (14).

• (13)

  

• (14)

  

Let us unpack these definitions. In essence, (13a) posits that an expression $E^{f}$ with feminine features gives rise to a presupposition failure when evaluated under a context c, an assignment function s and a world w unless E denotes a female individual in the world of the context c. When no presupposition failure is obtained, $E^{f}$ denotes whatever E denotes. Example (13b) and (13c) are analogous except that the presuppositional requirement is that E should denote the speaker or an addressee of the context c. Example (13d) is a standard interpretive rule for a pronoun $\mathit{pro}_{i}$ with index i, and it specifies that its denotation under an assignment function s is just s(i). Example (14) applies (13a) and (13d) to the case of a simple feminine pronouns, with $E^{f} = \mathit{pro}_{i}^{f}$.

It is immediate that phi-features on deictic pronouns are presuppositionally interpreted. For bound pronouns, the rules in (5) and (7) entail (for different reasons) that in (15a) the feminine feature of herself might be interpreted: for the morpho-syntactic analysis, this is so to the extent that the feature cannot be deleted under agreement;¹¹ for the semantic analysis, this is because the focus value plays no role in (15a). On either analysis, the desired inference follows from the Logical Form in (15b), together with the standard assumptions that (i) each object that satisfies the NP-restrictor must satisfy the presuppositions of the VP-nuclear scope, and that (ii) presuppositions project out of questions.

• (15)

  

By contrast, for (16), the morpho-syntactic analysis must appeal to the rule in (7) to ensure that the feminine feature of her remains uninterpreted; while the semantic analysis appears to (5b) to guarantee that features are ignored in the focus dimension.

• (16)

  

• Loci

We will now suggest that a similar semantics can be extended to loci. But whereas the features in (13) have a semantics which is only sensitive to the context parameter c, the contribution of loci is sensitive to the assignment function s—unless they are first and second person loci, in which case they too are sensitive to c only, as in (13b–c). A simple rule is stated in (17) and illustrated in (18). Example (17a) is parallel to the rules in (13a–c): a presupposition failure is obtained for $E^{a}$ unless its denotation is that of locus a, and when no failure is obtained it denotes what E denotes.

• (17)

  

• (18) Assume that a is a non-first, non-second person locus and that s(i) ≠ #. Then $[\!\![\mathrm{pro}_{\mathrm{i}}{}^{\mathrm{a}}]\!\!]^{\text{c,s,w}}$ ≠ # iff $[\!\![\mathrm{pro}_{\mathrm{i}}]\!\!]^{\text{c,s,w}} = \mathrm{s}(\mathrm{a})$. If $[\!\![\mathrm{pro}_{\mathrm{i}}{}^{\mathrm{a}}]\!\!]^{\text{c,s,w}}$ ≠ #, $[\!\![\mathrm{pro}_{\mathrm{i}}{}^{\mathrm{a}}]\!\!]^{\text{c,s,w}} = [\!\![\mathrm{pro}_{\mathrm{i}}]\!\!]^{\text{c,s,w}} = \mathrm{s}(\mathrm{i})$.

So far we haven’t said anything about loci that appear on expressions of quantificational type. Here the rules are somewhat more involved. We assume that in this case a DP associated with locus a introduces a λ-operator that binds variables with feature a (expressions of type e can but need not display this behavior). Rules are stated in (19)–(20) and schematically illustrated in (21). (We will ask in Sect. 3 whether in the end we need all the provisions made in (20), where the λ-operator simultaneously binds a variable i and the locus feature a that it carries.)

• (19)

  

• (20) For any expression E, any variable i of type e and any locus a ≠ 1,2,

  〚λ $\mathrm{i}^{\text{a}}$ f〛^c,s,w = λx. 〚f $]\!\!]^{\text{c,s}[\text{i}\rightarrow\mathrm{x}, \text{a}\rightarrow\text{x}], \text{w}}$ (with the standard convention that s[i→x,a→x] is the assignment function which is identical to s except that it assigns x to i and x to a)

• (21)

  

Finally, we assume that in the special case in which only associates with an expression E of referential type (individual or temporal, i.e. e or i), it is interpreted in accordance with the rule in (22); it predicts in particular a presupposition failure if an alternative to the denotation of E yields a failure when fed to the value of the sister of E. For instance, the sentence Only John continues to smoke presupposes that individuals that are alternatives to John used to smoke.¹²

• (22) For any expression e of type τ = e or i,

  〚only e $]\!\!]^{\text{c,s,w}} = \lambda \mathrm{f}_{<\tau , \mathrm{t}>}$. # iff (i) for some alternative $\mathrm{e}'$ to 〚e〛^c,s,w given by c, $\mathrm{f}(\mathrm{e}') = \text{\#}$, or (ii) $\mathrm{f}([\!\![\text{\textsc{e}}]\!\!]^{\text{c,s,w}}) = \text{\#}$; if ≠ #, 1 iff for every alternative $\mathrm{e}'$ to 〚e〛^c,s,w given by c, $\mathrm{f}(\mathrm{e}') =0$.

To illustrate, consider the first person example in (23a); to abstract away from irrelevant details, we treat finish-write-the-homework-of as an atomic predicate.

• (23)

  

Following (7a), we can apply the optional rule of feature deletion to the boxed expression, as in (23b), which is interpreted as in (24a) (for clarity, we include a barred version of the deleted feature, written as ; and we write $c_{a}$ for the speaker of c).

• (24)

  

Without feature deletion, we would obtain for (23c) the value in (24b), which yields a failure on all arguments except the speaker. Given the lexical entry of only in (22), this would incorrectly block the bound reading.

Consider now the third person example in (25), which favors a bound reading but also has a strict one:

• (25) 7 john _a mary _b sam _c the-three-a,b,c only-cl_one-c finish write poss-c homework

  ‘Of John, Mary and Sam, only Sam finished writing his homework.’ (ASL; 22, 13a; 3 judgments)

On the assumption that the λ-operator inherits the features of the DP that introduces it, we can derive two readings, as shown in (26). By the rule in (19), the subject DP may but need not carry the c feature, which we indicate by putting it in parentheses.

• (26)

  

Assuming that locus features can be deleted under agreement, we derive the bound reading, as in (26a). For the strict reading in (26b), the locus feature of the boxed pronoun does not hurt the interpretation.

• Variable capture

Now consider Kuhn’s example involving ‘variable capture’. As mentioned, (27a) can have four readings, including one on which is bound but POSS-b is free, and one on which is free but POSS-b is bound. Presumably Kuhn must assume that, by one mechanism or another, ONLY-ONE can inherit the locus feature of BILLY; and in fact, as J. Lamberton (p.c.) points out, the key is then to assume that variables can be bound by (λ-operators introduced by) $\mathit{BILLY}_{b}$ or by ONLY–$\mathit{ONE}_{b}$, but that in any event the feature b which they inherit need not be interpreted. The two mixed readings can thus be analyzed as in (27b–c). The key to obtain the desired reading is that the locus b that appears on the embedded pronouns is inherited through agreement and thus can be ignored by the interpretive component.

• (27)

  

• Locus re-use

Consider now Kuhn’s argument based on variable re-use. In (11), locus a was used to refer both to John and to Mary, while locus b was used to refer both to Billy and to Suzy. We could posit that locus features are inherited through agreement, as in (28):

• (28)

  

But this won’t account for the case in (29). Here THE-TWO-a indexes the position a, and it is not c-commanded by either antecedent DP.

• (29) 7 every-day, john _a tell mary _b love ix -b. bill _a never tell suzy _b love . the-two-a smart.

  ‘Every day John tells Mary he loves her. Bill never tells Suzy he loves her. Both John and Bill are smart.’ (ASL; 22, 72b, 3 judgments)

At this point we can make two choices.

(i) First, we may posit that variables can be deleted under agreement not just in case of standard binding, but also in case of dynamic binding. If so, we could argue that THE-TWO has split dynamic antecedents and inherits the features of both antecedents—and hence gets the feature a twice by way of agreement.

(ii) Alternatively, we may revise (17) to only require that an expression with locus feature a denote a part of what a denotes. This would make this rule particularly parallel to that in (13c), where we required that an expression with second person feature denote an addressee, and not necessarily the addressee.

• (30) [second version] For every locus a ≠1, 2, if e is an expression of type e, 〚e ^a〛^c,s,w = # iff 〚e〛^c,s,w = # or 〚e〛^c,s,w isn’t a part of s(a). If 〚e ^a〛^c,s,w ≠ #, 〚e ^a〛^c,s,w = 〚e〛^c,s,w.

In order to account for (29), all we need to posit is that a (and m) denotes the plurality John+Mary; and under this assumption, we don’t need feature deletion, as seen in (31).

• (31) John_a $\lambda \mathrm{i}^{\text{a}}$ Mary $\lambda \mathrm{k}^{\mathrm{a}}{}\ \mathbf{t}_{\mathbf{i}}{}^{\mathbf{a}}\ \mbox{tell}\ \mathbf{t}_{\mathbf{k}}{}^{\mathbf{a}}\ [\mathbf{pro}_{\mathbf{i}}{}^{\mathbf{a}}\ \mbox{love}\ \mathbf{pro}_{\mathbf{k}}{}^{\mathbf{a}}].\ \mathrm{pro}_{\mathrm{m}}{}^{\mathrm{dual}, \mathrm{a}}\ \mathrm{smart}$.

• Refining the analysis of deictic loci

As mentioned, the present analysis provides a simple account of deictic loci. Still, a refinement is needed, since deictic loci must usually correspond in signing space to the actual position of the objects they denote. Schlenker et al. (2013) posit a presuppositional rule that can be adapted to the present system, as seen in (32); in a nutshell, it posits that the position of deictic loci must roughly correspond to that of their denotations.

• (32) [third version] For every locus a ≠1, 2, if E is an expression of type e, 〚e ^a〛^c,s,w = # iff 〚e〛^c,s,w = # or〚e〛^c,s,w isn’t a mereological part of s(a) or 〚e〛^c,s,w is present in the situation of utterance in c and 1, 〚e〛^c,s,w and a are not roughly aligned. If 〚e ^a〛^c,s,w ≠ #, 〚e ^a〛^c,s,w = 〚e〛^c,s,w.

We leave it open whether conditions on first and second person pronouns should be made to follow as a special case of (32).

• Intermediate summary

Let us step back. While granting the correctness of ‘Kuhn’s Generalization’, we have proposed that loci may simultaneously display the behavior of variable and of features. The analysis makes loci particularly similar to person features, especially second person features. These are typically assumed to have a semantics; for instance, you is constrained to denote an addressee of the speech act. Still, in other cases second person remains uninterpreted, as in Only you did your homework on its bound reading. We took loci to be similar, with the difference that they are assignment-sensitive rather than context-sensitive. By extending to loci assumptions that are standard about person, we were able to solve Kuhn’s problem of ‘variable capture’. We then sketched two possible solutions to the problem of ‘variable re-use’. One posits that loci may remain not just under standard but also under dynamic binding. The other posits (by analogy with the interpretive rule for second person) that a locus a on an expression $E^{a}$ only comes with a requirement that E denote a part of what a denotes, not that E denote the same thing as a.

In Sect. 3, we turn to cases in which bound loci are arguably interpreted. In the Appendix, we discuss recent results that crucially hinge on a treatment of plural loci as variables, with interpretive properties that iconically mirror their realization—a result which is non-trivial to match in a feature-based analysis of loci.

3 Bound iconic loci

At this point, no example argues for the rule in (20), and thus we could have a system in which (i) deictic loci receive their value from an assignment function (which is itself provided by the context), and (ii) bound loci are features and are never interpreted, just as in Kuhn’s system. But as we will now see, some loci are both bound and interpreted.

Schlenker et al. (2013) discuss high loci, which can be used to refer to important, powerful, or tall individuals. In the latter case, they simultaneously display a variable-like and an iconic semantics: loci are structured areas rather than points in space, and they play the role of simplified pictures of their denotations (Liddell 2003). The position (up or down) indexed by a pronoun or by an agreement verbs turns out to have interpretive consequences even under binding. Thus in (33), LOOK-a_high and FILM IX-a_low index the same locus a, but the first expression indicates that one looks at the relevant individual while she is up, and the second that one films her while she is down. Example (33b) makes the further point that this feature does not undergo deletion under only, since the inference about the other gymnasts involves their particular positions as well.

• (33)

  

Crucially, the subject quantifier introduces a neutral version of locus a, and hence the high and low versions of the same locus cannot be disregarded under agreement.

With the binding rule in (20) and the iconic semantics in (34) (slightly simplified from Schlenker 2014), the Logical Form in (35a) correctly derives truth conditions of (33a) on which all instances of a are bound, but come with different positional restrictions: as desired, the occurrence that comes with LOOK-a_high is associated with a ‘high’ positional restriction, while the occurrence that comes with LOOK-a_low is associated with a ‘low’ positional restriction.

• (34) Let c be a context of speech, s an assignment function which assigns values to structured loci, and w a world (with c_a = the author of c; c_w = the world of c). We assume that c determines a projection $\pi_{\mathrm{c}}$ from the salient situations in $\mathrm{c}_{\mathrm{w}}$ to the signing space of $\mathrm{c}_{\mathrm{a}}$. If i is a point locus which is part of a structured locus I, and if and s(I) denotes a human being, 〚ix-i〛^c,s,w = # unless I is the projection of s(I) in the signing space of $\mathrm{c}_{\mathrm{a}}$ according to $\pi_{\mathrm{c}}$, and i is within I the projection of the upper part of the body of s(I)). If 〚ix-i〛^c,s,w ≠ #, 〚ix-i〛^c,s,w = s(I).

• (35)

  

Without the binding rule in (20), we would have to posit that a_high and a_low are made of two parts: a pure locus a which remains uninterpreted, and an iconic locus high or low which provides positional information, as sketched in (36). But appropriate interpretive rules for these truncated positional loci have yet to be investigated.

• (36) [all gymnast] λi^a $\mathrm{t}_{\mathrm{i}}$ ^a want [I look i^{<a, high>}] at-t and [film i^{<a, low>}] at-t+1

It should be noted that the semantic analysis of features under only fails to capture (33b), as it predicts that high and low features are obligatorily disregarded in the focus dimension. But a small modification might work: we could posit in (5b) that features are optionally disregarded in the focus dimension.¹³

It should also be noted that Kuhn’s system as it stands cannot account for examples such as (33), where the bound pronouns cannot inherit their features from their antecedent. Still, Kuhn might develop a version of the analysis in (36), where loci themselves are not interpreted, but the iconic restrictions they realize are; the ramifications of this move would need to be worked out, however.

4 Featural variables in spoken language?

Having argued that ASL loci are featural variables, we will now suggest that comparable examples can be found within the tense system of English.¹⁴

Following Partee (1973) and Heim (1994), Schlenker (1999) and von Stechow (2004) take the semantics of the past tense to involve overt time variables, which we write as $t_{i}$ below. But instead of adopting the standard rules in (37) (Heim 1994), for the past tense they adopt the more complex conditions in (38):

• (37)

  

• (38)

  

In words: the standard rules in (37a,b) posit that a time variable with present or past tense features is constrained to denote moments that count as present or past relative to the context of the speech act. Example (38) posits a more complex analysis for a time variable with past tense features; very roughly, its value is constrained to precede the value of some other time variable that carries present tense features.

The motivation for (38) over (37b) lies in part in (39a), with the highly simplified LFs in (39b) or (39c) ($t_{0} -1$ abbreviates one month before $t_{0}$; see von Stechow 2004 for a more thorough treatment, especially of the present tense). The key is that the denotation of a past tense variable need not be before the context of evaluation; in some cases, it is enough that its denotation be before a time denoted with a present tense variable.

• (39)

  • a. Whenever John changes jobs, he gets into a fight with people who were his best friends one month before. (modified from von Stechow 2004)

  • b. [$\forall^{\mathrm{D}}\mathrm{t}_{0}: \mathrm{t}_{0}{}^{\mathrm{pres}}$ Pierre changes jobs] [∃x: [$\mathbf{t}_{\mathbf{0}}\mbox{-}\mathbf{1}$] $\boldsymbol{\lambda} \mathbf{t}_{\mathbf{1}}$ Pierre befriend x][$\text{t}_{0}{}^{\mathrm{pres}}$ Pierre fight x]

  • c. [$\forall^{\mathrm{D}}\mathrm{t}_{0}: \mathrm{t}_{0}{}^{\mathrm{pres}}$ Pierre changes jobs] [$\exists\mathrm{x}: [\mathbf{t}_{\mathbf{0}}\text{-}\mathbf{1}]$ $\boldsymbol{\lambda} \mathbf{t}_{\mathbf{1}}$ Pierre befriend x][$\mathrm{t}_{0}{}^{\mathrm{pres}}$ Pierre fight x]¹⁵

Example (37) gives rise to the Logical Form in (39b); on the assumption that the domain restriction D on the universal time quantifier denotes an interval $\mathbf{D} \subseteq \mathrm{T}_{\mathrm{c}}$, the boldfaced part contributes the additional presupposition that for every time t in D, t-1 precedes the time of c—an incorrect result if D extends, say, several months after the time of utterance (so that the sentence states a broad regularity). By contrast, no problem arises with the Logical Form in (39c); briefly, we only get a presupposition that for every time t in D, t-1 before t-which is trivial.

For our purposes, what matters is that the element that spells out the past tense feature has a variable within it. We will now show that this featural and variable element can be semantically ignored under only, just like phi-features (and ASL loci). Consider the modified sentence in (40), with the addition in bold.

• (40)

  

Simplifying, we take at the time to be a time variable that ends up being (possibly dynamically) bound by a month before, hence the simplified LF in (40b). But the past tense features of the boxed time variable will have an undesirable consequence, namely to require that for every time t in D, every moment which is an alternative to t should be before t. As a result, we only get an inference that for every time t in D, the relevant people were not useful at times preceding t. By contrast, the desired inference is that for every time t in D, the people who had been useful at t-1 failed to be useful at t and later—which is the reason John could afford to get into a fight with them at t.

Getting the desired reading requires that the past tense features of the boxed variable be semantic ignored. Within their morpho-syntactic analyses, Schlenker (1999) and Stechow (2004) discuss simpler versions of the same problem, already seen in (9). The same mechanism can be extended to (40c): the adverbial at that time can receive the (underlined) past tense feature ${}^{<\mathbf{t}_{\mathbf{0}}{}^{\mathbf{pres}}}$ (with at that time just represented as $t_{1}$). While this feature does not hurt the interpretation of at that time, it can trigger deletion of the feature of the boxed variable—as is desired (these ideas could in principle be adapted to the semantic analyses sketched in (5)). But crucially the feature in question, namely ${}^{<\mathbf{t}_{\mathbf{0}}{}^{\mathbf{pres}}}$, includes a variable within it, which thus undergoes deletion as well.

If this analysis is on the right track, the past tense provides another example of variables that can be ignored in the environments in which phi-features are. Specifically, the past tense of were in (40a) was analyzed as the boxed variable in (40), and we saw that the entire past tense specification ${}^{<\mathbf{t}_{\mathbf{0}}{}^{\mathbf{pres}}}$, including the variable $t_{0}$, can be semantically ignored.

5 Conclusion

While the present system should certainly be constrained further,¹⁶ it preserves the main insights of the loci-as-variables approach while granting the correctness of Kuhn’s key intuition, namely that loci can also behave like features; synthesizing the two views, it proposes that loci are often featural variables. Our approach deals with the problems of ‘variable capture’ and ‘variable re-use’, and it also accounts for some cases in which iconic loci are bound and yield interpretive effects. While it makes use of multiple covert variables, it was never part of the ‘loci as variables’ view that all variables are realized as loci, only that loci spell out variables. We certainly have not refuted the variable-free view of loci laid out in Kuhn (2015); but we have shown that a variable-full analysis can account for the same data as well as well ones Kuhn doesn’t discuss, and we have suggested that the variable-free account would need to be extended not just to account for deictic loci, but also for the iconic bound loci discussed in Sect. 3. Finally, we have suggested that featural variables might exist in spoken languages as well, namely in the temporal domain.

Appendix:  Complement set anaphora and loci as variables

Part of this discussion borrows from Schlenker et al. (2013) and Schlenker (2015a, 2015b).

We briefly discuss below an interpretive property of plural loci that can be analyzed within a variable-full system, but might not be trivial to handle in a pure agreement-based analysis. Plural loci are realized in ASL (and LSF) as semi-circular areas, which closely correspond to plural variables. Now in some cases one plural locus a can be embedded within a larger plural locus ab, with the result that a ‘complement locus’ b suddenly pops into existence, and denotes the complement of the denotation of a within the denotation of ab. An ‘iconic’ analysis was offered for this phenomenon in Schlenker et al. (2013), but it hinged rather crucially on a treatment of plural loci as variables that have a denotation.

To introduce the issue, we start with the English sentence Most students came to class. Recent research has argued that it makes available two discourse referents for further anaphoric uptake: one corresponding to the maximal set of students, as illustrated in (41b) (‘maximal set anaphora’); and one for the entire set of students, as illustrated in (41c) (‘restrictor set anaphora’).

• (41)

  

By contrast, no discourse referent is made available for the set of students that didn’t come to class (‘complement set anaphora’, as this is the complement of the maximal set within the restrictor set); this is what explains the deviance of (41a) (see Nouwen 2003 and Schlenker et al. 2013 for further discussion of apparent counterexamples).

On the basis of ASL and LSF data, Schlenker et al. (2013) made two main observations.

Observation 1

When a default plural locus is used in ASL, data similar to (41) can be replicated—e.g. complement set anaphora with most is quite degraded.

Observation 2

When embedded loci are used, the effect is circumvented: one large locus (written as ab, but signed as a single circular locus) denotes the set of all students; a sub-locus (=a) denotes the set of students who came; and a complement locus (=b) thereby becomes available, denoting the set of students who didn’t come, as illustrated in (42) and (43).

• (42) 7 poss-1 student ix-arc-ab most ix-arc-a a-came class. ix-arc-a a-ask-1 good question.

  ‘Most of my students came to class. They asked me good questions.’ (ASL; 8, 196)

• (43)

Schlenker et al. (2013) account for Observation 1 and Observation 2 by assuming that (i) Nouwen is right that in English, as well as ASL and LSF, the grammar fails to make available a discourse referent for the complement set, i.e. the set of students who didn’t come; but (ii) the mapping between plural loci and mereological sums preserves relations of inclusion and complementation, which in (42a) makes available the locus b.

The main assumptions are that (A1) the set of loci is closed with respect to relative complementation: if a is a sublocus of b, then (b-a) is a locus as well; and (A2) assignment functions are constrained to respect inclusion and relative complementation: if a is a sublocus of b, the denotation of a is a subpart of the denotation of b, and (b-a) denotes the expected complement set. In (42a), where embedded loci are used, we can make the following reasoning:

• Since a is a proper sublocus of a large locus ab, we can infer (by assumption A1) that (ab-a) (i.e. b) is a locus as well.

• By assumption A2, we can also infer that s(a) ⊂ s(ab) and that s(b) = s(ab)-s(a).

In this way, complement set anaphora becomes available because ASL and LSF can rely on an iconic property which is inapplicable in English. For present purposes, what matters is that the locus b in (42a) and (43) is not inherited by way of agreement, since it is not introduced by anything. From a variable-full perspective, the existence of this locus is inferred by a closure condition on the set of loci, and its denotation is inferred by an iconic rule. But the latter makes crucial reference to the fact that loci have denotations. It is not trivial to see how this result could be replicated in a variable-free analysis in which loci don’t have a denotation to begin with. One possibility is that the complement set locus should be treated as being deictic (which is the one case in which the variable-free analysis has an analogue of variable denotations). This might force a view in which complement set loci are handled in a diagrammatic-like fashion, with co-speech gestures/diagrams incorporated into signs—something that would require a more detailed investigation.