2 + 2 = 3: Number contrasts in Blackfoot

Blackfoot nominals are singular, plural or general in number. The existence of languages with either a singular-plural opposition or a plural-general opposition is well-attested in the literature. Following Bliss (2013), we argue that Blackfoot has both systems, albeit in different contexts. We propose that the co-existence of these two systems in Blackfoot is due to the exceptional nature of plural specification in this language – it is alternatively realized as a head or modifying feature in the syntactic representation (in the sense of Wiltschko 2008). We show that regardless of whether plural is syntactically a head or a modifying feature, the semantic interpretation is the same. This is consistent with Rullmann & You’s (2006) finding that plural marking has the same denotation no matter whether it contrasts with singular or with general number.


A three-way number contrast in Blackfoot
In this paper we explore the syntax and semantics of number marking in Blackfoot, which shows some typologically rare patterns. 1 In particular, we observe that Blackfoot has nominals that are unambiguously singular, indicated by the singular demonstrative, as in (1a) and nominals that are unambiguously plural (marked by means of a plural suffix on both the demonstrative and the noun), as in (1b). In addition, Blackfoot has bare nominals that can be interpreted as either singular or plural, and hence can be described as displaying general number in the sense of Corbett (2000) and Rullmann & You (2006), as in (1c). 2 (1) a. Nit-a'pihkahtoo-'p-wa amo aipasstaam. 1-sell.ti-th-3sg dem.sg apple = 'I'm selling 1 apple.' ≠ I'm selling >1 apple.
(2) singular :: plural :: general On most analyses of feature specifications, the existence of a three-way contrast is surprising. Features are typically conceived of as either mono-valent (privative) or bi-valent (binary). On the assumption that features are mono-valent, contrast comes about as a result of distinguishing the presence of a feature from its absence, as in (3a). Thus, if [F] is plural, then the absence of F (i.e., the unmarked case) is interpreted as singular. In contrast, on the assumption that features are bi-valent, contrast comes about as a result of having a positive or negative specification, as in (3b). Thus, if [F] is plural, then [+F] will correspond to a plural specification and [-F] will correspond to a singular interpretation.
[+F] :: [-F] Neither of these approaches typically deals with three-way contrasts. On the mono-valent approach, one could assume two distinct features (i.e., plural and singular) each of which contrasts with the absence of that feature. On the bi-valent approach, one could assume that the contrasting feature specification [±plural] itself contrasts with the absence of this feature. In this paper, we develop an analysis that makes more explicit the ingredients of this particular three-way contrast. Specifically, the goal of this paper is to argue that this three-way contrast reduces to two two-way contrasts, as in (4).
(4) a. singular :: plural b. plural :: general The remainder of this paper is organized as follows: in Section 2 we show in detail how the three-way contrast results from two two-way contrasts. For objects of morphologically transitive verbs we observe a contrast between singular and plural, but for objects of morphologically intransitive verbs we observe a contrast between general number and plural. 3 In Section 3 we develop a syntactic analysis for these two two-way contrasts. We argue that plural marking in Blackfoot can either function as a syntactic head (contrasting with its singular counterpart) or as a modifier (contrasting with general number). We discuss the semantic consequences of our analysis in Section 4, and its typological consequences in Section 5. In Section 6 we conclude.

Two two-way contrasts: The role of the verbal predicate
The appearance of a three-way number contrast among Blackfoot object nominals can be reduced to two two-way contrasts: a singular vs. plural contrast, on the one hand, and a general vs. plural contrast, on the other. In this section, we shall see that the two contrasts are licensed in different contexts, and in Section 3, below, we argue that they are derived in different ways. We begin by demonstrating that the number options available with different object nominals are determined by the morpho-syntactic classification of the verb. In particular we show that the distribution of the two two-way contrasts is conditioned by the class of the predicate that the nominal is dependent on: objects of transitive predicates display a contrast between singular and plural number (5a), whereas arguments of intransitive predicates display general number effects (5b).
(5) a. object of transitive predicate: singular :: plural b. argument of intransitive predicate: plural :: general In the remainder of this section, we describe in more detail the morpho-syntax of predicate classes in Blackfoot. Like all Algonquian languages, Blackfoot classifies its verbs into four distinct classes based on the morpho-syntactic content of the final morpheme of the verb stem. The traditional Algonquianist terms for these classes and their defining properties are summarized in Table 1.
As the names of these classes and their characterization suggest, verb classification is based on two properties: (i) whether the verb is transitive or intransitive, and (ii) whether a particular argument (subject or primary object) is animate or inanimate. Both of these properties are morpho-syntactic in nature, as is evident from the fact that the classes do not always reflect the argument-structure properties of the verbs, or the semantic content of their arguments. We shall see below that morpho-syntactic intransitivity does not always align with semantic intransitivity, due to the existence of animate intransitive (AI) verbs that have a primary object. Similarly, we will see that morphologically animate nouns are a semantically mixed class, including both nouns with a human or animate referent and nouns that refer to inanimate objects, such as objects that roll, objects made of metal, some items of clothing, some berries, some plants, etc. (cf. Wiltschko & Ritter 2015 for discussion). 4 Note that for morphologically intransitive verbs, the designated argument whose animacy is specified is the subject, but for morphologically transitive verbs, the designated argument is the primary object. The primary object is either the direct or indirect object, and is determined as follows: if there is a semantically animate indirect object it will be the primary object; otherwise the primary object is the direct object.
All Blackfoot dyadic and triadic verbs have both transitive and intransitive variants belonging to different verb classes, and the choice among them correlates with properties of the primary object. If this object is specific and referential, the verb will be realized as morphologically transitive (TA or TI), but if it is non-specific or non-referential, the verb will be realized as morphologically intransitive (AI). 5 In other words, Blackfoot and other Algonquian languages have sets of two, three or four verbs in different classes that correspond to a single verb in English. In this paper we focus on sets of verbs that consist of an AI, TA and TI verb, as illustrated in Table 2. 4 Evidence that this is a morphological classification system can be gleaned from the fact that the form of the plural marker can reliably be used to determine the classification of nouns: animate nouns are always pluralized with the suffix -iksi and inanimate nouns with -istsi. 5 Blackfoot imposes a strict animacy requirement on external arguments of (di)transitive verbs. Consequently, there are no (di)transitive verbs of the inanimate intransitive (II) class in this language.  Frantz (2009: 41) uses the term para(di)transitive for AI verbs that have a primary object, because they lack object agreement. These verbs also lack theme marking (also known as direct/inverse marking). The theme marker is a suffix that appears immediately after the verb stem and indicates the relative status of the arguments based on a language-specific person/animacy hierarchy (cf. Bliss 2005Bliss , 2013. Primary objects of morphologically transitive (TA or TI) and intransitive (AI) verbs differ in three other respects, as illustrated in (6)-(8). First, objects of morphologically transitive verbs co-occur with a demonstrative or other determiner, but objects of morphologically intransitive verbs do not. Second, objects of morphologically transitive verbs may be either singular or plural, but not general, while objects of morphologically intransitive verbs may be either plural or general, but not singular. Finally, note also that the form of a morphologically transitive verb stem is sensitive to the animacy of its primary object, but the form of a morphologically intransitive verb stem is not. The examples in (8) demonstrate that morphologically intransitive verb stems are compatible with both animate and inanimate primary objects. These differences are summarized in Table 3.
1-eat.ta-th chicken Intended: 'I ate one or more chickens.'  We interpret this cluster of properties as evidence for a syntactic difference between primary objects of morphologically transitive and intransitive verbs. In the next section, we adopt the proposal of Bliss (2013) that primary objects of transitive verbs are full arguments while primary objects of intransitive verbs are pseudo-incorporated arguments. Following Bliss, we also assume that this difference in argument status correlates with a structural difference. However, we argue that Bliss' (2013) structures fail to account for the different number contrasts in the two kinds of objects, and develop an alternative proposal which addresses this issue. 6

Syntactic analysis
In order to account for the observed differences between the two classes of objects, Bliss (2013) argues that only objects of morphologically transitive (TA/TI) verbs are full nominals and are licensed by functional heads outside of the VP. Full nominals contain all functional categories, including PhiP, DP and KP or Link. 7 Objects of morphologically intransitive (AI) verbs, on the other hand, are small nominals, i.e., they lack some functional projections, and are pseudo-incorporated in the sense of Massam (2001). More specifically, Bliss proposes that small nominals come in two varieties: she analyses bare nouns as lacking K/LinkP, DP and PhiP, but plural-marked nouns as lacking only K/LinkP and DP. The structures she proposes are schematized below: One of the distinctive properties of Blackfoot is that both singular and plural nouns are overtly marked with a suffix that indicates number and either obviation and/or animacy, as shown in Table 4. 8 Animacy refers to a morphological classification in Algonquian languages. Obviation refers to a kind of marking of discourse salience that is pervasive in these languages -morphologically animate nominals are marked as either proximate (discourse salient) or obviative (non-salient). We are not concerned here with the precise conditions that determine obviation. For discussion of obviation in Blackfoot, see Bliss (2013), and references cited therein. Bliss analyses the suffixes that mark singular number (and obviation for animate nouns) as associating with K/Link, and the suffixes that mark plural number and animacy as associating with Phi. Since only full nominals contain the category K/Link, but both full and small nominals may contain Phi, her analysis accounts for the fact that both types of nominals may be marked plural, but only full nominals may be marked singular.
Since -wa and -yi instantiate K/Link, rather than Phi, more must be said in order to account for the fact that -wa and -yi are interpreted as singular. Bliss assumes a morphological spellout restriction that permits only one suffix per noun, with the result that the K/Link suffix is only overtly realized if Phi lacks an overt plural suffix. (See Bliss, Ritter & Wiltschko 2014 for arguments that a similar spell-out restriction is required for verbs in Blackfoot.) However, this leaves us with the following question: if some small nominals are PhiPs, why is it that these PhiPs can be plural, but not singular? In the remainder of this section we develop an alternative analysis that attributes the lack of singular small nominals to the presence of two kinds of number features in Blackfoot. Following Bliss (2013), we assume that objects of TA/TI verbs have a more complex structure than objects of AI verbs, and that objects of TA/TI verbs always contain PhiP, whose head bears a number feature that determines whether that object is singular or plural. However, rather than assuming an optional PhiP in objects of AI verbs, we propose that in Blackfoot number marking may serve not only as a head associated with Phi, but also as a modifier which can associate lower than Phi. We begin by motivating our hypothesis that Blackfoot has two qualitatively different number features. Our point of departure is Wiltschko's (2008) proposal that there are essentially two different kinds of formal syntactic features: head features and modifying features. A head feature determines the identity of the category it associates with. A modifying feature does not have this property. Rather, it optionally adjoins to a category whose identity is determined by a distinct head feature. This difference is schematized in (10). As shown in (10a), when [plural] is a head feature, it is associated with the functional category Phi; in contrast, as shown in (10b), when [plural] is a modifying feature, it is associated with a category whose identity is determined by some other feature. The two types of features differ in terms of their distributional properties: a head feature is obligatory, but a modifying feature is not. According to Wiltschko (2008), English plural marking realizes a prototypical head feature. Observe that English plural nouns must bear a plural marker, even if the plural interpretation could be deduced otherwise, e.g., from a numeral like three, as shown in (11a). In other words, in English an unmarked noun is not compatible with a plural interpretation. When the noun is not plural-marked, as in (11b), Wiltschko argues that the noun still has to be interpreted as marked for number, and that in this case it receives a silent singular marker (Ø).
Wiltschko interprets the contrast in (11) as evidence that an English nominal phrase is always associated with a number feature, and that the feature is interpreted as either singular or plural. She proposes that head features are bivalent, and specifically that English has a bivalent feature [±plural]. When a nominal is specified as [+plural], it is interpreted as plural, and when it is [-plural], it is interpreted as singular. As noted above, head features determine the label of the functional category they are associated with, and in particular, [± plural] associates with the category Phi. Unlike a head feature, a modifying feature is optional and monovalent. To see this, consider the Halkomelem example in (12), in which plural marking displays different properties than in English. In (12a), the nominal swíweles 'boy' is not marked with a plural affix, although the numeral preceding the noun indicates that a plural interpretation is intended. And indeed, plural marking (in the form of ablaut) is optionally possible, as shown in (12b).

(12)
Halkomelem (Wiltschko 2008: 642) a. te lhíxw swíweles det three boy 'the three boys' b. te lhíxw swóweles det three boy.pl 'the three boys' When a monovalent modifying [plural] feature is present the noun is interpreted as plural in number; when it is absent, the noun is interpreted as number-neutral, unless there is some other element that restricts the interpretation of the nominal, such as a numeral or another quantifier. For instance, in Halkomelem, the unmarked noun swíweles 'boy' can be interpreted either as singular or plural. 9 The diagnostics for head and modifying features are summarized in Table 5. As shown in Section 2, objects of AI verbs can be realized either as plural marked nouns or as bare nouns. The data are repeated in (13). We now show that the [plural] feature on objects of AI verbs is best analyzed as a modifying feature, and that the nominal complement of AI verbs lack PhiP.
(13) a. Nit-ooyi aipasstaam-istsi. 1-eat.ai apple-pl 'I ate apples.' b. Nit-ooyi aipasstaam. 1-eat.ai apple 'I ate [one or more than one] apple./*I ate one apple.' The properties of the plural suffix in the context of objects of AI verbs indicate that the plural feature on these objects is a modifying feature as diagnosed by Wiltschko (2008). 10 First, with objects of AI verbs, the plural suffix -istsi is optional, as shown by the fact that it is not required for a plural interpretation. A plural interpretation is possible, both in the presence of a plural suffix, and in its absence. However, in the absence of the plural suffix the nominal is number neutral, i.e., it is compatible with both a singular and a plural 9 Wiltschko (2008) argues that the modifying [plural] feature in Halkomelem is adjoined to a root, but speculates that there could be cross-linguistic variation in the category that the modifying [plural] feature adjoins to. We propose that Blackfoot also has an optional modifying [plural] feature, but that it is adjoined to Inner Aspect (I-Asp), rather than to the root (see the discussion below). See also Butler (2011) for additional evidence that the position of modifying features varies across languages. 10 Blackfoot appears to be different from other Algonquian languages in this respect. For example, Mathieu (2013) shows that Ojibwe, another Algonquian language, only has a two-way contrast in number, i.e., singular vs. plural. He observes that Ojibwe bare nouns are not interpreted as number neutral, but rather as singular. Moreover, unlike in Blackfoot, Ojibwe bare nouns can appear as the object of TA and TI verbs. Taking these facts into consideration, Mathieu argues that plural suffixes in Ojibwe always instantiate a head feature, not a modifying feature. interpretation. This pattern of interpretation indicates that the plural feature on nominal objects of AI verbs cannot be a head feature. If it were a head feature, we would expect the contrast to be between singular and plural, as in English (11). In other words, the availability of a number-neutral interpretation in the absence of plural marking indicates that the category Phi is not projected. If it were, we would expect that nouns lacking an overt [+plural] affix would be PhiPs bearing a contrasting null [-plural] affix and hence be interpreted as singular. The hypothesis that [plural] on small nominal objects of AI verbs is an optional modifying feature entails that there must be a distinct category-defining head feature. In fact, Ritter and Wiltschko (2014) develop just such an analysis. More specifically, they propose that the optional modifying [plural] feature on these small nominals is associated with I-Asp, a position which is associated with the head feature [±animate] (Wiltschko 2012). On this view, all small nominal objects of AI verbs are analyzed as instantiating nominal I-Asp -a constituent which does not include PhiP. The structures of plural and number-neutral objects of AI verbs are schematized in (14a)  Summarizing the results of this section, we have argued that objects of TA/TI verbs are full nominals, and that they include PhiP, a functional category whose head is specified for a bivalent number feature [±plural]. Consequently, full nominals are specified as either singular or plural number. In contrast, objects of AI verbs are small nominals. In Blackfoot, small nominals are I-AspPs, a category whose head is obligatorily specified for a bivalent class feature [±animate], and optionally modified by a monovalent number feature [plural]. When the optional [plural] feature is present, a small nominal is interpreted as plural and when it is absent the small nominal is number neutral.
Given that plural features occur in two different head positions one might expect that a single noun could have two plural markers. This is not possible, as a double plural would be semantically vacuous: a head feature that specifies a value for plural receives no additional information from a modifying plural feature.

Semantic consequences
In this section we turn to the question of the semantic content of the two plural markers we have identified in Blackfoot. Relevant in this context is Rullmann & You's (2006) analysis of number in languages like English with only a singular-plural opposition and languages like Mandarin with only a general-plural opposition. Here we suggest that their analysis can also be applied to languages like Blackfoot, which have both types of opposition, but in different contexts. This is a desirable result given that there appears to be no difference in interpretation between the two kinds of plural in Blackfoot. Rullmann & You (2006) follow Link's (1983) general approach to the semantics of number, in which the domain consists of an atomic join semi-lattice. The atoms in this lattice represent singular entities (e.g., Alex), and their (non-atomic) sums represent plural entities (a.k.a. pluralities; e.g., Alex and Bob). A singular count noun denotes a set of atoms, and the corresponding plural noun denotes the set of all pluralities that can be constructed from those atoms, as schematized in (15). Within this general framework, the proposal about Blackfoot we have put forward in this paper can be implemented formally as follows. As in Schwarzschild (1996), Chierchia (1998), and Rullmann & You (2006), we model entities (type e) as non-empty sets. Singular entities (atoms) are singleton sets (e.g., {a}) and plurals entities (sums) are non-singleton sets (e.g., {a,b}). Sum formation corresponds to set-theoretic union (e.g., the sum of {a} and {b} is {a}∪{b} = {a,b}). Common nouns are of type <e,t> and denote (the characteristic function of) sets of entities.
In Blackfoot (and perhaps in all languages), noun roots have general number, that is, they denote sets that include both atoms and their sums. The denotations of Blackfoot noun roots are therefore closed under union (sum-formation); that is, for any (atoms or sums) x and y, if x Î ⟦N⟧ and y Î ⟦N⟧, then also x∪y Î ⟦N⟧.
Adjunction of the [plural] feature restricts the noun denotation to the sums and eliminates the atoms. There are two ways in which this can be implemented formally. One option is to treat the [plural] feature as a function of type <<e,t>,<e,t>>, which has been the usual approach in the formal-semantic literature on number. Concretely, the feature [plural] would denote a function which takes a set of entities and maps it onto the subset containing only its non-atomic members. This function is defined in (17). 11 (17) Denotation of [plural] as an expression of type <<e,t>,<e,t>> (FA analysis) ⟦ [plural] ⟧ = λX . X -At where At is the set of all atoms in D e .
In this analysis, the feature combines with the noun (or nP) to which it is adjoined by means of Function Application (FA; cf. Heim & Kratzer 1998). There is, however, a simpler analysis, at least for the Blackfoot data under consideration. If the [plural] feature combines with the noun by means of Heim & Kratzer's operation of Predicate Modification (PM) instead of FA, we can treat it as an expression of type <e,t> (the same type as the noun), which denotes the set of all non-atomic entities, as in (18). If [plural] is a monovalent modifying feature (i.e., in objects of AI verbs), both the FA and the PM analysis yield the opposition between general and plural number depicted in (16). In the absence of the [plural] feature, the noun (both the root and its nP projection) will denote a set containing atoms as well as their sums (e.g., {{a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}), whereas when the [plural] feature is adjoined, the resulting denotation will include only the sums (e.g., {{a,b}, {a,c}, {b,c}, {a,b,c}}). However, things are complicated somewhat by the fact that, for syntactic reasons, we argued in Section 3 that the [plural] feature is not adjoined to the nP itself, but to the [±animate] feature in I-Asp, as shown in (14a). This complication can be solved more easily, and elegantly, under the PM analysis than in the FA analysis. We will simply assume that the feature [±animate] is also of type <e,t>, with [+animate] denoting the set of all entities that count as animate in Blackfoot, and [-animate] denoting the complement of that set (i.e., all non-animate entities). 12 The [plural] and [±animate] features are then combined via PM, forming the intersection of their respective denotations, which in turn is intersected (again via PM) with the denotation of the nP. 13 What about objects of TA or TI verbs, which have a bivalent head feature [±plural]? Again, it's easier to deal with this if we assume the PM analysis. We will treat the minus (-) value of the feature as denoting the complement operation (relative to D e ), whereas the plus (+) value is the identity function (or semantically vacuous  (15). 14 In sum, we have shown that the apparent three-way contrast in number marking reduces to two two-way contrasts. In the context of transitive predicates we have a singular vs. plural contrast but in the context of intransitive predicates the contrast is between plural and general number. Crucially, the denotation of general number includes the denotation for plural. What is interesting is that the same plural marker (animate -iksi or inanimate -istsi) is used regardless of whether it contrasts with singular or general number. This is consistent with Rullmann & You's (2006) finding that plural marking has the same denotation no matter whether it contrasts with singular or with general. 15

Typological consequences
Languages with a singular-plural opposition are extremely common; languages with a general-plural opposition are less common, but include a sizeable number of genetically and areally distinct examples. For example, Haspelmath (2013) lists 133 languages with obligatory plural marking on all nouns but only 55 languages with optional plural marking on all nouns. The attested options in his survey are in fact much more varied: Haspelmath (2013) also lists 20 languages with optional plural marking -but only on nouns that denote humans, and 15 languages with obligatory plural marking on nouns that denote humans and optional plural marking on nouns that denote inanimate objects. This last group would appear to be another type of language with both a singular-plural opposition and a general-plural one, with the choice determined by the lexical semantics of the noun, rather than the classification of the verb. Languages like Blackfoot with number marking on all nouns that is obligatory in some contexts and optional in others appear to be extremely rare.
According to the analysis we have developed here, this apparent three-way contrast in Blackfoot reduces to the unusual juxtaposition of two distinct systems, which are themselves attested elsewhere. This still leaves the question as to why this constellation of facts would be so rare. We speculate that this constellation comes about only when several properties conspire. And it is precisely these properties that conspire to produce this constellation that are typologically rare for independent reasons. Specifically, Blackfoot has an overt marker for singular as well as plural number. The existence of a singular marker means that the contrast between singular and general number will be morphologically marked.
In sum, while the surface three-way contrast in Blackfoot number marking is typologically rare, we have shown here that this three-way contrast reduces to two two-way contrasts which are typologically common. It is the co-occurrence of both these systems which is typologically rare because they depend on other typologically rare properties.

Conclusion
In this paper, we discussed the issues surrounding the existence of a three-way number contrast in Blackfoot. We argued that this contrast reduces to two two-way contrasts that are conditioned by the embedding predicates. This is summarized below. (20) Surface three-way contrast: singular :: plural :: general (21) Underlying two-way contrasts: a. object of transitive predicate: singular :: plural b. argument of intransitive predicate: plural :: general In terms of the semantics, we have adopted Rullmann & You's (2006) proposal according to which the semantic properties of plural marking are identical no matter whether it contrasts with singular or with general number. This is consistent with the fact that the plural marker used in Blackfoot is identical in both cases. What remains to be seen is whether all three-way contrasts attested in the languages of the world are amenable to a similar analysis.