Minimalist Grammars (MG) are viewed as a resource consuming system where syntactic operations are triggered when a positive form of a feature matches with its negative form. But a problem arises when a feature lacks a positive/negative value. For the latter case, we introduce a default inference rule in order to account for the underspecification of the feature in a lexical entry.
In Bantu syntax, the computational system only needs a noun class formal feature to proceed analysis. Noun classes are sets of words that trigger the same agreement schema. Ewondo (Bantu, A72a)1 has 14 noun classes2. The choice Gender
Classes Class morpheme Agr marker on verb Agr marker on adj of a noun class prefix indicates whether the noun is viewed as a unit or a set of units. Except for locatives (cl16, cl17), even-numbered noun classes indicate augmented (AUG) and odd-numbered are for minimal (MIN)3. As it can be seen in Table 1, each class has a different class morpheme that triggers a different agreement morpheme feature; except for nouns of classes 9, 10 that share the same class feature. (1) 1. m´Ong´O áku ám` boo m-´Ong´O á-a-ku á-m` boo 1min-child agr1-past1-fall down agr1-lay flat ‘the child falled down and laid flat’ 2. b´Ong´O b´@ku b´@m` boo b-´Ong´O b´@-a-ku b´@-m` boo 2aug-child agr2-past1-fall down agr2-lay flat ‘the children falled down and laid flat’ (2) 1. ñag yàdì bíl`Og ñag y`@-à-dì bí-l`Og 9min.cow agr9-Pres-eat 8aug-grass ‘The cow grazes’ 2. ñag yâdì bíl`Og ñag y´@-à-dì bí-l`Og 10aug.cow agr10-Pres-eat 8aug-grass ‘The cows graze’ In (1), the agreement class feature of the head noun (m´Ong´O, b´Ong´O) spreads on the verbs. In (2) we have the same form of the noun for both the minimal and the augmented. In fact, when standing alone, one can’t tell whether ñag is minimal (i.e class 9) or augmented (i.e class 10). It’s rather the agreement it triggers that helps to distinguish one form to another. As already mentionned, Bantu agreement phenomenon is characterized by the spreading of class feature of the head noun all over its dependents including the verb. Structure building rules (merge, move) in MG are defined in a directional process with a feature checking system that is a mechanism of resource consumption i.e each selector feature must match a selectee and each licensor match a licensee. [On1] proposed to formalize bantu multiple agreement in MG by Head Movement with Copying, the idea being that a selector is not end-consumed as the items that select it still exist in the derivations. The aim of this paper is to see how to deal with the balancing of ambiguity versus underspecification in the feature (2) in a resource consumption system like MG. Underspecification has being addressed in type-based grammars [Cr1,Dn1], in Type-Logical Grammars [He1], but never in MG. Here, we propose to associate a defeasible inference rule ( ) to lexical items with underspecified class feature. is based on Prototypical 3 Ewondo grammatical number has been redefined as Minimal (Min) and Augmented (Aug), thus we have one single feature [ aug] [On1] Reasoning [Re1,An1]. Section 2 proposes three ways that languages can have (or not have) noun classes. Section 3 presents the indeterminacy of class feature of nouns of class 9/10 in Bantu syntax. In Sect. 4 we show how the building of syntactic operations works in MG, Sect. 5 provides a new solution that could help to account for underspecification in MG after showing the limits of the first proposal made in [On1]. The paper ends with a conclusion. 2
Given examples (3, 4, 5) that show agreement phenomenon encountered in French, English and Ewondo. Imagine one removes maisons (houses) from (3a), then if a French speaker is asked to give the masculine form of the adjective belles (beautifulfem,pl), he would say beaux (beautifulmasc,pl) because gender is inherent in adjective in French. On the other hand in English (4), the adjective stays unchanged, gender (or number) feature is not inherent in adjective. (3) 1. toutes ces belles
maisons allfem,pl
thisfem,pl beautifulfem,pl housefem,pl ’All these beautiful houses’ 2. tous ces trois
jours allmasc,pl
thismasc,pl threemasc,pl daymasc,pl ’All these three days’ (4) 1. all the beautiful houses
allø theø beautifulø housepl 2. the desperate housewives
theø desperateø housewifepl For a Ewondo4 speaker, if he is asked to give the gender class of a determinative5, he will be unable to give one. He needs to know the syntactic context in which this determinative appears to tell what its class marker is. ‘All these three days’ 4 Unless specified, all the examples that aren’t French or English are from Ewondo language 5 The class marker allows to distinguish between substantives and determinatives.
Substantives are the set of nouns that [Gr1, p. 7] called inherent gender because this category triggers agreement. The second one he called derived gender is made of words that agree with the first one. In Ewondo (as in most Bantu languages), there are two nominal categories that share the fact to have the same nominal prefix. We term this second one as "determinative" 2. bi-soá bi-t¯@ bi-s@ bi-lá 8aug-plate agr8-this agr8-all agr8-three ‘All these three plates’ The following observations6 can be made: (i) adjective in French is an unmarked form that potentially agrees with the noun; (ii) in Ewondo, we can’t indicate the class marker of a determinative except it appears in a construction, that means we need the presence of a substantive that bears a specified noun class marker to tell what are the class markers of the others items. Determinatives don’t have pre-specified class marker, they inherit the class marker of the head noun; (iii) adjective in English is invariable. French and Ewondo speakers differ in whether they are able to produce a particular inflected form of an adjective in isolation. This is an experimental finding, and can be explained in many ways. One possible explanation is simply that speakers of any gendered language, when faced with such a task, think of an appropriate context and report the form the adjective takes in that context. The different behaviour of the French and Ewondo speakers is a result of there being only two genders in French, and thus that it is much easier to think of an appropriate context. 3 3.1
In Ewondo, nouns of classes 9, 10 are problematic if one wants to determine their respective noun class. In (6), the DPs subjects aren’t different as can be found (chicken vs chickens) in English. It’s rather the agreement class marker the noun triggers (agr9 y`@ and agr10 y´@) that differentiates kúb in (6a, b) [Ow1, p. 65] is 9min and 10aug respectively. (6) 1. kúb yàkOn.
kúb y`@-à-kOn 9min.chicken agr9-Pres-be sick ’The chicken is sick’ 2. kúb yâkOn.
kúb y´@-à-kOn 10aug.chicken agr10-Pres-be sick ’The chickens are sick’ As in most Bantu languages, it’s assumed their nominal class morphemes are originally homophones n- (see Table 1). It’s also difficult to say whether a given noun has a root /NCVC(V)/ or /CVC(V)/ with a class morpheme n-. Linguists usually argue by analogy to others noun classes: if most nouns of classes 9, 10 begin with a nasal7, and if there are less nouns in others classes with that 6 My thanks to Greg Kobele for valuable comments after my aviva. 7 and there is a high percentage of initials [nD] and [nT] (where [D] is a voiced occlusives and [T] is a non voiced occlusives). structure, then people assume that roots can’t generally begin with NC; therefore nouns in classes 9, 10 that always have a NC initial must actually have a prefix /n-/. An answer to this argument is the possibility to mark a contrast between roots with NC and C initials in noun classes 9, 10. In Ewondo, we have nouns with [nD] but also words with [T] and [z]; such thematic roots are ambiguous. Two explanations are possible : (i) there is a phonological deletion of /n/ in front of voiceless consonants and fricatives, and (ii) there is a real contrast between NC and C initials in theses noun classes. In Ewondo, the prefix n- is deleted when it’s followed by another nasal (7), an unvoiced consonant (8) or by a voiced consonant /z/ (9): (7) n+Nàk ! Nag: cow (8) n+tsit ! tsíd: animal (9) n+z@k ! z@g: pineapple
In short, nouns of classes 9, 10 are morphologically invariable et neutral for class distinction. One may think there is no difference between minimal and augmented number. 3.2
As noted, for those nouns that don’t change in minimal/augmented form, the distinction is made by the agreement they trigger (10, 11). (10) 1. kúb é-n`@ o-nˇOn
9min.chicken agr9-pres.be 11min-bird ’The chicken is a bird’ 2. kúb é-nˆ@ a-nˇOn 10aug.chicken agr10-pres.be 5aug-bird ’Chickens are birds’ 3. *kúb é-nˇ@ a-nˇOn 10aug.chicken agr10-pres.be 5aug-bird ’Chicken are birds’ In (10, 11), tone doesn’t help to distinguish the two noun classes. Originally, the augmented form is obtained by adjoining a suprasegmental High tone |´ | to the noun of class 9. This floating High tone8 attaches either to the noun or 8 Regarding the architecture of tonal representations, floating tones (not associated) are usually represented in a circle : spreading high tone , spreading low tone . (12) (13) h buk h buk (15) (16) h buk h buk to the verb. Nevertheless, it seems that the verb, each time it’s present, bears the floating High tone. The association of the High tone is done from left to right. Nouns and verbs that bear a Low tone on the last syllable (10a) yield, when a High tone is added to them, a High-Low tone on the verb (10b). If the association is made from the right to the left, then we get a Low-High tone on the verb, thus the ungrammatical (10c). Let’s take the subject and the verb in (10b), the High tone of kúb (chickens) spreads on the right : 9 It’s important to note that tone isn’t a distinctive feature as the word already has a high tone. The main point to look at is the (minimal) agreement on the verb comparing to (10b) that is an augmented form. That means the proposed analysis is the same for N with Low tone.
on the verbal prefix (é), there is no difference.
(chicken) attaches on the right yielding (16): This floating High tone of kúb (chicken) pushes the Low tone of the verbal prefix to the right, and we get (17): (14) kúb é-n`@
9min.chicken agr9-pres.be ‘The chicken is’ With nouns originally with a Low tone, the difference is made at phonological level with a raising pitch on the first syllable of the verb. This syllable should bear the Low or High tone to indicate whether the noun is minimal or augmented and also specify the class agreement feature. But, as the verb already has a High tone on its first syllable, the original tone of nouns of class 9, 10 spreads to the last vowel of the verb (11). Let’s take an example with a noun bearing a High tone9 We have a High tone on kúb (chicken), a Low tone of class 9 on the verbal prefix è and a Low tone on the verbal root n`@ that are shown below (15): (17)
And the Low tone goes on this verb root, as the latter already bears a Low tone, nothing changes. We can conclude that tonal distinction on the agreement feature can be useful to distinguish the covert class feature of nouns of class 9/10. 4
MG [St1] attempt to implement the so-called minimalist principles introduced by [Ch1]. A MG is a quadruplet (V,Cat,Lex,F): V = {P [ I}, set of non syntactic features (vocabulary) where P represents the phonetic features and I the semantics features; Cat = {base [ selector [ licensor [ licensee}, finite set of non syntactic features (categories) which are partitioned into four kinds (x : base (c, t, v, d, n, ...), =x : selector/probe, -x : licensee, +x : licensor (feature that trigger move)); Lex = finite set of expressions built from V and Cat (lexicon); F = {merge [ move} : set of generating functions. Merge and Move are built with trees where : (i) internal nodes are labelled with direction arrows (< or >) indicating where the head of the structure is, (ii) leaves are pairs h ; i with = vocabulary item and = set of features. Merge (or external merge) is a binary operation that takes two trees and puts them together. The tree whose first feature is =x merges with a tree whose category feature is x to built a new tree. Features =x and x are deleted after merging. (18) merge (t1=x, t2x) = merge (t1=x, t2x) =
< t1 t2 > Move (or internal merge) is a unary operation that targets (some part of) an expression to remerge it higher in the structure. Move is applied to a subtree with a feature -x. Given a subtree with -x written t2 x that appears in a tree t1+X , we write t1[t2 x]+X . t1+X is the maximal projection of t1[t2 x]+X i.e the largest subtree with -x as its head. After extraction, the subtree t2 x merges as specifier of the head of the tree, features served for Move operation are removed from the tree. The shortest move contraint (SMC) that applies to Move requires there should be exactly one maximal projection t1[t2 x]+X displaying a subtree t x 2 .
The original place of t2 x is then filled by an empty tree i.e a single featureless node. (19) move (t1[t2 x]+X ) = > There are few syntactic operations implemented in MG (Scrambling & Adjunction [FG1], Head Movement [St2], Copying [Ko1], Head Movement with Copying [On1]). In MG, Merge and Move, need a selecting feature matching a selected feature (both being of the same category) to drive derivations. Now, what’s happened if the selecting feature is un(der)specified? 5 5.1
Given the examples below where nouns of classes 9, 10 are in subject position (20) and in object position (21), the analysis developed in [On1] for (20) is based on the theoretical claim that nouns of classes 9, 10 are not lexically specified for their class. (20) ñag yàdì bíl`Og ñag y`@-à-dì bí-l`Og 9min.cow agr9-Pres-eat 8aug-grass ‘The cow grazes’ (21) ńsOmO áwé ñag ń-sOmO á-a-wé ñag 1min-huntsman agr1-past1-kill 9min/10aug.cow ’The huntsman has killed the cow/cows.’ These nouns enter the derivation with an uninstantiated variable x that will be valued through postsyntactic insertion of the class morpheme of the agreement feature +a!gr on TP. Variable x instantiation means to copy on the subject DP the value of the agreement feature on T head. (22)
> <
< / / -ag!r /ñag(x)/ d -q (n(x)) (22) is built in 3 steps: (a) merge(n(x)<= !cl -k / /, n(x) -a!gr /ñag/), (b) merge(=>+cl +k d -q / /,a), (c) move(b).
Postsyntactic insertion means the agreement class feature on the verb is substituting for the variable x yielding the corresponding noun class feature on the noun. As we said it’s the agreement feature on the verb that give the information about the nominal class morpheme of the DP. The substitution process is made in two steps : (i) covert movement then (ii) agree (for detailed step-by-step justification see [On1]). If (20) is appropriately treated in [On1], the solution provided is still problematic for (21) where noun of classes 9, 10 are in object position. To solve this problem, let’s try another approach. Following [Ro1], we distinguish three features: -features are specified class feature for inherent noun classes; -features are underspecified class feature for noun of class 9/10; -features are flexible and inherited class agreement feature found on derived nouns (i.e determinatives) and verbs. If we think of noun class feature as Attribute-Value feature system, we could say, noun of class 9, 10 has an Attribute specification "n" without a Value (i.e without a class number). That means, a word like ñag(cow) is represented with the feature n . The difference being that a noun with a specified noun class (say m-´Ong´O: 1min-child) will be represented with a specified class feature n1. -features’ transmission is done through HMC. The question now is how to formalize -feature in MG? A default rule will be used to model feature underspecification through prototypical reasoning, the latter is used when most instances of a concept have some property10. Default Logic [Re1,An1] is a nonmonotonic reasoning approach allowing to rely on incomplete information about problem. A default theory T is a pair (z, ) where z is a set of FOL sentences representing the background information, represents the defeasible information (i.e a countable set of defaults rule). interpreted as: given and as there is no information that : i, conclude default. A default rule is called normal if and only if it has the form: : (23)
by (24) 10 That means for us the case when most instances of noun class feature have Attribute
Value property.
A semantics for Default Logic is provided through the notion of extension [Re1,AS1,An1]. An extension for a default theory T (T = (z, )) is a set of FOL sentences E where: (a) z E; (b) E = (E) where denotes the deductive closure; (c) E should be closed under the application of defaults from i.e if : 1;:::; n , 2 E and : 1 2= E; : : : ; : n 2= E then 2 E.
Definition 2. For T = (z, ), let = ( 0; 1; : : :) be a finite or infinite sequence of default rules from without multiple occurrences. is viewed as possible order in which default rules from are applied, so a default rule doesn’t need to be applied more than once in such a reasoning. The initial segment of with length k is denoted [k]. Sets of first-order formulae, In( ) and Out( ) are associated to such sequence as : (a) In( ) = (z[fcons( )j occurs in In( ) collects the information gained by the application of the default in and represents the current knowledge base after the default in have been applied; (b) Out( ) = f: j 2 just( ) for some occuring in g, Out( ) collects formulae that should not turn out to be true i.e that should not become part of the current knowledge base even after subsequent application of the other default rules.
Definition 3. is called a process of T iff every k such that k occurs in . k is applicable to In( [k]), for
g), (25) 2 E and : 1 2= Definition 4. For a given process of T: (a) is a successful process iff In( ) \ Out( ) = ;, otherwise it is a failed process; (b) is a closed process iff every 2 that is applicable to In( ) already occurs in . Closed processes correspond to the desired property of an extension E being closed under application of default rules from .
Definition 5. For the application of a default rule, a consistency condition should be satisfied.
=
: 1; : : : ; n is applicable to a deductively closed set of formulae E iff E; : : : ; : n 2= E.
Proposition 1. The rule of thumb when treating nouns of class 9, 10 is to say
these nouns are of class 10 unless stated by the grammarian they are of class 911.
If the latter is done, the default rule (i.e the rule of thumb) already mentionned
isn’t rejected, it’s simply no more applicable as the missing information is now
known.In a classical logic setting12, we need to say what is the Value of Attribute
n for classes 9, 10 nouns. As we don’t know, no decision could be taken in such
a system. But in default reasoning, the previous rule of thumb can be applied.
11 That means, only the augmented is grammatically marked, minimal will have a zero
grammatical marker that is realised as a low tone by default. In fact, that’s the
strategy generally used in natural language that the minimal has a zero morpheme.
12 As well as in Stablerian MG
Proof. We write Attribute with indices to differentiate between n1 (that stands
for class 10 nouns) and n0 (that stands for class 9 nouns). So, for T = (z,
), let z = fn1; n0g and = n11:0:9 ; n0:9:10 . The default theory T is
represented as : T = fn1; n0g ; n11:0:9 ; n0:9:10 . Let also assume that 1 = n11:0:9 ,
0 = n0:9:10 and = ( 1; 0). As contains only (a finite number of) two
default rules, closedness doesn’t matter. We apply default rules as long as they
are applicable, and then we get a closed process. So, we apply the first default
1 and check default 0 with respect to the knowledge collected after the
application of 1. For [
[n " ] (!cl -k / / nx / ñag/ MG are by definition encapsulated, which means that they make reference only to their own internalised system, and not to any external formal system, such as a logic for general reasoning. This is intrinsic to the claim of the language faculty being prior, feeding into more general reasoning devices but separate from them. If the current proposal is in line with Stabler’s formalization, then we think MG clearly differentiate a Stabler form of minimalism with others. That might mean that the notion of encapsulation may be rather different for a Stabler form of grammar than others. 6
In this paper we attempt to introduce a rule in a Stablerian MG that could help to account for feature underspecification in a resource consuming system. The proposal rests on default reasoning that allows to deal with incomplete information about a problem.
We thank Stan Dubinsky, Ruth Kempson and two anonymous reviewers for their constructive comments, which helped us to improve the manuscript.