Semantic and Morphosyntactic Differences among Nouns: A Template-Based and Modular Cognitive Model

Abstract

The noun category exhibits diverse dissimilarities, characterised at the semantic (e.g., countable/uncountable) or/and morphosyntactic (e.g., determined/determinerless) level, which may be more or less important depending on languages. In this paper, we would like to discuss those differences, which we named inter-word and inter-process morphosyntactic variations. The Riffian language served us as a reference in our enquiries, before referring to other languages to show how our discoveries could be applied to them. By putting in perspective those aspects, this led us to propose a formal mathematical model denoted as a Template-Based and Modular Cognitive model. The latter is able to predict the nonlinear dynamic mapping of lexical items onto morphological templates. The aims of this article are thus manifold and cover theoretical issues. We demonstrate that nouns are organised and distributed in modular cognitive sets, having their own morphological template and unmarked forms. The extent of these sets and their number as well as the template, are specific to each language. All sorts of markers can compose with the template, but some, namely countability markers, are prevalent among several languages with no relationship. This approach allows us to explain the marking discrepancies existing between different kinds of nouns (borrowed, proper, countable and uncountable nouns) for a given linguistic variety or between languages. The main assumption of this model is that these irregular markings are caused by a template shift, occurring when items undergo a process of word and meaning formation. Our contribution represents an initial stride toward understanding the fundamental patterns of morphosyntax and opens venues for applying this mathematical model with other behavioural and natural phenomena.

1. Introduction

In synthetic languages1, nouns are equipped with markers that inflect to construct word-forms, but all nouns are not marked in the same way. Depending on morphemes that can be affixed to these lexemes, we can distinguish them and classify them into paradigms on the basis of this marking. For example, in many languages, we can define proper nouns (hereinafter, PNs) and common nouns (hereinafter, CNs) according to these criteria, but this is not a universal observation. Moreover, there is no sharp boundary; this marking may easily overlap between these categories. For example, in English, the definite article is not used with most PNs, but some of them must be marked by this morpheme (e.g., the North Pole).

This study was performed so as to contribute to general linguistic issues about the essence of nouns regarding their morphological and semantic properties. The aim was also to identify which linguistic traits characterise those nominal lexemes and to describe the morphosyntactic differences between these categories in order to discuss their nature from a theoretical perspective. We chose to concentrate on particular grammatical paradigms having similar irregular marking behaviour. While those morphemes are unrelated, our thesis is that the underlying pattern affecting these languages is identical. Thus, a theoretical model, based on a computational mathematical approach, is proposed to capture these phenomena. For instance, in Riffian, morphemes, known as the state, can generally mark the CNs, while the PNs tend not to have them. This behaviour is put in correlation with what happens in other languages, such as English.

Table 1. Morphosyntactic differences between PNs and CNs regarding the state and definite article.

	Riffian			English
	State	Radical		Article	Radical
CN	a	ɾgaz	man	the	man
	N/A	seksu	couscous	N/A	life
PN	N/A	muħend	Muhend	N/A	John
	a	ʒðiɾ	Ajdir	the	Arctic

, below, summarises the possible markings in Riffian and English.

As one can see, there is no clear evidence that the PNs and CNs are totally different. One feature may be shared by one or the other category. Nevertheless, these are exceptions in terms of the number of occurrences. Thus, the state or the article is not found to be used extensively with PNs.

While it is assumed that the PNs and CNs are regarded as subgroups of the noun category, these two classes can become the other by word-formation processes. When this happens (see (1)), a morphosyntactic shift results that surfaces only after a lexical derivation from a PN (1(a)) to create an eponym. With onymised2 CNs (1(b)), it is noticed that the definite article or state is preserved, while we might expect that these morphemes are not inserted, as they are supposedly not part of the features of PN.

(1)

(a) CN from PN

   In Riffian: a-ʒeħːa ‘mischievous person’, from ʒeħːa ‘Jehha’

   In English: the-diesel, from Diesel

(b) PN from CN

   In Riffian: a-zˤʁaɾ ‘Azghar (toponym)’, from a-zˤʁaɾ ‘plain/flat land’

   In French: la-manche ‘The English Channel (toponym)’, from la-manche ‘sleeve’

Thus, this morphological variation can be at the inter-word (see Table 1) or morphosyntactic shift (see (1)) level. Different explanations have been proposed for Semitic, Romance or Germanic languages to elucidate these morphological contrasts. One of the traditional views in this matter is based on the supposed semantic properties of those nouns. For example, for PNs, it is stated that they are definite by nature; therefore, according to them, the definite article becomes pointless. Other hypotheses of semantics-driven marking, as we may refer to them, are advanced, but they cannot describe all the irregularities mentioned before. They have the disadvantage of being language-specific, dedicated to one or a few categories of nouns (e.g., kinship, uncountable, PNs or countable nouns) or both. Their intent is not to conciliate these morphological realisations, but rather to justify why they are as they are.

There are also more global approaches based on syntactic structural properties, in conjunction, sometimes, with semantic features, proposed under the generativist framework. In contrast to the previous perspective, those researchers aim to unify, under a simpler cognitive system, the various manifestations found, in surface representation, in the natural languages. The definite article is generally not perceived as a constitutional element of the word, worthy of study in the field of morphology. Defined as part of a functional category, namely the determiner, they form with the noun a syntactic structure generated in D-structure in the syntax component. Moreover, the insertion of this morpheme is initiated by the semantic side (viz., categorial, case and φ-features); to this extent, they are also, though in a different way, a semantics-driven marking approach.

The two outlined approaches, based on syntax or/and semantics, are different in many premises from the theoretical analysis tackled in this study. An alternative proposition was considered, so as to shed light on those inter-word, inter-process or inter-language morphosyntactic variations. One of the central elements of our theory postulates that the forms of words are articulated around a pre-syntactic morphological template encoded in the lexicon as being conceived by the modern Word and Paradigm approach. Therefore, the state or the definite article is considered a piece of inflectional morphology. In this case, a morphological structure including these morphemes enters the syntactic derivation as such, by merging with other syntactic categories.

We provide a formal mathematical description of this theory, named Template-Based and Modular Cognitive model (hereinafter, TBMC model), and demonstrate how it can be used. The TBMC model is able to infer the nonlinear dynamic behaviour of the morphological system. Its purpose is to ensure the mapping of the lexical items onto the morphological templates. The contours of this article are, however, restricted to some topics. The proposed model will serve to analyse mainly two grammatical categories, that is to say, countability and definiteness, which are central in the analysis of the morphological variations presented earlier. It is assumed that the countability paradigm, especially the collective marker, is a universal category present in the synthetic languages. To draw attention to these particularities, we will focus first on the Riffian language as an empirical grounding. We will give a novel insight into the state that will throw a new perspective on our general problem. Such research work for Riffian has not been discussed in detail in the literature.

Predominantly, PNs use the bare form (in accordance with established practice, we also refer to nouns without the definite article as determinerless or undetermined). It could be argued that this is due to their Arabisation, but a finer analysis suggests that there is indeed a dissociation between the state and the PN. Moreover, the PN category works in the same way as the uncountable nouns with respect to the state. Borrowed PNs and uncountable nouns are marked analogously when they become countable nouns. The examination of these particularities reveals a scheme; loanwords have different morphologies according to whether they are countable or not. When they have the Berber morphology and are uncountable, we can predict that only the singular will be used and the state cannot be prefixed. From this, it has seemed obvious that the state answered to another function that is in link with countability.

In this vein, we extended the study of these markers to all nouns to explain the irregularities mentioned in Table 1. It was observed that their presence or absence was conditioned by which means these nouns were formed, like in (1). A nonlinear model of word and meaning formation, which is a partial implementation of the mathematical model, was established to show when and how the state appears or not on the surface. The processes at the origin of this fluctuation are the conversion and the semantic shift. This leads us to reconsider the concept of the state; these morphemes were labelled anew as the collective and singulative markers. In addition, it appears that they are mutually exclusive. For this reason, they have been put together into the same grammatical paradigm, to wit countability. The last mentioned proposal is introduced as a new feature in the description of Riffian, and, in the wake of this empirical observation, of other synthetic languages.

This unified view of morphological marking of nouns founded on a formalist approach posited as universal properties gives the possibility to justify other morphological phenomena equally. The aforementioned issues are used as an entry point to explicitise the conceptual notions of inter-word and morphosyntactic shift. Thus, in addition to the paradoxical marking of nouns regarding the collective, singulative or definite marker, it can account for the gender shift observed in various languages happening when a noun derives from another noun. Similarly, for the Riffian language more specifically, the non-presence of the feminine marker with some borrowed nouns (e.g., ɾχeðm-eθ/*ð-ɾχeðm-eθ, ‘work’) finds an explanation under this model by also taking into consideration the semiotic encoding. Thereby, different kinds of nouns (uncountable, proper, loanword, denominal, deverbal, address) with an irregular morphosyntactic marking are reanalysed.

The other topic broached in these pages concerns central issues of the morphological system captured by the TBMC model. Different theoretical notions are discussed and introduced. Beforehand, we formalise some cognitive attributes peculiar to the lexical derivation having implications in other parts of the system. Numerous findings come out of this proposal. Thus, a formal definition of the expletive morpheme is proposed, integrating, among others, the new mechanisms of template shift. Likewise, this formalism is decisive in resolving different singularities found in morphophonology, named irreversibility, and also historical lexicology, denoted as immutability. For each of them, their formal mental process is explained as a mathematical property of the TBMC model.

All these differences, discovered in various languages, might be considered as an indication that their dissymmetrical morphology is proof that, for instance, PNs and CNs are two distinct categories. However, from a general linguistic standpoint, there is no grammatical feature that is utterly excluded from one or another category. Our conclusion is that there is no evidence supporting that their distinction is projected at the morphosyntactic level. These differences are mainly paradigmatic in nature. In fact, we were able to bring those irregularities together into a minimalist framework. This theoretical cognitive model, which is empirically supported, argues that those nouns have the same morphological template, but different unmarked forms, especially regarding countability. Nevertheless, uncountable and proper nouns systematically share common unmarked forms. Consequently, it seems that semantic constraints must force their association. We propose seeing this observation as an identity property, which is given in the form of a mathematical expression.

In what follows, we will methodically examine what constitutes the particularities of the noun categories and their underlying representation. To perform this, two distinct research method strategies are used, namely analytic and synthetic methods. The combination of both is required to solve the morphological issues presented earlier. This will ensure that we are able to make the most of the observed data. The elements coming under consideration belong to different linguistic categories and disciplines. They are, moreover, interconnected and their interaction is nonlinear. It follows that it is essential to approach these characteristics as a whole, that is, as a system. It is impossible to strictly circumscribe the investigations to one element and isolate it from the other parts of the (sub-)system. From a rhetorical perspective, despite our efforts to simplify, to the utmost, the demonstration, the reader may be overwhelmed by the complex abundance of facts and terminologies motivated by this methodology, but this is necessary. Rigorous reasoning is required to tackle these fine points. We have specifically called attention to these matters, since the emphasis in traditional studies in language science is more oriented towards reductionism and less towards systems-based studies. Therefore, this study is positioned within the fields of systems theory.

To put in place such a philosophy, the article is divided into several parts that are not structured chronologically. First, in Section 2, we review related work on this topic addressing similar problems in different subfields of linguistics. Afterwards, we outline the theoretical framework adopted in our study. Section 3 describes the aspects of our data by highlighting the morphosyntactic differences between PNs and CNs with regard to the state. The aim of Section 4 is two-fold: to determine patterns from observed data that had been modelled in Section 2 and to infer fundamental properties from our mathematical model. We also discuss general cognitive linguistic questions that are raised by these data. On the basis of our previous results, in Section 5, cross-linguistic data are also mentioned to demonstrate the universal character of this model. Finally, we sum up our remarks in Section 6.

2. Literature Review and Theoretical Foundation

This section introduces the theories founding this study and reviews relevant literature concerning different topics raised in this article. First, there is no extensive study on Riffian3, which also implies that there is no specific study on Riffian proper nouns. This paper wishes to fill this gap. Notwithstanding, this Berber variety has been documented in the literature. We refer readers interested in having a broad view of the Riffian language to these publications, [3,4]. In the following sections, a literature review will be presented. The author will focus on general theoretical questions related to our objects of study and the Riffian system, especially the nominal morphology and proper nouns.

2.1. The Nouns and Cognition: A Review

Different paths were taken in the attempt to understand where the dissimilarity among nouns is placed. PNs were certainly the first kind of nouns highlighted for their difference. Since the Antiquity with the Stoicists [5], to whom we are indebted for this class, many essays were completed in distinct fields, including the philosophy of language [6], semantics [7], pragmatics [8,9], semiotics [10], syntax [11], etc. We will not go through all this literature that focuses on one category, as the PN is affected in the same way as other nouns with respect to morphology. Hence, we will mention some works concentrating on the cognitive reality of language and the morphosyntactic irregularities of the nominal category. This will be an opportunity to present the cognitive model that sustains the TBMC model.

The focus of this review will be only on the theoretical premises structuring the different approaches related to our object of study. What will matter the most are the fundamental mechanisms and the cognitive conceptual components manipulated, and how both are related. We see fit to regard them as a system and only the most relevant parts are presented. In addition to mentioning these cognitive building blocks, as a means of exposition, a general classification of these different theoretical frameworks is proposed. This categorisation can be viewed as being organised in a hierarchical setting. At each level, some branches will be cited but not reviewed. As for those that have held our attention, as we explore them, more details will be given.

2.1.1. Morphosyntactic Irregularities

One of the main topics analysed in morphosyntax theory that has triggered abundant discussions and has implications for our study is the question of head of noun phrases. The problem posed is to know whether the head is the noun or the determiner. These hypotheses are, respectively, named as NP or DP analysis. Both positions have been defended by linguists and this debate is still ongoing. Following Abney’s work [12], partisans of DP analysis have posited several assumptions. In summary, they state, notably, that, for certain languages, nouns in some syntactic positions or some nouns, in spite of lacking articles, have a DP node dominating the noun that is not projected on the surface. To account for this, it is postulated, when the DP node is empty, that a movement4 where a noun—PN [11] or CN [13,14]—or a grammatical morpheme—a demonstrative [15]—moves to the DP or an intermediary node by filling this position. The fact that different linguistic units can fill this syntactic position is in accordance with Abney’s theory. He has claimed that the DP node is a functional category able to include different morphemes; he, himself, has proposed that this node can be filled by the genitive case marker. As for the absence of the definite article in genitive constructions, he proposes two hypotheses, namely a co-occurrence restriction (i.e., ‘s and the cannot stand together) or an elision.

Another approach used to explain, this time, the presence of the article, when the NP has a generic meaning, instead of its absence, is based upon the concept of the expletive form ([16], followed by [11]). The last explanation is subject to discussion. On the one hand, in the generative framework, the notion of expletive, also termed dummy word, is indeed used [17], notably to describe the properties of it/there, but it seems that the use of this term is not employed in the same sense depending on researchers. The expletive dimension of the definite article, as used by Vergnaud, refers to a morpheme that has no representation in the semantic side (i.e., more precisely, the D domain [18]), whereas Chomsky and others’ use of this notion does not imply such a consideration since these pronouns bear some grammatical features (i.e., deictic meanings). Their expletive character alludes to their lack of referentiality, which is not similar to having no meaning. Therefore, it would be more convenient to distinguish those concepts terminologically, and we suggest calling the first one expletive (meaningless) form5 and the second one, (expletive) referentless form. Later in this article, when the word expletive will be mentioned, only the former will be considered. Furthermore, a formal definition of the expletive form in morphology will be detailed (see Section 4.5).

Let us note that, generally, no justification is given to support this movement to the DP site. However, some do provide a linguistic motivation for this phenomenon by opting for a semantic orientation. Thus, according to Longobardi, an N-to-D movement is caused by referentiality questions. This optic joins the traditional argument adduced for the PN category, in that they have no article because they are inherently definite. Chomsky ([17], p. 213), adopting the N-raising-to-D hypothesis by Longobardi, argues that the feature strength [±strong] of the functional category D is what causes or does not cause the movement. If the one of D-features is strong, then NP can realise this movement. The nature of the definite article, null or expletive, is also often explained by virtue of the semantic properties of those nouns. More precisely, the semantic properties of the noun govern its morphology. By simplifying their theories, we remark some similarities. By adopting a formal semantic analysis, they have tried to distinguish two classes, into which all sorts of NPs are divided, which denote different meanings, namely, object/kind [19], followed by [11], type/token [16], predicate/argumental [20], light noun/full noun [21]. The characteristics of each of these categories do not necessarily match each other; nevertheless, they all rely on the quantificational properties of those nouns to distinguish them. In their view, number and countability form a whole; they are not distinguished grammatically. It is also worth noticing that, for the proponents of these ontological and pragmatic approaches, the PN category is often considered a prototypical linguistic paradigm on which they build their categories. It derives from their analysis that the boundary is no longer between PN and CN, but the division is undifferentiated with respect to the parts of speech/categorial features. Two morphemes belonging to the same part of speech can be incorporated into either one or another supposed category. Thus, Montague, for example, even puts quantifier phrases and PNs in the same category [22].

As reviewed, there are globally two directions followed by researchers to provide a rationale for the morphosyntactic irregularities of nouns. After careful consideration, both will not be endorsed in this study. We do not question that semantics impinge, in several ways, on morphosyntax, but we are reluctant to admit semantic determinism in morphology. It cannot account for the morphological patterns present universally as those discussed in this article. Because, when semantics is implicated, the produced linguistic facts tend toward divergence; for instance, the relationships between form and meaning are arbitrary and conventional. This inevitably leads to the particularism and the development of linguistic theories prevailing only for a small sample of grammars, while what we observe is not language-specific and could be in UG. Likewise, the syntactic analysis of the definite article will not be adopted. It can partly account for the inter-word variations and only for some languages. Moreover, no thoughts are given to the inter-process morphosyntactic variations. As strong evidence against the syntactic approach, one of the reasons to consider these morphemes as affixes is their predisposition to be affected by morphophonological phenomena (e.g., in English, the is pronounced either ðiː before a vowel or ðə before a consonant; in Riffian, see Section 2.3). Furthermore, even the proponents of this hypothesis have not completely excluded that the lexicalist approach could underlie the encoding of some nouns (see Section The Lexical Approach and the Definite Article). We propose to unify the encoding of all nouns similarly. Thus, we state, instead, that this morpheme and others alike are entirely pieces of morphology.

In the next section, to situate our work within current scientific theories, firstly, we will broadly explore the landscape of morphological theories, and then we will later discuss the specifics of formal morphology.

2.1.2. Theories of Morphology

The first writings of the generative school did not give much importance to morphology. However, Generativism provided a basis for subsequent theories, either theoretical outgrowths of the generative grammar or competing theories, dealing more precisely with morphology ([23,24,25,26,27,28,29,30], [31], referred to as Distributed Morphology (henceforth, DM) and [32], among others). In interaction with this school of thought, different theories, which have been developed outside the generative framework, also propose their reading of the morphological system; they may be included in the field of cognitive linguistics [33] or theoretical classical linguistics [34,35,36,37]. The points discussed in these frameworks are beyond the scope of this article. Therefore, we will only mention aspects related to the topics being studied.

In general, the theories discussing the formation of words can be classed into two approaches, namely Item and Arrangement (Beard [32] calls this approach Lexical Morphology Hypothesis) and Item and Process [38]6. The former (e.g., [39]; Ref. [40] on the basis of the Mirror principle, Ref. [31]) advocates that the morphemes or meaning units are concatenated iteratively in a one-to-one and continuous way. In their view, the formation of words and syntactic constituents are generated in the same manner. DM has pushed the reasoning further by assuming that there is no intermediary cognitive component between syntax and lexicon. The lexicon, as described by Chomsky, is put aside; DM’s theory is purely anchored in syntax. Hence, they are opposed to the Lexicalist Hypothesis [41], defending, mainly, the idea that lexicon and syntax are two separate cognitive levels. The second approach, instead, proceeds in two steps and regards morphology as an autonomous subsystem having its particularities. Thus, the distinction between lexicon and syntax is perfectly delimited. The role of the lexicon is to build words/lexemes from morphemes before handing over the outputs to the syntax component to generate sentences. Therefore, the word/lexeme as a linguistic unit is a cognitive construct indispensable to describe languages. Furthermore, the concatenation of morphemic or meaning units is preceded by a process changing the properties of the stem morpheme. In this way, this allows the affixation of grammatical morphemes satisfying such properties. Moreover, they argue that the items being able to serve as a basis structure to form new words through word-formation processes (e.g., derivation) can correspond to different word units, for example, a free morpheme (e.g., free morpheme + bound morpheme: sell + er) or a word after a word-formation process (e.g., word [<free morpheme + bound morpheme] + bound morpheme: national + ity). An interesting observation, we emphasise for later discussion, is that among the bound morphemes considered, the definite article is not particularly analysed as a grammatical affix having its place in morphology.

Another debated subject amid linguists adopting the Item and Process approach is the scope of morphology; in other words, whether both inflected and derived words are the product of morphology, or inflection is rather handled by the syntax than the morphology [42]. The last assumption is referred to as the Split-Morphology Hypothesis [43] or the Weak Lexicalist Hypothesis in the literature. This hypothesis can only be accounted for if, universally, derivation precedes inflection in word formation systematically, which is not supported by evidence [44]. However, we can observe that derivation and inflection are generally mapped separately; to put it another way, the same marker used for derivation and inflection is rarely documented in natural languages. Yet, in certain linguistic descriptions, we may find reported examples of gender-based derivational markers, especially feminine derivational affixes (e.g., in German7, with -ung: zahl-en ‘to pay’/die Zahl-ung ‘payment.F’, with -schaft: Bürger ‘citizen.M’/Bürger-schaft ‘citizenry.F’). However, this gender shift also occurs in zero derivation (e.g., in French, le voile ‘veil.M’/la voile ‘sail.F’, le mort ‘dead person.M’/la mort ‘death.F’). If this change can happen without an overt derivational marker, then this provides substantive arguments to claim that derivation and inflection are indeed split, but not in the way of Anderson’s suggestion.

Nonetheless, even by considering this observation, a usual Item and Process approach is not currently in position to utterly explain the gender shift occurring at the same time as the derivation. To solve this, Beard [32] proposes to fuse the classical distinction between derivation and inflection by attributing the same processing abilities to them. The latter consists of a variety of processes changing the binary semantic distinctive features of words (e.g., ±singular, ±plural, ±noun, etc.) without paying attention to the form (cf. Separation hypothesis). This scenario is problematic in two ways. On the one hand, by putting these features into one category (G-features are either inflectional or derivational), he obliterates the concept of paradigm, while the linguistic facts show us that similar grammatical features have a propensity to share the same syntactic position. Additionally, they do not alter the referential properties of an immediate item and agree with the head, as the words do. Such an inflectional property cannot be disregarded. On the other hand, he implies that the set of G-features is a maximally specified set that is attached to any item; it is a finite set with no subset besides itself because nothing is inserted, and everything is transformed. Thus, a semantic feature specific to the verb is also included with the noun but neutralised according to Beard (actually, he tacitly assumes instead a system of trinary features: plus, minus and neuter). Moreover, this leads to an increasing creation of combinations of stepped processes in order to meet all the G-feature shifts existing in all languages. Most importantly, this approach is unable to model morphological indeterminism8 occurring in some languages (see example (2) in Section 2.2.1). In addition, the major drawback of the Separation hypothesis is its inability to give a rational reason for the presence of expletive forms.

Furthermore, it should be noticed that Anderson’s thinking about these inflectional morphemes is binary. Because we are entitled to question this arbitrary cleavage, we could wonder if some of them could be distinguished regarding their morphological properties. Explicitly, the question is, should we analyse, for instance, deictic markers in the same way as gender markers in terms of relation to the item? Thus, in keeping with the reflections on the particularities of inflectional morphemes, some researchers noticed that words come with a range of defined grammatical features that are not disposable. Those given credit to this observation consider that this canonical body of the word is set up in the lexicon [17]. Depending on the theoretical approach, it may have many names: $\phi$-features [17], Feature bundle [31,45], Inherent inflection [44], Paradigmic pattern [46]. For some following the DM thesis, this internal canonical schema may be determined at the semantic level, which is why even those following a non-Lexicalist Hypothesis approach include in their theory this concept.

This observation yields two important reactions. If this latter frame of reference suggests that some classes of grammatical features are packaged semantic properties, it should also assume an unbundling process letting these linguistic facts attach to other items when they are generated. The previous remark is elicited by the fact that the morphological system is, likewise, a dynamical system. Distinct cognitive operations can make this linguistic sub-system evolve into a new state. For instance, one of the manifestations of this dynamism is the inter-process morphosyntactic variations. As seen previously, the items resulting from a process of word and meaning formation acquire grammatical forms different from those of the base items. Therefore, we cannot assume they are inherited from the base items. Such grammatical markers can only be assigned to the resultant items; this happens when the latter is created in the lexicon. However, the preceding phenomena are not captured by the non-lexicalist theories or others as well (see Section The Item and Template Approach). This implies a computational tool operating before the syntax component, which is not admitted by these approaches. The second contention that we would like to mention follows from the first. As the pairing is not generated, it should be supposed that there exist different entries representing identical grammatical features each being bundled into its respective item. To illustrate this, the singular or any other grammatical feature is encoded in the mind countless times. As they are duplicated, their number would be overabundant, since their occurrence is correlated with the number of items. Thus, none of these hypotheses is economic and they are cognitively expensive.

The Item and Template Approach

In contrast, some propose that these packaged grammatical features are encoded in a template. The combination of an Item and Process approach and a templatic view of the word in morphology9 is categorised as a Word and Paradigm approach [38]. Some of the previously quoted theories are based on these axioms; for example, Chomsky refers to this template as the Complex symbol [47] or Lexical array [17]. Other researchers have suggested a similar concept: Morphosyntactic Representation [30] or Morphosyntactic array [26]. Although they bear different names, their attributes are very similar. All of them state that the template, when not instantiated, is deprived of forms or morphemes, what Anderson [30] has called A-morphous. It only encodes abstract (binary) morphosyntactic features. Thus, each stem is paired with a specific template having its morphosyntactic characteristics. Therefore, there are, by default, several modular templates with dissymmetrical features that are not, in other words, the outcome of a process (for a modular Item and Template approach without dissymmetry, see Beard [32]). The assumptions formulated by these approaches, in guises more or less subtle than are commented here, pose some challenges. Some reservations could be made regarding these theories.

First of all, we are not rejecting this approach; on the contrary, the framework used in this paper is based on the same principles developed by the different Word and Paradigm approaches. However, we dissociate ourselves from them in regard to the morphosyntactic features encoded in the template. We claim that some grammatical markers expressing meanings related to definiteness and countability should be specified in the template of the nominal category. The latter semantic paradigm is not recognised as a universal grammatical paradigm. Depending on languages, this class can include, at least, the singulative, the partitive and/or the collective. Such morphemes are little studied in linguistic theory and not pinpointed as forming a whole linguistic category like gender, number and so on. One of the positions argued in this article is to acknowledge this grammatical category and its importance in morphological marking. Moreover, let us note that these markers of countability are often assimilated to other grammatical categories (e.g., case markers as in Riffian, see next section) or omitted by researchers because they are null markers. This is particularly true for the collective marker that, we assume, exists in several synthetic languages including English, French, Riffian, etc. For lack of space, we do not intend to review all these languages to corroborate these affirmations, but, later on, we will bring empirical evidence for Riffian to show the factual bases supporting our claim (see Section 3).

As for the definiteness paradigm, the recognition of this class is not under debate between linguists. Nevertheless, the theoretical analyses carried out on this category diverge, especially for the definite article on which we will focus in this article. While, generally, it is stated that this grammatical feature does not enter the derivation with the lexical item, Chomsky [47] suggests that this category is encoded in the template. By taking into account, according to Chomsky, Matthews’ critics showing that the linguistic facts predicted by the traditional transformational rules go beyond what natural languages can generate, he proposes the concept of contextual features to deal with ungrammatical utterances. To put it simply, this new conceptual tool serves merely to filter the items that can merge together in syntax10. Inside this template, two sub-templates are distinguished, namely the strict subcategorisation and selectional restrictions. The former is proposed for both nouns and verbs; thus, it is in charge of allowing whether the definite article will be affixed to the noun or not. Therefore, this puts the definite article in equal position with the complement arguments of verbs. This has to be noted because this preludes the syntactic structural similarities assumed between VP and NP that will be developed from there on by Chomsky [41] and other researchers [12,49]. In line with this new device, the theta-theory [18] is proposed as an extension of the contextual features. Likewise, these extra configurations, added to describe the grammatical roles, are encoded in the lexicon. However, all these new theoretical apparatuses raise some problems, especially the lack of mathematical formalism and integration between those new elements and with the model of syntactic derivation [50].

In contrast, Matthews [51] established an alternative model incorporating a mathematical formulation to remedy the issues that he helped to identify in Chomsky’s theory, but it was not elaborated any further and the absence of the definite article with some nouns is not considered. His model focused more on the questions of transitivity. Although, in a less abstract way, this remark may be also applied to Williams’ concept of argument structure11 [54] following on from Chomsky’s framework [18]. What is relevant to note here is its formal contribution to lexical derivations. Thus, the transition, for instance, from a verbal template encoding thematic roles to a nominal one is modelled by a mathematical function. We mention these works because they are the first attempts trying to mathematically formalise the Item and Template approach using Set theory, even though formal morphology was not their central concern. We defend a thesis that is similar in spirit; strictly speaking, there is a set of templates, and each lexeme is indexed to one of them through a function. Nevertheless, the elements mapped are grammatical features, those of which include definiteness and countability, that are entirely operated in the lexicon; they are not committed to being used post-lexically as filtering features when instantiated, as outlined before.

The promotion of mathematical-like formalisation in morphology is not novel in itself. The use of functions is, first, proposed by Aronoff [26] and he is then followed by other researchers [37,55]. However, their functions do not define a relation from a lexeme set to a template set. In fact, all these works consider, following Beard [32] and the concept of Separation, that these relations merely involve the form of an item and those of its affixes. Such a relationship is called the Realisation rule (Beard named this kind of process the M-rule) or morphophonological function, and this appears to be a recursively defined model. The latter can be compared to Chomsky’s model [50] dedicated to the syntax module. While this is not explicitly asserted by these authors, we can deduce from their proposals that, in their view, the morphology component includes a computational device as well (see also Selkirk [48], Kiparsky [56]), which is generally not acknowledged in the mainstream generative theory. Similarly, in the domain of Natural Language Processing, let us also mention works [57] aiming to model with finite-state transducers the pairing of grammatical markers and items. Conceptually following these researchers, we admit that the morphology component is also composed of a generative device—to put it another way, a computative lexicon—enabling realisation rules, but also allowing, among others, engendering templates.

The main grievance that we can formulate about these models is the incompleteness of their formalism. For example, the concept of the Realisation pair [26] merging a Morphosyntactic array and an item is not defined as such. This inconsistency is probably due to the inadequacy of their suggestions with regard to relational mathematics. Such a pairing formula, permitting the modelling of many-to-many relationships, is not allowed in Set theory. Hence, with their proposal, the relation of an item with members of a set of morphosyntactic templates is not represented by a function. As a consequence, these pairings are not generated and are stored as they are. If no mapping functions link them, a template is listed in the lexicon as many times as there are nominal or verbal lexemes to which it is paired. Nonetheless, we could conceive functions to map the items and the templates, but we would have to model a function for each template, which will occasion the creation of several of them. Either way, such a system is not appropriate in terms of scientific and cognitive parsimony. The other issue, still concerning these models, that we would like to address is the non-consideration of inter- or intra-lexical derivations (intra-lexical derivation refers to a word formation implicating the same syntactic categories (e.g., from N to N), whereas an inter-lexical derivation implies different ones (e.g., from V to N)). Word formation also has implications in morphology, not solely on thematic roles [54], and nothing formal is proposed to tackle these phenomena. One of our goals will be to remedy this.

The Lexical Approach and the Definite Article

Let us come back to the encoding of the definite article. As said earlier, Chomsky suggests that this morpheme is handled by syntax and that PNs carry an empty feature—symbolised as [ __]—instead of the determiner feature—its symbol is [Det__]—in the contextual feature, which would explain why they cannot merge with this determiner. However, as noticed by Chomsky, some PNs can be used with the definite article (e.g., the Nile). In this case, he supports the idea that, with such PNs, the definite article is part of the nouns and not selected by the strict subcategorisation. Thus, for the PN category, Chomsky proposes to combine a lexicalist and syntactic approach; the choice between both will depend on whether they have or do not have a determiner. However, this asymmetrical encoding is questionable; one may wonder why the human mind would decide arbitrarily to encode only PNs differently, while nothing with respect to their linguistic function distinguishes them. As an alternative, we intend to unify the encoding of those nouns; whether for the common noun or the proper noun, we consider that the definiteness and countability markers are inherent features of these nouns.

Such a lexicalist approach was already proposed for some Scandinavian languages, that is to say, the definite marker is not associated with the lexeme post-lexically, but within the lexicon [56,58]. Nevertheless, Hankamer and Mikkelsen’s approach is unorthodox. According to them, the presence of the definite article is due to a word-formation process. The definite article is analysed as a derivational marker changing the noun category into a Det category. This line of thought is disputable since the Det category and the N category are placed on the same footing. Thereby, a noun can become a determiner in the same way as a noun can become a verb. The determiner is no longer perceived as an element governing or governed by a noun; by deduction, their approach inevitably leads to reject both the DP-analysis and NP-analysis. In addition, this hypothesis implies that the functional and lexical categories are no longer separated cognitively. While their purpose is strictly morphological, the definite article is not viewed as being incorporated in a template, which is a conceptual descriptor essential in our theory.

The review has surveyed the recent literature proposing different approaches to analyse the morphemes under study. Also mentioned were the numerous theoretical considerations discussed in the field of morphology, with specific reference to definiteness. From now onwards, we plan to address the theoretical premises of the TBMC model.

2.2. Theoretical Proposal

In the previous section, we tried to predispose the reader to view favourably some of the main ideas of our model by contrasting it with other approaches. That said, our theory shares with them different points of similarity scattered in various generative frameworks. Let us briefly recall (for a more thorough discussion, see Section The Item and Template Approach) some elements:

—

Computative lexicon: the lexicon has a cognitive generative device comparable with the syntax component.

—

Modularity of systems: the templates form a system, which is itself part of the morphological system, and they are independent of each other (i.e., elements of a set).

—

The Item and Template approach: items are paired with templates that include grammatical meanings.

—

Mathematical mapping: the pairing of linguistic facts is carried out by mathematical relations as defined in Set theory.

—

Integration of word formation with the template: lexical derivation contributes to template shift by attributing a different template to an item.

Henceforth, we would like to pursue here the development of the particularities of our formal model. To this end, these listed points will be extended in scope to include new theoretical systems and a more rigorous formalism.

2.2.1. A Mentalistic Algorithm of Lexical Derivation

In conjunction with this encoded information in the lexical entries, a supplementary element is required to adequately grasp the non-uniform marking affecting nouns. We argue that the processes of word and meaning formation12 are the main phenomena disturbing morphological marking. As a matter of fact, several controversial observations in morphology, such as the disruptive precedence of derivation over inflection, can be ascribed to these processes. To illustrate this, let us look at this example taken from Riffian implicating, at the beginning of the word formation, the feminine toponym t₁-masin-t₁ ‘Tmasint.F’ that serves to form, after derivation the masculine or feminine demonym, u-t₁-masin-t₁ ‘Tmasint man.M’/t₂-u-t₁-masin-t₁-t₂ ‘Tmasint woman.F’. The main aspect to notice here is that, following the use of the derivative marker u-, the feminine marker t₁ has been preserved by being nested inside the word and has, most importantly, become meaningless. Additionally, to retrieve the feminine sense, another marker t₂ can be added alongside the derived radical, in this case, the feminine marking occurs twice.

Such a phenomenon can be named stacked morphology. The cognitive process behind this coined concept is derivation. This terminology is necessary to distinguish simple derivation, where only the root is involved (e.g., the verb talk gives the noun talk), from complex derivation or stacked morphology where more than one morpheme, aside from the derivative marker, undergo this operation. A combination of different markers (inflectional or/and derivational) and syntactic units (word/lexical entry, phrase, clause, sentence) may be involved in this process; for example, this can include lexicalised VPs13. As, later in the paper, conversion is the cause of morphological discrepancies that preoccupy us (see Table 1), then we will restrict our analysis to this subprocess; let us set aside, for the moment, the derivation process in its entirety.

Conversion is a traditional linguistic descriptor for which a formal theory has never been proposed in the literature. Traditionally, this process is defined as the operation permitting the formation of new lexemes on the basis of an existing item without the affixation of a new morpheme. As it stands, we agree with this definition, but we would like to bring a more precise understanding of this phenomenon. We aim to introduce a cognitive mechanism of this process, especially regarding its interaction with the morphological template. First of all, conversion may operate on different types of linguistic units and its goal is to perform an item cloning, which explains why the feminine marker $t_1$ is joined after the derivation. When conversion targets the lexical entries—not only the root, which is a word shelled from its template and the smallest unit choosable by the derivation—some linguistic facts are necessarily selected. We defend the idea of an item cloning floor beyond which conversion cannot go down. This minimal scope includes the root and some grammatical features. For the languages of interest to us here, this can encompass the definiteness, countability, number and gender paradigms. For instance, in English, the definiteness, number and countability markers are systematically cloned at the level of the lexical entry when subjected to this process (see Section 1 and Section 5). In Riffian, countability and number are the only required categories in the sway of conversion (see Section 3). Second of all, the newly formed item can undergo a template shift that is contingent on the original item (e.g., in French, deadjectival nouns invariably have the feminine gender: la beauté ‘beauty.F’ from the adjective beau ‘beautiful’).

In addition to this item cloning floor and template shift, the semiotic pairing of items must be taken into account. Thus, depending on items, the range and the kinds of meanings (i.e., lexical or grammatical) mapped to the form of an item may diverge. More concretely, some grammatical features can be associated with the items in a different way, instead of being solely encoded in the template set. These specific grammatical features may also construct a pair with the form of an item in combination with other meanings. This singularity, which can be named morpheme flow, appears when a linguistic unit is the object of a process of word and meaning formation. In particular, two of them are the driving force of this transfer, namely conversion and loanword. Said differently, in the case of a lexical item, the template of a base word is reassigned to the resultant word. Thereby, two templates compete for the spelled-out form, namely the covert meanings (i.e., the reassigned template) and the overt meanings (i.e., $h\left(w\right)$, see Section 2.2.2) (additional details will be provided in Section 4.1).

To embody the various algorithmic properties of lexical derivation outlined previously, we symbolise the cognitive process of conversion by the following function symbol: →. It must be viewed as an n-ary function symbol $\to (x_1,\dots ,x_n)$. While the mathematical definition of this function is not detailed in this article, this section explains its complex underlying cognitive process. To deal with additional word formations, other function symbols will be used to represent, respectively, borrowing, suppletion and morphological derivation14: ↰, ↛, ↬. The tail of the arrow represents the initial form and the head the final one.

Therefore, this dimension allows us to shed light on the inter-process morphosyntactic variations of the state (see Section 2.3) met in Riffian when a countable noun derives from an uncountable noun or a proper noun from a countable noun (see (1)). In (2), just to mention a case among others—more will be described and explained in Section 3—the derivation leads to different morphological realisations, whereas the morphological input is supposedly identical according to the traditional view (see Section 2.3). In (2(b)), after the conversion, we should have expected to see the state affixed, but this did not happen. Such a pattern may be denoted as an overt anti-morphosyntactic shift (in (1(b)), the proper noun keeps the initial vowel and in (2(b)), the countable noun, instead, preserves the form without the initial vowel. It may seem like two different behaviours, but our thesis is that they are governed by the same underlying phenomenon. Other situations of overt anti-morphosyntactic shifts are discussed in Section 4.1).

(2)

(a)   With the vowel: ða-ɾeʃːint ‘an orange’ from ɾeʃːin ‘oranges’;

(b)   Without the vowel: t-bujːut ‘a bun’ from bujːu ‘bun’.

However, this apparent morphological indeterminism15 that we are raising here is resolved thanks to the algorithm of lexical derivation, but also the semiotic encoding and the mathematical model presented in the next sections.

2.2.2. Mathematical Foundations of the Cognitive Model

As said earlier, we do not subscribe to several aspects that were developed by some frameworks of the Item and Template approach, especially with respect to template encoding and the limits of mathematical formalism. For this reason, we wish to give a more formal formulation to our model. Let us recall that the latter is strictly confined in the lexicon and does not interfere with the syntax. Additionally, let us clarify that this model is not intended to manage the pairing of morphemes with other morphemes so as to establish the word structure.

To begin with, let us include some relevant concepts. We claimed that different templates, autonomous to each other, exist in the lexicon as postulated by the Word and Paradigm theory; however, we would like to specify that their relationship with the item can diverge among this linguistic unit. Our position makes use of the markedness theory to dissociate them. We consider that certain templates should be viewed as unmarked templates and, by association, that the grammatical features of these unmarked templates are also unmarked. Such an assumption is not held by other researchers who generally associate this concept merely with the grammatical features without extending it to the templates. On that account, any item is mapped to an unmarked template, but not all the templates are mandatorily paired to an item. The templates that could form a pair can be, at least, a singleton subset (i.e., with one grammatical feature that could be a null marker), but never an empty set. Thus, unpaired templates should be regarded as the marked templates. The latter notion should be understood as relating to an item. This signifies, in mathematical terms, that, for a given function, the unmarked templates coincide with the elements of an image set that is a proper subset of the co-domain set, namely the set of all the templates. The pre-images of these unmarked templates are the items corresponding to the elements of the domain set. Therefore, we define the notion of templatic markedness in relation to the image set of a function. Additionally, as explained before, when an item is chosen to form a new word, a template shift can emanate from this process. Regarding an item, one of the marked templates picked will be, in turn, the unmarked template. Due to this cyclic process, a template may be unmarked and marked at the same time.

Furthermore, as said previously, a linguistic system can have several templates and each one of them can be composed of different binary grammatical features, but the items related to a particular categorial feature are not mapped arbitrarily to these templates. In order for a relation to be valid, it must satisfy two conditions. That is, a particular syntactic category must be mapped to subsets of grammatical features, first, having an identical inventory of grammatical features, which, second, are constituents of the same sets of grammatical features. Therefore, any function mapping the items and the templates together is constrained by these conditions. It ensues from this that the templates are closely intertwined with the syntactic categories.

The mathematical formulation of these proposals is introduced in Definition 1. We will confine ourself to the formal description of the template and the item–template pairing.

Definition 1. 

Let us consider the following definition of the TBMC model of lexicon:

• Let L be the set of the languages;
• Let W be the set of the morphemes or items with $\varnothing \subset W$;
• Let M be the set of the meanings;
• Let G be the set of grammatical meanings with $G\subset M$;
• Let ${}^{\neg }G$ indicate the set {¬g | g $\in G\}$, that is, ∀ ¬g, a negative element of ${}^{\neg }G$ is defined as an element assigned to the empty set of W;
• Let Γ be a set composed of binary grammatical features such that $\mathrm{\Gamma }=G\cup$${}^{\neg }G$. We can write $\mathrm{\Gamma }=\{$+g₁, −g₁, +g₂, −g₂, …, +g_n, −g_n with +g_j$\in G$, −g_j$\in {}^{\neg }G$, and ∀$j\in \mathbb{N}$.

Definition 2. 

Let us consider the following definition of the template:

• Let S be a set of syntactic categories;
• Let P be a set of grammatical paradigms generated by the power–set function $\mathcal{P}\left(\mathrm{\Gamma }\right)$ and $P\subset \mathcal{P}\left(\mathrm{\Gamma }\right)$. Let Φ be a family of subsets of a set P;
• Let $e:\mathrm{\Phi }\to S$ and $\exists {\varphi }_{s\in S}\in \left\{\varphi \right|\varphi \in \mathrm{\Phi },s\in S,e\left(\varphi \right)=s\}$=$e^{-1}\left(s\right)$$\subset \mathrm{\Phi }$. The function e holds if only if it satisfies the conditions as follows:

  (a)

  If $\forall {\varphi }_{s\in S}\in \mathrm{\Phi }$, its cardinality is equal to a defined number: $\#\varphi =k_{s\in S}$, such that ${\left(k_s\right)}_{s\in S}$ is a set consisting of a sequence of integer numbers and, depending on the set S, a function maps S into K.

  (b)

  If ∃ subsets ${\varphi }_1$, ${\varphi }_2\in e^{-1}\left(s\right)$$\subset \mathrm{\Phi }$, such that ${\varphi }_1\ne {\varphi }_2$ and $e\left({\varphi }_1\right)=e\left({\varphi }_2\right)$, then their properties include that the absolute value of ${\varphi }_1$, ${\varphi }_2$ are improper subsets of each other: $|{\varphi }_1|=|{\varphi }_2|$, and their symmetric difference is an empty set: $|{\varphi }_1|∆|{\varphi }_2|=\varnothing$.

• Hence, let T be a family of subsets corresponding to the templates such that: $T=\{\bigcup (\varphi ,e\left(\varphi \right)\left)\right|\exists l\in L,\forall \varphi \in \mathrm{\Phi }\}$.

Definition 3. 

Let us consider the following definition of the item–template pairing:

• Let V be the set of processes of word and meaning formation;
• Let K be an unordered set defined as the set of the retrospective determinants of shifting template such that $K\subseteq V\times T\times M\times W=$$\left\{\right\{v,t,m,w\left\}\right|$$v\in V,t\in T,m\in M,w\in W\}$ and the empty subset ∅ is also a subset of K;
• Let $f:W\to K$ be the backward recursive function that maps an item to a subset of K, that is, each subset includes the antecedent shifting factors of an item, and let $g:K\to T$ be the gradient function that maps an element of K to the output template. Hence, let $h:W\to T$ be the morphosyntactic transfer function defined by the composition function $g\circ f$ that maps an item to an output template, such that $\exists l\in L,\exists w\in W$:
                            $h\left(w\right)=g\left(f\right(w\left)\right)$16

The resulting model is general for all kinds of items stored in the lexicon (i.e., verbs, nouns, pronouns, etc.) and all synthetic languages. The supplementary advantage of this theory is its minimalist design providing a great gain in parsimony without loss in generality, since only a few functions are required to establish the entire mapping from the grammatical features to the items. In that way, a lexical entry is encoded in the following manner: $(w,h(w\left)\right):w\in W$. We intend, later, to demonstrate the applicability of this parsimonious model, in a less abstract fashion, to the readers. To illustrate this, we will restrict ourselves to the nominal category and, more particularly, to the proper nouns and uncountable nouns. Moreover, as the properties of these pairings between the sets and the elements of these respective sets depend on each language, we cannot go into the details of all the linguistic varieties. This will be an impractical task to accomplish without immeasurably extending this article. Therefore, the Riffian language will be the main sample of analysis.

In conjunction with these functions, all the pairs of elements from those universal sets satisfy a gradient condition or gradient conditions that are specific to the language under consideration. Nonetheless, some of them are manifested in numerous languages. To underline some salient morphosyntactic shifts found cross-linguistically, the universal canonical form of those gradient conditions (i.e., the simplest form that can be materialised in a particular language) is given in Section 3 and Section 4 (see Gradient condition (1) and (2)). Such a parametric conditional factor, for the item–template pairing, is related to the set K, and, more specifically, to the subset of meanings M, which of elements are likewise subsets of the elements of the set of items W and templates T; in other words, for $w\in W,\exists {\left\{m_i\right\}}_{i\in \mathbb{N}}\subset M,{\left\{m_i\right\}}_{i\in \mathbb{N}}\subset w$, or for $t\in T,\exists {\left\{m_j\right\}}_{j\in \mathbb{N}}\subset M,{\left\{m_j\right\}}_{j\in \mathbb{N}}\subset \left|t\right|$. In regard to this matter, later, we will add to this generic model other conditions that are relevant for the language sample used in this study17 in order to ensure that the derived item falls in the correct template. However, this may not be a universal feature if the collective form is never employed as an unmarked form like in Greek or if all the nouns are countable in a particular language. Likewise, other semantic features may be found to be used in other languages.).

Furthermore, the item–template pairing also depends on a specific meaning encoded in K, such as for $k\in K,\exists m\in k$. The role of this meaning is to assign the items into cognitive sets when they undergo a category change. The following Proposition 1 expresses, in a formal way, the notion of cognitive set and is written as follows:

Proposition 1. 

In this context, let us define a cognitive set as the inverse image of f:

• Let ${\left\{K_m\right\}}_{m\in M}$ be an exhaustive disjoint collection of subsets, such as $\exists \mu ,\nu \in M$, and $K_{\mu },K_{\nu }\subset K$, then, $K_{\mu }\cap K_{\nu }=\varnothing ,\mu \ne \nu$, and ${\bigcup }_{m\in M}{K}_m=K$;
• Let $K_m$ be a subset: $\left\{k\right|k\in K,\exists m\in M,m\in k$};
• Hence, let the cognitive set be $f^{-1}\left[K_m\right]=\left\{w\right|w\in W,\exists f\left(w\right)\in K_m,K_m\subset K\}$.

As a supplement to Definition 1, for the purpose of expository analysis, let $\mathscr{H},ℱ,𝒢$ be families of partial functions with disjoint domains; then, $\mathscr{H}=ℱ◦𝒢=\{f_i◦g_i|\forall f_i\in ℱ,\forall g_i\in 𝒢\}$ such that, for all $h_i\in \mathscr{H},h_i:W\to T,{\bigcup }_{i=1}^{n}dom{h}_i\subset domh$, and ${\bigcup }_{i=1}^{n}ran{h}_i\subset ranh$. Each of these functions may be defined in linguistic terms as a prototypical process chain of template shifts—these partial functions can be viewed as a manifestation of self-regularisation [59]. Each chain starts from an input head assuming the starting conditions and it illustrates successive directional derivatives along a given grammatical element. These mappings are separated into two parts. Figure 1 focuses on the proper noun category and Figure 2 on the uncountable noun one. We listed these partial functions that are denoted with Greek letters. Examples of pairing are proposed for each of them. If we take the partial function $\gamma$, written also as $h_{\gamma }$, four pairings are given in illustration. For instance, the word-form t-samːeɾθ is an unmarked pair formed by $\gamma ($samːeɾ₁$)=$$\{N,+SG,-PL,$$-M,+F,+COL,$$-SING\}$. Let us provide another example, emphasising a template shift issuing in samːeɾ, which derives from t-samːeɾθ; here is the mathematical representation of this word-form: $f_{\gamma }($samːeɾ₂$)=$$\{CONV,$$\{N,$$+SG,-PL,-M,$$+F,+COL,-SING\}$, C, samːeɾ₁$\}$, then, from this subset, we obtain the template of samːeɾ: $\gamma ($samːeɾ₂$)=$$g_{\gamma }\left(f_{\gamma }\right($samːeɾ₂$\left)\right)$$=\{N,+SG,$$-PL,+M$, $-F,$$+COL,$$-SING\}$.

For each pair presented in Figure 1 and Figure 2, we decided not to unroll the complete template, but only the spelled-out form. The template of an item may be reconstructed thanks to

Table 2. Number and gender inflection in nouns.

aqzin	dog.SG.M
ð-aqzin-t	dog.SG.F
iqzin-en	dog.PL.M
ð-iqzin-in	dog.PL.F

and Table 8. Similarly, it is assumed for the European languages cited in the article, that the template of each item is a matter of common knowledge for the linguists adopting an Item and Template approach; hence, they will not be reproduced systematically. We depart from the traditional template by adding the countability and definiteness features. Therefore, for instance, a template for the English nouns should look like the following: $h\left(john\right)=\{N,+SG,-PL,+COL,-DEF\}$ or $h\left(book\right)=\{N,+SG,-PL,-COL,+DEF\}$.

We will not pursue the inquiry further. We have left several topics unaddressed, but we will not explore them, as they do not directly relate to our analysis. These other aspects of morphology will not concern us here. We may assume any of the different views on these matters; the choice will not matter for the issue treated here. Our purpose is to present some concepts in a formal form as a theoretical basis and a prelude to the more detailed description presented later.

2.3. Riffian Nominal Morphology: Traditional Theoretical Background

Proper and common nouns are usually considered a part of the nominal category. This lexical category is believed to have other subclasses. Additionally, adjectives and numerals are also part and parcel of this category. This aspect should be noted because CNs, adjectives and numerals can also become PNs. We will come back to this point later on. Now, it is important to give an understanding of Riffian morphology. Riffian is a synthetic language and tends to be a fusional language. That means that Riffian resorts to lexical derivation and inflection to form new words or word-forms. The general canonical form of a noun, generated in accordance with the formal cognitive model (see Definition 2 in Section 2.2.2) and symbolised by ⨂, is composed of the following elements:

The masculine and singular form is generally the unmarked form, as one can see below in Table 2 (all examples have been transcribed with the IPA notation). The same morpheme may be used several times, especially for the singular feminine marker positioned at both ends of the radical. In terms of terminology, the radical is different from the root. It must be understood as opposed to the inflectional morphemes. A radical can be composed of a root morpheme, a scheme morpheme and derivational morphemes. As to the state, it is not a universal linguistic category used in the description of other languages. Many discussions have been held about the nature of these morphemes [60]. They are named either by case or state. We do not use the former denomination, because we think that this grammatical class has nothing to do with case markers [61], as we will demonstrate later (see (3).).

In Berber linguistics, this term is used to describe a grammatical class, or many according to the analysis considered. It can be viewed either as one morpheme or two [62] that is affixed to a nominal base—nouns and adjectives only—and some grammatical morphemes. During this study, we will consider the hypothesis of two forms independent from each other, but acting in tandem. Depending on different syntactic constructions ([4], pp. 113–115), the nominal paradigm can take mainly two states: free state (FS) and annexation state (AS)—the terminology used by some authors can be annexed, annexation or bound. The state corresponds to the initial vowel (V) and the consonant preceding the radical. This grammatical paradigm also varies/agrees in number and gender (see

Table 3. State inflection in nouns.

Free State	Annexation State
a-qzin	u-qzin ⇐ /we-qzin/
ða-qzint	ðe-qzint
i-qzinen	je-qzinen ⇐ /we-qzinen/
ði-qzinin	ðe-qzinin

). Moreover, in the annexation state, this vowel is elided and, only in the masculine gender, a semi-vowel appears. A central vowel may take the place of the V-initial, but it does not contrast with it; the presence of this epenthesis is due to syllabification reasons. In brief, here are the main characteristics of the state. As we intend to speak of the state later, we would, as of now, like to introduce new concepts. In order to discuss the particularities of these morphemes, we will have to distinguish them and use specific terminology, that is to say, C-state for the consonants (w/ð) and V-state or V-initial for the initial vowels (a/i). When the V-state is in use, the C-state is mandatory; the alternative case is not true; the V-state is expendable whether it be in FS or AS. This vocabulary is provisory, it will help us to ascertain the essence of these linguistic facts when we will tackle this subject (see Section 4.5).

Thus, we believe that the state is a marker that encodes features that have nothing, or not only, to do with syntactic alignment/grammatical function marking as often suggested. In some cases, the annexation state does not mark a relationship of dependence with an overt verb as in (3). In the following sentence, at the surface level, there is neither verb nor copula and both nouns are in the annexation state:

(3)

mana u-miʁis-a u-ħenʒiɾ            ([63], p. 283)

What AS-clever-this AS-teenager

‘What a clever teenager!’

Alternatively, some researchers prefer to speak of construct state. It would seem erroneous to use this terminology as some researchers do [60,64,65]. According to Creissels [66], the annexation state is not comparable to what we observe in Semitic. In this family language, the construct state marks the head; obviously, this does not conform to what we find in Berber where it is the dependent that is marked. We can argue that no matter whether it is a head-marking or dependent-marking, we cannot disqualify a cross-linguistic comparison based on this kind of marking; the genitive is a perfect example [67]. In Semitic, two particularities appear on the NP head; if the word is feminine, it will contain a final dental consonant and, for both genders, no definiteness marker can be used [13]. The first phenomenon, often neglected in morphology studies, can be observed in Berber. It is crucial to clearly discern what is in the purview of morphology or phonology.

Hence, a part of the construct state may be analysed as a phonological phenomenon operating at word or morpheme boundaries. Thus, if the conditions are satisfied, a phoneme may take form in this environment. In such a case, it is possible to say that what happens in Berber and Semitic, and, of course, other languages knowing such a morphophonological process, might be identical at some point. It has been observed that the semi-vowel in AS shares the same syntactic position as the feminine marker. Therefore, it was proposed [68] to regard this consonant as the masculine marker that is only overtly marked at word boundaries (AS: w-∅-radical), otherwise it is a zero morpheme (FS: ∅-a-radical). In fact, the semi-vowel might have played a functional role in an ancient stage of the Berber language (FS: *w-a-radical/AS: *w-∅-radical); today, we should admit that it is just a relic of the past. However, we shall point out that this consonantal epenthesis is not always inserted. In some cases, in heteromorphemic sequences, the AS will not appear:

(4)

a-ħaɾmuʃ-a ð w-{}-ɾgaz-a

boy-this and w-{}-man-this

‘This boy and this man’

(5)

a-ħaɾmuʃ-a ð-{}-a-ɾgaz

boy-this ð-{}-a-man

‘This boy is a man’

In these two sentences, at the underlying phonological level, they are supposedly identical: /ð/ + /aɾgaz/. However, the noun aɾgaz, at the surface level, has different forms according to the grammatical words that precede it. In (4) and (5), they are two distinct morphemes with the same phonological properties. In (4), ð is a coordinator that triggers the annexation state (the empty set {} symbolises the disappearance of the V-state). Then, in (5), it is a marker that does not induce the state change (the non-presence of the C-state is noted by the following character: {}). This simple example demonstrates the complexity of the annexation state. The phonological phenomenon illustrated before is only activated when the V-initial is lenified. Therefore, this example shows that the phenomenon in question is not a purely morphophonological issue; it also has to do with morphosyntax.

It seems that the previous explanation could properly fit into the description of the construct state. However, on the other hand, the V-initial does not bear any definiteness values like in Semitics; therefore, the construct state is not comparable with the Berber data. Moreover, AS is not only triggered in genitival construction. As things stand now, by considering these previous remarks, we would rather not assimilate the Semitic and Berber situation. We can even conjecture that, in Berber studies, the use of this terminology is more of a terminological borrowing without implications for scientific inquiry. Other researchers, in contrast, are more explicit in their claims. For example, Longobardi [69] has made the same comparison for Romance and Germanic languages; but, according to him, this only operates on some nouns being the head of such a construction, especially the words casa/home. He suggests that this kind of noun should be understood as having some sort of genitive/possessive meaning formulated in different ways. Thus, their vision of the construct state has limited productivity and their analysis differs from the traditional view.

Upon examination of these works, it seems that there is a lack of consensus on this specific conceptual descriptor and, outside of Berber studies, it is associated with syntactic constructions distinct from those in Riffian. The concept of the state should be revisited and expanded to include new evidence from other languages. As discussed, the state in Riffian is an intricate process at the interface of morphophonology and morphosyntax. Then, it is preferable not to conclude, for now, with a final and general answer about the linguistic nature of the state. Later on, the assumption made in this study is that the phonological process triggering the appearance of the semi-vowel depends on a null morpheme named collective marker (see Section 4.5) prompted in particular syntactic contexts (like in (4)). Its (non-)presence is also determined by the noun categories in which the collective marker appears (see Figure 1 and Figure 2, Section 4.4). Hence, the mechanisms of these phenomena are not considered by the computational framework of the TBMC model (see Section 2.2.2) and are managed by other computational devices specific to phonology.

That being said, we will not further analyse in detail this phenomenon altogether; it is outside of the scope of the present paper. However, it was necessary to give an insight into this topic to the readers who are not familiar with Berber studies. The morphology that we just described should be carefully delimited and not be extended for all the nouns. Because, as stated by Penchoen ([70], p. 12), for the Berber of Ayt Ndhir (The Middle Atlas/Central Morocco), “Nouns in Berber differ morphologically according to whether they are Berber or berberised common nouns, unberberised loan nouns, proper nouns or kinship nouns”. This statement is also true for Riffian. Due to language contacts, mostly Arabisation18 and Romanisation (French and Spanish) and some phonetic changes, some nouns do not shape the ancestral Berber morphology anymore. Furthermore, the morphology of these nouns has also been disrupted through the cognitive operations of processes of word and meaning formation. We will see, in the next sections, how this evolution has an implication regarding the difference among nouns.

3. Data and Analysis of Empirical Observations

The current section aims to re-examine the assumptions adopted by different researchers about the state by analysing the relationship between the PN/CN and the state. We propose an exhaustive analysis of this topic. The numerous examples displayed may be, for some, regarded as overloaded details, but we justify this extensive study, first, because of the richness of morphological realisations in Riffian and, second, in order to avoid confirmation bias by retaining only data that comfort our assumptions. For this purpose, we gathered a sample of data emanating from our knowledge, as a speaker of this language—Ayt Waryaghel variety—and the Serhoual’s dictionary [63]—Ayt Said variety mainly. From this book, we collected more than 200 PNs of all sorts (anthroponym, toponym, etc.) and CNs. By taking different regional sources into account, we ensured a dialectal heterogeneity. However, as Riffian is not a standardised language, some of our data may be divergent from usages in other Riffian varieties. However, we do think that this sample is quite representative.

3.1. Preamble

As noted earlier, the state is assigned to nouns. However, most of the anthroponyms do not bear the state. From this, we could generalise and assert that PNs are stateless. One may presume that this generalisation is overly precipitated. At first glance, many pieces of evidence go against this affirmation. On the one hand, this pattern might be explained by the provenance of these PNs. It should be noted that those personal names, notably, are Arabic in origin. On the other hand, a few Berber personal names also ignore the state (e.g., miʁis, muðɾus, medːuɾ, ħemːu, hemːut, etc.), while, with others, its presence is required (e.g., a-mzˤjan, iðiɾ, a-mʕiʃ, etc.). It is worth mentioning that some CNs and adjectives also completely leave out the state or the V-state (e.g., fað ‘thirst’, seksu ‘couscous’, mizˤið ‘sweet’, ðe-mzˤi ‘youth’, etc.). From these observations, one would conclude that the state is not a category proper to the CN and appears arbitrarily on some nouns and not others. This assertion would be correct if the distribution was homogenous and not biased. Despite this appearance, it is statistically obvious that there is a clear distinction between the PNs and the CNs concerning the state. Mentioning that most of them are foreign names as being the cause of that deficiency is not enough to justify this disparity.

First, we would like to expose why some Berber PNs have the state and others do not have it. Riffian names, in most cases, are onymised CNs or adjectives. For reasons not elucidated so far—we will come back on this point infra—a few lexemes belonging to these paradigms do not accept the same morphology as the majority. For those nouns, the place of the state is left empty. It ensues that when they are onymised without morphological derivation, the morphology is preserved as it is. Thus, if the lexeme had the state, the PN would also have it, and if it did not, the PN would do the same. For those PNs, the state usage is contingent on the starting morphology. The onymisation does not intervene in this situation with a morphological derivation. However, that does not mean that PNs are bound to the state as any other nominal lexemes. The following question arises: why do the CNs have different morphological forms? This will be outlined in the next sections by empirically introducing the countability paradigm.

3.2. Countability and Proper Nouns

We claimed that Arabic loan names (we refer to direct loanwords from Arabic names such as jusef, ʕali, fatˤima, etc. That does not include Berber names with Arabic etymology such as ðamimunt that is an onymised berberised adjective (↰ in Arabic: maymun ‘blessed’).) never get the state. The most striking fact, in regard to this matter, is the constant refusal to Riffanise those PNs19. This constancy is not observed with other nouns. Although not all Arabic loanwords adopt Berber morphology, a large number of them, when they enter the Riffian language system, will be integrated into varying degrees. If they are entirely assimilated, the state is affixed. The data show us that borrowed words can inherit Berber declension forms, therefore, why PNs are not able to fit in. In fact, they can, but only if they go through a deonymisation process—by becoming ethnonyms for instance—then, they will get inflectional affixes. This signifies that the state is related to linguistic features that are absent in the PNs, but present in the CNs. This factor demonstrates that the state is indeed an alien category for PNs. Some PNs resulting from a word-formation process keep the state, because of a particular morphological formation phenomenon, which is called stacked morphology (see Section 2.2.1). The underlying process causing the onymisation is, therefore, the conversion that imposes, after any category change, conserving the initial morphology. This phenomenon will be illustrated many times in the sections to follow.

What we observed with PNs was also identified in another noun category. Uncountable nouns, borrowed from Arabic (e.g., tːefːaħ ‘apples’, ɾeʃːin ‘oranges’, sːˤaħafa ‘journalism’, etc.) or other languages (e.g., from French ɾpuɾis ‘police’) or, originally, in Riffian words (e.g., seksu ‘couscous’), also have a different morphology compared to their counterpart, the countable nouns. Thus, there is no state inflection with those uncountable nouns. However, some can become countable nouns by conversion, if it occurs in Riffian, or by morphological derivation (e.g., using the suffix-ier in French, or-i in Arabic), if it occurs in the donor language before entering the recipient language. In this case, the state has to be affixed20 to the noun (see

Table 4. Countabilisation of borrowed and native uncountable nouns.

		Uncountable Nouns		Countable Nouns
Arabic	↱	tːefːaħ ‘apples’	→	ða-tːefːaħθ ‘an apple’
		ɾeʃːin ‘oranges’		ða-ɾeʃːint ‘an orange’
		sːˤaħafa ‘journalism’	↱	a-sˤaħafi ‘a journalist’
French	↱	ɾpuɾis ‘police’		a-puɾisi ‘a policeman’
Riffian		seksu ‘couscous’	→	ða-seksuθ ‘a couscoussier’

). As observed, when countabilisation—the passage from uncountable noun to countable noun—is trigged by the Riffian linguistic system, the Arabic definite article is preserved (ɾeʃːin/ða-ɾeʃːint, ɾ corresponds to the Arabic article al- that adapted to the Riffian phonological/phonetic system). In the inverse situation, it means that the uncountable and countable nouns have been borrowed separately and the latter have been Riffanised (e.g., ɾmeʃmaʃ (↰ in Arabic: l-meʃmaʃ) ‘apricots’/ða-meʃmaʃt (↰ in Arabic: meʃmaʃ-a) ‘an apricot’, see Table 5).

As a consequence, we cannot conjecture that the reason for the absence of the state with these borrowed uncountable nouns is due to this article. A word like ɾmeʃmaʃ should not be analysed as being composed of two full morphemes (ɾ + meʃmaʃ). The state is not there, simply, because, in agreement with our theory, it has no reason to be affixed, since the words are uncountable. It should be then emphasised that there is no phonological motivation behind this empty marking. Furthermore, it should be noticed that even the Arabic feminine marker, included in PNs, is borrowed in its current form while, normally, it can be replaced by the native one (e.g., ɾχeðm-eθ (↰al xadm-a) ‘work.F’). An Arabic feminine name, like fatˤim-a, does not take the Berber feminine marker -θ/t (*fatˤim-eθ), even if its phonology is adapted to the Riffian system (e.g., faðˤm-a/*faðˤm-eθ, the Berberised form of fatˤim-a). However, native feminine names can use the Berber marker (e.g., muna-t, hemːu-t, mamːa-t). Such an observation is not generalisable with uncountable nouns, some may have the ending -θ/t (e.g., ɾeksiβ-eθ ‘cattle’, ɾχeðm-eθ, etc.). Nevertheless, borrowed PNs may exhibit Berber hypocoristic morphemes—besides other morphological processes or sound changes for instance reduplication, apheresis, apocope, etc.—to form hypocorisms (e.g., ħmid-uʃ (↰ probably from ħamid in Arabic)).

Application of the Formal Model to PN in Relation to CN

As described, borrowed PNs do not take the Riffian nominal morphology in any way; most importantly, they ignore the state like other uncountable nouns. However, PNs, like uncountable nouns, can become countable nominals. When that happens, the state will be prefixed to the radical. That occurs especially when a demonym21 (hereinafter D), which is countable, derives from a PN, either a toponym or a personal name. In Figure 1, we have established a model22 of formation of PNs in relation to Ds that allows us to identify the template shift23 occurring during the derivation and implicating either the countability or/and number—the arrows show the direction of category change. Demonyms can be used as an adjective or a noun. Riffian has two demonymic affixes, the prefix u- (the process χ), from Berber origin, and the suffix -i (the process ψ), borrowed from Arabic—designated in Arabic as nisba. They are used concurrently, but, currently, we observe that the Berber morphology still has the speakers’ preference when used with Riffian toponyms. Let us highlight that Ds are not necessarily constructed from these affixes. Quantitatively, most are direct Berberised loanwords from other languages (the process ω); there is no derivation between the toponym and the D, both are borrowed separately24.

Finally, some Ds derive from personal names (the process τ). Two subtypes can be met that are identical to the processes ω and χ in terms of derivational markers. Furthermore, in contrast to other toponyms, the toponyms of this process will obtain the plural form of D. Let us draw special attention to this process. We would like to focus on the mathematical facet of the partial function $h_{\tau }$ to stress that the template shift is not random in this case and is indeed deterministic. Thus, in addition to fulfilling the geographical and animate meaning requirements, the gradient conditions allowing these items to be paired to a template are as follows:

$$\begin{array}{cc}\hfill \mathrm{Gradient}\mathrm{condition}1:& \exists w\in W,\mathrm{if}\exists t,\mathrm{such}\mathrm{as}t\subset k=f\left(w\right)\in K,N\in t,\mathrm{and}\mathrm{if}\exists t,\mathrm{such}\mathrm{as}\hfill \\ \hfill & t=h\left(w\right)\in T,N\in t,\mathrm{then},\mathrm{their}\mathrm{symmetric}\mathrm{difference}\mathrm{must}\mathrm{be}\hfill \\ \hfill & t∆t=\{-PL,+PL\}\hfill \end{array}$$

(1)

Therefore, a new PN (e.g., a toponym) can be created from another PN (e.g., a personal name) after it has been deonymised by becoming a countable nominal (i.e., a demonym). The deonymisation and onymisation processes follow the following derivation chains: PN ↬ D → PN or PN → D → PN. Moreover, the directional derivatives shown in Figure 1 are not the only ones possible. Given this, it may be opportune to give a few more details. Thus, it is also worth noting that when the onymisation process may start directly from a PN, either PN → PN or PN ↬ PN, like with hypocorisms, the state is never prefixed. Likewise, a D from ω, ψ (in singular) and χ (in plural) can become another type of PN (e.g., a family name). In such a scheme, the state will be preserved; this is caused by the conversion as already advanced previously. Hence, except for the Ds formed with the latter prefix u-—a segment of two consecutive vowels is impossible in a word-form—all other types of Ds are necessarily accompanied by the state. To date, we have put the focus on PNs, but a CN can also become a D; in this case, it is another subtype of process τ. If it is a non-Riffanised CN (see Table 5) or an uncountable noun without the V-state (see Table 5), then it will obtain the state when it becomes a demonym (e.g., D: a-qeɾʕ-i ‘an Aqerai man’ ← CN: lqeɾʕa (↰ in Arabic: l-qelʕa) ‘citadel’; D: a-qubiʕ ‘a Qubia man’ ← CN: qubiʕ ‘eurasian hoopoe’).

Figure 1. Formation model of PNs in relation to Ds in Riffian.

Figure 1. Formation model of PNs in relation to Ds in Riffian.

From these different functional oppositions that we put forward, it seems, then, that the state shall be used with countable nominals (adjectives and nouns) and is unnecessary when the nouns are uncountable—here, PNs are regarded as such. However, not all uncountable nouns, including PNs as already specified in the preceding paragraphs, are, in this respect, concerned by this lack of state inflection. The word a-ʃefːaj ‘milk’, for instance, does have the state although it is a mass noun. However, compared to the uncountable nouns cited before, a-ʃefːaj has a plural i-ʃefːajen meaning ‘different kinds of milk’, which is not the case for seksu. Additionally, some countable nouns are used without the V-state that reappears, notwithstanding, in the plural (e.g., fuð ‘knee’/i-fadːen ‘knees’). Thus, the suffixation of the state is not carried out in a linear way. Again, it appears that arbitrariness is which prevails, but we will demonstrate that it is not the case. For PNs, we proposed that the conversion was responsible for the presence of the state with some PNs. We think that the CN follows an identical situation. To ensure that this hypothesis and our analysis concerning the grammatical function of the state are accurate, we must accomplish a more global study on this grammatical morpheme.

3.3. Countability and Common Nouns

Uncountable nouns, like the PN category, are not a category that has attracted particular attention in Berber linguistics. Therefore, most of the topics will be covered here for the first time. Up to this point, it has been shown that the state is related to the countable noun category. The next sections will be dedicated to the study of the relationship between CNs and countability. In addition, this will be also the opportunity to exemplify further the algorithm of lexical derivation (see Section 2.2.1) to show its relevance with other examples implicating other grammatical features than the state, such as the number and the gender.

3.3.1. Countability and Loanwords

The state is more contrasted when we are dealing with loanwords. That allows us to understand what may be the function of these morphemes. In Table 5, we present the morphology of nouns in relation to their degree of Riffanisation and their countability. The cross-tabulation of these two variables (i.e., countability and Riffanisation) delineates in what conditions the state is licensed or omitted. It is highlighted that the state is intimately related to the countability meaning and is subject to word formation, in which case, it also partakes in category change. To a greater extent, we provide evidence that word formation takes specific grammatical elements into account (i.e., by retaining the uncountable sense and not the countable one) so as to construct a new item in the lexicon. Before referring to Table 5, we would like to introduce a set of data extending and reinforcing the results displayed in this table.

Thus, non-Riffanised nouns, as expected, will never integrate any Berber inflectional morphemes, including the state. For non-Riffanised countable nouns, since they are countable and can be declined for number, the plural form—along with the gender marker as well—can also be borrowed. Hence, the Riffian plural is completely disregarded (e.g., lizikuteʁ (↰ in French: les écouteurs) ‘earphones’)25. As for Riffanised nouns, the uncountable nouns are the only ones that do not affix the state or the number, but they can suffix the feminine marker. Borrowed pluralia tantum are never Riffanised (e.g., ɾeħwajeʒ (↰ in Arabic: l-ħwajeʒ) ‘belongings’), while borrowed masculine singularia tantum are difficult to differentiate; they can be Riffanised or non-Riffanised uncountable nouns (e.g., ɾmaɾq (↰ in Arabic: l-maɾq) ‘broth’). Masculine is not overtly marked in Riffian like in the languages with which it is in contact. Thus, Riffanised uncountable nouns are always singularia tantum. The last category, Riffanised countable nouns, may assume any Berber nominal marker, especially the state with or without the V-state. That being said, we shall distinguish the uncountable nouns constructed internally with a foreign etymology from uncountable nouns borrowed, directly, as it is. Intra-uncountable nouns are formed from a countable noun (e.g., a-feʤːaħ (↰ in Arabic: felːaħ) ‘a farmer’ →ða-feʤːaħθ ‘farming’) or are deverbal nouns (faq (↰ in Arabic: faq) ‘wake up’ ↬ a-faqi‘waking’). Their morphology and phonology are different from their counterpart in the other language (e.g., in Arabic: lfilaħ-a/in Riffian: ða-feʤːaħθ—if it is was a direct loanword, its form would be *ɾfiɾaħeθ or *ɾfiɾaħa), but, more importantly, they have the state. As uncountable nouns, they are not supposed to have the state and this theoretical dilemma will be dealt with in the next section.

Let us note another interesting point; the Riffanisation of these nouns can be at different stages for the same word if this one includes both countable and uncountable meanings. When the loanword accepts an uncountable interpretation, it will be Riffanised without integrating the V-state or the state in its entirety, whereas for the countable sense, no Riffanisation process is applied except for a phonological sieve (e.g., Riffanised uncountable meaning: t-sːaʕeθ26 (↰ in Arabic: sːaʕ-a) ‘the time of the day and night’; ɾefðˤiħ-eθ (↰ in Arabic: l-fdˤiħ-a) ‘shame’/non-Riffanised countable meaning: sːwajeʕ (↰ in Arabic: sːwajeʕ) ‘hours’; ɾefðˤajeħ (↰ in Arabic: l-fdˤajeħ) ‘scandals’). This uncountable interpretation, not necessarily in use in the original language, should be in the singular; otherwise, it could not be Riffanised (e.g., Riffanised countable meaning: a-ħβiβ (↰ in Arabic: l-ħbib) ‘loved one’/non-Riffanised uncountable meaning: ɾeħβaβ (↰ in Arabic: l-ħbab) ‘loved ones’)). Additionally, the loanwords may merely have a countable meaning in the borrowed language. However, in Riffian, the singular meaning will be Riffanised and interpreted with an extra uncountable sense; as for the plural, it will remain as it is (e.g., sːˤiniθ (↰ in Arabic: sːˤinija) ‘tray/trays’/sːˤwani (↰ in Arabic: sːˤwani) ‘trays’). Another noticeable phenomenon is that, even when a word is plural but perceived as uncountable (e.g., pairs of objects), it will have a singular meaning in Riffian and the foreign plural form will be preserved as it is (e.g., wantˤis (↰ in Spanish: guantes ‘gloves’) ‘glove’/iwantˤisen ‘gloves or pair of gloves’).

Table 5. The morphology of borrowed nouns vis-à-vis the state.

	Riffanised	Non-Riffanised
Uncountable	stateless + feminine marking (e.g., ɾfahmeθ ‘understanding.F.SG’)	stateless + inflectionless (e.g., ɾqahwa ‘coffee.F’)
Countable	state + all declensions (e.g., a-mkan ‘place.M.SG’/i-mukan ‘place.M.PL’)	stateless + inflectionless (e.g., ɾmaʕna ‘meaning.F.SG’/ɾemʕani ‘meaning.F.PL’)

Table 5. The morphology of borrowed nouns vis-à-vis the state.

	Riffanised	Non-Riffanised
Uncountable	stateless + feminine marking (e.g., ɾfahmeθ ‘understanding.F.SG’)	stateless + inflectionless (e.g., ɾqahwa ‘coffee.F’)
Countable	state + all declensions (e.g., a-mkan ‘place.M.SG’/i-mukan ‘place.M.PL’)	stateless + inflectionless (e.g., ɾmaʕna ‘meaning.F.SG’/ɾemʕani ‘meaning.F.PL’)

As seen in Table 5, it becomes apparent that the state is inherent to the countability. If the noun is uncountable and Riffanised, the state is not prefixed. Moreover, it appears that a constraint imposes a determined morphology27 (i.e., a template) on these loanwords. The latter can only be singular and reject their integration or reanalyses them if they violate this imposed morphology, as analysed earlier. The formal formation model of this kind of noun is presented in Section 3.3.3. The present classification scheme applies to any Riffian or Northern Berber varieties. However, depending on the variety, the same loanword may fall into either of the categories. Nothing will prevent that; the process is language-specific. The degree of Berberisation will be based on when the word—not the language of contact—came into contact with this variety; this situation is very variable. This may not be the only explanation; other factors may be attributed to this point at issue. Therefore, it is expected to find a lot of discrepancies (e.g, ɾqahweθ instead of ɾqahwa)28. In the next section, we will analyse the reason for the appearance and disappearance of the V-state.

3.3.2. Countability and V-State

The lack of state, especially the V-state, in Berber, is a subject of study that has been little examined. There is no thorough research on this question. The more accepted explanation given in Berber studies on the presence of the state on some nouns and not on others is that of André Basset ([71], p. 161). According to him, the reason is of phonological nature. If a radical begins with the sequence CV, then, in this phonological environment, the V-state will not stand. We do not share this hypothesis. This phenomenon cannot be accounted for on the basis of phonological rules. There are too many examples contradicting it. Nouns having two consecutive consonants can ignore the state (e.g., lːʁa ‘singing’, ðe-mzˤi ‘youth’, etc.) and, conversely, nouns with a CV segment may have the state (e.g., a-bːuʕ ‘an insect’, a-muni ‘union’, etc.).

Our theory is that the V-state fulfils a grammatical marking, and its presence depends on the NP’s semantic values, namely countability. In Riffian, uncountable nouns have different morphologies. In Table 6, we sum up the different types met. However, before discussing in detail the data of this table, we would like to address a couple of issues about the initial vowel. We must distinguish two kinds of V-initial words. According to our assumption, the first one is part of the radical and does not correspond to the state. This vowel, which could be /a/, /i/ or /u/, will remain in its place when the noun is in annexation state, but a semi-vowel/C-state will, even so, be prefixed if it is not a feminine word (e.g., aman ‘water.FS’/w-aman ‘water.AS’/*u-man ‘water.AS’). If these words do not derive from a verb, especially visible in synchrony, where the initial vowel is already part of the verbal radical/word-form (e.g., iʁis ‘be smart’/ð-iʁist ‘intelligence’), then the origin of this vowel might also be diachronic. The loss of phonetic segments make the V-initial displace inside the radical. For phonological reasons, Riffian does not accept a V-V sequence within the scope of a word-form. As the V-state cannot stand side-by-side with another vowel, this one is overtly not expressed. Due to this apheresis, this causes the vowel to remain as it is and not vary, even in number (e.g., awaɾ ‘word.FS’/w-awaɾ ‘word.AS’/awaɾen ‘words.AS’; afeɾ ‘wing.FS’/w-afeɾ ‘wing.AS’/afɾiwen ‘wings.FS’; aʤːun ‘drum.FS’/w-aʤːun ‘drum.AS’/aʤːunen ‘drums.FS’—see Table 2 for comparison).

A comparative study in Berber demonstrates that some of these words are indeed the consequence of a morphological erosion (e.g., in Riffian: adːud/in Middle-Atlas Berber: a-baddud ‘stature’ ([72], p. 15); in Riffian: ðaɾa/in Tuareg: ta-hala ‘spring’ ([73], p. 183); in Riffian: aɾen/in Ghadamis: a-βaɾen ‘flour’ ([74], p. 121)). Other phonetic/phonological changes may also be at the origin of that, an instance among others (vocalic harmonisation, resegmentation, etc.) that can be mentioned is the vocalisation of glides (e.g., in Riffian: iɾem/in Ouarsenis: a-jlim ‘skin’ ([75], p. 103).

However, for a large part of these words, this evolution occurred in an early stage of the language, because in most of the Berber varieties, we retrieved, for a given word, the same inventory of sounds with slight phonetic differences (e.g., aman/imin ‘water’; aðan/adan ‘bowel’; agla/ajla/agɾa ‘wealth’). For these cases, we presume that they underwent the same phenomena as those shown before. In this case, we can conceive to reconstruct their hypothetical forms from a few existing words such as in the instances we provided. This will be only hypothetical in reality; we may never know if this remaining V-initial was the process of a linguistic change or contained in the verb stem—the verb is now lost or has changed—their obscure etymology will probably persist. Another case of lenition, very specific to the Riffian language, that creates a false V-initial is the phonetic mutation of trills [r] and [rˤ] into, respectively, [aɾ] and [aɾˤ] (e.g., aɾiʃ (↰riʃ⇐ /l-ɾiʃ/) ‘feather’; aɾˤuz (↰rˤuz⇐ /l-ɾˤuz/) ‘rice’). Some loanwords also have to be regarded as having false V-initial (e.g., amaɾika ‘America’, uʒda ‘Oujda’); this vowel was already present in the source language. Let us note, as to these two last illustrations, that the semi-vowel is not materialised in annexation state (e.g., *w-amaɾika; *w-aɾiʃ, see also the discussion on the nouns of address further in Section 4.4). Thus, when considering these lexemes, and if they are uncountable, it is expected that they conserve the V-initial (e.g., imendi ‘barley.M’, ð-aguθ ‘fog.F’, ð-uðeɾθ ‘life.F’, etc.).

The second kind of V-initial words corresponds to the true V-initial (see Table 3). Depending on its state, the vowel will either be present or not. With uncountable nouns, its presence is not required as demonstrated earlier, but a few of them resist our claim. The table below puts together the different types of uncountable nouns existing in Riffian according to their morphology and meaning. We will review them one by one to show their differences. We have taken into account five linguistic descriptors to identify them, namely the countability, the singular (SG) and the plural (PL) meaning, and the state in the singular and plural form. According to the presence or the absence of one of these linguistic facts, we determined seven types of uncountable nouns. First of all, Type I29 includes the borrowed uncountable nouns previously described (see Table 5) and typical Riffian uncountable nouns. Those uncountable nouns will never take, under any circumstances, the state and are singularia tantum if they are Riffanised. This means that neither the C-state nor the V-state is never found used with these uncountable nouns. The suffixed feminine marker, not the feminine C-state, is only used with Arabisms because, in this language, these uncountable nouns have a feminine meaning. A few Berber uncountable nouns, such as dːaħiθ ‘pride.F’ or dːist ‘pregnancy.F’, might be thought to belong to this category. It seems, rather, that the feminine C-state is absent because a sequence of two different dentals is not allowed in Riffian. If we accept this phonological constraint as a valid argument, then it will be preferable to categorise them in Type IIIb. The next class, the Type II, encompasses three subtypes. We put them together because all have the V-state. The first subclass30 consists of nouns having a meaning that is either uncountable or countable, especially in the plural, and signifies ‘different kinds of X’. The last subclasses are solely uncountable and are employed in one number-form only; the nouns of Type IIb31 have the pluralia tantum forms and those of Type IIc32 have the singularia tantum ones. Let us move on to Type III; the uncountable nouns of this class are marked by the C-state, but not by the V-state in the singular form. Type IIIa33 has a countable meaning—the plural will get the V-state (e.g., i-ʃaɾen ‘different kinds of soil’)—on top of an uncountable sense, while Type IIIb34 is merely singular in number. The last type35, the IVth, is not uncountable, except for this, it has the same characteristics as Type IIIa. Type IV has been listed in this table because we assume that they were former uncountable nouns. Having now catalogued the uncountable nouns into several classes, in the section that follows, our results will be drawn from this typology (Table 6).

Table 6. Typology of uncountable nouns.

	Countable	SG	PL	State SG	State PL
Type I (seksu ‘couscous’)	−	+	−	−	N/A
Type IIa (a-ʃefːaj ‘milk’)	−	+	+	+	+
Type IIb (ði-mekɾaðˤ ‘scissors’)	−	−	+	N/A	+
Type IIc (a-ʁi ‘whey’)	−	+	−	+	N/A
Type IIIa (ʃaɾ ‘soil’)	−	+	+	±	+
Type IIIb (t-fujθ ‘sun’)	−	+	−	±	N/A
Type IV (fus ‘hand’)	+	+	+	±	+

Table 6. Typology of uncountable nouns.

	Countable	SG	PL	State SG	State PL
Type I (seksu ‘couscous’)	−	+	−	−	N/A
Type IIa (a-ʃefːaj ‘milk’)	−	+	+	+	+
Type IIb (ði-mekɾaðˤ ‘scissors’)	−	−	+	N/A	+
Type IIc (a-ʁi ‘whey’)	−	+	−	+	N/A
Type IIIa (ʃaɾ ‘soil’)	−	+	+	±	+
Type IIIb (t-fujθ ‘sun’)	−	+	−	±	N/A
Type IV (fus ‘hand’)	+	+	+	±	+

3.3.3. Application of the Formal Model to U in Relation to Other Lexical Categories

The purpose of this overview is to understand what might explain why do some uncountable or countable nouns bear the state, or do not? To answer this issue, the solution to that is to fathom the processes of word and meaning formation undergone by these nouns. Each type described before is distinguishable according to the morphological and semantic operations to which they are submitted. The model proposed (see Figure 2) justifies the views defended in our article on the role of the state and its existence with uncountable nouns. The parts of speech considered are the verb and the nominal. The subparts of nominals taken into consideration are the noun of action (NA), the countable nominal (C) and the uncountable noun (U). The passage from one category to another one is carried out either by word formation (morphological derivation or conversion) or semantic change (symbolised by the superset symbol ⊇). In Figure 2, the category on the left is either the starting or ending category; the arrows specify the direction of derivation. We established twelve different chains followed by the different types of uncountable nouns. For each of them, some examples have been given, but they are just instantiations of a word and meaning formation model that is far more varied. The most important for us, within the scope of this study, is to highlight the directional derivatives, as defined below in Figure 2, that led to the formation of uncountable nouns. Aside from this, other configurations can be identified between V, NA and C for the same process. For example, inside the process δ, we could have included another one (see below (6)) that is different from those given in Figure 2, but identical concerning the formation of U:

(6)

V: mzˤi ‘be small’   →U: ðe-mzˤi ‘youth’ ⊇NA: ðe-mzˤi ‘fact of being young’

             ↬C: a-mezˤjan/i-mezˤjanen ‘small.SG/small.PL’

Hence, other patterns could have been added, but this is out of the reach of the present work. We are not proposing a general model ofword and meaning formation in Riffian, but merely a model for the uncountable nouns. This model does not apply to all the uncountable nouns as well; some U from Type I (e.g., wuħβeɾ ‘soul’) are in relation with no process; it was thereupon decided not to include them in the model. Thus, Type I is a native category; it does not arise from another lexical class (e.g., the processes ν, ξ, π and $\mathsf{\rho }$); they are uncountable from the beginning. Evidence in favour of this assumption comes from the Riffanised uncountable loanwords (see Table 5). When entering the Riffian system, they are natively uncountable and do not have any mark of the state. Yet, as seen in the previous section, some Riffanised uncountable nouns also have a countable meaning. The singular countable sense has the same form as the uncountable one, the bond between them results from the same process in λ. However, the transition from the singular to the plural is done by suppletion—its symbolic notation is ↛—because the first has a Riffian morphology but the second does not (e.g., C or U: ɾχeðmeθ ‘work’/C: ɾeχðajem ‘works’). Since the plural uses a non-Riffian morphology, the state is not affixed. Ergo, we have chosen to put these words in a different process. If the plural was also Riffanised, then, we should have expected to run into the state (see Figure 2λ). Similarly, the process π goes from U to V (i.e., a denominal verb) and not the other way around like with the processes γ, δ and ε. This assertion is supported by the lack of the C-state. We postulate that if U would be derived from V, the affixation of the C-state should have been feasible. The other types, on the other hand, are the consequence of a category change, even several. Type II may stem from different processes enumerated as follows: α, ζ and η. Types III and IV may be set up from the following processes: γ, δ, ε, λ and μ.

We would like to discuss in more length in the next part several specificities outlined in this section to explain their importance and highlight their relevance.

Figure 2. Formation model of uncountable nouns in Riffian.

Figure 2. Formation model of uncountable nouns in Riffian.

4. Discussion and Properties of the TBMC Model

Several of these processes involve the same input head, but as the subsequent iterative transfer operations (viz. $h\left(w\right)$) map the items to different syntactic categories and morphological templates, it was decided to highlight distinctly their particularities by means of disjoint functions.

Thus, the processes α, γ, δ, ε and η are different from the other ones in that the starting category of U is necessarily a verb. The processes δ and ε have in common an identical word-formation process, but the process δ is made to show the conversion that operates between V and U, while the process ε exemplifies the morphological derivation. Moreover, the derivation illustrated in the process γ and ε will not form the same type of U. With the process γ, U is a classic uncountable noun, but in the process ε, a noun of address is created instead, that is why they were separated.

As for the other processes, such as ζ, λ or μ, they do not originate from a known verb in synchrony. Several particularities distinguish these processes. The process μ is typical of a morphological derivation from an adjective to a noun of address regarded as U. While their input head starts with the same antecedent factors, the processes ν and ξ have been disassociated. As a matter of fact, there is no relation of derivation between the items of the process ξ. Each item is its own starting category in the recipient language, but they are related in the donor language. In contrast, this is not the case for the process ν and the other processes. The processes λ and $\mathsf{\rho }$ are similar in terms of lexical derivation; however, the countable noun created in $\mathsf{\rho }$ does not inflect in number. As said before, a lexical suppletion is instead used to express the plural.

To deepen even further our perspective, let us look over the case of deverbal NAs (process α). This cognitive set contains the state in a similar manner as deverbal countable nouns (process η). Thereby, one could ask why the V-state is present, although we said, previously, that this morpheme is linked to the countable meaning. NA is, indeed, unaffiliated to the countability (i.e., countable or uncountable sense), but this is not the only aspect to underline.

NA and U are related in many aspects of their semantic peculiarities. For instance, they are not discretisable; nonetheless, they must not be confused. The most fundamental difference between NA and U is that the latter can be modified by a noun or a determiner; this is easily observable when they are two different lexemes (e.g., *a-ndah-inu/ða-nedhiwθ-inu ‘my driving’). While NA does not convey countability regarding its intended semantic purpose and can be interpreted as closer to the uncountable category, the V-state will be even so prefixed.

This morpheme is the unmarked form of this cognitive set and has to be present by default; in the same way as the singular and masculine meanings that are, however, unformed. Since it belongs to the syntactic category of N, its unmarked template comes with paradigmatic markers (see Definition 2 in Section 2.2.2) that cannot be disregarded. In other words, these morphemes share with the radical a particular relationship that other inflectional morphemes do not have (e.g., deictic markers). Hence, not all the grammatical markers are concerned by markedness. To articulate this in a different manner, a marker of each of these paradigms must be assigned to the item depending on its affiliation to a cognitive set. The marking is not necessarily driven by the semantic properties of the item (see also Section 5).

One can also notice, in Figure 2, that features range across the entire model. They will be mentioned in detail in the following sections. Accordingly, we wish to take occasion to refer to different cognitive properties of the TBMC model at the interface of morphology, phonology, semantics and semiotics.

4.1. Semantic Widening Versus Conversion and the Overt Anti-Morphosyntactic Shift

Word formation is not the only strategy employed to widen a language’s semantic catalog; meaning formation is also used for this purpose. The use of either formation can produce an identical spelled-out form between the base word and the resultant word. This is notably true for the processes associated with U. When the countability is in play, the V-state can be used with C (see Figure 2η) or cannot if C derives from U (see Figure 2γ). As seen, both have the same form as their base word regarding the state. Such a pattern corresponds to what this article calls an overt anti-morphosyntactic shift (see Section 2.2.1). However, our theory is that, covertly, those types of derivatives have different cognitive structures. We claim that the inter-process morphosyntactic variations can also be detected in the mental states. Even if the spelled-out form is unchanged, internally, they have rather different cognitive representations.

Thus, the prefixation of the V-state with uncountable nouns is acquired when the word to which it is linked also has the initial vowel. C or NA is necessarily the source of the uncountable nouns having the V-state. The link between them may operate by conversion or semantic change. In Figure 3, we described the cognitive contrast existing between the semantic change (viz., semantic widening) and the conversion. Their semiotic relationship [76] between the form and the meaning is dissimilar, one will have a lexical entry for each meaning, while the other will add both meanings under one form. In other words, when conversion acts, a new item will be stored in the set W (see Definition 1 in Section 2.2.2).

In terms of lexical inventory, the fact that no new entry is engendered in W during the semantic widening also has consequences for the semiotic relationship of the grammatical markers. If no template shift intervenes in the course of the meaning formation, the morphological pairings of forms and meanings remain unchanged. Several remarks can be inferred from this observation. Thus, it implies that the morphemes of the resultant word must be analysed as being always computed in a deterministic way, that is to say, morphological indeterminism cannot happen. Another implication of this finding is that non-deterministic morphological realisations only surface when a template shift occurs.

Figure 3. The semiotic relationship before and after the category change.

Figure 3. The semiotic relationship before and after the category change.

After this clarification, let us proceed to further determine the cognitive properties that lie behind this anti-morphosyntactic shift. Despite the fact that no additional item is generated with a semantic widening, the TBMC model enables, in theory, to carry out a template shift after a meaning formation, for example when an uncountable sense is appended to a countable one. In this case, a new template could be assigned to the resultant item by $h\left(w\right)$. However, this is not the choice opted for Riffian; no template shift is realised. For this language and others, the gradient function assigns the same template as the one used by the base item to the resultant item. To do so, the gradient condition responsible for keeping the template unaltered after a semantic widening is equal to Ø (i.e., $t∆t=\varnothing$). That said, it would be erroneous to suppose that all languages avoid computing a template shift with semantic widening or any semantic change.

We can conclude that the overt anti-morphosyntactic shift of both formations conflicts. In one case, a template shift is carried out and, in the other case, the template is conserved. To solve this contradictory situation, it must be assumed that, at the cognitive level, their encoding is dissimilar and has different attributes. Ergo, for the semantic widening, there is a covert anti-morphosyntactic shift; the surface and underlying representation are similar as expected. While with the conversion, conversely, there is a covert morphosyntactic shift. However, the template emanating from the covert morphosyntactic shift does not automatically construct a sign with the forms of the spelled-out form, but the latter does so with a second template inherited from the base word (see the algorithm of lexical derivation in Section 2.2.1).

Thus, as theorised, two templates are in competition with each other. The grammatical elements of these templates are named the overt meanings and the covert meanings. The latter meanings have the word and meaning formationminence to build, with the spelled-out forms, a sign. On the other hand, the overt meanings are sent backward and blocked by the covert meanings to establish a sign. It is under these circumstances that the expletive form is formed. The formal required conditions leading to this morphological specificity are the non-symmetrical features between the overt and covert meanings in combination with the semiotic relationship linking the covert meanings and the spelled-out forms. Hence, if a grammatical element, such as the state or the definite article, is present in the covert meanings and absent in the overt meanings, then this grammatical element must be regarded as an expletive form (see Section 4.5).

As demonstrated, both are covertly different; nonetheless, it is not the only characteristic distinguishing the semantic widening and the conversion. With conversion, the overt anti-morphosyntactic shift is only partial (on condition that the item is not associated with the animate meaning (see Section 4.2)) and does not encompass gender. If, in surface, the morphosyntactic shift is only limited to the gender without affecting the other grammatical markers (except for the Type I items (see Section 4.5)), this must be imputed to the item cloning (see Section 2.2.1). Therefore, the criterion retained to determine when we are facing either cognitive operation is the gender shift occurring when passing into another nominal category. This will be analysed in the next section.

4.2. Gender Shift

The gender marking appears or disappears when the conversion is carried out. If the starting category, which can be C or U, has the feminine marker, then the derived category will lack the feminine marker (see Figure 2γ), and the same goes the other way around (see Figure 2ζ). We designated this phenomenon as inverse gender marking (different from gender polarity where two distinct genders are present in one word-form). If that does not happen, we estimate that we are dealing with a semantic change. However, if there is a category change without gender shift (Figure 2δ, noun of address36: miʁis ‘clever one’ → Adjective: miʁis ‘clever’), we may assume that the conversion is also at work, but, as these words refer to animate referents, they cannot shift. If, in addition to this, there is a state shift, then we can predict that a morphological derivation is instead employed by cloning a root instead of lexical entry (see Figure 2λ, Adjective: a-kːuħ ‘small’ ↬ Noun of address: kːukːuħ ‘tiny one’).

For some authors [77], the feminine marker should be considered a derivational marker, but, as stated before, we disagree with this affirmation. In our analysis, this marker is still inflectional, as is the V-state. The word-formation process that contributes to the template shift is the conversion and not the morphological derivation. The gender shift is manifested when the countability is implied in both directions C → U (e.g., a-gezːaɾ ‘butcher’ →ða-gezːaɾθ ‘butchery’) or U → C (e.g., ɾχuχ ‘plums’ → ða-ɾχuχt ‘a plum’). Thus, it is inconceivable to imagine that a derivational marker can do two contrary cognitive operations. This contradictory situation led us to propose another more suitable word-formation process In Riffian, such an alternation is also found in connection with other category changes. This is regularly observed when we can form evaluative nouns (diminutive or augmentative) from non-evaluative nouns.

These numerous irreconcilable configurations37 demonstrate that the feminine marker still serves as an inflectional marker. Thereby, if it is not a derived noun, one of gender markers is assigned randomly to the item, but if a new noun is built by conversion from another noun, it will take the opposite gender of the base noun involved. This rule is possible only if the drifted nominal (i.e., $w\in W$) or antecedent nominal (i.e., $w\in k\subset K$) does not refer to animate objects whether it be directly or via concord. In such a case, the animate referent acts as a constraint on the underlying operation, the gender shift will not be allowed (e.g., U: ð-awmatt ‘brotherhood’ → Noun of address: ð-awmatt ‘dear (for a man)’; U: ɾqahwa ‘coffee’ ↬ Adjective: a-qahwi ‘brown’). We also already observed this situation when that implicated PNs (PN to CN or CN to PN, see Figure 1), so we will not linger over them here. As suggested by our theory, this gender shift is the consequence of a template shift. If the items entail non-animate meaning, then the pairs to be computed must satisfy some gradient conditions as follows:

$$\begin{array}{cc}\hfill \mathrm{Gradient}\mathrm{condition}2:& \exists w\in W,\mathrm{if}\exists t,\mathrm{such}\mathrm{as}t\subset k=f\left(w\right)\in K,N\in t,\mathrm{and}\mathrm{if}\exists t,\mathrm{such}\mathrm{as}\hfill \\ \hfill & t=h\left(w\right)\in T,N\in t,\mathrm{then},\mathrm{their}\mathrm{symmetric}\mathrm{difference}\mathrm{must}\mathrm{be}\hfill \\ \hfill & t∆t=\{-F,+M,+F,-M\}\hfill \end{array}$$

(2)

Hitherto, our analysis was concerned with conversion, but the same observation can be made about morphological derivation. The derivational markers, such as those in German (e.g., -schaft, -heit) or in French (e.g., -té), may also, to all appearances, engender a gender shift. It is often suggested that the feminine meaning is attached to these markers, which would explain the gender of the resultant word. However, this hypothesis is unsubstantiated and, alternatively, we suggest that a template shift also occurs in such a circumstance. This proposition leads us to conclude that the derivational markers are grammatically agnostic and only the inflectional markers bear the grammatical meanings (i.e., those being part of the morphological template, for instance, number, gender, countability, etc.). Let us underline that a parallel must be drawn between the nominal and verbal systems, where we can notice similar patterns. Thus, it should be assumed that the derivational markers are grammaticalness-blind markers (see also the case of the derivational marker m- in Section 4.4) as we may name them for convenience. The second implication that can be drawn from this observation is that these markers are not determined by the TBMC model.

In the next section, we will take into account the gender shift from another angle. It will be considered under a historical perspective. In doing so, this allows us to discern another property of the morphological system, to wit immutability.

4.3. Diachronic Considerations and Immutability

By considering only the V-initial, we observe that a large part of these uncountable nouns, such as a-ʁi ‘whey’, no longer have any relation with a NA or C. Given that they have the state, if we rely on our model that is itself based on synchronic data, we may expect that these nouns followed the same process, but we cannot say with certainty which one, among the processes α, ζ and η, they underwent. Moreover, some uncountable nouns with the V-initial may suggest that they are derived straightforwardly from verbs (e.g., V: smeðˤ ‘to be cold’ ↬ U: a-semːiðˤ ‘low temperature’). However, we would like to bring into discussion that these nouns were, in the past, compatible with our prototypical process chains of template shifts illustrated in Figure 2 (Stage 0: η: V ↬ C → U or ζ: V ↬ C ⊇ U or α: V ↬ NA ⊇ U). Lexical loss/obsolescence has modified this archetype; thereby, this gives the impression that they are direct deverbal nouns (Stage 1: V ↬ _____ U).

That hypothesis is confirmed by the morphology adopted by these uncountable nouns. A word such as a-semːiðˤ has the same morphology (scheme: a-C₁C₂iC₃) as the following countable nouns: V: ngez ‘to jump’ ↬ C: a-negːiz/i-negːizen ‘jump/jumps’; V: χβeʃ ‘to scratch’ ↬ C: a-χeβːiʃ/i-χeβːiʃen ‘scratch/scratches’; V: ʃeɾːeg ‘to rip’ ↬ C: a-ʃeɾːig/i-ʃeɾːigen ‘rip/rips’. Moreover, it is not just a conjectural idea because, in other Berber varieties, the plural of a-semːiðˤ exists, but with a different meaning (a-semːiðˤ/i-semːiðˤen ‘cold wind/winds’; see [78], p. 643 and [79], p. 36).

From this specific instance, we can extend the principle to other kinds of uncountable nouns of that sort. Notwithstanding, in any case, we are insinuating that the uncountable and countable nouns have necessarily different schemes. Deverbal nouns either C or U may possess the same morphology (e.g., the scheme C₁C₂C₃ is identical to verb and noun: V: wsːeɾ ‘to be old’ → U: ðe-wsːeɾ ‘old age’; V: mʒeɾ ‘to reap’ → C: a-mʒeɾ/i-megɾan ‘billhook/billhooks’). Neither we affirm that a-semːiðˤ was previously a countable noun on the basis of other examples38. The direction of category change, for these very specific cases, is irrelevant to analyse for the points raised here. The true state of affairs is that those nouns have the same scheme and template, nonetheless, they belong to different cognitive sets. Therefore, it should be viewed as proof that this paradoxical pairing is caused by the processes of word and meaning formation. By taking into consideration this and the lexical loss, we can observe that the evolutions affecting these items do not have an impact on the item–template pairing in surface. This conservative force is predominant over everything else.

The diachronic dimension also has to be taken into account with Type IV. They are morphologically identical to Type III but semantically countable. We claimed that a semantic shift is at the origin of the passage from the uncountable to the countable class. Due to this cognitive operation, we are in a position to justify the non-inclusion of the V-state with these nouns. The semiotic relationship between Type III and IV is defined in Section 4.5. Cross-linguistic studies enable us to lay the groundwork for this hypothesis [80]. The nouns known to have particular meanings and refer to different groups are often associated with the uncountable category in several languages. They are such nouns in Riffian that lack the V-state, for example, animals/insects (e.g., jis ‘horse’, kuz ‘weevil’, fidʒːus ‘chick’, kuɾðu ‘flea’, etc.); pair-wise objects/body parts (e.g., fus ‘hand’, ðˤaɾ ‘foot’, fuð ‘knee’, t-siɾiθ ‘footwear’, ð-juga ‘harness’, etc.), long cylindrical shaped objects (e.g., ð-miʒːa ‘throat’, qabu ‘stick’, t-sudʒːeθ ‘basket’, etc.), plants/fruits (e.g., zumbi ‘corn’, βaw ‘broad bean’, etc.).

It appears, then, that such items have indeed experienced a process of word and meaning formation and even several. If we admit that $h\left(x\right)$ is infinitely differentiable (i.e., $C^{\infty }$) and that the resultant word is independent of the base word (i.e., $\exists w_1$, $w_2$$\in W$, $w_1$≠$w_2$, if $w_2$ is formed from $w_1$), therefore, we can postulate that intermediary stages could have existed between stage 0 and 1 implicating different processes successively (lexical losses, meaning formations and word formations). The formation model presented earlier in Figure 2 gives significance to the first apparition of the uncountable noun during successive directional derivatives along NA, C or U. Thereby, stage 0 represents only the first and second derivatives; higher derivatives are not studied. Some of the items in W are, however, the consequences of more than two derivations and other directional derivatives as well (e.g., successive directional derivatives along NA then C, or C then C). This is noticeable from the template with which they are paired. Hence, other grammatical features of the unmarked template, like the state, are also indicators of the degree of anomalous morphology39. Thus, if we analyse the word a-semːiðˤ again, even by assuming that its primitive is a countable noun, this does not completely explain why its form is masculine and not feminine (see Figure 2η). Due to this, more directional integrals must be implemented to reach one of the cases of the stage 0.

In essence, despite the successive derivatives (i.e., k-times differentiable in W) and the paradoxical morphology induced by these processes, the morphological system is able to show conservation. Subsequently, it is also of particular significance to note that the determinacy of these semantically shifted items is ensured by the set K, denoted as the retrospective determinants of shifting template, in relation to the backward recursive and gradient functions, symbolised, respectively, by f and g. In this way, paradoxical pairings are never computed by h. The model is able to predict this surface paradox (e.g., the non-gender unmarked form of asemːiðˤ) by also considering as a shifting determinant the processes of word and meaning formation. Hence, even when the originating item is no longer listed in the lexicon, the item–template pairing remains immutable. Since languages evolve constantly over time, the immutability preserves the languages from severe lexical imbalance.

This feature can be thought of as a structure-preserving mapping between the anterior and posterior states of the morphological system after a lexical loss. Such a phenomenon affecting the set W and the morphosyntactic transfer function $h\left(w\right)$ can be formulated mathematically as follows:

Property 1. 

• Consider the following retraction mapping (or lexical loss function) $r:W\to \mathrm{\Omega }$, such as Ω is the set of items of the posterior state and W is the one of the anterior state. Thus, r is a surjective map, such as every ʊ $\in \mathrm{\Omega }$ has one and only one pre-image $w\in W$.
• Therefore, there exists an injective mapping $i:\mathrm{\Omega }\to W$, such as $i\left(\mathrm{\Omega }\right)\subset W$ and $r\circ i=1_{\mathrm{\Omega }}$. Hence, ∀ ʊ, $i($ʊ)= ʊ and $r($ʊ$)=$ ʊ.
• Let the 2-tuples ${\mathbb{M}}_n$=$⟨W,h^n⟩$ and ${\mathbb{M}}_{n+1}$=$⟨\mathrm{\Omega },h^{n+1}⟩$ be two instances of the TBMC model representing, respectively, the anterior and posterior states, such as $h^n:W\to T$ and $h^{n+1}:\mathrm{\Omega }\to T$.
• There exist the homomorphism mappings $l_r:$${\mathbb{M}}_n$→${\mathbb{M}}_{n+1}$ and $l_i:$${\mathbb{M}}_{n+1}$→${\mathbb{M}}_n$, such as ∀ ʊ $\in \mathrm{\Omega }$, $l_r\left(h^n\right($ʊ$\left)\right)$=$h^{n+1}\left(l_r\right($ʊ$\left)\right)$ and $l_r($ʊ$)=$$r($ʊ).
• Therefore, ∀ ʊ $\in \mathrm{\Omega }$, we have $l_r\left(l_i\right($ʊ$\left)\right)=1_{\mathrm{\Omega }}$, $l_i(l_r\left(h^n\right($ʊ$\left)\right))=1_T$, and, hence, $h^n($ʊ)=$h^{n+1}($ʊ). Thus, the homomorphic map $l_r$ preserves, in case of lexical loss, the relation $(W,T,h)$ as well as the relations $(W,K,f)$ and $(K,T,g)$40.

In the upcoming part Section 4.4, we will continue the study of the particularities of the morphological system through the lens of the TBMC model. We will uncover a fundamental property observed at the morphophonological level.

4.4. The Noun of Address, Morphophonological Avoidance and Linguistic Irreversibility

Amongst the uncountable nouns of Type III (the process δ), the deverbal nouns of address stand out (see Table 7). Only the feminine C-state can be affixed, while, in the masculine gender, the semi-vowel is absent in AS (e.g., FS: mizðið/AS: mizðið (*u-mizðið) ‘willowy one.M’). On the other hand, it should be expected that the masculine C-state be affixed to the radical since the feminine C-state is able to do it. Nouns of address are, in fact, similar to PN and, in consequence of this factor, the morphophonological rule existing for CNs is violated. This is not a unique case; other morphophonological/phonological rules escape the PN category. For example, still in Riffian, the sequence /ð/ + /θ/ or /ð/ + /t/ is realised [tː], but with PNs, the surface and the underlying representation match (e.g., /ð tiʒani/ = [ð tiʒani] ‘it is Tijani’ vs. /ð tsa/ ≠ [tːsa] ‘it is a liver’). The avoidance of phonological rules by PNs is also found in Arabic [81] as well as in French [82]. Thus, this aspect, which can be labelled morphophonological avoidance, is one of the characteristics distinguishing the deverbal nouns of address from other nouns, nonetheless they share also similar patterns with the nominal lexemes. Hence, as these nouns are the consequence of a morphological derivation, the place of the state is joined with the radical. In addition, as a term of address refers obviously to animate referents, the feminine C-state can be assigned.

Nevertheless, when we are not dealing with deverbal nouns of address, onymised CNs can have an identical morphology to Type II; they bear the V-state that will withdraw in AS (e.g., FS: ða-feɾsiθ/AS: t-feɾsiθ ‘Tafersit’; FS: a-ʒdiɾ/AS: u-ʒdiɾ ‘Ajdir’). However, other typical PNs know the same form in FS and AS (e.g., FS/AS: nkuɾ (*u-nkuɾ) ‘Nekor’, ʃeqqɾˤan (*u-ʃeqqɾˤan) ‘Chakrane’). Therefore, we may assume that the particularities of deverbal nouns of address and onymised CNs are incumbent on the process they have undergone. From this observation, several deductions can be formulated to resolve these contradictions. One of the postulates is that during the cloning process in addition to the morpheme flow (see Section 2.2.1), the phonological rules applying to CNs are also transferred into the new item. That said, it is assumed that this cloning is not intentionally fashioned. This cognitive process is not able to overrule this phonological rule on account of another cognitive fact. The specificities of the latter may be understood as a sort of restrictive access level.

More precisely, any performed cognitive operation cannot be undone by an incoming process of the same or different nature; in other words, the integrity of an item cannot be violated when generated. This characteristic is comparable to the conception of strict cyclicity [17] or bracketing erasure [56]. This explains why this phenomenon is not neutralised with some onymised CNs. As a consequence of that, we can predict that the PNs not affected by this phonological rule (e.g., nkuɾ) are identical to uncountable nouns of Type I (see Section 3.3.2) that totally ignore the state. Thus, the difference between stateless or stateful onymised CNs is imputable to the theory of lexical derivation of the TBMC model (i.e., the cloning of morphemes and phonological rules) and item integrity—in a larger perspective, this cognitive fact may be named linguistic irreversibility. As for the deverbal nouns of address, since they are not the result of an intra-lexical derivation, they can overrule the phonological phenomenon while inheriting the state.

Table 7. Summary of the differences between the deverbal noun of address and some noun categories.

		Lexical Derivation	Covert Meanings	Overt Meanings	Morphophonological Avoidance	Morphophonological Cloning
PN	Deverbal N-of-A	V ↬ N	N/A	COL, F or M	$\underset{(\mathrm{C}-\mathrm{state})}{+}$	−
	Onymised CN	NP → N	SING or COL, F or M	COL, F or M	+	$\underset{(\mathrm{C}-\mathrm{state})}{+}$
	Onymised VP	VP → N	N/A	COL, F or M	+	+
CN	Deverbal U	V ↬ N	N/A	COL, F	−	−

Table 7. Summary of the differences between the deverbal noun of address and some noun categories.

		Lexical Derivation	Covert Meanings	Overt Meanings	Morphophonological Avoidance	Morphophonological Cloning
PN	Deverbal N-of-A	V ↬ N	N/A	COL, F or M	$\underset{(\mathrm{C}-\mathrm{state})}{+}$	−
	Onymised CN	NP → N	SING or COL, F or M	COL, F or M	+	$\underset{(\mathrm{C}-\mathrm{state})}{+}$
	Onymised VP	VP → N	N/A	COL, F or M	+	+
CN	Deverbal U	V ↬ N	N/A	COL, F	−	−

While on the subject, in order to give a full picture, we will carry on to compare the deverbal nouns of address to other types of nouns. Then, deverbal nouns of address and deverbal uncountable nouns (the process γ/Type III) are different categories on two grounds: gender and morphophonology. The latter observation was just described, but it is also interesting to note another property associated with this phenomenon. Thus, the conversion of a deverbal U along any direction into another noun will not change the morphophonological structure. This implies that, as long as it is not a feminine PN, the masculine C-state is expected to appear in AS. Conversely, directional derivatives of a noun of address will not cancel the morphophonological avoidance acquired during the first derivative from the input head. We argue that these two different patterns are equivalently the result of linguistic irreversibility. As linguistic irreversibility is not impaired in the course of a chain of recursive word formations, we can state that intra-lexical conversion is locally idempotent. This specificity is defined formally as follows:

Property 2. 

• If there exist $ite{m}_0$ and $\to \left(ite{m}_0\right)$ = $ite{m}_1$ such that $Morphophonological$ $Structur{e}_{\mathbf{1}}$⊂ $ite{m}_1$.
• If there exists $\to (\to \left(ite{m}_0\right))$ = $ite{m}_2$ such that $ite{m}_1$ and $ite{m}_2$ belonging to the same syntactic category (see Property 3).
• Then there also exists $Morphophonological$ $Structur{e}_{\mathbf{1}}$⊂ $ite{m}_2$.

As to the former observation regarding gender, a rule commands that deverbal uncountable nouns should be always marked by the feminine C-state, which is not the case for deverbal nouns of address, both genders may be found due to their animacy. To a lesser degree, it is also worth mentioning the particularities of onymised VPs, which are likewise impacted by morphophonological idempotence and linguistic irreversibility. While deverbal nouns of address and onymised VPs are both PNs, we should be careful not to confuse them. The last one bears all the possible verbal markings, because of the conversion (VP → PN), whereas the first one has been subjected to a morphological derivation (V ↬ PN). Moreover, onymised VPs may clone any morphophonological rules embedded in VP.

Lastly, let us note that the derivational marker m- used with deverbal nouns of address also serves to form patient or agent nouns that are countable (e.g., zguɾ ‘get ahead of’ ↬ a-mezgaɾu/i-mezguɾa ‘first.M/firsts.M’). We cannot then attribute the lack of state to this morpheme, since it has no role in the treatment of grammatical markers (see the notion of grammaticalness -blind marker in Section 4.2). The V-state is absent because PNs, included nouns of address, are strangers to the cognitive set of the countable category.

The forthcoming section is dedicated to introducing a new grammatical paradigm in the analysis of languages. Furthermore, the morphosyntactic encodings of these morphemes will be explored. Such semiotic encodings are crucial features enabling to dispel the irregular morphosyntactic markings saw in Table 6.

4.5. Towards an Alternative Theory: The Countability Paradigm and the Morphosyntactic Encoding

The uncountable nouns with the V-state do not differ semantically from the ones with solely the C-state. As has been demonstrated, the V-state presence is caused by the processes of word and meaning formation, namely the semantic change and the conversion. Now besides the conversion, which was the first argument that we proposed, the semantic change is also another one that allows elucidating the existence of the initial vowel with U. Unlike the latter, uncountable nouns without the V-state are never the outcome of a change category that emerges from C or NA; this explains why the V-state is absent. Moreover, on a semantic level, the uncountable nouns without the state (i.e., Type I, see Table 6) are not distinguishable from the others. Types I and III ignore the V-state, but the C-state has to be affixed with Type III. The prefixation of the C-state without the V-state is only found with deverbal nouns (the processes γ, δ and ε) or/and have also a countable meaning that is encoded with the Berber morphology (the process λ) —let us recall that if it is a foreign morphology, the state cannot be used (see Section 3.3.1).

It is worth asking why the C-state is used with Type III, even though the V-state is not present, notably in the singular, but, also, why it is absent with Type I. To answer this problem, we must enquire into their semiotic relationship. Thus, the distinction between these two types lies in morpho-semantics. To date, we have only announced that the state is in relation to countability without further details. It is time now to address the issue of the semantic values of the state. First of all, we need to identify the encoding scheme of these types. It is assumed that the state meanings are not linked semiotically (see the mental model in Figure 4) to the same category of morphemes (i.e., affix or radical). Two different encodings are distinguished and underlie these words. Words of Type I do not need an additional morpheme to mark state meanings; it is a fusional encoding because the meaning is bound to the radical. Whereas Type III has to affix a morpheme to express the state meanings, the encoding is rather analytic.

For completeness of the analysis, we also bring into the discussion the Types II and IV. Like Type III, both have an analytical encoding; however, they are distinct from each other. Our thesis is that their difference is merely paradigmatic. Instead of considering that no marker (i.e., V-initial) is affixed to the radical, we argue that a null morpheme is prefixed between the C-state and the radical. Thus, Types III and IV have a null morpheme. Moreover, it is assumed that the V-initial marker and this null morpheme are organised into a paradigm. In addition, from a semiotic standpoint, the paradigmatic markers of these types of uncountable nouns differ. The relationship between the spelled-out form and the overt meaning (see Section 4.1) is dissimilar. Type II and Type IV have an expletive form, and, in contrast, Type III has a regular form. It means, for the former, that the overt meaning of a grammatical element coincides with the radical form, and the covert and overt meanings of a marker (whether null or not) are unsymmetrical. Figure 4 symbolises schematically how the overt meanings and the spelled-out forms are related for each type of uncountable nouns.

Figure 4. The semiotic relationship of the fusional and analytical encoding in relation to the category change.

Figure 4. The semiotic relationship of the fusional and analytical encoding in relation to the category change.

These semiotic archetypes are not specific to Riffian, the uncountable nouns or/and the state. This proposal, in theory, applies to all sorts of items (verbs, nouns, etc.), cognitive sets (NA, U, C, etc.), grammatical markers and synthetic languages. Another general property that can be put forward is regarding derivation. These three categories of semiotic relationship (see Figure 4a–c) are also related to each other. When an item possessing one of these semiotic archetypes is the object of conversion, it will follow a determinate chain of semiotic encodings. Thus, the chain of semiotic encodings is as follows: the semiotic archetype A (Figure 4a) is followed by the semiotic archetype B (Figure 4b)41, then the semiotic archetype B will acquire the semiotic archetype C (Figure 4c)42. Thereupon, the semiotic archetype C is invariable and keeps its semiotic encoding as it is; the successive derivatives do not affect it from the second derivative.

After mentioning the main properties of these semiotic archetypes, we would like to look closer at the particularities of the morphological marking in Riffian. In Figure 4, ${ℳ}_1$ represents state meanings that can be linked to the radical (Figure 4a) or the affix (Figure 4b). If ${ℳ}_1$ is rendered only by the form ${ℱ}_1$, then ${ℳ}_2$ and ${ℱ}_2$ solely depict, respectively, the meanings and the form of the radical. Another configuration is when the semantic change or the conversion preserves the form ${ℱ}_1$, but will detach or add the state meanings ${ℳ}_1$ to ${ℱ}_2$. In this context, the sign will no longer be conserved between ${ℱ}_1$ and ${ℳ}_1$. This situation is illustrated in Figure 4c with Types II and IV, where ${ℱ}_1$ is just an expletive form (i.e., ${\varnothing }_{{ℳ}_1}$). Moreover, as Figure 4b may characterise the countable nouns as well, we need to divide ${ℳ}_1$ and ${ℱ}_1$ into two other labels, that is to say, ${ℳ}_{1a}$/${ℱ}_{1a}$ and ${ℳ}_{1b}$/${ℱ}_{1b}$. Let us adjust our different types to the freshly created labels. Types I, II and III have ${ℳ}_{1a}$, whereas Type IV has ${ℳ}_{1b}$. Type II has ${ℱ}_{1b}$, while Types III and IV have ${ℱ}_{1a}$.

Hence, let us ascertain the semantic values of these labels. As said previously, we assume that the state works with an inflectional paradigm (see Table 8) constituted of two markers that denote, namely, the collective (i.e., ${ℳ}_{1a}$) and the singulative (i.e., ${ℳ}_{1b}$). The collective marker is a zero morpheme (i.e., ${ℱ}_{1a}$ or ${\varnothing }_{{ℱ}_{1a}}$) that agrees in gender (i.e., C-state) and the other form is a fusional marker (i.e., V-state/${ℱ}_{1b}$) that indicates both number and singulative and that inflects in gender (i.e., C-state). When the countability, especially the collective, is not marked analytically, the C-state cannot be present, since its meaning is borne by the radical (i.e., Type I).

We decided that Types I and III have different encodings, while they are both not derived from countable nouns, on the grounds of the C-state presence. Let us recall that the masculine C-state is not a morpheme as is the feminine C-state. In this article, they were considered in common for simplification purposes, but our thesis is that w-/j- is just a phonological artefact (see Section 2.3). C-state is present with Type III even when the V-state is absent. The only way to resolve this dissimilitude is to postulate an analytic encoding with an unformed morpheme, in other words, a null marker (i.e., ${\varnothing }_{ℱ}$) that expresses the uncountable meaning. It is this unformed morpheme that allows the prefixation of the C-state. This claim is supported by the impossibility of affixing the singular singulative marker on a countable noun that derives, after a conversion, from an uncountable noun of Type III (i.e., the overt anti-morphosyntactic shift, see Figure 2γ). The collective marker, although unformed, is preserved owing to the conversion. Whereas, when the countable noun comes from an uncountable noun of Type I the singular singulative marker a- can be prefixed since this position is not occupied (see Figure 2ν and Table 4), as explained before in more general terms (see the notion of chain of semiotic encodings). Let us also underline that the feminine marker is affixed to the countability markers; it inflects inside of the countability paradigm. The collective marker does not agree in number since it is unformed, but the singulative does. Both of them constitute a distinct constituent from the nominal. This is similar to what happens in Romance languages where the gender markers are affixed to the definite article.

Table 8. The countability paradigm in Riffian.

	Singular		Plural
	Feminine	Masculine	Feminine	Masculine
Uncountable/Collective	t-${\varnothing }_{ℱ}$	(w/j)-${\varnothing }_{ℱ}$	N/A	N/A
Countable/Singulative	t-a	a	t-i	i

Table 8. The countability paradigm in Riffian.

	Singular		Plural
	Feminine	Masculine	Feminine	Masculine
Uncountable/Collective	t-${\varnothing }_{ℱ}$	(w/j)-${\varnothing }_{ℱ}$	N/A	N/A
Countable/Singulative	t-a	a	t-i	i

In order to validate the theoretical proposal presented in Section 2.2.2 and Section 4, the dataset will be widened by adding another grammatical feature from other synthetic languages. We will also seize the opportunity to observe another property of the morphological system referred as morphological anti-asymmetry.

5. Cross-Linguistic Perspective and Model Validation

In Riffian, the collective marker is chosen, in place of the singulative marker, to be the unmarked form of PNs and we claim that other languages, such as English or French, encode similarly PN with a null marker by opting for the collective instead of the definite marker. However, with other linguistic varieties, like in Modern Greek, the definite article is, conversely, preferred as the unmarked form of PN. The fact that some languages can affix the definite article to PN pleads in favour of our hypothesis that other languages are favouring the collective meaning with PN; which means a commutative relation exists between the collective and the definite marker in the template. Thus, the markers of PN are not reduced to gender and number in these languages; countability is also included. One may find this contradictory, since, according to them, the representation of the non-cognitive world should be the same for any person and, therefore, languages must encode the mental representations in the same way. However, we would like to underline for their attention that the grammatical markers do not conform to reality. Let us refrain from looking for a perceptible relationship with tangible materiality. Any item has to be paired with a template. This morphological template takes precedence over any other aspect of the linguistic system and shapes the morphology.

Thus, in keeping with the inter-word variations, let us consider also the inter-process morphosyntactic variations. As seen in Riffian, depending on the encoding and the direction of derivation, a proper noun, even an uncountable noun can be marked differently from its congeners. A similar pattern is found in other languages, but the unmarked and marked forms at stake are not identical. If we take as an example Germanic and Romance languages, the definite article can appear with PN (e.g., the Pentagon), while most PNs are undetermined. Similarly, deonymised PNs (e.g., In French: la bougie ‘candle’ from the toponym Bougie, a Berber city in Algeria) bears the definite article, whereas the original PNs do not have one.

The presence of this morpheme with these different nouns is caused by different factors. First of all, we should admit that CN, especially C, and PN are two different subcategories having distinct unmarked templates in these languages. The unmarked form of C is the definite marker. Concerning PN, its unmarked form is the collective marker. The collective meaning attached to this kind of PN (e.g., Bougie) is unformed and has a fusional encoding (see Figure 4a and

Table 9. Examples of inter-process morphosyntactic variations in some European languages vis-à-vis the semiotic encodings.

	Semiotic Archetypes
	A	→	B	→	C
PN	Bougie				The-Pentagon
CN			la-bougie the-pentagon

). Such semiotic encoding allows the definite marker to be affixed to the CN when a PN is deonymised. As for the PNs in possession of a definite article (e.g., the Pentagon), they are systematically onymised CNs. The word-formation process existing between these two subcategories is the conversion. As the underpinned process is from CN to PN, the different unmarked forms present in the template bound to the CN will be transmuted into the PN (viz., the cloning process and morpheme flow, see Section 2.2.1). This marker in such circumstances is expletive (see Figure 4c and Table 9). This explains why some PNs have the definite article even though the unmarked form of PN should be the collective meaning only.

As seen here, there are languages where PNs and CNs are marked in the same way with respect to definiteness and countability (e.g., in Modern Greek, the definite article is the unmarked form for all the nouns). Other languages consider (some) U and PN alike, and the latter classes contrast with C. However, these are not the only possible types of markedness overlapping. Another interesting typological example is when U and C have the same unmarked form and PN differs from them. In such a case, there are exceptions to the rule. Some PNs may be marked as U and C (e.g., in French, some toponyms, especially the country names, bear the definite article). From these observations, several facts emerge. Firstly, some languages may have different unmarked templates for PN that are not interferences caused by a process of word or meaning’s formation as described before (i.e., CN → PN). Moreover, it entails from this typology that between U and PN a cognitive junction exists. Thus, a PN category cannot have a grammatical marker that is not used with U—the forms may be different, but they express the same sense. For instance, the case where U uses, as an unmarked form, only the collective sense and, in contrast, PN, only the definite meaning is unattested. This particularity may be named morphological anti-asymmetry; in other words, they can be identical, dissymmetric (e.g., U_col & U_def vs. PN_col; U_col vs. PN_col & PN_def), but never asymmetric. Therefore, we may postulate that some semantic constraints bring U and PN closer. This anti-asymmetric specificity may be conceived as an identity property, of which the mathematical formulation is presented below:

Property 3. 

• There exist $w_1$ and $w_2$ forming a pair with morphological templates, such as $h\left(w_1\right)$=$t_i$, $h\left(w_2\right)$=$t_j$ and $i\ne j$.
• There exist $w_1$ and $w_2$ belonging to two different linguistic categories, such as $w_1\in W_X\subset W$, $w_2\in W_Y\subset W$, $W_X\cap W_Y=\varnothing$, and both are related to the same syntactic category $s\in S$, such as $s\in t_i$ and $s\in t_j$.
• Let the set ${\left\{g_i\right\}}_{i\in \mathbb{N}}$ correspond to the attributes in $t_i$ and $t_j$ that must be identical, such as $g_i\in \mathrm{\Gamma }$ is a binary grammatical feature and ${\left\{g_i\right\}}_{i\in \mathbb{N}}$⊄$t_i$Δ$t_j$
• There holds that $|t_i\setminus {\left\{g_i\right\}}_{i\in \mathbb{N}}|$−$|t_j\setminus {\left\{g_i\right\}}_{i\in \mathbb{N}}|$= 0, such that the set ${\left\{g_i\right\}}_{i\in \mathbb{N}}$ contains at least one element.

Finally, this study has reached several conclusions. Our theory is based on evidence coming from different languages. This model, expressed mathematically, assumes the modularisation of the noun category into cognitive sets (U, C, NA, etc.) that have each their unmarked template. Furthermore, the processes of word and meaning formation interfere with the morphological template as a whole and not with individual markers. Its advantage is to offer a minimalist framework enabling us to carry out a unified analysis of the irregular markings met in distant and very different languages.

6. Conclusions

This work addressed different topics that started with the dissymmetrical marking met among nouns. Throughout this paper, differences were shown to occur between CNs and PNs, but also between uncountable and countable nouns. These results are crucial as there have been few extensive studies on the PN or uncountable category in Berber varieties. We have posited fundamental elementary criteria that allow us to resolve this morphological dilemma. One of the goals of this study was dedicated to clarifying the particularities of the Riffian morphology and questioning the traditional view of the state. This was necessary in order to resituate these morphosyntactic attributes within a broader cross-linguistic perspective.

By cross-referencing different facts in Riffian, it seems that the initial vowel a- prefixed to the radical is related to nouns that can express individualisation and pluralisation of a referent. It was demonstrated that this morpheme is a singulative marker and that, when it is not present, the collective meaning is conveyed instead. Only countable nouns (and nouns of action) can be affixed with this grammatical feature, while proper nouns and uncountable nouns are categories that ignore this marker. This is particularly obvious, especially, with uncountable loanwords. The latter can never get this marking. However, when they undergo a countabilisation process by becoming countable nouns, the singulative marker is required to inflect with the radical. The collective meaning can be an unformed marker, this particularity allows us to shed light on the existence of the feminine marker with some uncountable nouns in initial position. Moreover, we propose to incorporate both markers into the grammatical class of countability. The general canonical form of a noun43 is rewritten as follows:

Despite that, this marking can cut across these separated categories, notably with non-borrowed nouns. From this empirical observation, we implemented a theoretical model that is aligned with the modern Word and Paradigm approach. This model is based on a mathematical formalism. The original aspect of this formal computative lexicon is that the morphosyntactic transfer function is regulated by the agency of the set of retrospective determinants of shifting template (or K). The latter ensures that any item is paired to a template in a deterministic way. This is carried out by the backward recursive and gradient functions that structure the mapping between radical morphemes and their morphosyntactic features. Another non-negligible element of this model that contributes to understanding certain morphological evidence is the theoretical algorithm of lexical derivation. It was mainly postulated that, during a lexical derivation, a cloning process occurs that leads to the transfer of the morphological (and phonological) features of the input item to the output one.

Based on these latter dispositions, we established a model of word and meaning formation for proper and uncountable nouns to explain this morphological abnormality. Several prototypical process chains of template shifts were put forward. Each of them illustrates successive directional derivatives at a given item. It was discovered that the conversion and semantic shift are the cause of the presence of the singulative marker with some uncountable nouns and the collective with a few countable nouns. The reflection was also extended to other languages to demonstrate that what happens in Riffian is not isolated. What may appear as an insular feature among the world’s languages is, in fact, commonly attested. It was claimed that other synthetic languages have in their grammatical features the countability paradigm.

The minimalist framework is proposed to unify the diverse morphological realisations found in synthetic languages. These variations that can be situated at the morphosyntactic shift or inter-word level have been clearly distinguished. The inter-word variations are mainly provoked by the modular templates to which items are attached, while the inter-process morphosyntactic variations are caused by the encoding of some PNs and borrowed Us having a fusional encoding instead of an analytical one. Thus, we were able to fuse different grammatical markers/paradigms, observed universally, that can be predicted by the Template-Based and Modular Cognitive model. Definiteness and countability were the main linguistic features analysed to highlight the power of this model. It was also advanced that countability should be regarded as a grammatical paradigm used cross-linguistically. In addition to the formal mathematical expressions presented in Definitions 1–3, Proposition 1 and Properties 1–3, this model comprises several key elements:

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Unmarked modular template: the template set can encode several templates imposing the grammatical forms that the words must contain.

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Modular cognitive set: the items are assigned to cognitive sets that regulate the pairing of items and templates.

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Formedness: a marker can be formed or unformed (no phonological representation).

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Semiotic encoding: a meaning can be fusional, i.e., it shares a form with another meaning or can be analytic, i.e., it is bound to a different form.

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Prototypical process chain of template shifts: the establishment of such a model clarifies and defines the unmarked form of each modular category without paradox.

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Template shift: depending on gradient conditions, a template shift occurs when passing from one cognitive set into another one.

The purpose of this paper was to discover why different lexemes among nouns stand out in various languages. In light of this study, our findings revealed that we can reconcile those nominal categories under a theoretical model of morphosyntactic marking. It was demonstrated that the nouns share the same morphological template. Some linguistic varieties prefer to use, for example, with the PN category, the collective as the unmarked form instead of the definite or singulative marker. This distinctiveness with respect to markedness is not unique to some PNs. As predicted by this model, other nouns may have different unmarked forms, especially regarding gender. Thus, this alternative approach enables the unification of different morphosyntactic realisations used by synthetic languages. Moreover, it was discovered that multiple cognitive properties shaping the morphological system. We have chosen to focus on some grammatical paradigms; by retaining these linguistic units, we do not pretend to have exhausted the subject. Other particularities might have been added, but that would have lengthened the present paper considerably. Our contribution, which relates to relatively uncharted themes, is a first step towards understanding the underlying patterns of morphosyntax. This mathematical model offers invaluable insights and paves the way for future modelling of different nonlinear dynamical phenomena in other domains of study.