2.1 Paradigms

Paradigms represent information about the pairing of grammatical properties and linguistic forms. Thus, a paradigm can be seen as a list of cells representing an inventory of linguistic forms ⟨x,y,z,…⟩, each paired with a unique property or combination of properties or features (Stump 2016). As already mentioned in the introduction above, we explore an approach where the order that the cells are presented in plays no role in the morphology. That is our proposal applies to any set of cells regardless of whether a one-dimensional linear order of the cells is assumed or some multidimensional arrangement of the cells. For presentational purposes, we assume there is a conventional linear order for a given inventory of cells, and thus for the corresponding paradigms (lists of forms). As a simple illustration, partial case paradigms for selected German pronouns can be given as in (3), where the cases are presented in the order in (2):

The German pronouns further illustrate a property that is central to the study of paradigms, namely, syncretism, the many-to-one mapping from features or properties (like case) to exponents (phonological forms) seen in (3b)–(3c). Where the first person singular pronoun is characterized by a three-way contrast, the first and third person plural forms each show only a two-way distinction in form, corresponding to a three-way distinction in grammatical properties. Describing these patterns as syncretic constitutes a claim that the identity of form is represented grammatically, and is thus distinct from accidental homophony (see Harbour 2008; Sauerland & Bobaljik 2013). In the notation of the previous section, a syncretic pattern ABB, as in (3b), has only two listed forms (A=wir, B=uns), and the grammar codes the fact that the B form occupies the second and third cells of the paradigm. Accidental homophony is the state of affairs where the grammar lists three forms (ABC), but two of those forms simply happen to have the same phonology. Our investigation concerns only syncretism, although we recognize that it is in practice a notorious analytical challenge to identify a sharp dividing line for the analysis of specific data points.³

It is widely held that syncretism is central to investigating the nature and inventory of features as in (2). That the accusative and dative forms of the 1PL pronoun are syncretic suggests that those two cells share a feature, a property not represented in (2).⁴ In this manner, comparing the range of attested and unattested syncretisms over some sufficiently large sample may reveal the underlying inventory and organization of the relevant features.

When we abstract away from the details of particular forms, the study of syncretic patterns is at its core the study of partitions of a set, a well-defined mathematical concept. The number of distinct partitions for an n-celled paradigm is the Bell number: B_n. For a three-celled paradigm, the B₃=5 distinct partitions are listed in (4a). The same information is displayed graphically in (4b) to emphasize that the absolute values (A, B, etc.) are not relevant; all that matters is sameness or difference of cell contents:

(4)	a.	AAA, AAB, ABA, ABB, ABC
	b.

The Bell number grows very fast. There are B₈ = 4,140 logically possible partitions of an 8-cell paradigm, and a 10-celled paradigm space (arguably, the number of cases in Russian, see Corbett 2008) has 115,975 possible partitions. In medium- to large-scale studies of syncretism (Cysouw 2003; Bobaljik 2012; Baerman et al. 2005; Harbour 2016), it is commonly observed that only a subset, often only a very small subset, of the theoretically distinct partitions are attested. For example, Cysouw (2003; 2011) considers a sample of person paradigms drawn from more than 250 languages, characterized as an 8-cell paradigm space,⁵ but finds only 60-some-odd distinct partitions from among the more than 4,000 logical possibilities. The *ABA generalizations, mentioned above, make the same point: over some sizeable range of data, where 5 patterns are possible, only four are actually found in the world’s languages: AAA, AAB, ABB, and ABC, but not ABA. Typically, studies of syncretism seek explanations for such typological patterns—i.e. develop theories that predict only a subset of partitions to be possible. In what follows, we address exactly this problem but one level of generality higher—we investigate how general assumptions about morphological analysis restrict which subsets of partitions can arise as typological predictions. For example, we show that a restriction to the partition set {AAA, ABB, ABC} cannot be derived solely within our general assumptions, while the *ABA condition can be derived. In this, we hope here to make new contributions in the formal investigation of the restrictiveness of competing models.

2.2 Features

In order to make headway on these issues, we propose to start by presenting a largely theory-neutral means for representing features. Our notation allows us to express any of numerous competing assumptions about features and feature-logics, allowing us to then compare them directly. We start by recognizing that at its most basic, a feature is a name for individual cells or sets of cells in a paradigm. With reference to an n-celled paradigm, we write a feature as f indexed with a binary vector, where 1 indicates the cell or cells that feature names. Thus, one way of naming features that generate a 3-celled paradigm is as in (5), with a unique feature naming each cell.

We define a model of a given paradigm as having two components: an inventory of features, and rules of exponence, which relate features to form. Alongside the simple feature inventory in (5), we may state the rules of exponence in (6):

Each model is a grammar (fragment), generating one paradigm. In the trivial example just considered, the model comprising (5) and (6) generates an ABC partition—a three-celled paradigm that is maximally differentiated, i.e., in which each cell has a distinct form. Any number of examples of such an approach can be found in morphological descriptions. The description of the German 1SG pronouns in (3a) could be expressed in these terms. Three unanalyzable case features are assumed: f₁₀₀= “nominative”, f₀₁₀ = “accusative”, etc., and each one is associated with exactly one exponent, yielding the maximally differentiated paradigm. Analysis of personal pronouns that use three unanalyzed features like “first person” (f₁₀₀), “second person” (f₀₁₀), and “third person” (f₀₀₁) also instantiate this schema.⁶ Any maximally differentiated paradigm can be expressed in these terms. In fact, from the inventory in (5), the maximally differentiated partition ABC is the only complete partition that may be generated. (By complete, we mean that a phonological form is assigned to every cell of the paradigm space.) Using only rules of exponence of the format in (6), only maximal differentiation is possible, because the cells share no features in common.

But as we have already seen, maximal differentiation is by no means the only way in which an n-celled paradigm space may be partitioned. Thus a feature inventory that is restricted to generating the maximally differentiated partition provides no purchase for an account of syncretic patterns, such as ABB, AAB and the like, other than via accidental homophony.⁷

Characterizing syncretic partitions such as the ABB pattern seen in (3b) thus requires features that name (contain) more than one cell of the paradigm such as f₀₁₁. Consider, from this perspective, the inventory in (7), which represents the standard approach to *ABA generalizations in the notation used here:

This encodes the same relationship among paradigm cells as (1). One feature is shared by all three cells ((7c)—this constitutes the default), one by two, and one is unique to a single element. On the assumption that the feature inventory in (7) remains constant across languages, but that the rules of exponence may vary from language to language or even from lexeme to lexeme, a variety of different paradigms (partitions) may be generated from this single, shared inventory of features. Rules of exponence for two models sharing the inventory in (7) are given in (8) and (9).

As the reader may verify, the model in (7) + (8) derives a maximally differentiated, ABC paradigm, one in which each cell is distinct from the others. The model consisting of (7) + (9) derives an ABB paradigm, with syncretism of the last two cells. In these models, rules of exponence are ordered sequentially (read by convention from top to bottom), and disjunctively – the first rule of exponence specified for any given cell must apply to that cell, and once one rule has applied, no other rule may apply. In both (8) and (9), the final exponent (A) is the default – in principle it is compatible with all three cells – but it does not appear in any but the first cell because the rule introducing the default is ‘blocked’ by the application of the more specific rules.

Returning to our German pronoun example, the feature inventory in (7) (unlike the one in (5)) could then support the description of each of the pronouns in (3); the 1SG pronoun using rules of exponence corresponding to (8) and the 1PL to (9). We may represent this outcome more compactly, by listing the features realized by rules of exponence in the feature-based morphological analysis of a given data set as a sequence of features (ordered left-to-right, rather than top-to-bottom for compactness). (10a) represents the ordered rules in (8) as a sequence and (10b) that in (9) ((10c) derives (3c) from the same feature inventory). The string to the right of each sequence characterizes the partition defined by that sequence.⁸

The presentation in (10) expresses the fact that several different partitions are derivable from the common feature inventory in (7), by invoking different sequences of features in the rules of exponence.

2.3 Partition Sets

This now gives us the tools we need to introduce the main object of inquiry, namely partition sets:

As we have seen, the partition set of (5) is trivial: PS₍₅₎={ABC}. That is, the inventory in (5) will generate all and only maximally differentiated partitions. The partition set of (7) is more interesting, and (10) represents only a subset. To see this, consider all the sequences generable from (7). Since there are 3 features, there are 3! = 6 (total) sequences to consider, as in (12) (we explain the use of blue font presently):

Collecting the derivable partitions, we find that PS₍₇₎={AAA,ABB,AAB,ABC}. Notably, of the B₃ = 5 possible partitions of the three-celled space, one is missing: ABA is not included in the partition set of (7).

A different feature inventory may yield a different partition set. For example, adding the default feature f₁₁₁ to the inventory in (5) renders the inventory unrestrictive: any logically possible partition may be derived. The following partial list of sequences, from among the 4!=24 possibilities, demonstrates this point:

In this way, we see the general logic that relates typological generalizations to conclusions about features in Universal Grammar. The data we have are the attested partition sets in some domain—the range of partitions that are (un)attested cross-linguistically. The explanans is then the feature inventory: Domains in which the *ABA generalization holds are domains in which one logically possible partition is not attested. The gap is explained if Universal Grammar admits only the feature inventory in (7)—as (12) shows, the unattested partition is not in the partition set of this inventory.

Our aim in this article is to attempt to approach the issues here from the other direction. In the following, we use the notation introduced here to explore the consequences of various kinds of formal restrictions one could conceivably apply to models of this sort (inventories of features and associated rules of exponence). We do so in the first instance entirely in the abstract, with no connection to substantive features or empirical data. Our goal is to better understand some of the formal properties of feature logics, and to compare the ways in which various intuitively plausible assumptions do and do not restrict the combinatorics.

That is, rather than starting with some observed partition sets and attempting to infer the feature inventory (a task for which there are often multiple solutions), we investigate here ways in which general constraints on feature algebras do (or do not) restrict the hypothesis space. In other words, rather than arguing that something like (7) is a plausible feature inventory in some domain because it derives the observed facts, we will look instead for general reasons which may favour an inventory like (7) over other inventories on a priori grounds.

One reason to pursue this exercise is that, as our notation calls attention to, absent any prior assumptions about the content of features, the number of possible features that can be defined grows quickly. For a paradigm of n cells, there are 2ⁿ – 1 non-empty features that may be defined. For a three cell paradigm, the 7 definable features are these:

If features could be freely chosen to form inventories, then 128 distinct feature inventories could in principle be constructed from these features (including the empty set). For a four cell paradigm, there are correspondingly 15 features and 32,768 possible inventories to consider. As we have seen above, from each inventory, a number of distinct models can be constructed. That is, each inventory can be mapped to one or more sequences, thus yielding a variety of partition sets. If rule ordering is unconstrained, then from a single inventory with n features, there are n! distinct sequences that may be so constructed.⁹ The number of possible models (and thus grammars) thus quickly becomes astronomical, and we suggest it is therefore important to ask whether there may be some universal constraints that drastically restrict the classes of possible models to be considered. Thus, we will spend a fair part of the following discussing the combinatorics involved. We will approach this as follows: Using the understanding of features, paradigms, and models outlined above, we will set out to explore in quantitative terms various conditions that may be imposed, and show explicitly how they do and do not restrict the space of possibilities. Many of the numerical results are non-obvious, and we provide the code in on-line supplemental materials for this paper.

Before proceeding further, by way of a brief housekeeping remark, we note that some of the features in some sequences are redundant. The redundant features in (12) are indicated in blue. Because sequences are ordered, once each cell has been assigned an exponent, all further features will have no effect in characterizing the partition. That is, a feature is redundant in a sequence if it is bled by earlier rules of exponence, and thus in principle cannot be exponed. Eliminating a redundant feature from a sequence is indistinguishable from the sequence with that feature (compare the notion of inessential feature in Kracht 1997; Pullum & Tiede 2010). The sequence (12d), for example, is formally indistinguishable from the partial order in (10b), since the first two rules are sufficient to cover all of the cells.¹⁰ If no feature in a sequence is redundant, we call it redundancy-free.

The careful reader may have noticed that in presenting the range of sequences generable from one inventory in (12) we gave only sequences that represent total orderings among the features of the inventory. However, in our exposition above, we also included sequences that contained only a subset of the features (as in (10)). It turns out that consideration of the total sequences is sufficient for calculating the range of partitions generated by an inventory, under the assumption of completeness, which we may define as follows:

The partial sequence in (10) is complete, but the partial sequences in (16) are not (each leaves one cell without an exponent):

In general, as a simplification in what follows we will consider only non-redundant complete sequences, since this class is sufficient to exhaustively characterize the partition set of any inventory.¹¹

2.4 Restricting the hypothesis space: the minimal valid inventory

Here, we define briefly the two conditions, and note a third assumption, that will be central in the investigation that follows. We suggest these are a priori plausible conditions to restrict the class of possible inventories, and we will work through their consequences in detail in sections 4 and 5 and appendix A below.

The first, basic condition is that an inventory be valid.

The maximally distinct partition of a paradigm (space) is the partition in which each cell is distinct from every other cell. In other words, for a three cell paradigm, a valid inventory is one for which there is some set of rules that will derive ABC. Note that Validity is a property of inventories, not models (grammars). The rules of exponence in (8) demonstrate that (7) is a valid inventory, but we do not require that every grammar (model) generate ABC; syncretism by definition precludes there being such a requirement for every model.¹² The model consisting of (7)+ (9) is perfectly well formed (and examples apparently conforming to such ABB patterns are widely instantiated).

The second, and more interesting condition on inventories is Minimality:

In Section 4 and Appendix A, we will work through these conditions for various sizes of paradigms, starting with 2-celled and then 3-celled paradigms. One finding in this paper is that the two simple assumptions on inventories just noted – that inventories use the minimal number of features to describe a paradigm space – have the curious effect that in certain paradigm spaces, notably those with three cells, certain patterns of syncretism become unstatable. In a sense to be made clear below, ABA patterns of a certain type are indescribable. More accurately, no minimal valid inventory yields a paradigm set that includes all three bifurcations of the three celled space: {AAB,ABA,ABB}. If two are included, the third is not. Since we have treated order in a paradigm as arbitrary, all of the results we describe hold only up to linear permutations in this way. This result is of interest, because it arises without the nesting/containment assumption that plays a central role in other treatments of *ABA generalizations (Bobaljik 2012; Caha 2009; Starke 2009). Another result is a curious pattern in the nature of the restrictiveness that these assumptions create.

Lastly, at least to start, we will assume following standard practice in morphology that intersection of features is available. If f_a and f_b are in an inventory, then f_a ∩ f_b ↦x is a well-formed rule of exponence. (I.e., in more standard notation, if [F] ↦ A and [G] ↦ B are well-formed rules of exponence, then so is [F,G] ↦ C.) Most feature-based morphological analyses invoke this (for example, if [FEMININE] and [PLURAL] are features in the inventory, then there can be an unanalyzable exponent of [FEMININE,PLURAL], without needing a separate feature [FEMPL]). We assume that intersection is the only Boolean operation on features that is available (but see below for further discussion).

Adding intersection is not innocuous. Because of the way we have defined features and inventories, intersection intersects with Minimality in a non-trivial fashion. Intersection, like rule ordering, allows for exponents that do not directly conform to features that are in the inventory. If an inventory consists only of f₁₁₀ and f₁₀₁, a rule can be stated referring to: f₁₁₀ ∩ f₁₀₁ = f₁₀₀. While this generates an exponent that only expresses the first cell, it does so without the feature f₁₀₀ being contained in the inventory. This will play an important role in the discussion of 3-cell paradigms. Of course, it is worth considering the consequences of minimality and validity without the additional assumption that intersection is available. We do so in section 5.4 below. Note that the core result holds either way, but the inclusion of intersection is a widespread assumption in morphology, so we consider that scenario first.¹³

Before proceeding to the discussion of our main results, we will present a number of additional concepts and assumptions which we hope will allow for more familiarity with the general notation. In particular, we offer some remarks on how various current ideas can be rendered in our notation, allowing for commensurability among analyses or frameworks. The reader interested primarily in the consequences of the assumptions just made can skip ahead to section 4.

3.1 Extrinsic Order and Pāṇinian Sequences

Above we have noted that sequences (i.e., rules of exponence) must in some cases be ordered. The sequences in (19) contain the same features, but the difference in order alone yields different partitions:

As is well known from early discussions of rule systems, rule ordering may be extrinsic (a stipulated language-particular order, as in (19)) or intrinsic, i.e., such that more specific rules automatically bleed more general rules. A specific formulation of the intrinsic Pāṇinian ordering principle or Elsewhere Condition is as in (20) (after Kiparsky 1973):

We translate the operative notion into our set up as in (21), which picks out the class of sequences for which any reordering that does not introduce redundancy has no effect on the partition set. That is, a Pāṇinian sequence is not necessarily a total order, but all order with any consequence is determined by (20).

The sequences in (19) do not satisfy (21). Since (19a) and (19b) are permutations of one another and yield different partitions, neither of them is Pāṇinian. Other than (12a), the sequences in (12) likewise cannot be Pāṇini-sequences since they are not redundancy-free. But the redundancy-free sequence in (22a) is Pāṇinian because it and its only redundancy-free permutation in (22b) yield the same partition: ABC.

This raises two points. First, Pāṇinian ordering is the kind of general condition one could entertain as a restriction on rule systems. As the comparison of (19) and (22) shows, imposing a condition that sequences must be Pāṇinian may reduce the partition set for some inventory I by excising all partitions that are derived only by non-Pāṇinian sequences. Rather than build this assumption in, we see our goal as investigating the effects of assumptions, since we have a notational apparatus that allows us to directly compare systems with and without such an assumption. As it happens (we will present this in more detail below), imposing Pāṇinian ordering will have a drastic effect on minimal feature inventories that are closed under intersection, essentially preventing analysis of syncretism in the three-cell cases.

Now consider again the observation that the sequences in (19) are not Pāṇinian, but those in (22) are. The features invoked in these sequences are not unrelated. The features f₁₁₀ and f₀₁₁ are common to all of these sequences, and the relation between them is that they have partial overlap, but neither is contained in the other. Generally such a relationship between two features f_a and f_b is what makes a sequence non-Pāṇinian, unless there is another feature or other features fc₁, …, fc_n that cover the intersection of f_a and f_b and are contained within both f_a and f_b. One easy case is that there is only one other feature f_c (i.e. n = 1), namely the intersection or conjunction of the two features f_a and f_b. This is what we see in (22): adding the feature that corresponds to the intersection of the two features in (19) renders the sequences Pāṇinian. Generally, if a redundancy-free sequence is closed under intersection, then it is Pāṇinian. This provides another reason for us to include intersection: allowing intersection makes it easier to compare intrinsically ordered and Pāṇinian analyses. It does, though, raise the question of whether any other Boolean operations on features should be countenanced.

3.2 Feature Algebra: Privativity, Defaults, Containment, and Intersection

While all Boolean operations are generally assumed to be available in semantics, most work in morphology assumes that the feature algebra is restricted and that e.g. the union operation is not available.¹⁴ As noted, we adopt the assumption that intersection, but no other algebraic operation is part of morphology (we do however consider systems without this assumption in section 5.4). We briefly mention some alternatives in the following subsections.

5.1 Restrictions

We now consider the degree to which the assumptions mentioned above restrict the space of possible grammars (analyses).

5.1.1 Validity

The first restriction we impose is Validity, as in (17). For example, the inventory f₁₀₀, f₀₁₀, f₀₀₁ is valid, in that it describes a three-way contrast, while the inventory f₁₀₀, f₁₁₀, f₀₁₀ is invalid – it is not complete, as it provides no means to describe the third cell. It turns out that 96 of the 128 possible inventories of active features are valid in this sense in the three cell case (see Table 3 below). With four cells, the ratio is 31,962 out of 32,768 (see Table A2 below). Validity thus restricts the number of feature sets, but the restriction is not particularly strong.

Table 3

Table showing which 12 of the 16 logically possible three cell partition sets can be generated by feature inventories (applying intersective closure) and by how many inventories. Partition sets related by cell permutation are grouped together.

2 features, order	–	–	–	–	–	1	1	1	–	–	–	–	–	–	–	–
3 features, order	1	2	2	2	–	3	3	3	–	–	–	1	3	3	3	3
4 features, order	–	1	1	1	–	3	3	3	–	–	–	3	2	2	2	14
5 features, order	–	–	–	–	–	1	1	1	–	–	–	3	–	–	–	15
6 features, order	–	–	–	–	–	–	–	–	–	–	–	1	–	–	–	6
7 features, order	–	–	–	–	–	–	–	–	–	–	–	–	–	–	–	1
total with order	1	3	3	3	0	8	8	8	0	0	0	8	5	5	5	39
2 features, Pāṇini	3	–	–	–	–	–	–	–	–	–	–	–	–	–	–	–
3 features, Pāṇini	4	4	4	4	–	–	–	–	–	–	–	1	3	3	3	3
4 features, Pāṇini	–	3	3	3	–	1	1	1	–	–	–	3	2	2	2	14
5 features, Pāṇini	–	–	–	–	–	1	1	1	–	–	–	3	–	–	–	15
6 features, Pāṇini	–	–	–	–	–	–	–	–	–	–	–	1	–	–	–	6
7 features, Pāṇini	–	–	–	–	–	–	–	–	–	–	–	–	–	–	–	1
total Pāṇini	7	7	7	7	0	2	2	2	0	0	0	8	5	5	5	39

5.1.3 Order and the Effect of Pāṇini

Note that in order to describe the ABC pattern, the rules of exponence must be (partially) ordered, such that the exponent of the conjoined features takes preference over the rules in (29b)–(29c) (this holds for any of the three inventories in (28)). The property of Order was not relevant in the two-cell paradigm, but the Pāṇinian order condition has a strong effect with the valid three cell inventories.

Because each of the inventories in (28) has two basic features that may be conjoined to define a third feature, the number of possible sequences for each inventory is 16, although many of these sequences will be redundant or incomplete. The order (29) is in addition to deriving the complete partition also Pāṇinian as defined above. To see this consider first that if the order of (29b) and (29c) is changed as in (30), the resulting sequence still derives the complete partition ABC.

(30)	a.	f110 ∩ f011 = f010 ↦ B
	b.	f011 ↦ C
	c.	f110 ↦ A

Other orders of the rules in (29) render feature f₀₁₀ redundant. For example, if the order of the first two rules in (30) is exchanged, then the rule f₁₁₀ ↦ A will assign the exponent A to the first two cells, bleeding rule (31b) (i.e., rendering rule (31b) redundant).

(31)	a.	f110 ↦ A
	b.	f110 ∩ f011 = f010 ↦ B
	c.	f011 ↦ C

As a general property (well understood from studies of Rule Ordering), ordering f_a before f_a ∩ f_b will render the conjunction redundant, and is thus equivalent to not selecting (or having no rule referencing) the conjoined feature. This corresponds to an intrinsic order: if the conjoined rule is active, it must be ordered before its individual conjuncts.

If the conjoined rule is omitted or not ordered first, it can be omitted and only the order between the two rules referring to the basic features matters. The resulting sequences are not Pāṇinian, but require extrinsic order. (32) yields AAC while (33) yields ACC.

(32)	a.	f110 ↦ A
	b.	f011 ↦ C
(33)	a.	f011 ↦ C
	b.	f110 ↦ A

The following table shows, for one inventory, the six possible sequences (six distinct orders of three rules) and the three corresponding partitions that are derived. (As before, redundant elements in the sequences are in parentheses). The analogous table for the other two choices can be readily constructed. As an expository device, we use green text to indicate a feature that is derived as the intersection of the two basic features. As explained in section 2.4, the green features are not part of the feature inventory, but are a convenient abbreviation for rules of exponence that make reference to the intersection of two features in their structural description.

(34)	inventory	sequences	partition
	f110, f011	f010, f110, f011	ABC
		f010, f011, f110	ABC
		f110, (f010), f011	AAC
		f110, f011, (f010)	AAC
		f011, (f010), f110	ACC
		f011, f110, (f010)	ACC
			3

What (34) shows is the following: There are (only) three minimal valid feature inventories that can generate a maximally differentiated three-celled paradigm space. One such inventory is {f₁₁₀, f₀₁₁}. From that inventory, 6 (=3!) sequences may be formulated, where each sequence is a distinct, total ordering of rules of exponence for the two features and their intersection.²⁰ While there are six rule orderings possible, only three distinct partitions are generated. The first two lines in (34) derive the same surface patterns (partitions), since the ordering of the last two rules is irrelevant.

As the reader may verify, the other two minimal valid inventories (in (28)) have the same properties as (34). The three inventories amount to permutations in the order of the cells, but are otherwise identical in their formal properties. Each inventory generates a partition set that contains only two of the three logically possible bifurcations of the paradigm. Since we have not stipulated a meaningful order of the paradigm cells, the three are equivalent, up to linear order.

The information in (34) is represented graphically in (35):

(35)	universal features	sequences	partition

5.2 Result: Generalized *ABA

At this point, we note two properties we believe to be of theoretical interest. For a three-celled paradigm space, there are B₃ = 5 distinct partitions. However, imposing the conditions of Validity and Minimality on the UG feature inventories restricts the expressive power of the system, such that each inventory generates only 3 of the 5 possible partitions. The three inventories that are permitted are moreover linear permutations of one another. We believe this is of interest since it appears to be true at least in some domains that the number of attested partitions is a small subset of the logically possible ones. The example we noted above was that in the 8 cell division of the person/number space, only 60-some-odd distinct partitions, out of B₈ = 4,140 possibilities, are attested in Cysouw’s 250+ language sample. Being able to predict restrictions on the space of possibilities is thus of potential theoretical interest, if the restrictions indeed line up with the data. In the case at hand, the following restrictions obtain:

Of the five possible partitions of a three-cell space, four show some differentiation among the cells. However, each of the inventories in (28) generates only three of those partitions. As in the case of inventory #3 in the two-celled paradigms, we are now able to connect our formal results to potential empirical evidence. If there is, as we have hypothesized, a fact of the matter for some domain, such that UG contains only one of the inventories in (28), then this should show up as the following empirical generalization: across the relevant domain, only three of the four possible patterns of differentiated partition should be attested. In (34), we show that the inventory f₁₁₀, f₀₁₁ generates the partition set {ABC, AAB, ABB}; that inventory does not generate ABA. No sequence from that inventory will generate a pattern in which the first and last cell share an exponent, to the exclusion of the middle cell.

The same holds for the other two inventories, up to the linear order of the cells: each inventory will fail to generate exactly one of the possible partly syncretic partitions. Inventory f₁₁₀, f₁₀₁ in (28)a generates the partition set {ABC, AAB, and ABA}, but it does not generate ABB. Similarly, the inventory f₁₀₁, f₀₁₁ in (28b) generates the partition set {ABC, ABB, and ABA} but does not generate AAB. As discussed above, since the linear order of the cells is arbitrary, these inventories and partition sets are permutations of one another, and thus each can be reduced to the first via permutation of the cells. For example, syncretism of the accusative and dative, to the exclusion of the nominative (as in (3b)) would be described as an ABA pattern if the order of cases were ACC-NOM-DAT, but if we permute the order of cells, giving the list NOM-ACC-DAT as in (2), then the same pattern is described as an ABB pattern. In this way, there is, in what we develop here, a formal equivalence among partition sets that differ only as a function of linear permutations of the cells. We cannot, in principle, say that *ABA is excluded absolutely (rather than *AAB, for example, since what counts as ABA under one order counts as AAB under a linear permutation), but what we have found is instead a generalized version of *ABA: the three inventories in (28) all exclude precisely one pattern in which two cells are syncretic and one distinct. They either exclude *ABA or are reducible to this by linear permutation alone.

This result is noteworthy in the current context, since it provides a means of characterizing the absence of *ABA patterns without assuming featural containment. Existing accounts of *ABA patterns invoking containment are all built on what, in our terms, is a non-minimal feature structure, with strict nesting of features – some version of: f₁₀₀, f₁₁₀, f₁₁₁.

In other words, what we have just shown has two parts. The easy part is a demonstration that it is possible to derive a *ABA generalization for some domain without invoking containment. We have just done so. The slightly harder part was the demonstration that the type of feature inventory that derives *ABA without containment is not only possible, but is in fact preferred (over containment), if UG makes use of Minimal Valid feature inventories. We postpone until the next section some speculative remarks on whether this result constitutes a plausible alternative scenario for the account of *ABA generalization examples in the literature.

Before that discussion, we note one further point about these inventories. No valid, minimal inventory for a 3-cell paradigm space generates the maximally undifferentiated partition AAA. Curiously, it is not a general property of our assumptions that such undifferentiated partitions are universally excluded in the minimally valid inventories, and we show below that it does not hold for four cells. We can say that at this point that the undifferentiated partitions are excluded when the number of cells is from the sequence 2ⁿ – 1 for n ≥ 2, i.e. 3, 7, 15, …. We note this, but leave it as an unexplored aspect of the system. Total syncretism appears to exist, of course, and we do not exclude it across the board. We return to this issue again in section 5.4, where we show that giving up the assumption that intersection is always available will preserve the generalized *ABA result considered here, but will admit AAA patterns. The upshot of that section will be that the (equivalent of the) 3 inventories considered to be minimal valid inventories with intersection become three among a larger class of minimally valid inventories (including the containment patterns). Some inventories from among the larger class permit AAA, but the general result holds: no member of that larger class admits all three bifurcations of the paradigm space: any minimal valid inventory whose partition set contains ABB and AAB will necessarily exclude ABA.

5.3 More on the 3-cell space: Non-Minimal sets

Thus far, we have examined only the three minimal valid inventories that generate a three-cell paradigm. To evaluate the effect of minimality, we now look also at non-minimal inventories. In the two-cell case, we were able to present a complete discussion of all the possible inventories and of the partition sets described by each inventory. There were only 8 possible inventories for the features definable over a two-cell paradigm, and 4 inventories were invalid. But for a three cell space, there are 128 inventories, and numerous sequences to consider.

Table 3 provides a summary of important aspects of the grammar of three-celled paradigms and the models that generate them. In the next paragraphs, we walk through this table in some detail, identifying various properties that are of potential interest. Among these, we note that imposing Pāṇinian ordering—limiting all models to intrinsic rule ordering—turns out to have rather drastic consequences. Possibly of more interest, we note that there are some partition sets that do not arise under any constellation of the assumptions considered here. Even without Minimality, for example, feature inventories turn out to be somewhat restrictive.

Table 3 is divided horizontally into two halves. Each half tabulates all the valid feature inventories, and counts inventories grouped by the number of features they contain (y-axis) × the partition sets that may be generated from them (x-axis). The two halves of the table differ as follows: In the top half, it is assumed that extrinisic order of rules of exponence is permitted, while in the bottom half, we add the additional assumption that only intrinsic (Pāṇinian) ordering is permitted. We discuss the differences below.

The columns in Table 3 represent possible partition sets of a three-cell paradigm space, using colour instead of letters, as in (35) above: the same colour in two cells indicates the same exponent (syncretism). There are B₃ = 5 distinct partitions (the rightmost column) and 16 different subsets of partition that contain the maximally differentiated partition (ABC = dark orange, light orange, light purple).

The header of each column represents a distinct partition set, and the number in a given column represents the number of formally distinct (valid) inventories that can in principle generate that set. The three minimal valid inventories that we have discussed above are in the top row of the top half of the table (columns 6–8). These are the only three valid, two-feature inventories. But the table provides a range of information about what happens if we do not include the minimality requirement.

In the leftmost column of the line “3 features, order”, one finds the number 1. Assuming extrinsic rule ordering is allowed, there is exactly one choice of an inventory with three features, from among the 7 possible features, which yields only an ABC partition. We have seen that already; it was the inventory in (5). If that inventory is chosen, from among the 128 possible inventories, then the only partition that can be generated is ABC.

On the same line, the number in the rightmost column is 3. There are (exactly) three distinct choices of feature inventories from each of which all five logically possible inventories can be derived. One such inventory is f₁₁₀, f₁₀₁, f₁₁₁, i.e. it is derived from a valid two-feature inventory by adding the default f₁₁₁. The other two inventories are also of this type; i.e., the two linear permutations of this inventory.²¹

This line also shows that there are 3-feature inventories that generate a partition set which excludes ABA. For example, the third columnn from the right notes that there are three inventories whose partition sets contain ABC, AAB, ABB, and AAA, but not ABA. One of the three inventories which generate this partition set is f₀₀₁, f₀₁₁, f₁₁₁ as we saw above already (the containment inventory). A second possibility is f₁₀₀, f₁₁₀, f₁₁₁ (a linear permutation of the previous one). Finally, the inventory f₁₀₀, f₀₀₁, f₁₁₁ also generates this partition set, but without containment. Furthermore, all three inventories exclude *ABA from their corresponding partition set regardless of whether extrinsic ordering is allowed or not.

Bear in mind that the numbers in this table do not count models or sequences, but count inventories. Other than those in the leftmost column, each valid inventory in the table may be contained in multiple models, thus yielding sets of generable partitions. For example (34) (= (35)) is here coded by the number 1 in the top line, column 7. This is a two-feature inventory that generates the partition set at the top of column 7; moreover, this is the only choice of (two) features which generates that exact partition set (and requires extrinisic rule ordering to do so).

5.4 Deriving *ABA in Intersection Free Systems

As we mentioned in section 3.2 above, assuming that intersection of features is available to rules of exponence accords with standard practice in morphological theory. Systems where the feature set is closed under intersection, combined with the assumptions of minimality and validity, yield tight restrictions on paradigm sets, including one that seems to be of special interest in current morphology and therefore we have focussed so far on systems with intersetcion. In this section we discuss what happens if we drop this requirement. In particular, we show that these allow a different route to derive a generalized *ABA constraint. Table 4 shows an overview of the possibilities for deriving the 16 valid partition sets for the three cell case when feature intersection isn’t available.

Table 4

Table showing which 12 of the 16 logically possible three cell partition sets can be generated by feature inventories (not applying intersective closure) and by how many inventories. Partition sets related by cell permutation are grouped together.

2 features, order

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

3 features, order

1

2

2

2

–

3

3

3

–

–

–

–

3

3

3

–

4 features, order

–

1

1

1

–

3

3

3

–

–

–

3

4

4

4

7

5 features, order

–

–

–

–

–

1

1

1

–

–

–

3

1

1

1

12

6 features, order

–

–

–

–

–

–

–

–

–

–

–

1

–

–

–

6

7 features, order

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

1

total with order

1

3

3

3

0

7

7

7

0

0

0

7

8

8

8

26

2 features, Pāṇini

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

3 features, Pāṇini

4

4

4

4

–

–

–

–

–

–

–

–

3

3

3

–

4 features, Pāṇini

–

4

4

4

–

1

1

1

–

–

–

–

4

4

4

7

5 features, Pāṇini

–

–

–

–

–

2

2

2

–

–

–

–

1

1

1

12

6 features, Pāṇini

–

–

–

–

–

–

–

–

–

–

–

1

–

–

–

6

7 features, Pāṇini

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

1

total Pāṇini

4

8

8

8

0

3

3

3

0

0

0

1

8

8

8

26

One consequence of assuming that intersection is not available for morphological feature algebras is that a valid inventory must contain at least three features.²² This can be seen as follows: the minimally valid inventories with two features when intersection is available contain two features such as f₁₁₀ and f₀₁₁ which each contain two cells. But if intersection is not available, such a system cannot describe the maximally differentiated paradigm and thus is not valid. In this case, {f₁₁₀, f₀₁₁, f₀₁₀} is available only as a three feature inventory – the feature f₀₁₀ corresponding to the intersection of the other two features must be included explicitly.

Generally, any partition set that can be generated in a morphology that allows intersection can also be generated in an non-intersective one by explicitly adding the features intersection would derive. In the three-cell case, this relationship holds also in the reverse direction as the comparison of Table 4 with 3 shows: any partition set that can be generated non-intersectively can also be generated in a system with intersective closure.²³

Returning to the *ABA constraint, and related considerations of restrictiveness, there are results that we believe should be of interest here. First, we note that admitting or banning intersection has no role on the overall impossibility of the four partition sets identified in section 5.3.1 as ungenerable. But rejecting intersection does make a difference in the definition of minimal valid inventories. In (the top half of) Table 3, where intersection is admitted, there are exactly three minimal valid inventories, which are linear permutations of one another, all of which derive a partition set with exactly three members. When intersection is not admitted, the number of minimal valid inventories increases to 25. Even so, the system is restrictive: without the Pāṇinian restriction, only 10 of 16 conceivable partition sets are generable.²⁴ None of the minimal valid inventories generates the unrestricted partition set (the rightmost column in Table 4) and none generates the partition set that excludes only AAA (column 12). Note that that pattern is derivable from non-minimal feature inventories, as indicated in the table. Hence it is Minimality that is playing a key role in excluding that partition set.

In other words, we include the following among our results. Regardless of whether intersection is admitted, and regardless of whether Pāṇinian ordering is enforced, the assumption of Minimal Validity as a condition on feature inventories ensures the following:

These results amount to a generalization of the *ABA generalization up to linear permutation. A special case of (36) is the rightmost column of Tables 3 and 4: no minimal valid feature inventory generates an unrestricted partition set. The assumption of minimality always entails a restriction. Another special case is the implication that if any two bifurcations of the three-celled space are in the partition set of (minimal valid) inventory, then the third is not. Up to linear permutations of the cell orders, this is the *ABA generalization: if AAB and ABB are admissable paradigms, then ABA is not.²⁵