A Heuristic 
for Paradigms 
Joseph E. Grimes 
Cornell University and 
Summer Institute of Linguistics 
This paper helps clarify one of the 
pervasive problems of linguistic analysis: 
the interaction between the paradigmatic 
and syntagmatic dimensions of language. 
Paradigms are sets of alternatives: the 
speaker must decide on one member of the 
set to use, and the hearer must figure out 
which he used. In a syntagm or 
construction, an element chosen out of one 
paradigm is put together with elements 
chosen out of others. Thus far all 
grammars of all languages agree. 
The problem comes when we put the 
grammar together. The choices available 
in one paradigm turn out often to be 
limited by those made in some other 
paradigm that is part of the same 
construction. Grammar is never as simple 
as a Cartesian product of paradigms. 
Various forms of grammar have various 
means, none of them quite satisfying, to 
express these limitations. A common one is 
footnotes about irregularities; ad hoc 
features to trigger or block special rules 
when needed are also used. 
Grammar ought to highlight the mutual 
constraints between paradigms and 
constructions, not downplay them. 
Halliday's systemic grammar has done well 
in this regard (Halliday 1961, Hudson 
1971) . It is already known to 
computational linguists through Winograd's 
work (1972). The heuristic, based on work 
by Lowe, Dooley, and myself (in press), is 
expressed within Halliday' s framework 
here, though it is applicable within any 
other model of language as well. 
In systemic terms a paradigm is known 
as a 'system'. A choice in one system can 
be the entry condition for another system, 
one part of a system can have different 
properties of combination from another 
part, and two or more systems can be 
activated together as the basis for a 
construction. The heuristic is intended to 
clarify something that is more often 
guessed at than proved: what element 
belongs to what system. 
What I find, on looking at languages 
other than English, is that membership in 
a Hallidayan system is by no means obvious 
in all cases. This is true for two 
reasons: first, some elements have 
properties that permit us to assign them 
to more than one system, and second, some 
elements are artifacts of the mapping 
relation between systems and forms, rather 
than direct manifestations of choices 
within systems. 
The Data 
Table (i) gives some data which 
illustrate this general point by means of 
a limited example. It reports 
cooccurrences among a particularly complex 
subset of the prefixes to the verb in 
Huichol, a Uto-Aztecan language spoken in 
the Mexican Sierra Madre. A 1 in the table 
means that the prefix at the head of the 
column has been observed in the 
combination that the row reports. For this 
language there are exactly 15 observable 
combinations of these prefixes, each 
represented by one row in Table (i). The 
order in which the rows are written down 
makes no difference, nor does the order in 
which the columns appear, kal- and ka2- 
are homophonous forms that occupy 
different positions in the prefix string 
and have different meanings. ~ stands for 
a high back unrounded vowel. 
(i) kalke p& m& ka2ni 
i 0 1 0 1 0 
1 0 1 0 0 0 
1 0 0 0 1 1 
1 0 0 0 0 1 
0 1 0 0 0 1 
0 1 0 0 0 0 
0 0 1 0 1 0 
0 0 1 0 0 0 
0 0 0 1 1 1 
0 0 0 1 1 0 
0 0 0 1 0 1 
0 0 0 1 0 0 
0 0 0 0 1 0 
0 0 0 0 0 1 
0 0 0 0 0 0 
The simple fact that two forms cannot 
cooccur with each other is the most 
obvious basis for saying that those two 
232 
are members of a single system, that they 
are in opposition as alternatives in a 
paradigm, that the choice of one as over 
against the other has linguistic 
significance. In Table (I), for example, 
p&- does not occur in any combination 
where ni- occurs, and vice versa. 
Noncooccurrence patterns 
The patterns of noncooccurrence are 
derived from Table (i) by a simple 
algorithm: 
For each column: 
Create a vector of as many 0's as 
there are columns 
For each row that has a 1 in the 
column in question: 
Unite that row with the 
vector. 
Complement the vector. 
Each of the uncomplemented vectors 
represents the union of all the 
combinations which the form at the head of 
its column enters into. The l's in its 
complement therefore identify the elements 
with which it cannot cooccur. 
The Huichol data -- and this is true 
of other languages, possibly of all 
languages -- do not allow us to draw 
immediate conclusions about mutual 
exclusiveness or simple comembership in 
systems. The prefixes represented by the 
complement vectors of each form are 
(2) kal: ke, m& 
ke: kal, p&, m&, ka2 
p&: ke, m&, ni 
m&: kal, ke, p& 
ka2: ke 
ni: p& 
A form like p&- can be assigned to 
one system in opposition with ni-, and to 
another in opposition with ke- and m&-; 
but kali-, which could also g--o into--a 
system with __ke and m_&&-, cooccurs with p&- 
and therefore cannot represent an 
alternative to it. The logic of systems 
in grammar is more complex than 
independent commutation, with the 
Cartesian products that that implies, in 
which each form of one set cooccurs with 
every form of another. 
Decomposition 
The true interdependency of a 
systemic network can be captured in a 
cooccurrence graph by first decomposing 
Table (I). The most manageable 
decomposition strategy found so far is to 
start with the column that minimizes the 
number of l's that would be removed from 
the table if all the rows that have l's in 
that column were removed. We convert those 
rows into a component subgraph, then 
continue recursively on the table minus 
those rows until no rows are left, or 
until the zero row is left; then we also 
convert the zero row if there is one into 
a component subgraph. In the final step of 
the heuristic, the component subgraphs are 
united to give the complete cooccurrence 
graph. That graph of forms is the aim of 
the heuristic. It is not a systemic 
network diagram itself, but is rather a 
statement of a major constraint on the 
semantic systemic diagram that accounts 
for the forms. 
Component subgraphs are formed by 
putting alternatives vertically in any 
order within square brackets, and 
connecting forms that cooccur in any order 
by horizontal lines. Absence of any form 
in a particular combination is represented 
by ---. 
In Table (i) the two rows that 
contain l's for ke- have a total of only 
three l's in them; so those two rows are 
taken out for the first subgraph: 
(3) ke 
This subgraph, like the two rows of Table 
(i) that it represents, says that ke- can 
occur with or without ni-. 
The full 
derived from Table 
simple alternatives 
products: 
set of component subgraphs 
(i) contains only 
and their Cartesian 
Eni (4) (a) ke , --- 
Ek~ ~ Re2 
E ka2 (c) kal ni 
(d) m& ka2 E::_\] 
(e) ka2 
(f) m& .... E::_7 
(g) ni 
(h) 
Union of component subgraphs 
We unite these subg raphs by 
conflating what they have in common and 
symbolizing their differences as 
alternatives, by the distributive 
property. Four of the subgraphs, (a), (f), 
(g) , and (h) , can be combined without 
changing the picture of simple systems and 
233 
Cartesian products: 
ni 
(5) ke ___ 
A restriction on Cartesian products 
appears, however, when we expand the 
composite diagram further. (d) has three 
out of four of its elements in common with 
elements already in the composite diagram 
(5) . The fourth element, however, has 
nothing to do with ke- or its absence, but 
only with m&-. Here is where the 
discrepancies in noncooccurrence 
properties of different forms come into 
the picture, and here is where the 
Hallidayan device of linked brackets is 
needed in order to show up those 
discrepancies. The elements in (6) are 
reordered to disrupt the graphic shape 
given by (5) as little as possible: 
Ika2 I~__m& ~ ~ni 
(6) ke 
Cooccurrence graph 
The complete cooccurrence graph is 
built up by continuing in the same way 
until all the component subgraphs are in 
it: 
2 ~ P&- 1 
The use of two null symbols in a 
single set of alternatives does not mean 
that Huichol has two zero prefixes that 
contrast with each other, but rather that 
the graph is essentially nonplanar. 
Redundant nulls could be eliminated by 
crossing lines in an equivalent graph. 
This diagram now shows all the 
constraints on cooccurrence that there are 
for these Huichol prefixes. It is not yet 
a systemic diagram, because systemic 
diagrams give differences in meaning and 
this one gives only cooccurrences of 
forms. The systemic diagram we come up 
with will, however, have to account for 
each of the constraints on cooccurrence 
given by this diagram. 
Our scrutiny of cooccurrences and 
noncooccurrences has shown us what forms 
might be in opposition with each other in 
a semantic system, and how those forms 
interlock. That is as far as our explicit 
heuristic take us; but it narrows the 
field for semantic investigation 
considerably. 
Computational aspects 
Before I go on to show the payoff in 
terms of systems of meaningful choices, 
let me sketch the computational aspects of 
the heuristic. For a small problem like 
the one in the example, of course, no 
computing is needed. But were we to take 
in all 42 verb prefixes of Huichol, and 
state how they combine with suffixes and 
different stem types as well, the 
heuristic would never get off the ground 
with pencil and paper. It is a good 
example of how a computationally simple 
process, actually a twist on concordance 
generation, can bring order into an area 
where a linguist is otherwise all too 
likely to shrug his shoulders and define 
oversimplified systems, then write 
interminable footnotes about why they 
don't quite combine as he says they do. 
A linguist in the field needs a 
three-step computational aid. Step One is 
data entry: take in occurring combinations 
of forms, which could as well be function 
words or suffixes or any combination of 
closed class phenomena, and develop a 
table like Table (i). Step Two is union: 
read the table and develop a vector for 
each form that shows the union of all its 
combinations. Step Three is decomposition: 
segregate out from the table the subsets 
of its rows that facilitate making its 
component subgraphs. 
These three steps are easy to 
implement. The fourth step of the 
heuristic, forming the cooccurrence graph 
by uniting the component subgraphs, is at 
least an order of magnitude more complex, 
and may not be feasible for a small field 
computer. 
Systemic diagram 
After the heuristic procedure is gone 
through, whether with pencil or by 
computer, the construction of a semantic 
hypothesis rich enough to account for all 
the patterns of cooccurrence can go ahead. 
This is a standard linguistic undertaking, 
and has two sides. The first is to 
investigate the reasons why one or another 
member of a noncooccurring set like the 
ones in (2) gets chosen. The reasons for 
choosing either member of a pair may not 
be the same in the context of one pattern 
of choices made in other systems as it is 
in other contexts. The second part of the 
semantic inquiry is to identify or 
combinations of forms whose presence is an 
artifact of the mapping between meaning 
and form, and not an assertion of a 
particular meaning. 
This arbitrariness in the mapping 
relation shows up in two places in the 
example. When p&- is present, k__aal- has 
either a tentative or a very strong 
negative meaning: kaalp&m{e means 'he 
might not go' or 'he shall not go!' (the 
234 
meaning split is not too different from 
that of English terribly in terribly 
disfigured vs. terribly nice). With ni-, 
however, __kal- has to be there when k_~a2-, 
the ordinary negative, is present, and may 
or may not be there when ka2- is absent. 
The requirement that kal- always go with 
ka2- in the presence of ni- eliminates 
the possibility of the two homophones ever 
being opposed to one another, with 
resulting confusion between negative and 
tentativn between negative and tentative 
meanings. 
The other arbitrariness turns up on 
trying to relate m&- with ni-. m&- by 
itself is the sign of a dependent verb, 
and ni- by itself of an independent verb 
at a partice combination m&ni, however, 
has nothing to do with either of these 
meanings; it makes a statement of the 
speaker's opinion. I take it to be a 
morphologically complex expression of a 
separate term of the modal system. 
Taking these discrepancies into 
account gives us a systemic diagram: 
(8) Ftentative- 
~assertive| kal 
p& \[_definite 
negative-- narrative --- 
ka2 ni 
evaluative !. 
dependent 
. m& positiv conjunct \[~ 
imperativ ni 
ke eneral 
\[narrative\] implies kal 
obligatory with \[ne-gative\] 
optional with \[positive\] 
\[evaluative\] realized as m&+ni 
It is by straightening out kal- and 
m&ni- that it becomes possible fo-{-- us to 
give a systemic diagram plus a set of 
realization rules for it. The 
straightforward realization rules are 
written right into the diagram: for 
example, if you choose \[negative\] , utter 
ka2-. The more complex realizations are 
given at the bottom of the diagram. 
The terms of the systemic diagram are 
labels for semantic choices that have been 
explained elsewhere and do not concern us 
now; they do not constitute explanations 
in themselves. Once the arbitrary mappings 
are defined in realization rules, the 
diagram embodies only one real restriction 
on Cartesian products of paradigms, in 
that Huichol has no special negative 
imperative form. (It uses the negative 
declarative p&ka2 in its place.) The 
completeness o--f---the analysis is supported 
by the fact that the fnterconnected 
paradigms of (8) have exactly 14 paths 
through them, and that together with the 
optional rule for the realization of kal- 
with ni-, these yield exactly the 15 --lows 
of Table (i) with which we began. 

Bibliography 
Grimes, Joseph E., Ivan Lowe, and Robert 
A. Dooley. in press. Closed systems 
with complex restrictions. 
Anthropological Linguistics. 
Halliday, Michael A. K. 1961. Categories 
of the theory of grammar. Word 
17:241-292. 
Hudson, R. A. 1971. English complex 
sentences. Amsterdam: North-Holland 
Publishing Company. 
Winograd, Terry. 1972. Understanding 
natural language. New York: Academic 
Press. 
