Models of Simultaneity and Sequentiality in Human Cognition [Reprint 2020 ed.] 9783112327562, 9783112327555

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PdR Press Publications in COGNITION 2

MARSHALL D U R B I N

Models of Simultaneity and Sequentiality in Human Cognition

THE PETER DE RIDDER PRESS

MARSHALL D U R B I N

Models of Simultaneity and Sequentiality in Human Cognition

LISSE

THE PETER DE RIDDER PRESS 1975

© Copyright reserved No part of this text may be translated or reproduced in any form, by print, photoprint, microfilm, or any other means without written permission from the author. ISBN 90 316 0040 7

The text of this article is reprinted from Linguistics and Anthropology: In Honor of C. F. Voegelin edited by M. Dale Kinkade, Kenneth L. Hale, and Oswald Werner (Lisse: The Peter de Ridder Press, 1975) pp. 113-135

Printed in Belgium by N.I.C.I., Ghent

CONTENTS

INTRODUCTION

5

IDEAL PARADIGMATIC RULES

6

IDEAL TREE RULES

10

IDEAL TAXONOMIC RULES

18

SEQUENTIAL RULES

24

NOTES

26

REFERENCES

27

INTRODUCTION

One of the goals of social science has been to seek models which adequately represent the organizing principles of human behavior.1 In this paper, I would like to discuss this search for models from the standpoint of the sequential organization vs. the simultaneous organization of behavior. No unit of behavior can occur in isolation as a representative of a set of simultaneous features without being influenced by surrounding sequences of behavior; similarly, sequences of behavior cannot occur without being composed of simultaneous elements within the sequential units. Obviously, this is not the only way to view human behavior, though it has been one of the most popular over the past few years, especially in the field of linguistics. One reason for looking at behavior from these two contrastive points of view is that it allows for a kind of dissection which no other model can provide. This kind of dissection permits us to see the constant interaction of simultaneous features from one sequential unit to the next. Also, an analysis based upon simultaneous features and sequential units can be set into two kind of rules: 1) rules on one hand which account for simultaneous events which are timeless and occupy no space, and 2) on the other hand rules which account for the sequential arrangements of these units over spans of time and space. In language, rules which account for simultaneous events are encountered in the area of phonology, while semantics and syntax have traditionally been accounted for by a different order of rules which map sequences into a right-to-left order representing linear time and occupying real or imagined space. The purpose of this paper is to examine the models currently used to explain these two kinds of behavior. Let us look at simultaneous first, the more complex of the two behavior types. For this type of behavior I have pointed out elsewhere (Durbin, 1966) three different types of rules: 1) An ideal1 paradigmatic set of

6

MARSHALL DURBIN

rules; 2) An ideal tree set of rules; and 3) an ideal taxonomic set of rules. A portion of this paper will be devoted to showing that an ideal taxonomic rule is a combination of certain characteristics of ideal paradigmatic and tree rules. These sets of rules are labelled ideal because there is no known human behavior such that one could map the behavior onto the rules according to any morphism. Rather, as we shall see below, it is the combination of these types of ideal rules which account for most simultaneous behavior.

IDEAL PARADIGMATIC RULES

A special feature of this type of rule is that there is no requirement on internal ordering. Ordering is lacking both horizontally and vertically. Another way of stating this is that a sequential ordering of the outputs of the rules is totally lacking as shown in Figure 1. One reason for this lack of sequential ordering is that the rules are not analytic, i.e., the rule does not state that A is composed of a B and a C, rather that A merely leads or maps onto B and C. Further, all nodes on any one level of mapping are expanded in the same way after the first level of expansion has taken place. However, each output is uniquely distinguished from all other outputs. Note that outputs 1, 2, 3, 4, 5, 6, 7, and 8 are defined by ABDF, ABDG, ABEF, ABEG, ACDF, ACDG, ACEF, and ACEG, respectively. Because the initial expansion derives two different nodes, an infinite number of features should be employed such that each output

A

D F

G

1

2

F

B

C

E

D

G 3

F 4

F.

G 5

F 6

G 7

8

Fig. 1. An Ideal Paradigm

6

MARSHALL DURBIN

rules; 2) An ideal tree set of rules; and 3) an ideal taxonomic set of rules. A portion of this paper will be devoted to showing that an ideal taxonomic rule is a combination of certain characteristics of ideal paradigmatic and tree rules. These sets of rules are labelled ideal because there is no known human behavior such that one could map the behavior onto the rules according to any morphism. Rather, as we shall see below, it is the combination of these types of ideal rules which account for most simultaneous behavior.

IDEAL PARADIGMATIC RULES

A special feature of this type of rule is that there is no requirement on internal ordering. Ordering is lacking both horizontally and vertically. Another way of stating this is that a sequential ordering of the outputs of the rules is totally lacking as shown in Figure 1. One reason for this lack of sequential ordering is that the rules are not analytic, i.e., the rule does not state that A is composed of a B and a C, rather that A merely leads or maps onto B and C. Further, all nodes on any one level of mapping are expanded in the same way after the first level of expansion has taken place. However, each output is uniquely distinguished from all other outputs. Note that outputs 1, 2, 3, 4, 5, 6, 7, and 8 are defined by ABDF, ABDG, ABEF, ABEG, ACDF, ACDG, ACEF, and ACEG, respectively. Because the initial expansion derives two different nodes, an infinite number of features should be employed such that each output

A

D F

G

1

2

F

B

C

E

D

G 3

F 4

F.

G 5

F 6

G 7

8

Fig. 1. An Ideal Paradigm

7

MODELS OF SIMULTANEITY AND SEQUENTIALLY

would be uniquely distinguished from all other outputs by at least one node. Burling (1964) has already pointed out that since order is lacking, this type of rule can be partitioned in a very large number of ways. In Figure 1, since there are no internal requirements on ordering we can reorder the rules hierarchically (vertically) in five other ways (for a total of six representations of the same set of rules) as shown in Figure 2 and attain the same outputs in each case. That is, the rule set given in Figure 2 has three levels of expansion, seven features, 14 nodes (not counting the starting node A), eight outputs, and six possible representations. If we were to add another level of expansion, we would then have the possibility of 24 representations of the rule set, or an increase of four times. An ideal paradigm in any of its representations can also be ordered horizontally so that in Figure 3 we see an ideal paradigm ordered such that in effect we can potentially derive a very large number of representations of this paradigm, especially if we include all possibilities of the combinations of horizontal and vertical ordering. One can notice the ordering changes that have occurred in the rules by noting the sequential order of the outputs. But even though there is a total lack of ordering it is obvious that some sequential constraints

B

F

G

F

G

D

AE DAE D AE D AE

I

I

I

3

2

4

1. A -> B C (B) 2

-

|

c

h

F

G

( F

3 . j

G

J - > D E

I

5

I

7

I

6

I

8

D

E

AB C AB C AB C BAC

D

1

E

I

I

I

I

1

1 5 3 7

I

2

I

6

1. A - * F G (F) 2

3

-

|

G

( D

- J

E

h

h

D

E

B C

Fig. 2. Some Vertical (Hierarchical) Reorderings of an Ideal Paradigm

I

4

I

8

8

MARSHALL DURBIN

B

D I 1

B

E D E D E D E I I I I I I I 3

1. A 2

5

2

4

6

8

-+FG

-!G!-

3.

7

B C

L U D E

D

/I K A A A A A A A A F

B I 1

G

G

B

C B C B C B C I I I I I I 5 2 6 3

F

1. A D) Ej F) G

-»DE FG BC

F

C

G

F

I I 1 2

I 5

1. A

DE

D) E) B) Fig. 2. (continued)

B

G

I 6

BC FG

F

C

G

F

G

9

MODELS OF SIMULTANEITY AND SEQUENTIALLY

D

B

A A A A A A A A

B

C

C

I

I 8

I

1. A 2.

3.

E) D) G)

B

C

I

B

I

C

I

B

I

E

I 1

I

D

I

E

I 8

ED

1. A

CB

GF

2.

C) Bj

FG

BC

3.

F)

D

I 6

E

I

D

E

D

I 1

I 4

I 2

DE

Fig. 3. Some Horizontal Reorderings of an Ideal Paradigm

are present since we cannot have an output beginning with 2187 ... although ... 2187 ... may occur medially. Thus, in spite of the fact that horizontal and vertical order are totally lacking, there exist sequential constraints and restrictions on the linear outputs. These sequential constraints are a product of the fact that the first level of expansion includes two nodes which are differently labelled; thus outputs 1 and 8 differ by a maximum number of features (ABDF and ACEG, respectively). Because of this maximal differentiation, the two outputs can never occur adjacent to each other. I know of no data in human behavior which conforms perfectly to rules of this set type. Rather, we can view the ideal paradigm as a type which human behavior approximates at times. For example, there are portions of a phonological analysis which approximates an ideal paradigm by deviating from it in certain ways. One of the most common of these deviations is by imperfection which gives us an imperfect paradigm as shown in Figure 4. Again, the types of imperfect trees which could be represented here are of a very large number. Further, it is to be noted that an imperfect tree begins to place partial ordering on the tree where none existed

10

MARSHALL DURBIN

A

B

C

B

D

D

F

G

i

:

F

G

F

E

G

1 2

1. A

BC

2. B

DE

3. ' D

FG

D

3

1. A C

A

F

G

4

BC DE

D) E)

(FG 10

Fig. 4. Some Imperfect Paradigms

before. Thus, in Figure 4 A -> B C and can never be written as *A-> D E since the output of C has the history (or is dominated by) A C (in that order hierarchically) and only by expanding A into B C this can be achieved. However, B D E FG can still be ordered in any fashion. The less 'perfect' the paradigm, the greater the hierarchical (vertical) ordering restrictions. It should be noted that a paradigm (ideal or imperfect) can be represented as a matrix or a graph as seen in Figure 5. It can be immediately seen that the only redundant feature in a paradigm is the starting node A. All other nodes are distinctive and nonredundant, and necessary to distinguish any output from all others. That is, a paradigm represents maximal efficiency in the employment of its features and its rules, but sacrifices ordering possibilities.

IDEAL TREE RULES

The opposing case of an ideal paradigm is an ideal tree rule in which all nodes at all levels must be expanded differently as shown in Figure 6.

10

MARSHALL DURBIN

A

B

C

B

D

D

F

G

i

:

F

G

F

E

G

1 2

1. A

BC

2. B

DE

3. ' D

FG

D

3

1. A C

A

F

G

4

BC DE

D) E)

(FG 10

Fig. 4. Some Imperfect Paradigms

before. Thus, in Figure 4 A -> B C and can never be written as *A-> D E since the output of C has the history (or is dominated by) A C (in that order hierarchically) and only by expanding A into B C this can be achieved. However, B D E FG can still be ordered in any fashion. The less 'perfect' the paradigm, the greater the hierarchical (vertical) ordering restrictions. It should be noted that a paradigm (ideal or imperfect) can be represented as a matrix or a graph as seen in Figure 5. It can be immediately seen that the only redundant feature in a paradigm is the starting node A. All other nodes are distinctive and nonredundant, and necessary to distinguish any output from all others. That is, a paradigm represents maximal efficiency in the employment of its features and its rules, but sacrifices ordering possibilities.

IDEAL TREE RULES

The opposing case of an ideal paradigm is an ideal tree rule in which all nodes at all levels must be expanded differently as shown in Figure 6.

MODELS OF SIMULTANEITY AND SEQUENTIALLY

A

D

B

C

E

D

E

AG FAG F AG

F

I 1

I 2

I 3

I 4

B X

X

X

X

C D X

X

E

X

F X

X X

G

A G

F

I 5

I 6

I 7

I 8

X

X

X

X

X

X X

X

X

X

X

X

X

X

Fig. 5. Ideal Paradigm Represented as a Matrix (Graph)

A 1.

B

D

A A A A

H I I 1

I 2

J

K

L

M

N

O

I 3

I 4

I 5

I 6

I 7

I 8

Fig. 6. An Ideal Tree

2. 3. 4. 5. 6. 7.

A B C D E F G

-» B C -»DE ->FG -•HI ->JK ->LM -•NO

11

12

MARSHALL DURBIN

The units in the outputs 1, 2, 3, 4, 5, 6, 7, and 8 contain features ABDH, ABDI, ABEJ, ABEK, ACFL, ACFM, ACGN, and ACGO, respectively. Each output differs from all others by at least one feature. In an ideal tree, the features which produce the outputs occur simultaneously as in an ideal paradigm. It is to be noted that this is so in spite of the fact that an ideal tree has strict vertical (hierarchical) ordering. That is, we say that in an imperfect paradigm (Fig. 4) hierarchical ordering was imposed at the point of perfection. Now we see that in an ideal tree hierarchical ordering is strictly imposed due to the fact that each node is expanded differently, but it should also be noted that horizontal reordering is still possible in an ideal tree (as in an ideal paradigm). Figure 7 shows some horizontal reorderings which are possible in an ideal tree. In spite of the fact that an ideal tree has more features, it has less representational possibilities than an ideal paradigm since hierarchical reordering is not permitted in the tree. For this ideal tree with three levels of expansion, 15 features, 14 nodes (not counting starting node A), and eight outputs, there are fourteen ways of reordering it horizontally.

A

G A 0 N 1 i 1 1 8 7 1. 2. 3. 4. 5. 6. 7.

F D A A L M H I 1 I 1 i 1 1 1 1 5 6 1 2

A CB C -> GF B ->• DE G - ON F LM D -* HI E - JK

A

E A J K 1 I 1 1 4 3

F A M L 1 1 1 1 6 5

G E D A A A 0 N K J H I 1 I 1 ! 1 1 1 1 1' 1 1 1 8 7 4 3 1 2

1. A CB 2. C FG 3. B ED 4. F -> ML 5. G ON 6. E KJ 7. D - HI

Fig. 7. Two Horizontal Reorderings of an Ideal Tree

MODELS OF SIMULTANEITY AND SEQUENTIALLY

13

If we add another level of expansion which would give us 30 features, 29 nodes (not counting starting node A) and 16 outputs, we would then have 30 different representations of the tree. For example, in an ideal tree we could not arrive at the sequence 13572468 which is a possible output of a paradigm. That is, all sequential possibilities found in a paradigm are not allowed in a tree. This brings us to the realization that the number of features or the number of rules employed in a behavioral system has little to do with the complexity of the system. Rather, complexity in the system is created by the model into which the features are cast. An ideal tree may also be imperfect as is the case shown in Figure 8. However, imperfection does not affect a tree as it does a paradigm since the possibility of hierarchical reordering is already excluded in a tree. Again, there appears to be no human behavior which corresponds exactly to an ideal tree but phonological analyses again resemble ideal trees. One of the ways in which phonological analyses deviate from ideal trees is by a hierarchical mixing of features (a false tree) as seen in Figure 9 where F and G are employed at two levels. These rules are ordered and can be cyclical as shown in Figure 10. This type of mixing cannot occur in a paradigm since each level of expansion must be expanded in the same way. This sort of mixing is sometimes called embedding which can lead to infinite recursiveness as shown in Figure 10. The rules are applied again in an infinite cycle. As with a paradigm, a tree can also be presented by a matrix or a graph as shown in Figure 11. Out of 112 cells only 24 are specified, or approximately 21%, a much lower figure than for an ideal paradigm (48%).

A

B

D

C 1. 2. 3. 4.

E

F

G

H

I

1

2

3

4

5

Fig. 8. An Imperfect Tree

A B C B -> D E D F G HI E

MARSHALL DURBIN

B

1.

A

-> B C

2.

B

- D E

4.

D

-»HI

5.

F

- J K

6.

G

D

H

I

F

J 1

G J

K

L

K

L M

L M

M

2 3 4 5 6 7

8

9

10

Fig. 9. A Mixed Tree

1.

A

-» B C

F D

A A A A

F

D


D E 3. C -> F G 4.

H (g)

-> H 1

6. I ->

T U V W X Y 11

12

" JK NO RS VW ~ LM PQ TU XY

13

14

// /

/"

/

/"

/ /

Fig. 13. The Merging of a Tree and a Paradigm

15

D E F G D E F G

16

18

MARSHALL DURBIN

constraint on reordering which is present in a tree. Horizontal reordering, however, may occur. Merging, then, introduces a new feature into our rules; the context or the environment. These types of rules are usually called context-sensitive rules. Chomsky (1956) has shown that language, as one example of human behavior, requires context-sensitive rules. Merging allows for the same type of redundancy specification as seen in an ideal tree. For example, if an output in Figure 13 is specified for feature J, then must also be specified for H, D, and B, etc. When we take into account merging and partial merging of trees and paradigms, imperfections in each of them and their merged products, and the mixing of hierarchies in the portions of a merger (i.e., the tree portion) where it is allowed, we have structures which closely correspond to human behavior and we can begin to map this behavior into the types of rules discussed above. This is especially true in the phonological analyses which linguists have traditionally carried on. If we look at a phonological analysis we notice that the upper portion resembles a paradigm (each node expanded in the same way at a given level). On the lower levels we find a tree (where each node is expanded differently at a given level). We can also see a great amount of imperfection. There is a great amount of mixing (same features used on different levels) which indicates to us that a tree rather than a paradigm is present indicating various degrees of merging. The products of such combinations of trees and paradigms require rules that are context-sensitive and that eliminate redundancies to generate them properly and economically.

IDEAL TAXONOMIC RULES

But it is not only in phonological rules in linguistics that we can see the effect of the tree and paradigm rules. It also is found in taxonomic rules. Generally, taxonomic rules are thought to be the same as trees or paradigms (and their combinations) except that each node in a taxonomy corresponds to a lexeme rather than to a distinctive feature (Kay, 1966:83) with the implication being that the only difference between a paradigm and a key on one hand and a taxonomy on the other is names given to the nodes (cf. Kay 1966). I have already pointed out (Durbin 1966:37) that a taxonomy is hierarchically but not horizontally ordered like a tree but that mixing can occur as in a paradigm. Tyler (1969:10) has distinguished a taxonomy from a paradigm in that the

18

MARSHALL DURBIN

constraint on reordering which is present in a tree. Horizontal reordering, however, may occur. Merging, then, introduces a new feature into our rules; the context or the environment. These types of rules are usually called context-sensitive rules. Chomsky (1956) has shown that language, as one example of human behavior, requires context-sensitive rules. Merging allows for the same type of redundancy specification as seen in an ideal tree. For example, if an output in Figure 13 is specified for feature J, then must also be specified for H, D, and B, etc. When we take into account merging and partial merging of trees and paradigms, imperfections in each of them and their merged products, and the mixing of hierarchies in the portions of a merger (i.e., the tree portion) where it is allowed, we have structures which closely correspond to human behavior and we can begin to map this behavior into the types of rules discussed above. This is especially true in the phonological analyses which linguists have traditionally carried on. If we look at a phonological analysis we notice that the upper portion resembles a paradigm (each node expanded in the same way at a given level). On the lower levels we find a tree (where each node is expanded differently at a given level). We can also see a great amount of imperfection. There is a great amount of mixing (same features used on different levels) which indicates to us that a tree rather than a paradigm is present indicating various degrees of merging. The products of such combinations of trees and paradigms require rules that are context-sensitive and that eliminate redundancies to generate them properly and economically.

IDEAL TAXONOMIC RULES

But it is not only in phonological rules in linguistics that we can see the effect of the tree and paradigm rules. It also is found in taxonomic rules. Generally, taxonomic rules are thought to be the same as trees or paradigms (and their combinations) except that each node in a taxonomy corresponds to a lexeme rather than to a distinctive feature (Kay, 1966:83) with the implication being that the only difference between a paradigm and a key on one hand and a taxonomy on the other is names given to the nodes (cf. Kay 1966). I have already pointed out (Durbin 1966:37) that a taxonomy is hierarchically but not horizontally ordered like a tree but that mixing can occur as in a paradigm. Tyler (1969:10) has distinguished a taxonomy from a paradigm in that the

19

MODELS OF SIMULTANEITY AND SEQUENTIALLY

former orders its labels by contrast and inclusion (in the formal logical sense) whereas a paradigm is arrayed so that its features are multiple and intersect. Tyler (1969:10-11) further distinguishes trees from paradigms and taxonomies in that the former's features are ordered by sequential contrast of only one feature at a time such that they are based on successive choices between only two alternatives. First of all, we can see that Tyler's differentiation of paradigm and taxonomy is a false one since in his presentation of taxonomies he uses a branching diagram labelled with lexemes as seen in Figure 14 whereas in his presentation of paradigms he uses a matrix and distinctive features as seen in Figure 15.

FURNITURE

chairs

sofas

desks

tables

end tables

dining tables

Fig. 14. Branching Diagram Representing a Taxonomy (Tyler 1969:8)

SEX

> H — t 1 «5 D H < S

adult M-l

male $

female ?

stallion boar

mare sow filly gilt

adolescent M-2 child M-3

colt shoat

baby M-4

foal piglet

neuter 0

gelding barrow

Fig. 15. Matrix (Graph) Representing a Paradigm (Tyler 1969:10)

20

MARSHALL DURBIN

If we convert either into the same system we see they are similar as seen in Figures 16 and 17; similarly, Tyler's tree can be cast into a taxonomy or a paradigm as seen in Figure 18. As can be readily seen, then, the difference which Tyler wishes to draw between a taxonomy and a paradigm and a tree depend not upon any inherent difference in the data but whether the analyst wishes to utilize lexemes or distinctive features for his analysis (bearing out Kay's

FURNITURE

CHAIRS

SOFAS

DESKS

SMALL

LARGE

I

I

END

DINING

TABLES

TABLES

FUNCTION +

SITTING

— SITTING — WORK

+

WORK

S

I z

SMALL

CHAIRS

END TABLES

E

DESK LARGE

SOFAS

DINING TABLES

Fig. 16. Conversion of Taxonomy in Fig. 14 into Features and a Matrix (Graph)

21

MODELS OF SIMULTANEITY AND SEQUENTIALLY MATURATION

AGE CAPABLE OF

NOT AN AGE CAPABLE OF REPRODUCTION

REPRODUCTION

SEX

CHILD

BABY

Colt Shoat

Foal Piglet

MALE

Gelding Barrow

Stallion Boar

ANIMALS

Stallion

Piglet Shoat Gelding

Gilt

Fig. 17. Conversion of Paradigm in Figure 15 into a Branching Tree and a Taxonomy

distinction mentioned above). By comparing Figures 15, 16, 17, and 18 we can see that the distinctions which are traditionally drawn between taxonomies and other branching diagrams is not due to any inherent differences in the data but whether we, as analysts, choose to label the nodes of the branches with lexemes or with distinctive features. Further-

22

MARSHALL DURBIN

flowers spurred

+ flowers regular

+

I

delphinium

aquilegia

ranunculus

involucre

+

I

clematis

Anemone

Clematis

anemone

Delphinium

Aquilegia

Ranunculus

\

Fig. 18. The Conversion of a Tree into a Taxonomy and a Paradigm

more, it is of little value to distinguish between those data which are cast into branching diagrams vs. those which are cast into matrices since each system is convertible onto the other (as mentioned earlier). The success of one system over the other often depends upon how much we know about the distinctive features underlying the lexemes we are studying. It seems obvious then that what are traditionally called taxonomies (branching diagrams with most or all nodes labelled with a lexeme which is the object of our study) are of little value to us if we wish to understand the cognitive processes behind the lexical system we are studying. A traditional taxonomy is of little value to anyone except to a native speaker, since it leaves unmentioned the features which distinguish one lexeme from another.

MODELS OF SIMULTANEITY A N D SEQUENTIALLY

23

Base of Flowers Flowers with Spurs

m

u -

O a

Symetrical

C3

U

Delphinium

Asymetrical

X

>> Iu h E >>

Flowers without Spurs

Aquilegia iii

3 O > c

Petals

a

Ranunculus

Absent

o o. E o U

Clematis

Present

'