ON THE STRUCTURE AND CIRCULARITY OF COUNTABILITY WITHlN THE NOMlNAL DOMAlN

ON THE STRUCTURE AND CIRCULARITY OF COUNTABILITY WITHlN THE NOMlNAL DOMAlN

Judy Yoneoka

1.0 INTRODUCTION

The concepts of mass and count, or things and stuff, have long been recognized and accepted as traditional grammatical distinctions of countability within the nominal domain. However, as linguistic distinctions they have consistently encountered difficulties in acceptability. The difficulty accounting for the usage of many nouns in both count and mass senses (exemplified par excellence by Quine's and Mother Goose's "Mary had a little lamb") has traditionally been the largest problem with the count-mass distinction. Recently, other challenges have developed from much of the recent work in plural analysis (Link. Landman, etc.). Specifically, it is fast becoming recognized that many plurals are more than just sets of single elements. This poses the special problem of whether to treat plurals as count nouns (as has been traditionally assumed) or as mass nouns, as has been suggested by Cherchia (1982) for the case of bare plurals, or both.

This study challenges the traditional count-mass distinction based on the above criticisms and offers as a substitute a circular structure based on three binary features: unity, boundedness and focus. Through these features are defined six functional nominal classes, and shifting between these classes is accounted for through the definition of three shifting processes: induction, deduction and unit interchange.

2.0 LIMITATION OF THE COUNT-MASS ONTOLOGY

2.1.a. Difficulty in classification

The most widespread argument against the count-mass distinction as a linguistic concept is the fact that many, even most, nouns can be used in both count and mass contexts, as in the following:

1, a.) There is apple in the salad, but there are more apples on the shelf.

b.) We each ordered an ice cream and a coffee.

c.) When he finished eating, there was egg all over his face.

How are we explain the count-mass ambiguity within most nouns, or the manner in which this ambiguity occurs? More importantly, if so many nouns have both count and mass usages, how are we to justify at all the use of the count-mass ontology as a valid linguistic of cognitive distinction?

Of course, there, have been many methods proposed for classification of the MOST: COMMON sense of a noun as either count or mass; the most elementary of which are:

1. Acceptance of quantifiers, denumerators (including the indefinite article)

COUNTa, one, each, every, any, either

cat

three, many, both, a few

cats

MASS* a, one, each, every, any, either

water

* three, many, both, a few

waters

2. Acceptance of amount expressions:

COUNT*little, much, a cup of, 2 centimeters of dog

MASS little, much, a cup of, 2 centimeters of rice

There have been other proposals. For example, Bunt (1979) tentatively, classifies adjectives such as “big, small, heavy…..” as count adjectives due to their unacceptability in sentences such as:

2 a) *Gold is big.

b) *I have small coffee every morning.

and proposes that these may hold another clue for count-mass, distinction. However this is refuted by Allen (1980), who cites sentences such as “The equipment will be too large to fit inside this room.” And “Mankind is small compared with the blue whale.”1

All of these are only attempts at capturing the MOST COMMON countability sense of a noun. They do not give an answer as to why many nouns can seemingly change their classifications at will (as in 1 a) – c) above) or why, as pointed out by Ware (1979), many nouns are used equally in both count and mass senses, such as 'iron' or 'paper', and other nouns seem to be neither, such as 'brotherhood', 'nearness' or 'brevity'. Indeed, it has been suggested that all count nouns may be made into mass nouns (for example, with a "universal grinder" such as that suggested by Pelletier (1979)), and conversely that all mass nouns may be understood as containing a "hidden" elliptical counter, such as "a scoop of, cup of, cone of . . . ice cream".2

2.1.b. Analysis of plurals

Recent work with plurals further challenges the count-mass distinction with respect to plural nouns. Intuitively, a plural noun should represent the combination of several singular count nouns, and therefore should be countable as well. However, the question of HOW plurals should be that is, whether they should be counted as I or more sets counted of some singulars, or as the number of singulars themselves, has posed a tricky problem.

Consider the following. If I add 3 apples to 4 apples, am I adding seven apples or two sets ? If the 3 apples are small and green and 50 cents apiece and the 4 apples are big and red and 65 cents apiece, it is probably quite important to preserve the internal set structure. However, if the apples are all the same, and I am gathering them to feed my seven children, the set structure seems to lose its importance in favor of the necessity of knowing the exact number of individual apples. This example represents the essence of what is called the group-sum distinction (Link, Landman). Let us review another example:

3 a) The mothers of the PTA rejected the plan.

b) The mothers of the PTA are Mary, Sue, Jill, Louise…

c) Mary, Sue, Jill, Louise…rejected the plan

Of course, 3 c) does not necessarily follow from 3 a) and 3 b). Mary, Sue and Jill could be against the plan and even if Louise were emphatically for it, the plan would still be rejected by the mothers of the PTA. The reason for this is that the "the mothers of the PTA" does not denote the sum of all mothers, but the group consisting of all mothers. The difference is, as Landman (1987, p.52) puts it:

Groups . . . are individuals collected under a certain aspect…Through this aspect (which is an intensional notion), the group is individuated as an entity that is more than the sum of its parts, that is in certain respects independent of its parts; in the group, so to say, the part structure is wiped out.

In other words, the physical structure of the group is unlike the mathematical structure of a set in that 1+1 does not equal 2. Rather, it equals I group. Furthermore, the properties of this group as a whole may not necessarily apply to each member of the group, as in the example above.

We may then ask whether a "sum" distinction is well-motivated at all, or whether all plurals should be regarded as aspectually different from the sums of their singular members. What of the possibility that ANY plural brings together and strengthens certain common aspects of the single elements involved in it?

Perhaps the most obvious candidates for a "sum" interpretation would be spelled-out plurals such as in the example below.

4) Jim, Mark and John stole my wallet.

Say that Jim, Mark, John and Lester are the students in my English 3 class. If I were asked whether the students in my English 3 class stole my wallet, I would be more likely to respond with a simple affirmative than with something like "Yes, all except Lester" (unless I were in a court of law or especially worried about Lester's reputation). Moreover, if I knew 4) but had no information as to the complicity of Lester, I would be much more likely to assume that he was, if not an accomplice, at least a witness, rather than assuming the same of Fred in my English 4 class (unless Fred had some special loyalties to the three guilty students that I happened to know about).

This shows that even in extreme cases such as the spelled-out plurals of the above discussion, there seems to be a certain cohesion which should not occur in a simple set. This cohesion comes about from ABSTRACTION of a common property (an important point to be discussed later).

It is not my intention, however, to completely abandon the possibility of "sum" plurals. In fact, I feel there is a perceived difference in the following examples, which the group-sum distinction:

5 a) ?John and Joe live in Kansas and Kentucky.

b) The boys live in Kansas and Kentucky.

In 5 a), the hearer accepts the information given, but tends to want further clarification of the spatial identity of each unit (such as "respectively", or "Kansas m the summer and Kentucky m the winter") Thus, when listed separately, the two boys retain their separate spatial identities. However, in 5 b), the boys as a group lose their individual spatial identities and therefore can be in two places at once. Perhaps this issue of spatial identity may be used to shed more light on the group-sum distinction.

2.1.c. Analysis of Bare Plurals

The question of analysis of bare plurals, such as "books" or "people", is particularly difficult from the standpoint of the count-mass distinction. Take, for example, the following sentences from Cherchia (1982, p. 243):

6 a ) Dogs are mammals. (all)

b ) Dogs are intelligent. (most)

c ) Dogs give live birth. (most female dogs)

d ) Dogs are barking in the courtyard. (some)

e ) Dogs are numerous. (???)

and this one from Partee3:

7 a ) Dogs must be carried. (at least 1 per person present)

b ) Dogs must be carried. (all present)

Common sense tells us that plurals should be based on singulars and therefore should be countable. If this is the case, we should be able to answer the question "How many dogs?" for the above examples. As can be seen from the attempted responses shown in parentheses, however, the "number" of dogs in each case differs, or is impossible to arrive at. Does this mean, then, that the bare plural should be considered a mass term? Cherchia draws a clear analogy between mass nouns and bare plurals and cites various similarities (1982, p. 246) between the two to support this analogy. A few of these are shown below:

Opacity － Mary wants to find gold. (opaque reading of gold)

Mary wants to find football players (opaque reading of football players)

c.f. Mary wants to find some gold / football players. (opaque and transparent readings)

Scope ― Gold was found and gold was not found. (contradictory) Boys were found and boys were not found. (contradictory) c, f, Some gold was / boys were found and some was / were not found.

(two readings: contradictory and non-contradictory)

Anaphora － Mary is looking for gold and Bill is looking for it too. (opaque)

Mary is looking for policemen and Bill is looking for them too.

(opaque)

c.f. Mary is looking for some gold / a policeman and Bill is looking

for it / him too. (transparent)

This analogous analogy seems to those in 6) and 7) using mass nouns:

8 a) Gold is metal. (all)

b) Gold is beautiful. (most)

d) Gold can be found in San Francisco. (some)

e) Gold is scarce. (???)

9 a) Gold must be worn. (some per person present)

b) Gold must be worn.

However, the bare plural-mass analogy doesn't seem to explain the four different functional uses of the term "dogs"(or "gold") in the above examples: i.e., the full universal of 6a), the partially quantified universal of 6b) and c), the existential of 6d), and the type of 6e). These four functions are syntactically different. For example, existential and partially quantified universal bare plurals may be distinguished by their acceptability in "There" constructions, whereas full universals and types are- not acceptable:

10 a) *There are dogs who are mammals.

b) There are dogs who are intelligent.

c) There are dogs who give live birth.

d) There are dogs (who are) barking in the courtyard.

e) *There are dogs who are numerous.

Further, although universals may occur in singular constructions, type bare plurals may not:

11 a) This dog is mammal.

b) This dog gives live birth.

c) This dog is barking in the courtyard.

d) *This dog is numerous.

It is clear that the distinctions between these different functions of "dogs" will remain unclear within a simple count-mass framework. Indeed, the general confusion regarding count and mass, particularly with respect to analysis of plural nouns, suggests that this distinction may be the wrong vehicle to be using to analyze the structure of the nominal system. What seems to be needed is a finer-grained ontology which can account for group and sum plurals as well as the several types of bare plurals already discussed, and which allows for most nouns to have both count and mass functions. I will attempt to propose such an ontology in the following section.

3.0 THE CIRCULAR BINARY COUNTABILITY ONTOLOGY: A NEW CONCEPT OF NUMBER

The count-mass distinction, based on concepts such as singular and plural, countable and uncountable, suggests to us a number line such as often found in mathematics representing the count domain, where rules

∞・・・・・・・・・・・・・・・1・・・・・・・・・・・・・・・∞

←　5　4　3　2　1　O　1　2　3　4　5　6　→

Fig. 1. The familiar mathematical infinite whole number line (unbounded).

of basic mathematics apply and plurals are formed by adding sets of singular elements. However, the problems recently exposed in plurality and discussed in the previous section show this concept to be lacking with respect to plurals in natural language. That is, if plurals are nothing more than sums of 1+1+･･･+n, how are we to explain sentences such as 3 a,) in which the predicate does not apply directly and equally to each member of the plural ? Furthermore, it is unclear how the mass domain is the uncountable should be interpreted within this framework something nearing infinity, or is it something which comes about when we enter the domains of fractions or decimals? Is it even interpretable with respect to this framework at all? In order to try to solve these puzzles, let us try a new approach to nominal countability. approach to nominal countability.

3.1. The "unity" feature

This approach will be based on a binary system of number. When we use numbers, we employ them in one of two ways. We talk about positive or negative units which may be added and which have definite end points, similar to the number line in Fig. I above. Alternatively, we speak of divisions of a bound unit (say, the unit between O and 1), each division of which may be redivided infinitely:

0・・・・・・・・・・・・・・・・｜・・・・・・・・｜・・・・｜・・・・1

1/2 1/4 1/8…∞…

Fig. 2. The infinite fractional number line between O and I (bounded).

In the former number line conception, "I" is a unit; the whole (from = to ∞) is a unit; and any combination of I (i.e. 2, 3, 4 . . . ) is a non-unit. Any manipulation of number within this line is basically a form of positive or negative ADDITION. Division is not allowed; for example, if we were to try to divide 3 by 2, we would find it impossible to conceive of the answer within this framework.

For any work involving positive or negative division, we must turn to the latter concept of number represented in Fig. 2. Here, the whole (from 0 to 1 inclusive) is a unit; any fraction thereof is a non-unit. Manipulation within this framework occurs using a form of DIVISION, but we are not allowed to add; the answer to 3/4+3/4 lies beyond the scope of this concept.

In this study, the two different types of unit manipulation shown in Figs. 1 and 2 are blended by asserting the following propositions:

Prop. 1) The infinity between 0 and 1 (Fig. 2) and the infinity between "the greatest negative number" and "the greatest positive number "(Fig. 1) are equivalent,

Prop. 2 ) the units represented by the supreme combinations of each of the two number-lines are equivalent, and

Prop. 3 ) the two equivalent numerical infinities described in Prop. 1 ) are essentially different from the units described in Prop. 2 ) in that they can be mathematically manipulated (added to, divided, etc.) whereas units do not have this feature.

Therefore, our approach consists of two different number features －UNIT as defined in Prop. 2 and NON-UNIT as defined in Prop. 1. A NON-UNlT may be derived either by division within or by addition to a UNIT, and becomes mathematically manipulable due to Prop. 3. NON-UNITS include any traditional count plural as well as most mass nouns, and therefore this concept differs from the concept of plural.

3.2. The "boundedness" feature

There are different subtypes of both units and non-units which we will discuss in the next sections. One of the differentiating factors between these subtypes is BOUNDEDNESS. In the traditional mass-count context, count nouns are always bounded: that is, they have definite ending points, exist in a certain time, and consistently refer to the same "amount" of whatever is named. On the other hand, mass and abstract nouns have no definite end points and may refer to different "amounts" over different periods of time.

However, there are problems with this. How do we interpret, for example, role interpretations of count nouns such as "president" in the "The president is elected every 4 years" or "temperature" m "the temperature rises"? They are obviously not bounded in the senses described above: we cannot answer the question "Which president?" or "Which temperature?". Like mass nouns, their references change over time. Should they, then, be interpreted as mass terms? In that case, we would have to abandon our traditional ideas about mass terms not being able to take quantifiers and denumerators, since combinations like "a president" and "every temperature" are of course possible.

Clearly then, we must use boundedness as a defining feature. This feature is relatively transparent within the unit context, so let us start our discussion from there.

3.2.a. Boundedness within units

Remember that we defined two equivalent UNIT types in 3.1, Prop. 2, represented by the number lines shown in Figs. 1 and 2. The difference between these UNIT types is easily understood from the number lines: the former unit has no defined end-points, and therefore is UNBOUNDED, whereas the latter has the end-points of 0 and 1 (or any alternative end-points one may wish to designate), and is BOUNDED. The former may be referred to as a TYPE. The latter UNIT is an INSTANCE. Examples of INSTANCES are single objects or individuals such as the following:

12 a) John

b) Mary

c ) my present house

d ) the water in my glass

These bounded INSTANCES are all physically as well as perceptually contiguous, and cannot include more than what they already denote without becoming a non-unit (or a type). On the other hand, a TYPE UNIT (unbounded) such as the following:

13 a) chocolate (is loved by children everywhere)

b ) kindness (is appreciated universally)

or 6e) dogs (are numerous)

has no end-points, either physical or perceived. It may also expand to cover any new instantiations of itself, without losing its unity; if I were to lend you a book tomorrow, you would consider it an example of 'kindness', without having to change the definition or conception of 'kindness' in any way. These two features of dl EXPANDABLE and i PERCEIVED END POINTS are the primary methods of distinguishing between BOUNDED and UNBOUNDED nominals.

Another important feature between BOUNDED and UNBOUNDED terms is that of TEMPORALITY. For a BOUNDED term such as John or my chair, the value of the term is relatively constant over a certain period of time, and may be specified at any moment during that period of time. John may break his arm or the chair n}ay get a fresh coat of paint, but the essence of either of them remains the same. An UNBOUNDED term, on the other hand, is impossible to specify over time; the number of divisions within it (such as the number of chocolates which are loved by children) constantly changes.

3.2.b. Boundedness within non-units

Let us now turn to the discussion of BOUNDEDNESS within NON-UNIT nominals. BOUNDED NON-UNITS, like BOUNDED UNITS described above, are temporally constant, have perceived and definable end points and cannot expand to include new instantiations. This group is mainly comprised of what we traditionally consider count plurals. In contrast, the references of UNBOUNDED NON-UNITS vary over time, and are expandable. Within this group we will find most of our interpretations of traditional mass nouns. Examples of BOUNDED NON-UNITS would be:

14 a) John and Mary

b) six cats

c) the Beatles

e) the ladies of the PTA in June 1988.

Each of these non-units is spatially definable; that is, each has defined end-points. They do not change in essence over time: the ladies of the present PTA may become more obese or cease to be involved in the PTA, but that does not change the fact that they belong to the group (as defined in June 1988) of ladies of the PTA. Similarly, the Beatles may break up, go their separate ways; one of them may even be assassinated but that does not cloud the fact that they are still (as we know and love them) the Beatles.

In contrast, examples of the latter UNBOUNDED NON-UNITS are:

15 a ) the President of the United States (of which there have been 40 and will be more)

b) the Rolling Stones

c) furniture

d) bees in my back yard

These non-units lack any temporal definition － the "number of bees" in my back yard may change from day to day and season to season; the President of the United States (usually) changes in essence every time election year comes around.

Contrastmg 14 c) "the Beatles" and 15 b) "the Rolling Stones" may shed some special light on this problem. At first glance, these seem as if they should have identical features, but they actually quite have different functions. "The Beatles", as described above, will never cease to be the famous group of 4 from Liverpool; our grandchildren in the next century will come to know them exactly as we do now - as John, Paul, George and Ringo.

The situation for "the Rolling Stones" is quite different, however. The internal structure of this group is variable and dependent on time. The members of "The Rolling Stones" of 20 years ago were different from the members today, and if a new member entered tomorrow it would change nothing about the denotation of the name or our acceptance of its appropriateness. On the other hand, if suddenly a group appeared and called themselves "the Beatles", even if it were composed of Paul, George, Ringo and Ernie, it would have a very difficult time being accepted as either a continuation or a replacement of the original group.4

Thus, as with the boundedness feature among units, the salient differences between the former and latter examples are ± TEMPORALITY, ± EXPANDABILITY and ± PERCEIVED END-POINTS.

3.3 A feature for non-units only: the "focus" feature

The problems brought up through recent work with plurals and discussed already in previous sections have formed part of the basis for doing away with the count-mass distinction. Specifically, the question of sums vs. groups within non-units has not yet been accounted for within this ontology. This contrast is analyzed as a difference of FOCUS in the present paper.

Intuitively, the difference between a sum and a group plural, both BOUNDED NON-UNITS within the context of this paper, is the position of FOCUS or attention of the interpreter. If, as in the case of "John and Mary each ate a pear", the focus is on what occurred equally and identically to each individual member of the nominal, then this can be regarded as a sum. On the other hand, if the focus of the interpreter is directed more towards the characteristic(s) shared by each member of the non-unit (as, for example, in "the ladies of the PTA voted down the proposal") rather than towards the individual actions of each member separately, then this points to a group-type nominal. Recall that in 2.1.b we attributed the cohesion of groups to ABSTRACTION. The sum NON-UNlT is -FOCAL (-ABSTRACTION) ; no common characteristics are abstracted and each member receives a separate and specific share of the interpreter's Contrarily, the group NON-UNIT is + FOCAL (,+ attention. ABSTRACTION); the interpreter's attention is undividedly given to certain properties of the group as a whole, and individual members of the group cannot be singled out.

Let us take another approach. Recall the apple example in 2. 1. b, in which the question of importance of having either a -FOCAL sum of 7 apples (all the same) or a +FOCAL group of 2 sets of 3 (small, green) apples and 4 (large, red) apples was largely a matter of situational context. If we preserve the internal set structure, we must regard each part within the NON-UNIT as unequal or HETEROGENEOUS. On the other hand, if we do not preserve the internal structure, we can consider each portion essentially the same, or HOMOGENEOUS.5

This issue of HOMOGENEITY can be especially useful in distinguishing focus or non-focus within UNBOUNDED NON-UNITS. If we compare 8 c) "furniture" and 8 d) "bees m my back yard" we can see that although the internal structure of "furniture" is quite relevant ("Furniture" may be heavy, but that baby chair is furniture and it's quite light) the internal structure of "bees" is much less so. In other words, every single bee is pretty much the same, but every single piece of furniture may differ from other pieces in many ways. Therefore, we cannot regard "furniture" as homogenous. A lack of HOMOGENEITY then is an indicator of a presence of FOCUS.

To illustrate the FOCUS property within UNBOUNDED NON-UNITS, let us take the following sentences as examples of the three different UNBOUNDED nominal types: mass, property (non-unit) and type (unit):

16a) Gold is heavy. (property)

b ) Gold is metal. (mass)

c ) Gold is scarce. (type)

If we ask the question "Is this instance of gold (say a necklace) so ?" we will get three completely different responses:

17 a ) No, this instance is not particularly heavy.

b ) Yes, of course. ALL gold is metal.

c ) What ?

You can't ask whether one instance of gold is scarce! ?

Note that a property is always prefaceable by "As a general rule . . . " and may admit exceptions. (c.f. "As a general rule, gold is heavy.", " *AS a general rule, gold is a metal.")

Thus, we see the same distinction regarding FOCUS between properties and masses: a property functions as a general characteristic which is not necessarily true of each instance of itself. A mass is similar to a sum in that the interpreter's attention as well as the predicate applies equally to each and every instance of the mass. A type, of course, has no individual member structure at all, and thus is indivisible with regard to the predicate.

3.4. The complete structure of nominal countability categories.

Fig. 3. Tree structure of countability categories within the nominal domain.

Altogether we have defined six categories of noun functions, defined by the three binary features of unity, boundedness, and focus. Examples of each of these categories are:

Instance:Ronald Reagan

My chair is broken.

Did you see that lady's red hat?

Type:

mankind

Chairs are alike all over the world.

Red hats are available in many stores.

Group:

the ladies in the PTA, the Beatles

The chairs in the museum are old.

The red hats in this store are (mostly) fashionable.

Sum:

each of these men

These chairs sell for $30.00 each.

All of those red hats match my dress.

Property:the Rolling Stones

Chairs have four legs.

Red hats are usually worn in the summer.

Mass:

water

Chairs are to sit in.

All red hats are red.

4.0 INDUCTION, DEDUCTION AND INTERCHANGE: THE VEHICLES OF CIRCULARITY

We have already defined the features and categories involved in the circular binary countability ontology, but we still must account for the processes of shifting between these different categories. Here, we will attempt to provide the explanation for the variation in countability shown y many English nouns which is lacking in the count-mass system. This variation will be accounted for by postulating 3 different shift processes － INDUCTION, DEDUCTION and INTERCHANGE.

4.1. INDUCTION (the process of abstraction)

It is no accident that the first two processes we are about to define share the same name as the logical functions of deduction and induction. When we perform inductive logic, we compare different objects and abstract from them common properties; "jumping to conclusions" on the basis of a certain number of examples. This is essentially what happens, too, in our version of induction. Let us imagine ourselves in a car, on a U. S. highway for the first time. Say that we are in the habit of noticing the cars passing by us. The first car we see is a white Toyota. The second and third happen to be white Suzukis. At this point, we may say to ourselves, "Those 3 cars were white Japanese cars", and conclude that "cars on the U. S. highway are (generally) white Japanese cars". This may be true even if the next car is a blue Ford. Assuming that it is, if the fifth car is again a white Nissan, we may say to ourselves, "Boy, white Japanese cars are really numerous!". What we have done here is move from one INSTANCE, to a GROUP of instances, abstracted a PROPERTY from them through induction, and finally unified and erased all the individual instances within that property. This process is diagrammed in Fig. 4 below.

Flg.4. Mechanism of Induction, or abstraction.

INDUCTION may be further understood by referring to the number line in Fig l. It is this whole number line which is used in the INDUCTION process thus, the process proceeds on the basis of ADDITION; starting with the unit “1" and progressing to the unit of "∞"

4.2. DEDUCTION (the process of distinction)

DEDUCTION, or the process of focussing and distinction, may be compared with the view of, say, a field of flowers through the lens of a camera. First, if the camera is way out of focus, we are likely to see nothing but a blur. Then, as we start to rotate the focus ring, the blur begins to separate and blotches may be distinguished. At this stage the blotches are divisible, but they are not yet countable. As the view becomes clearer, we begin to distinguish separate flowers, and to be able to count their numbers. If we then use our zoom lens to concentrate on just one of these flowers, we will have completed another cycle: from TYPE, to MASS, to SUM, to INSTANCE. As another example, let us consider a bowlful of marbles. If we look at it from a sufficiently far-away distance, we will find that it indeed looks like a bowlful of something, but not know what that something is. If we get closer, we may realize that that something is marbles. We may also discern that there are different colors of marbles: green, yellow, red, etc. However, we may not yet count the marbles, until we turn the bowl and spill out the contents. After the count is finished, we may find one particular marble that we are attracted to. And thus, again, the cycle from type to instance has been completed.

Fig. 5. Mechanism of Deduction or distinction

Note that this cycle, too, is based on the logical process of deduction. We start out with all the information we need; however, we must look at it carefully and sort it out in order to draw conclusions from it. We will never come to a conclusion about something which may be refuted with the next example, since we have all examples at hand. Furthermore, the conclusions which we draw will be equally applicable to any portion of the original type. DEDUCTION begins from the non-unit recognized as the infinity between O and I in Fig. 2; thus the mode of mathematical manipulation is DIVISION.

4.3. INTERCHANGE: Modus for circularity

Let us consider what would happen if we were to try to continue either abstraction (INDUCTION) from a pure type or of these processes distinction (DEDUCTION) of an instance. In the former case, we would take the type "Japanese cars (a-re numerous)", shift it to an instance, group it with others having similar features (for example, 'American cars', 'European cars', etc.) and abstract a new property from these such as 'the world's cars'. We can then form the type by "erasing" the separate values of the property.

This may be referred to as a TYPE-INSTANCE SHIFT, and may be compared to taking something completely shapeless or timeless, say. air, and putting it in a box or canning it. We have not changed its physical structure or altered it in any way. We have only "bounded" it so that it may be compared with other similar bounded objects.

A parallel phenomenon occurs with distinction. If we look closely at any object, say a flower, we can distinguish the petals, leaves, stem of the flower, and we can start to count each of the members of these subgroups and perhaps focus on one petal or one leaf, starting the process again. In other words, the instance of one 'flower' has shifted to a type 'flower', which is first sorted and then counted in a repetition of the DEDUCTION process.

This INSTANCE-TYPE SHIFT may be compared with the act of taking a single object and placing it under a microscope. No addition, division or any other physical alteration has occurred to the object. It has simply been reoriented with respect to its surroundings, so that we are now in a position to begin a distinction process.

The above discussion has highlighted two further important features of this ontology: 1) the CIRCULARITY of the induction and deduction processes and 2) the INTERCHANGEABILITY of types and instances. The TYPE-INSTANCE and INSTANCE-TYPE shifts are motivated by the equivalency of the two unit types involved, as defined in 3. 1, prop. 2. A completed diagram of the interrelation between the categories and shifting processes is shown in Fig. 6 below.

It is the constant motion and interplay of these three processes INDUCTION, DEDUCTION and INTERCHANGE which leads to the possibility of one sememe having many different usages. The INDUCTION and DEDUCTION processes are both unidirectional that is, INDUCTION takes place only when an INSTANCE noun is raised to a, GROUP (e.g. through grammatical pluralization), PROPERTY (through ABSTRAPTION) or TYPE. Similarly, DEDUCTION only occurs from TYPE to MASS, SET or INSTANCE. A TYPE may never directly deduce to a PROPERTY; likewise an INSTANCE may never directly induce to a SET. On the other hand, INTERCHANGE is bidirectional and provides the setting for indirect induction or deduction.

Fig.6. Compelete mechanism for circularity and shifting within nominal binary countability system.

5.0 APPLICATIONS

Now, armed with the framework just described, let us return to our original problems and see how it may be applied. Repeating our first examples of count-mass ambiguity here:

1 a) There is apple in the salad, but there are more apples on the shelf.

b) We each ordered an ice cream and a coffee.

c) When he finished eating, there was egg all over his face.

We may now describe the change between apple and apples in 1 a) as a deduction from a sum of x apples (depending on how many there are in the salad) to an instance of “apple”, which interchanges with a type “apple”, and then deduces to a mass (the apple in the salad). In 1 c), which is analogous to 1 a), we are treated with a wonderful mental picture of the whole process: imagine our sum of eggs, broken and mixed into an instance of egg mixture, cooked into a beautiful omelet, and then gleefully crushed and mashed into a mass of egg by a baby’s hand.

In 1 b), the ice cream and coffee have been taken from their original status of mass, and simply deduced down to instances. This is an example of a DEDUCTIVE shift from mass to count. Here, in traditional terms, the count noun is interpreted as a instance of the mass noun. In addition to this type of mass-count shift, it is often pointed out (e.g. in Burge, 1972) that traditional mass nouns may always take a “kind” interpretation, at which time they act like count nouns. For example, consider the following:

16) We compared five different rices in a taste test.

The general mechanism for creating “kind” interpretations is illustrated through this example: firstly, rice as property is shifted to a type, which is interchanged with an instance. Then this instance can be combined with other instances and induced up to a group. This is an INDUCTIVE mass-count shift.

We are now, too, in a position to interpret and classify Cherchia’s different “dogs”:

6 a) Dogs are mammals. (all) －MASS

b) Dogs are intelligent. (most) －PROPERTY

c) Dogs give live birth. (most female dogs) －PROPERTY

d) Dogs are barking in the courtyard. (some) －GROUP

e) Dogs are numerous. (???) －TYPE

7 a) Dogs must be carried. (at least 1 per person present) －GROUP

b) Dogs must be carried. (all present) －MASS

Further, we may trace the processes by which each of these was derived: taking the type "dogs" 6 e) as a unit, we deduce therefrom the mass "dogs" of 6 a) and 7 b). On the other hand, the group "dogs" and property "dogs" were induced from combinations of single dogs. We may confirm these observations by applying an "each and every" test to each sentence:

18 a) Each and every dog is a mammal.

b) *Each and every dog is intelligent.

c) *Each and every dog gives live birth.

d) *Each and every dog is barking in the courtyard.

e) *Each and every dog is numerous.

19 a) *Each and every dog must be carried.

b) Each and every dog must be carried.

We see that, because mass "dogs" are deduced from a type, they lack focus and are therefore HOMOGENOUS, or treated equally. The other "dogs" being induced from instances, are +FOCUS and therefore the predicate does not distribute over each part, but is true only of the whole.

6.0 CONCLUSIONS

The inefficiency of the count-mass distinction as a linguistic criterion for discussing countability has led to the development of the new modality discussed in the previous sections. This circular binary countability ontology is based on three features (UNITY, BOUNDEDNESS and FOCUS) and defines 6 categories of countability. It includes two circular routes for mathematical manipulation: an INDUCTION process for pluralization and abstraction, and a DEDUCTION process for focussing and distinction. It also allows for free shifting between the unit categories so that INDUCTION and DEDUCTION processes may be carried on infinitely.

It is hoped that the circular binary countability ontology may prove to have a sound physical, mathematical and philosophical base and thus prove to be linguistically effective in the treatment of nominal countability in English. What has been developed so far is no more than a theoretical framework, thus there is much work yet to be done in providing axiomatic theory to the framework as well as putting it to the test in actual language analysis. Specifically, it should be provided with constraints to explain the relative mobility of certain nouns (e. g. 'iron') over others (e, g. 'equipment') as regards count-mass shifting.

NOTES

2. 1. In the same article, Allen also comes out against the dual mass-count classification system and suggests an arrangement of 7 different levels of countability based on possibilities of occurrence in different syntactic environments. His representative nouns for each of the 7 levels were car (completely countable), oak, cattle, Himalayas, scissors, mankind, admiration, and equipment (completely uncountable). Note, through that we may find plausible contexts for mass interpretation of even the most countable level (e.g. at an automobile graveyard － “I was shocked at the huge mountains made up entirely of car.” Or for count interpretation of the least countable level (at a sporting goods rental shop － “These guys want two tennis equipments － can you bring out some more from the back?”).

3. Pelletier (1974) reviews and points out difficulties with this position. As an example of mass nouns being elliptical count nouns, he says, "is water' might be elliptical (in some circumstances) to 'is a body of water' or (in other circumstances) to 'is a kind of water"'. He refutes this idea, however, with the sentence "What Jeff spilled is the same coffee as what he wiped up". Although this sentence states a true identity, the 'coffees' involved may not be counted in the same manner (a puddle of coffee may not be spilled; a cup of coffee may not be wiped up).

4. This example was given in a class lecture, 1987 Linguistic Institute, Stanford. The ambiguity here is interesting: 7 a) (analogous to "Hats must be worn,") means that "if you are here, you must carry a dog", while 7 b) means "if there is any dog here, it must be carried". Note that these two meanings do not have the same truth value if (a) is true, (b) may be false (there may be dogs running around without people to carry them); Iikewise, (b) may be true and (a) false (even if all dogs are being carried, there still may be people left over sans dog).

5. Fauconnier (1985) treats this problem of "role" vs. "name" proper nouns with an example of "Fido", naming, on one interpretation, a neighbor's specific dog, and on another interpretation, the role of the neighbor's watchdog (the role continues, even if the actual dog is replaced with another dog). In our analysis, the first interpretation would be an INSTANCE; the second would correspond to a PROPERTY.

6. Langacker (1987, p. 64-65) uses the terms "heterogeneous" and "homogeneous" somewhat differently than they are used here he speaks of "conceived homogeneity" as a factor for distinguishing count and mass nouns, but admits that it "is not an invariant feature of count nouns" and "is not self-evident" in mass nouns. Contrary to the present paper, he regards furniture as homogeneous, since "speakers aware of the internal diversitv_ of a substance are nevertheless capable of construing it as homogeneous". It is unclear how plurals are to be treated in his analysis.

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