Equality and Ordering - Comparing Adjectival Quantities

We now turn our attention to the words less than or fewer than, not equal, or not the same number as, equal or as many as, more than and the mathematical symbols that correspond[1] to them

•        comparing a discrete quantity to a discrete quantity
•        comparing a continuous quantity to a continuous quantity
•        comparing a discrete quantity to a continuous quantity

These cases need to be investigated separately.

a. Comparing discrete quantities

If we wish to compare two discrete quantities to one another we must consider several possibilities. These are

•        the referents of the quantities are identical
•        the referents of the quantities are distinct (and disjoint)
•        the referents of the quantities are neither identical nor disjoint

          i.                identical discrete referents - apples and apples

For example,    Six apples are fewer than eight apples.      {6, apples} < {8, apples}.

These sets have different cardinalities but the same set definition, and the equality and inequality symbols are used in the same way they are used for nominal quantity.

ii.              distinct discrete referents - apples and oranges

For examples, suppose we wish to compare   Six apples and Eight oranges.

iii.             discrete referents that are neither identical nor distinct - apples and fruit

For example, suppose we wish to compare     Six apples and Eight pieces of fruit.

Consider the question “Are there more apples than fruits in the fruit bowl?” The confusion surrounding the answering of this question is traceable directly to the sub-acts involved in the act of counting.  Specifically, one has to consider all the attributes of the entity to be counted and to decide whether it can be included in the set one is attaching magnitude to. 

Suppose, for example there are six apples and two oranges in the fruit bowl. The word fruit in the question may be interpreted by some as meaning non-apple fruit. In that case one can expect a quite different answer that if the respondent considers the word fruit to refer to both apples and oranges. The difficulty probably lies with the way we use language. In normal discourse the word or is used as if it were part of the implicit phrase either...or, that is to say, to separate possibilities that are disjoint.  We are embedded in a linguistic surround that abounds in the use of such phrases as hot or cold, tall or short, sedan or station wagon etc.[2]

b. Comparing continuous quantities

If we wish to compare two continuous quantities to one another we must consider several possibilities. These are

•        the comparison of identical attributes (e.g. weight) of the two quantities - within this case we must consider two cases, i.e. identical and distinct units of measurement
•        the comparison of distinct attributes of the two quantities

i.                identical attributes of referents with identical units

For example, suppose we wish to compare       six lbs. of peas and  eight lbs. of paper.

Here we are dealing with distinct magnitudes but the same attribute measured in the same units, and the equality and inequality symbols are used in the same way they are used for nominal quantity, e.g.

The 1.4 kg. of water in the brown pitcher is heavier than the .7 kg piece of meat.

ii.              identical attributes of referents with distinct units

Here we are dealing with distinct magnitudes and distinct units of the same attribute. This case raises a set of issues that forces us to delay resolving this question until after we have introduced the notions of extensive and intensive quantity and have considered some of the mathematical operations that transform the referents of adjectival quantity.

iii.             distinct attributes of referents

The maximum speed of the car is greater than its weight

The girl is younger than her height.

Occasionally, we permit ourselves a somewhat poetic violation of this apparent rule with the our use of such statements as

He has more luck than brains.[3]

• taller, shorter                         height
• heavier, lighter                       weight
• longer, shorter                       time duration, length
• wider, narrower                      horizontal extent   

In this section we begin the consideration of how the arithmetic operations that we know how to carry out with nominal numbers may or may not be extended to adjectival quantities. The operations we refer to are addition and subtraction. We will deal with multiplication and division in the next section. We will discover that there are systematic differences between the way a given operation works with nominal quantity and the way it works with adjectival quantity.

For example, a problem that has been studied extensively in the mathematics education research community is the following:

                    How many school buses are needed to take all the students in the school on a field trip if there are 500 students in the school and each bus carries 30 students?[5]

The expected answer is 17 buses even though a straightforward computation leads to the result 16 and 2/3 buses.  Students are expected to recognize that buses are count nouns and that the quantity {2/3 bus} is not acceptable.

How do arithmetic operations with adjectival quantity work? We observe that in the case of nominal numbers, the arithmetic operations provide ways for us to take two numbers and combine them so as to produce a new number.  If we try to extend these operations to adjectival quantities we must face the question of what happens to the referents of the quantities being combined and how is the referent of the resulting quantity determined. As in the case of comparing adjectival quantities that we considered in the previous section we must take into account the nature of the quantities being combined, i.e. are they discrete, continuous or both.

Broadly speaking, we consider two quite different kinds of operations. The first kind of operation, which I call referent-preserving, allows us to combine two quantities with the same (or equivalent) referents and produce a new quantity with the same referent. A simple example might be

                  {3, apples} + {6, apples}.

In this case the result {9, apples} has the same referent as each of the quantities that were composed in order to construct the result. Addition and subtraction are the most common examples of referent-preserving operations.

The second kind of operation, which I call referent-transforming, allows us to combine two quantities, with the same or differing referents, to produce a new quantity whose referent differs from either one or the other or both of the referents of the original two quantities.  A simple example might be

                  {30, apples/bushel} x {10, bushels}.

In this case the referent of the result, i.e., bushels, is not the same as either as the referent of either of the quantities that were composed in order to construct the result. Multiplication and division are the most common examples of referent-transforming operations.

a.               addition of discrete adjectival quantities

Addition would seem to be a referent-preserving operation. We are willing to say

                  Two apples and three apples are five apples.

and thus define an addition for adjectival quantity so that formally it corresponds to the addition of nominal quantity.

However, people are also willing to say

                  Two apples and three oranges are five pieces of fruit.

Clearly, the referents of the two and the three are not identical. In fact, they are disjoint, i.e. no apple is an orange and no orange is an apple.  On the other hand, both apples and oranges are pieces of fruit. In this case, when the referents of the two quantities are disjoint and there exists a superordinate class to which both referents belong, we can extend the operation of addition with nominal number to adjectival quantity in a straightforward fashion.

If the superordinate class is too remote from the referents being composed then it is not likely that we will find people “adding” them. For example, people do not say

                  Two elephants and three computers are five things.

One must be cautious about judgments in this area because statements of the following sort might clearly be made in the appropriate context.

                  Two toasters and three overcoats are five inventory items.

Finally, people do not often say

                  Two apples and three pieces of fruit are five pieces of fruit.

This statement is clearly ambiguous. The referents of the two and the three may or may not be disjoint, i.e. some, all or none of the apples may be among the pieces of fruit and we have no way of telling from the utterance alone. 

It seems to be the case that the operation of addition that we know for nominal numbers may be extended in a reasonably straightforward way to the addition of discrete adjectival quantities, provided we are careful about disjointness and superordinate classes. Does it all go as easily for continuous adjectival quantity?

b.              addition of continuous adjectival quantities

If we combine two volumes of water, 100 cm3 and 50 cm3, we obtain a new quantity of water whose volume is the sum of the two original volumes computed as if the quantities were nominal numbers. In this case we have identical referents, and we are considering common attributes measured in the same units

The first problem we encounter when we seek to move beyond this case is one that we encountered before when we analyzed the comparison of adjectival quantities. Suppose we combine two volumes of water, 0.1 liters and 50 cm³. If we physically do so, we obtain a new quantity of water, 0.15 liters or 150 cm³, whose volume is the combined volume of the two original volumes.  But how do we compute the magnitude of the combined volume?  As before, we must beg off at this point and delay resolving this question until after we have introduced the notions of extensive and intensive quantity and have considered some of the mathematical operations that transform the referents of adjectival quantity.

The second problem that we encounter when we try to extend our notion of addition to continuous adjectival quantity arises when we seek to combine a common attribute of distinct referents, for example, 100 cm³ of water and 50 cm³  of alcohol. The volume of the resulting fluid is in fact smaller than 150 cm³ !  If this example seems esoteric, consider combining 100 cm³  of water and 50 cm³  of air.

It is not the case that the common attribute of distinct referents can never be added algorithmically in the same way that nominal numbers can.  One can add one's shoulder height and the length of one's arm and the height above the ground of the ladder rung that one is standing on in order to determine just how high up the wall one can reach with a paintbrush.

Is there a clear, procedural way of deciding when and under what circumstances one can extend the operation of addition with nominal numbers to the combining of common attributes of distinct referents for continuous quantity?  It would seem not. The detailed nature of the referent situation must be taken into consideration and a judgment made in each instance. I suspect the ability to exercise reasonable judgments about such matters is part of what we regard as common sense.

c.               subtraction of discrete adjectival quantities

The situations that people try to describe and analyze using the operation of subtraction are for the most part situations in which there is a prior state, and then something happens to change it resulting in a final state. For example, Jimmy has five apples. He then gives two apples to Joey. Jimmy now has three apples. When cast in the form of a school mathematics problem, e.g. Jimmy has five apples. He then gives two apples to Joey. How many apples does Jimmy now have? such situations are referred to as cause-change problems.[6]

Consider the statement

                  Removing two apples from five apples leaves three apples.

This suggest that we can define a subtraction operation for discrete adjectival quantity if the quantities in question have common referents.          

On the other hand, how shall we treat the statement

                  Removing two apples from five pieces of fruit leaves....?

People are willing to end the sentence with the phrase three pieces of fruit.  This suggests an interesting distinction between the operations of addition and subtraction applied to nominal numbers and the same operations applied to adjectival quantity. In the case of nominal numbers, subtraction and addition are inverse operations.  If subtraction with adjectival quantity were the inverse of addition with adjectival quantity, then people would be willing to say that

                  Two apples and three oranges are five pieces of fruit.

and

                  Removing two apples from five pieces of fruit leaves three oranges.

are equivalent statements.  Formally it would seem that the allowable inverse operations to

                  Two apples and three oranges are five pieces of fruit.

are

                  Removing two apples from five pieces of fruit leaves three pieces of fruit.

and

                  Removing three apples from five pieces of fruit leaves two pieces of fruit.

For subtraction of discrete adjectival quantity with disjoint referents to make any sense, one referent must be superordinate to the other referent, e.g. apples are pieces of fruit as are oranges.

If the referents are distinct and neither is superordinate to the other, there does not seem to be any reasonable definition of subtraction.  For example,

                  Removing three apples from five oranges leaves...?

d.              subtraction of continuous adjectival quantities

If we remove one volume of water, say 50 cm³ , from a larger volume of water, say 150 cm³ , we obtain a new quantity of water whose volume is the difference of the two original volumes computed as if the quantities were nominal numbers. In this case we have identical referents, and we are considering common attributes measured in the same units.                   

As in the case of addition with continuous adjectival quantity, we encounter several problems when we seek to move beyond this straightforward case. The first is one that we encountered before when we analyzed the comparison of adjectival quantities. Suppose we remove a volume of water, 50 cm³ say, from a larger volume of water, 0.15 liters say.  If we physically do so, we obtain a new quantity of water, 0.10 liters or 100 cm³ , whose volume is the difference in the volumes of the two original volumes.  But how do we compute the magnitude of the remaining volume?  As before, we must beg off at this point and delay resolving this question until after we have introduced the notions of extensive and intensive quantity and have considered some of the mathematical operations that transform the referents of adjectival quantity.

The second problem that we encounter when we try to extend our notion of subtraction to continuous adjectival quantity arises when we seek to find the difference of two quantities that have a common attribute of distinct referents, for example,  150 cm³  of fluid and 50 cm³  of alcohol.  As was the case with the subtraction of quantities, if one of the referents is superordinate to the other, it is possible to make sense of the operation. 

Is there a clear, procedural way of deciding when and under what circumstances one can extend the operation of subtraction with nominal numbers to the problem of finding differences between continuous adjectival quantities with common attributes of distinct referents?  It would seem not. The detailed nature of the referent situation must be taken into consideration and a judgment made in each instance.

Notes