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Lecture 21, Concepts and Categories. As we've seen, learning, perceiving and remembering
require more than just forming associations between stimuli and responses, extracting
information from environmental stimuli, and reproducing information stored in memory traces.
Rather, learning is a process of generating and testing hypotheses, in which the learner
tries to figure out what predicts what, and what controls what, in the environment. Perception
is a constructive process, in which the perceiver tries to figure out what is out there, where
it is, what it is doing, and ultimately, what it means. And memory is a reconstructive process,
by which the rememberer tries to figure out what happened in the past, when it happened
and where and to whom -- and again, what it means. Thus, the basic functions of perceiving,
learning, and remembering depend intimately on reasoning, inference, problem-solving,
judgment and decision making -- in short, on thinking.
Nowhere, I think, is this clearer than with perception, which is dependent on the receiver's
repertoire of concepts stored in memory. As Jerome Bruner has noted, every act of perception
is an act of categorization. In identifying the form and function of objects, which is
after all the basic task of perception, stimulus information has to make contact with preexisting
knowledge. In perceiving we decide how, in general, to think about the objects and events
we perceive; how they're similar to other objects and events that we've encountered
in the past. When we classify two objects or events as similar to each other, we construe
them as members of the same category, an act which reveals a concept which is part of our
semantic memory.
A category may be defined as a group of objects, events, or ideas which share something in
common. A concept is the mental representation of a category, abstracted somehow from particular
instances. Concepts serve important mental functions. They group related entities together
into classes. And provide the basis and language, for things like synonyms and antonyms. And
also allow us to understand the implications, of the things that we see. Now, technically
categories exist in the real world while concepts exist in the mind. However, this technical
distinction is difficult to uphold, and so psychologists commonly use the two terms interchangeably.
Some categories are defined through enumeration -- an exhaustive list of all the instances
of a category. A good example is the letters of the English alphabet, A through Z. These
have nothing in common except their status as letters in the English alphabet, and if
you want to identify this category, there's really nothing else to do except reel off
all 26 letters. A variant on enumeration is to define a category
by a rule which will generate all instances of the category, an example is the concept
of integer in mathematics, which is defined as the numbers 0, 1, and any number which
can be obtained by adding or subtracting 1 from these numbers one or more times. If you
follow that rule, you'll generate all the integers and you'll only generate integers.
But by far the most common definitions of categories are by attributes -- properties
or features that are shared by members of the category. Thus, to anticipate an example
I'll give later, birds are warm-blooded vertebrates with feathers and wings, while fish are cold-blooded
vertebrates with scales and fins. All birds share those attributes in common and all fish
share those attributes in common. There are three broad types of attributes
that are relevant to defining a category. First, there are perceptual features, physical
features of the object -- features that you can see, or hear, or whatever. Then, there
are functional attributes, including the operations that can be performed with the objects, or
by the objects - the uses to which they can be put. Functional attributes are often used
to define categories of artifacts, like tools or vehicles. And finally, there are relational
features, which specify the relationship between an instance and something else. Relational
features are used to define many social categories like aunt, which is the sister of a father
or a mother; or stepson, which is the son of one's husband or wife by a former marriage.
Not all aunts look alike, nor do all stepsons. But all aunts share the relational feature
that they are the sister of a father or a mother. And all stepsons are the sons of one's
husband or wife by a former marriage. These categories are defined by relation.
Categories are groups of objects that share something in common. But as psychologists,
what we're really interested in is the mental representations of these categories or the
structure of concepts. The classical view of categorization was given to us by Aristotle
in his Treatise on Categories, part of the Organon from the 4th century B.C. It was Aristotle
who gave us the view of categories as proper sets. That's Aristotle on the right, next
to Plato on the left, from Raphael's School of Athens. According to the view of categories
as proper sets, the objects in a category all share the same set of defining features, defining features
which are singly necessary and jointly sufficient to define the category. By singly necessary
we mean that every instance of the category possesses that feature. The feature must be
present for the object to belong in a particular category. By jointly sufficient, we mean that
every entity that possesses the entire set of defining features is an instance of the
concept. The entire set of defining features is all you need to identify an object as
part of a category.
Examples of classification by proper sets include geometrical figures. Remember from
your high school geometry, that all triangles have three features in common: they're two-dimensional
geometric figures with three sides and three angles. And all quadrilaterals also have three
features in common: they're two-dimensional geometric figures with four sides and four
angles. Animals are also classified by proper sets: all birds are warm-blooded vertebrates
with feathers and wings, while all fish are cold-blooded vertebrates with scales and fins.
In each case, the features in question are defining features that are singly necessary
and jointly sufficient to identify an object as a member of a category. If you're a cold
blooded vertebrate with scales and fins, you're a fish, you're not a bird. And if you're a two-dimensional,
geometrical figure with four sides and four angles, you're a quadrilateral, you're not
a triangle.
According to the proper set view, categories can be arranged in a hierarchal system that
represents the vertical relations between categories, producing a distinction between
high-level superordinate categories and low-level subordinate categories, sometimes known as
supersets and subsets. These hierarchies of proper sets are characterized by perfect nesting,
by which we mean that subsets possess all the defining features of supersets.
Put another way, we create subsets by adding one or more defining features to the set of
features that define a superset. In this case, subsets contain all of the defining features
of supersets, plus new defining features.
So, in standard Euclidean geometry, the kind you learn in high school (and that's a picture
of Euclid there, from Raphael's School of Athens), there are four types of geometrical
figures. Points are geometrical figures in no dimensions; lines are geometrical figures
in one dimension; planes in two dimensions; and solids in three dimensions. So at this
level of categorization, the geometric figures are classified depending on the number of
dimensions they consume. Nested underneath the superordinate category of planes that
is, 2-dimensional figures, are subordinate categories defined in terms of the number
of lines and angles they have: triangles have three sides and three angles; quadrilaterals,
four sides and four angles; pentagons, five sides and five angles; hexagons, six sides
and six angles; and so on all the way up.
Nested underneath the category of triangle are additional categories defined in terms
of equality of sides and angles. For example, classifying triangles by the length of their
sides, we have equilateral triangles, which have three equal sides and three equal angles;
isosceles triangles, that have two equal sides and two equal angles; and scalene triangles
that have no equal sides and no equal angles. Alternatively, triangles can be sub-classified
in terms of the size of their internal angles: right triangles have one internal angle of
90 degrees; oblique triangles have no 90 degree angles. And oblique triangles can be further
subdivided into acute triangles, that have all their internal angles less than 90 degrees;
and obtuse triangles, that have one internal angle greater than 90 degrees. Notice that
what we've done is to create subcategories by adding a defining feature. All triangles
have three sides and three angles, but there are subordinate categories of triangles that
are defined either by the length of their sides or by the size of their internal angles.
And we could do exactly the same thing for four sided figures, quadrilaterals. All quadrilaterals
have four sides, but the sub-category of trapezium consists of quadrilaterals that have no sides
parallel; trapezoids have two opposite sides parallel to each other, but not the other
two; and parallelograms have both pairs of opposite sides parallel. Parallelograms can
be further subdivided into rhomboids, which have no right angles; rhombuses, a subcategory
of rhomboids, have all four sides equal in length; and rectangles that have four right
angles; and,squares, a sub-category of rectangles, have all four sides equal in length. Again,
we've created sub-categories of four-sided figures, quadrilaterals, just by adding another
defining feature: whether their sides are parallel or not; if they are, how many sides
are parallel; whether there's a right angle or not; and if so, how many there are.
Another example of classification by proper sets is the biological taxonomy set down by
Carl Linnaeus, a Swedish naturalist in the eighteenth century. Linnaeus first divided
the natural world into three great kingdoms, animal, vegetable, and mineral. The animal
and vegetable kingdoms were then divided into phyla; each phylum divided into classes; each
class divided into orders; each order divided into genera; each genus into species; and
each species into sub-species. In the Linnaean system, every living thing is known by its
genus and species name. Thus modern humans are known as *** sapiens, our genus and our
species -- technically, *** sapiens sapiens. Each level of the hierarchy is created by
adding one or more defining features to the level immediately above it. In this way, every
species, and every subspecies, is identified by a set of defining features that are singly
necessary and jointly sufficient to identify an individual as a member of one or another
species, so all proper sets can be arranged into hierarchies of supersets, superordinate
categories, and subsets, subordinate categories.
Within each level of the vertical hierarchy, there are also horizontal relations between
adjacent categories, or between subcategories, within a particular level of the vertical
hierarchy. These horizontal relations between adjacent categories are governed by an all-or-none
principle. Because category membership is determined by a finite set of defining features,
and an object either has the features or it doesn't have them, then an object is either
in a category or not. There's no halfway. There are sharp boundaries between adjacent
categories, and there's no straddling the fence.
So, for example, referring back to high school geometry, if you're a geometrical figure,
you're a point, a line, a plane, or a solid. There's no in-between. If you're a plane,
you're either a triangle or a quadrilateral or whatever; if you're a triangle, you're
either an equilateral triangle or an isosceles triangle or a scalene triangle; if you're
a scalene triangle, you're either a right triangle or an oblique triangle; and if you're
an oblique triangle, you're either an acute triangle or an obtuse triangle. You're in
one place or the other, because each of these categories, and each of these subcategories,
is defined by a unique set of singly necessary and jointly sufficient defining features.
You either have the features or you don't: if you do, then you're in the category; if
you don't, you're not in the category.
It also follows, from the classical view of categories as proper sets, that categories
have a homogeneous internal structure. Because all instances of a category share the same
set of defining features, it follows that all instances are equally good representatives
of the category. They're all alike with respect to their pertinent defining features.
So, for example, here's the entire set of quadrilaterals, or four-sided planes. As far
as the classical view of categories as proper sets is concerned, each of these examples
is just as good a quadrilateral as the others. They are all equal because they all share
the same feature of being two-dimensional, with four sides and four angles.
So, the classical view of categories as proper sets yields two procedures for categorization.
If we want to define a category of objects, then we have to determine the set of defining
features that are shared by all members of the category. On inspection, all triangles,
no matter what the kind, will have two dimensions, three sides, and three angles. And on inspection
all birds, of whatever kind, will be warm-blooded vertebrates with feathers and wings.
And if you want to categorize a new object into a familiar category, the procedure is
only slightly more complicated. First, you use your perceptual apparatus to analyze the
features of the object. Then, you retrieve from memory the defining features of various
candidate categories: is the object a triangle or a quadrilateral? Is it a bird or a fish?
Then you match the features of the object to the defining features of the candidate
categories, and if you find a match you simply assign the object to the appropriate category.
So if you encounter a new animal and you want to determine whether it's a bird or a fish,
you first determine whether it's a vertebrate or not and whether it's warm- or cold-blooded,
whether it has feathers or scales and whether it has fins or wings. If it's a warm-blooded
vertebrate with feathers and wings, you call it a bird. If it's a cold-blooded vertebrate
with scales and fins, you call it a fish. It's just that simple.
The classical view of categories came down to us from the time of the ancient Greeks,
and it's lasted a long time. It's certainly the logical way to go about creating categorizes
and categorizing objects. But however logical it may seem, it apparently is not how the
mind categorizes objects, because there are some problems with the classical view of categories
as proper sets that seem to call for another view of categorization. For example, some
concepts are disjunctive. That is, they are defined by two or more quite different sets
of defining features. An example is the concept of a strike in the
game of baseball. A strike is defined as a pitched ball at which the batter swings, but
does not hit. But a strike is also defined as a pitched ball which crosses home plate in
the strike zone, which is defined as the area over home plate, roughly between the batter's
shoulders and the batter's kneecaps. But a pitch hit in to the foul zone also counts
as a strike -- except for the third such hit and any subsequent foul ball. And finally,
a strike is a strike if the umpire calls it a strike. And for that matter, even if a pitch
falls within the strike zone, it's not a strike either -- unless the umpire says it is. So
we have at least four different definitions of the strike in baseball. And they don't
have anything to do with each other. There are four different kinds of strikes, if you
will. But these strikes have no defining features in common.
Another example is jazz music which comes in two broad forms, blues and swing.
But these two musical forms are completely different from each other: blues has one definition,
swing has another definition and they don't really overlap. Jazz can't be defined simply
as improvisational music because there are other forms of music like Bach's organ pieces
and even some rock music, which also involve improvisation. And you can't define jazz
in terms of its African-American heritage either because there are lots of white jazz
musicians; and there are some forms of African-American music, like gospel, that aren't jazz at all.
Disjunctive categories are problematic for the classical view because objects are in
the same category even though they don't share the same set of defining features.
Another problem is that many entities have unclear category membership. According to
the classical proper set view of categories, every object should belong to one category
or another. But is a rug an article of furniture? Is a potato a vegetable? Is a platypus a mammal?
Is a panda a bear? Is a pickle a vegetable? It's not at all clear. The problem is
we use categories like furniture without being able to clearly determine whether every object
is a member of the category.
This matter of defining categories actually turned into a Supreme Court case. In 1883,
the United States Congress enacted a Tariff Act, which placed a 10% duty on vegetables,
but permitted duty-free import of fruits. The customs collector for the Port of New York
declared that tomatoes were vegetables, and therefore taxable. But the International Tomato
Cartel -- which really, there was such a thing -- sued. And the case, known as Nix v. Hedden,
eventually reached the United States Supreme Court. The Court, in turn, unanimously declared
the tomato to be a vegetable, while knowing full well that tomatoes are fruits. Justice
Gray, writing for the Court, noted that, "botanically speaking, tomatoes are the fruit of a vine,
just as are cucumbers, squashes, beans, and peas. But in the common language of the people,
whether sellers or consumers of provisions, all these are vegetables, which are grown
in kitchen gardens. And which, whether eaten cooked or raw, are, like potatoes, carrots,
parsnips, turnips, beets, cauliflower, celery, and lettuce, usually served at dinner in with
or after the soup, fish, or meats which constitute the principal part of the repast and not,
like fruits, generally as dessert." Put more succinctly, you might say, if you put it on
your cornflakes, it's a fruit. Tomatoes don't go on corn flakes, so they're not fruits.
In 2005, the state legislature of New Jersey, considered a proposal to have the Rutgers
Tomato, a variety of tomato, declared the official state vegetable. Advocates for the
tomato, which included a class of New Jersey fourth graders, actually cited the Supreme
Court case as justification.
Which brings us back to a related problem of the classical view of categories as proper sets
which is that it's very difficult sometimes to specify the defining features of many of
the concepts that we use in ordinary life. Precisely because we use these categories
in our ordinary language, you'd think we'd be able to define how they're structured.
A favorite example, from the philosopher Wittgenstein, Is the concept of "game". A standard dictionary
defines a game as "a competitive physical or mental activity engaged in for amusement".
But there are lots of games that don't fit this definition. In solitaire, there's no
competition; in ring-around-the-rosy, a children's game, there's no winner; and in professional
football, played and enjoyed by adults across the country, there's really no amusement.
Professional football is work. So what makes solitaire, ring-around-the-rosy,
and professional football games like poker, cribbage, and monopoly? It's hard to say.
Here's another more recent example, the concept of planet in astronomy. The standard dictionary
definition of a planet is a celestial body that revolves around a star. And we all know
the names of the planets in our solar system: Mercury, Venus, Earth, Mars, Jupiter, Saturn,
Uranus, Neptune, and Pluto. But the problem is that there are lots of other celestial
bodies that revolve around the sun, but we don't call them planets,
for example, meteors and comets. Now, meteors and comets are typically pretty small, smaller
than a planet, but there are some objects in the Kuiper Belt, between Neptune and Pluto,
that are almost as large as Pluto is, like Ceres or UB313, sometimes known as Xena. We call Pluto a planet
but we don't call Ceres a planet: why not? To make things worse, astronomers discovered
another object in the solar system, known as UB313, sometimes called Xena, that's larger
than Pluto, and even more distant from the sun than Pluto is, but we don't call Xena
a planet either. So the question is, if a planet is a celestial body that revolves around
a star, what makes Pluto a planet but not Ceres or Xena?
Admittedly, Pluto has some planet-like features: it revolves around the Sun like the other
planets, and it even has its own moon, Chiron, just like most of the other planets have moons.
But in the first place, Pluto is really small: it's bigger than Ceres, but it's not bigger
than Xena. Moreover, Pluto has a very eccentric orbit: the other eight planets all orbit the
sun in a particular plane, known as the ecliptic, but Pluto's orbit is tilted; it's
in a different plane. Finally, Pluto's orbit is highly elliptical, much more elliptical
than that of the other planets -- so much so that it actually crosses the orbit of Neptune
at one point; none of the other planets do anything like that.
The situation was resolved in 2006, though frankly not to everybody's satisfaction, when
the International Astronomical Union voted to approve a new definition of planet that
reclassifies Pluto so that it's not a planet anymore. Note that the new definition is
organized around classical proper set lines. A planet, as now defined, orbits the sun;
A planet is also round. It has enough gravity to compress its mass into a sphere. This aspect
of the definition gets rid of Ceres as a planet because Ceres isn't spherical in shape. Moreover,
according to the new definition, a planet has to have enough gravity to clear its orbital
neighborhood of other objects. This feature also gets rids of Ceres because Ceres is
in the asteroid belt, in an orbital neighborhood that's full of other stuff. It also gets rid
of Xena, which is part of another set of objects known as the Kuiper belt, beyond Pluto, and
it also gets rid of Pluto because Pluto's orbit crosses that of Neptune; it obviously
hasn't cleared its orbital neighborhood. So the problem of defining a planet, and figuring
out whether an object like Pluto, or Ceres, or Xena, is a planet, gets solved by this
new definition. Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune are in.
Ceres, Xena, and Pluto are out. But note that the new definition creates its own problems.
We now know that there are lots of other stars that have celestial bodies orbiting around
them. We'd like to call them planets, but we can't call them planets because the way
planet has been redefined -- all planets orbit the Sun -- the definition of a planet applies
only to our own solar system. So what do we call these other things?
Pluto and the planets are a case where unclear category membership and defining features
conspire to make life very difficult for astronomers. But these are problems that really shouldn't
occur at all, if concepts and categories were really structured according to the classical
proper set view.
Yet another problem with the classical view is known as imperfect nesting. Remember, from
our discussion of the vertical arrangement of categories into subsets and supersets,
that subcategories are defined by adding one or more defining features to superordinate
categories -- with the result that subordinate categories are perfectly nested underneath
their superordinate categories. But that turns out not always to be the case, so that
the vertical relations between superordinate and subordinate categories is actually more
accurately characterized as a tangled hierarchy -- not unlike the tangle of wires that probably
exists behind your computer or your stereo system.
Here's a concrete example -- in fact, two concrete examples. Imagine a three-level hierarchy
in which animals are subcategorized into birds and mammals, and then birds are subclassified
into sparrows and chickens and mammals are subclassified into dogs and pigs. In a perfectly
nested hierarchy, sparrows and chickens would have more features in common with birds than
they have with animals. And dogs and pigs would have more features in common with mammals
than they do with animals in general. So, sparrows and chickens ought to be judged as
more similar to birds than they are to animals in general. And dogs and pigs should be judged
as more similar to mammals than they are to animals in general. But that turns out not
to be the case. To begin with, most people judge chickens to be more similar to animals
than they are to birds, even though they judge sparrows to be more similar to birds than
they are to animals. Now, everybody knows that a chicken is a bird, of course, but people
just think of chickens as more similar to animals than they do to birds. That shouldn't
happen. Another anomaly comes from examining response
latencies in a category verification experiment. In such studies we ask subjects simply, is
a sparrow a bird, yes or no; is a sparrow an animal, yes or no? Is a dog a mammal, yes
or no? Is a dog an animal? Yes or no? The classical view of categories as proper sets
predicts that it ought to take less time to verify that a sparrow is a bird than to verify
that a sparrow is an animal because the category verification process has to go through only
one level rather than two. Same goes for verifying that a chicken is a bird, as opposed to a
chicken is an animal. Or a dog is a mammal, as opposed to a dog is an animal, or a pig is
a mammal, as opposed to a pig is an animal. It works out just that way for birds. It takes
less time to say that a sparrow is a bird, or a chicken is a bird, than to say that either
of them is an animal. That's just the way it should be. But the response latencies reverse
themselves in the mammal case. It actually takes subjects less time to say that a dog
or a pig is an animal than to say that a dog or a pig is a mammal. And again, this just
shouldn't happen if categories are represented in neatly, perfectly nested proper sets arranged
hierarchically into superordinate and subordinate categories.
This brings us to the last and, in some ways, the most severe problem with the classical
view of categories as proper sets: the fact that some instances of a category are simply
better instances of that category than others are. This is known as the problem of typicality:
sparrows are rated better instances of the category bird than are penguins or
chickens or ostriches, sparrows are just more birdy birds. And within a culture there is
a tremendously high degree of agreement about these typicality ratings. The problem for
the classical view is that all the instances in question share the features that define
the category in question. And thus from the classical view they must all be equivalent.
Whether you are a sparrow or a penguin or a chicken or an ostrich, you're a warm-blooded
vertebrate with feathers and wings; you're a bird. And with respect to the classical
view of categories as proper says, all birds are created equal. But obviously some birds
are better birds, more birdy birds, than others are.
Here are some examples of variations in technicality taken from the study by U.C. Berkley's professor
Eleanor Rosch in 1975. The category of furniture is defined as movable articles that make a
room suitable for occupancy or use. All articles of furniture do that. But subjects rate some
articles of furniture, like chairs and desks, as more typical of the category than others.
Rugs are not very typical, and ashtrays are very atypical, members of the category furniture.
Sparrows are rated as highly typical birds -- much more typical than owls, and certainly
much more typical than chickens or penguins. Peas and corn are rated as typical vegetables
compared to pickles, and oranges and cherries are rated as typical fruits compared to pickles.
Note that pickles are equally bad instances of both the category vegetable and the category
fruit. And in line with Justice Gray and the United State Supreme Court, tomatoes are rated
more typical of vegetables than they are of fruits.
You can see variations in typicality even when the category in question has a clear
and obvious proper-set structure. In a study by Armstrong and colleagues,
subjects rated the number 4 as a very typical example
of the concept of even number and numbers like 18 and 106 as less typical. The number
3 was rated as a much more typical odd number, compared to numbers like 501 or 447. Mothers
and housewives are rated as more typical of the category female than princesses or
even policewomen. And, to put the icing on the cake, subjects rate squares and rectangles
as more typical of the category "plane geometry figure" than they do circles or ellipses.
All of these categories have a proper-set structure, a classical structure. Everybody
knows how each of these categories is defined. Yet within each category there are differences
among the instances in terms of how typical the instance is of the category in question.
This, again, simply should not happen in the classical view of categories as proper sets.
These variations in typicality have a clear effect on the process of categorization. Here
are the results of a category verification experiment in which subjects were presented
with instances and simply asked whether they belonged in a particular category. Some of
the instances were highly typical instances like sparrows being highly typical birds;
others were low in typicality like ostriches and penguins, not very typical birds; and
other instances fell in between.
Response latency, the amount of time it took subjects to verify that each object was a
member of the category varied clearly depending on the typicality of the instance. It took
less time to verify highly typical instances then it did to verify highly atypical instances.
But all these instances shared the same set of singly necessary and jointly sufficient
defining features. If all subjects were doing were paying attention to these defining features,
there wouldn't be any typicality effects in categorization.
So there we have a whole host of problems posed by the classical view of categories as proper
sets. And if categorization proceeded by comparing the attributes of the object to some list
of singly necessary and jointly sufficient defining features, we wouldn't encounter any
of them. Therefore, categories need to be represented some other way -- some way other
then as proper sets -- and people must do something else when they induce concepts or
deduce category memberships.
Recently, another view of categorization, known as the prototype view of categories
as fuzzy sets, has been introduced -- originally by the philosopher Ludwig Wittgenstein, simply
on philosophical grounds. But it's gained status within psychology, now, on empirical
grounds. According to this view, categories are not proper sets, but rather fuzzy sets,
in that there's only a probabilistic relationship between possessing any feature and being a
member of a particular category. No feature is singly necessary to define a category.
And no set of features is jointly sufficient. Some of these features are central in that
they are highly correlated with category membership. Most birds fly, but a few, like ostriches,
don't. Most non-birds don't fly, but a few, like bats, not to mention Rocky the Flying
Squirrel, do. Central features are found in many instances of a category, but in few instances
of alternative categories. Other features are peripheral, in that there is a low correlation
with category membership. The feature occurs with approximately equal frequency in a category
and in its contrast. For example, birds have two legs, but so do apes and humans.
Category members appear to be joined together, not because they share a single set of defining
features in common, but rather by a principle of family resemblance. Members of a family
share lots of features with other family members. You may have your father's nose, your mother's
eyes, and your aunt's hair. Similarly, members of a category seem to share a family resemblance.
Some instances look like other instances, but not all instances share the
same features in common. In this view, categorization is based on the
similarity of an object to the category prototype. Objects are assigned to categories, if they
have many features that are central to category membership, and few features that are central
to membership in contrasting categories. The fact that category members bear a family resemblance
to each other, and that assignment to a category is based on similarity to some prototype,
permits a considerable amount of heterogeneity within a category. And it means that some
category members will be more typical of the category than other category members are.
The prototype view of categories as fuzzy sets goes a long way towards solving many
of the problems that beset the classical proper set view. You can have disjunctive categories,
because the probabilistic view allows you to have instances of categories that don't
share any features in common with another instance of the category. That's an extreme
case, and we'll return to the problem of disjunctive categories later. But at least the probabilistic
view is not as troubled by the existence of disjunctive categories, as the proper set
view is. Unclear category membership is not a problem, because, well, in the fuzzy-set
view, the categories are fuzzy. The boundaries between categories aren't clear and sharp.
It's a matter of judgment. A tomato can be a fruit from one point of view, but it can
also be a vegetable from another point of view. The difficulty in specifying defining
features also flows naturally from the probabilistic view because it's in the very nature
of the probabilistic view that different instances of a category are not going to have the same
features in common. There aren't any defining features so it's not surprising that it's
difficult to specify them. Imperfect nesting occurs because of variations in the number
of central features possessed by various instances of a category. Some instances will possess
relatively few of these central features, and they'll actually begin to look like instances
of another category. And the most important problem of all, variations in typicality is
explained easily by the revisionist view of categories as fuzzy sets. If there's no set
of singly necessary and jointly sufficient defining features shared by each member of
a category, if category instances vary in terms of the number of central features they
possess, then of course there are going to be variations in typicality. A sparrow is
going to be a birdier bird than an ostrich is because it possesses more of the features
that are central to membership in the category bird.
The prototype view solves many of the problems that beset the classical view, but it has
some problems of its own. And for that reason, other theorists have proposed an alternative
exemplar view of categorization. According to the exemplar view, concepts aren't defined
by lists of characteristic features any more than they're defined by lists of defining
features. Rather, concepts are defined by lists of their members, with no defining or
characteristic features to hold the entire set together. It's a little like defining
a category by enumeration. The members may have some features in common -- or, at least
probabilistically speaking, in common. But according to the exemplar view that's not
important for categorization. What's important for categorization is that the members are
examples of the category. So according to the exemplar view, concepts
are represented as lists of their members -- especially the very salient examples of
a category. And when we want to categorize an object we don't compare its features to
a list of defining or characteristic features; we compare the object to the exemplars themselves.
We call an item an instance of a category if it resembles, if it's similar to, some
other object that we already know is an instance of a category.
The exemplar view is particularly good at addressing the problem of disjunctive categories
-- categories whose members don't seem to have anything in common, like the various
kinds of baseball strikes or the two kinds of jazz, blues and swing. That's because, in the exemplar view,
there isn't a single, unitary representation of the category that fits all category members
to a greater or lesser extent. Instead, there are multiple examples of a category and they
don't really have to resemble each other. As long as the new object resembles at least
one exemplar stored in semantic memory, you can place the object in the category. If it
sounds like blues or it sounds like swing, you can call it jazz, even though blues and
swing don't resemble each other.
The prototype view and the exemplar view are great advances in the common set view of
categorization, but each of them has some problems. And, from time to time, psychologists will try to propose some other type of categorization to cope with them.
This debate is likely to go on for a while. But, this debate over the structure of categories illustrates an important difference between the
purely philosophical approach to the mind and the empirical approach represented by psychology. Logically, perhaps, the best way to think about
categories is that they're structured as proper sets represented by lists of defining features. That was certainly the way that Aristotle thought
about categories. And that way of thinking carried the day for roughly 2500 years, a pretty good run.
But, psychologically, categories seem to be structured quite differently, perhaps as fuzzy
sets, represented by prototypes or perhaps by collections of exemplars. In fact, we may have both prototypes and exemplars in our heads at the same time.
There is evidence, for example, that novices in some domain tend to classify in some domain tend to classify new objects by category prototypes
rules, and so novices in a domain may represent categories in terms of prototypes, whereas
experts in that same domain will classify by comparing new objects in terms of category exemplars.
That's interesting, but the really important point is that the principles of reasoning do not necessarily follow the dictates of formal logic.
You can't discover how the mind works by reason alone. You need empirical evidence.
And when you collect empirical evidence, you're often surprised to discover that the mind works in a very different way entirely
than what you'd expect, based on logical reasoning. We'll see more examples of this general principle in the next lecture.