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This is to certify that the

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Structural Semantics:
A New Picture Theory of Representation

presented by

Heather E. Johnson

has been accepted towards fulfillment
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Ph.S. Philosophy

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6/01 c:/CIRC/DateDuo.p65-p.15

STRUCTURAL SEMANTICS: A NEW PICTURE THEORY OF REPRESENTATION
By

Heather Elizabeth Johnson

A DISSERTATION
Submitted to
Michigan State University
in partial fiilfillrnent of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Philosophy

2002

ABSTRACT
STRUCTURAL SEMANTICS: A NEW PICTURE THEORY OF REPRESENTATION
By

Heather Elizabeth Johnson

Structural semantics is a theory that purports to explain how mental states come to
represent or “be about” external objects and states of affairs. According to this view, both
mental states and the things they represent can be described as sets of elements structured
by relations. It is in this sense that mental states and external states of affairs are what we
call “relational systems.” A mental state, or “cognitive relational system” represents some
external state of affairs, or “external relational system,” when and only when there is a
homomorphic fimction mapping the elements of the cognitive relational system to those of
the external relational system.

Because the only necessary condition for representation is the existence of a
homomorphism between the representation and thing represented, structural semantics
may be criticized for failing to specify adequately which of any number of external
relational systems a given representation is about. In other words, insofar as
representations will always map homomorphically to more than one external relational
system, structural semantics cannot support the assignment of representations to unique
objects or states of afi‘airs in the external world. This is referred to as “the uniqueness
problem” for structural semantics.

This dissertation examines various ways of raising the uniqueness problem for

structural semantics including a version of the problem expressed through Quine’s

arguments regarding translational indeterminacy and a persistent version of the problem
introduced by Quine in Ontologi_cal Relativity a_n_d Other Essay_s.1 In addition to providing
an exposition of the “mechanics” of structural semantics as a theory of mental
representation and to formulating a response to the uniqueness problem, this dissertation
attempts to evaluate the ability of structural semantics to satisfy a number of basic
conditions commonly thought to be required for any adequate theory of mental

representation.

 

‘ Quine, W.V.O. Ontological Relativinr. and Other Essays. New York: Columbia University
Press, 1969.

This dissertation is dedicated to my husband Serge,
who made himself available on many occasions as
an editor, mathematics consultant, and coach.

iv

ACKNOWLEDGMENTS

The original idea for this topic arose in connection with a seminar on mental
representation offered at Michigan State University in the Fall of 1998 by Professor
Richard Hall. My experience in this seminar fostered an interest in mental representation
that proved enough to sustain the completion of this work almost four years later. During
that time, I’ve consulted a lot of people about the ideas that are described and defended
here. Among the most significant contributors has been my husband, Serge Canizares,
who has served as an invaluable resource and has been willing to drop in on the middle of
a half-formed train of thought, doing his best to make sense of it and provide critical
feedback.

I would also like to thank my mother, Jacqueline Johnson, who supports me in
everything I do, and has been an enormous encouragement at the times when I most
needed to be encouraged. Were it not for her love and support, I would not be in a
position to accomplish the goals associated with this work.

Finally, I’d like to express my deepest appreciation for my advisor, Richard Hall,
who has spent many precious hours evaluating my work and providing insightful criticisms
and suggestions for improvement to it. Our discussions about the ideas defended here
have always revealed aspects of the problem which I had not even imagined before, and

evidence of his contributions is found in every page of this dissertation.

TABLE OF CONTENTS

LIST OF TABLES ................................ ‘ ................... ix
LIST OF FIGURES ................................................... x
KEY TO SYMBOLS AND ABBREVIATIONS ............................. xii
INTRODUCTION .................................................... 1
CHAPTER 1
Robert Cummins’ ‘Interpretational Semantics’ ................................ 6
1.1 Cummins’ Positive Account of Representation ............................. 7
1.1.1 The Picture Theory of Meaning ................................... 7
1.1.2 Isomorphism and Multiple Levels of Representation ................... 9
1.2 Cummins’ Theory of Error ........................................... 12
1.2.1 An Example of What Can Go Wrong: Fodor’s Causal Theory ........... 12
1.2.2 How Cummins Handles Error ................................... 18
1.2.2.1 Targets vs. Contents: An Example ........................... 18
1.2.2.2 Performance Error vs. Representational Error .................. 26
1.2.2.3 How Are Targets Fixed and Can They Be Naturalized? ........... 28
1.3 The Uniqueness Problem ............................................ 37
1.4 Two Problems About Targets ........................................ 46
CHAPTER 2
The Picture Theory of Meaning .......................................... 51
2.1 An Introduction to the Picture Theory of Representation .................... 51
2.2 Interpretational Semantics and the Picture Theory of Representation ........... 53

2.3 Common Criticisms of Picture Theory and Responses From the Perspective of

Interpretational Semantics .......................................... 54
2.3.1 The Problem of Abstraction Via Daniel Dennett ...................... 55
2.3.2 The Problem of Abstraction Via Jerry Fodor ........................ 61
2.3.2.1 Non-Representational Properties of Pictures and Sentences ........ 67

2.3.2.2 Can Discursive/Symbolic Devices Uniquely Specify A
Subject’s Properties? ..................................... 69

2.3.2.3 Can Iconical Devices Uniquely Specify A Subject’s Properties? ..... 72

2.3.3 The Problem of Uniqueness for Picture Theory ...................... 75
CHAPTER 3
Representation As Homomorphism And The Problem of Uniqueness .............. 79
3.1 Cummins’ Isomorphic Mapping Relation ................................ 80

3.2 Understanding the Representation Relation as a Homomorphism: Structural
Semantics ....................................................... 88
3.2.1 How Homomorphism Accounts for Paradigm Cases of Representation . . . . 88

3.2.2 How Homomorphism Accounts for Representation Under Degraded

Conditions .................................................. 92

3.2.3 How Homomorphism Accounts for Substitutivity of Identity Failure ...... 95

3.3 A Final Form of Non-Uniqueness ...................................... 99

3.3.1 Suppes’ and Zinnes’ Project ..................................... 99

3.3.2 The Problem of Representation in Measurement Theory ............... 99

3.3.3 The Problem of Uniqueness in Measurement Theory ................. 103
CHAPTER 4

Withstanding Quine’s Criticisms of Picture Theory .......................... 109

4.1 The “Museum View” of Meaning ................................. 109

4.2 Quine’s Translational Indeterminacy ............................... 113

4.2.1 Meaning and Translational Indeterminacy ...................... 113

4.2.2 Translational Indeterminacy and Non-Uniqueness ............... 117

4.2.2.1 Two Cases of Indeterminacy ......................... 120

4.2.2.2 Quine on Language and Meaning ...................... 125

4.2.2.3 Two Kinds of Non-Uniqueness ....................... 132

4.3 Where Does This Leave Structural Semantics? ....................... 136
CHAPTER 5

Formal and Metaphysical Criticisms of Structural Semantics ................... 141

5.1 The Lowenheim-Skolem Theorem ................................ 142

5.2 The Pythagorean Problem ....................................... 146

5.3 The Role of A Background Theory Revisited ........................ 155

CHAPTER 6

The Problem of Error and Indirect Representation ........................... 158
6.1 The Problem of Misrepresentation ................................ 159
6.2 The Problem of Representing Non-Existent Objects ................... 169
6.2.1 The Axiom of Referring ................................... 169
6.2.2 Indirect Representation .................................... 171
6.2.2.1 The Classical Empiricist View ........................ 172
6.2.2.2 Representation of Fictions ........................... 176
6.2.2.3 Representation of Non-Physical Objects, Contradictory Objects,
Hallucinations and Ideas .................................. 177
6.2.3 Meta-Mental Operators ................................... 180
BIBLIOGRAPHY ................................................... 186

LIST OF TABLES

Table 1-A: Trace of Average Application .............................. 21 & 22
Table 3-A: Relational Database Table .................................... 106

LIST OF FIGURES

Figure l-A: Grading Algorithm .......................................... 20
Figure l-B: Non-Uniqueness—Cummins .................................... 39
Figure l-C: Non-Uniqueness—Standard .................................... 39
Figure 1-D:Non—Uniqueness—Measurement ................................. 39
Figure l-E: Autobot .................................................. 40
Figure l—F: Non-Uniqueness in the Autobot ................................. 42
Figure l-G: Isomorphism ‘Modulo’ A Substructure ........................... 49
Figure 2-A: ‘Sephra Approaching the Food’ ................................ 58
Figure 2-B: Man Climbing/Descending A Mountain ........................... 72
Figure 2-C: Man Climbing A Mountain .................................... 74
Figure 2-D: Standard Non-Uniqueness Via Fodor’s Critique .................... 76
Figure 2-E: The Duck-Rabbit ........................................... 76
Figure 2-F: Standard Non-Uniqueness Via the Duck-Rabbit .................... 77
Figure 2—G: Cummins’ Non-Uniqueness Via the Duck-Rabbit ................... 78
Figure 3-A: Mainfiame Database Tables ................................... 85
Figure 3-B: Failure of One-to-One ........................................ 86
Figure 3-C: A Representational Network ................................... 93
Figure 3-D: Substitutivity of Identity Failure ................................ 97

Figure 3-E: Non-Uniqueness in Measurement Theory ........................ 104

Figure 4-A: Terms As Labels for Mental States ............................. 111
Figure 4-B: Translational Indeterminacy .................................. 118
Figure 4-C: n Translated As e 1 .......................................... 119
Figure 4-D: n Translated As e2 .......................................... 119
Figure 4-E: Translational Indeterminacy for ‘ne rien’ ......................... 121
Figure 4-F: ‘ne rien’ Treated Atomistically ................................. 122
Figure 4-G: ‘ne rien’ Treated Holistically .................................. 122
Figure 4-H: Standard: Rabbit vs. Undetached-Rabbit-Part ..................... 133
Figure 4-1: Cummins: Rabbit vs. Undetached-Rabbit-Part ..................... 134
Figure S-A: The Pythagorean Problem .................................... 144
Figure 5-B: Construction of A Proxy Function for DLS ....................... 147
Figure 5-C: Interpretation of Numbers as ‘Objects’ .......................... 150

Figure 6-A: Representation of Co-instantiated and Non-Co-instantiated Objects . . . . 174

Figure 6-B: Gabriel’s Horn ............................................ 179
Figure 6-C: Meta-Mental Operators As Functions ........................... 185

Images in this dissertation are presented in color.

KEY TO SYMBOLS OR ABBREVIATIONS

Z - a cognitive system possessing mental representations

| X | - mental representation of X

(5, 9i, 3 - structured sets of elements, e.g.-relational systems or structures
P" - the predicate P in the structure 9t

f” - the function f in structure 3i

P - a set offormulas in a language

INTRODUCTION

If a person wants to know how to make a cheese souffle, one of the things she/he
might do is take a trip to the local library in search of relevant information. Today, one can
also find such information on the intemet, or perhaps by watching one of those Saturday
afternoon cooking shows on cable. Suppose I want to learn the art of souffle-making, and
I choose to visit the library to begin my tutelage. How will I find the information that I
need? Assuming that I am familiar with the library’s indexing system, I should be able to
narrow my search to the part of the library containing the book or books I want to read.
When I arrive at the right shelf, I should be able to recognize my book by reading the
titles. The words in the title, and later, the words contained in the text, convey information
to me. The title of the book reveals that it is about the process of making soufllés and the
words it contains collectively represent a set of instructions for making souflle's.

What is it that makes this book about souflle's and not chocolate mousse or
basketball? The immediate answer is that the book contains sentences which mean
propositions concerning the making of soumés and not anything else. But this answer only
defers the question: we can still ask what it is that makes these sentences mean what they
do, and in general, in virtue of what it is that language has meaning.

But ultimately, the question of why a set of representations is about soufllés and not
something else cannot be reduced to a question about the meanings of sentences anyway.
This is because language is not the only way of representing the process of soufilé-

making. The same information contained in the book might be accessible online,

via a computer, or embedded in a series of digitized photographs. When representations
are conveyed via the internet, they are not ultimately reducible to sets of sentences, but to
sets of electronic states of a system which hosts them. This collection of system-states
represents the process of soumé-making too, and by hypothesis, represents the very same
process as that represented by the library book. Or again, if I were to commit the contents
of the entire library book to memory, the process of soufiElé-making would be represented
in yet another medium, namely, in my own head. In this scenario, my own mental states or
states of the brain, somehow represent soufiés and soufllé-making.

The question we want to answer is this: in virtue of what can things like words, states
of a machine, and states of mind represent objects, processes, or states of afi'airs in the
external world? This dissertation concentrates on the last of these: it focuses on how states
of mind can represent, or be about, another thing (e.g., an external state of affairs). I will
propose and defend a theory called “structural semantics” which is an attempt to answer
this question.

Chapter 1 of the dissertation is devoted to the critical exposition'of a view very
similar to the one which I will defend, namely, Robert Cummins’ interpretational
semantics. Careful consideration of Cummins’ view will reveal a potentially serious
problem with the way in which he explains mental representation, a problem that applies to
my own view as well, and which I will refer to as “the problem of uniqueness.” Much of
the dissertation is an effort to explain the problem of uniqueness and respond to the threat
it presents.

Interpretational semantics and structural semantics both bear a strong resemblance to

a class of theories about mental representation known as picture theories. In Chapter 2, I
describe some of the features of picture theories of representation and compare their
strengths and weaknesses to those of interpretational and structural semantics. The
examination of the picture theory reveals that the uniqueness problem is not a new
concern. Opponents of the picture theory of mental representation have posed objections
to picture theory which are strikingly similar to the challenges the uniqueness problem
poses. In Chapters 2 and 3, I begin to suggest how to refine Cummins’ view in such a way
as to respond to the problem of uniqueness. This refinement is what I call structural
semntics, and is the view I defend for the duration of the dissertation.

Chapters 4 and 5 are both devoted to explaining and responding to specific ways of
raising the problem of uniqueness. Chapter 4 reveals how Quine’s arguments concerning
translational indeterminacy are really a version of the uniqueness problem. Consideration
of the uniqueness problem in this context helps to refine the sort of response which I can
offer to it on behalf of structural semantics. And Chapter 5 describes a persistent version
of the problem which resists any of the responses which I have defended until that point.

Most of the dissertation is devoted to explaining the mechanics of structural
semantics and to defending this view against the problem of uniqueness. However, many
philosophers agree that a good theory of mental representation needs to meet at least three
conditions in addition to providing a plausible account of how mental states come to have
representational content. Therefore, in both Chapters 3 and 6, I will attempt to state how
the theory I defend can meet these three basic conditions.

In Chapter 6, I attempt to explain how structural semantics can account for mental

representations of things which do not really exist. Explaining representation of non-
existent objects is the first of the three conditions which theories of mental representation
are typically required to meet. Representations of unicoms, of the boogie man,
hallucinations, or of your imaginary fiiend when you were five are all real enough as
mental phenomena; a good account of mental representation ought not to characterize
representation in such a way as to rule out these sorts of representations. Second, a theory
of representation ought to allow for misrepresentations. For example, it ought to be
possible for one to represent the state of affairs expressed by the sentence, ‘The cat is on
the mat’ even when the cat is in fact nowhere near the mat and the sentence is therefore
false. Since we are capable of representing things in a way that is inconsistent with the
facts, we need an account of representation which is compatible with the possibility of that
inconsistency. Finally, in Chapter 3, I discuss how a good theory ofmental representation
will be consistent with the widely held view that “substitutivity of identity” fails where the
representational content of mental states is concerned. For example, in Star Wars, it is true

that Luke Skywalker believed:

(s) I want to be a Jedi like my father.

Despite the fact that, as later movies revealed, Darth Vader and Luke’s father are one and

the same, we would not conclude that Luke therefore believed:

(s') I want to be a Jedi like Darth Vader.

The content of mental states does not necessarily remain the same when we substitute one
description of a thing for another description of the same thing. A proposition that
expresses the content of Luke’s representation is true under one of these descriptions (s),
but false under another (s’). This is a peculiar characteristic of the contents of mental

states which one does not see in non-mental contexts. For example,

(t) Anakin Skywalker was Luke’s father. and

(t’) Darth Vader was Luke’s father.
are the same statement save for the substitution of the co-referential terms, and this
substitution does not change the veracity of the proposition. A successful account of
mental representation will explain how the content of two representations can be different,
even when the things which they represent in the external world are exactly the same.

I believe that structural semantics meets all three of the conditions for a successful
account of mental representation and also, offers many additional advantages for an
analysis of representational content. I also believe that it can be successfully defended
against the uniqueness problem. In many ways, structural semantics is a new and
somewhat unconventional approach to the problem of explaining mental content.
However, the view has recognizable ties to traditional approaches, and in one notable
case, to a recent theory offered by Robert Cummins called “interpretational semantics.” I

begin the analysis of mental representation with an examination of Curmnins’ view.

CHAPTER 1: Robert Cummins’ ‘Interpretational Semantics’

Robert Cummins’ “interpretational semantics” is a close relative of structural
semantics—the view that I will defend here. Cummins has written two books describing
and defending interpretational semantics. The first, Meaning and Mengl Represeiation2
was published in 1989 and provided a basic overview of a number of traditional and
modern theories about representation. After detailing the difficulties with each one,
Cummins concludes the book with a characterization of his own view and the ways in
which it avoids falling prey to the shortcomings of other accounts. In 1996 Cummins
published Representations, Targets, and Attitudes.3 This work went timber in its defense
of interepretational semantics by attempting to address a number of criticisms that had
been raised against it since the release of the first book. In it, Curmnins provides a detailed
response to the problem of representational error and tries to address a number of
concerns regarding whether or not his theory of representation can be fully naturalized.

In this chapter, I will describe Curmnins’ positive view of representation as it is set
forth both in Mgnpng‘ and Mental Represepgtion and in Representations, Targets, and
Attitudes. I’ll also examine the way in which Cummins addresses the problem of
representational error for interpretational semantics and will consider whether or not there
are any other difficulties facing interpretational semantics which require attention.

Ultimately I will argue that there are a number of important problems which

 

2 Cummins, Robert. Meaning and Mental Representation. Cambridge: The MIT Press, 1989.

3 Cummins, Robert. Representations, Targets and Attitudes. Cambridge: The MIT Press: 1996. p.
96.

interpretational semantics leaves unsolved. The uniqueness problem is arguably the most
important of these as it provides much of the grounds for moving to a significantly
different account of mental representation, namely structural semantics.

Cummins’ positive account of mental representation changes very little fiom his 1989
work, Mean—mg' and Mental Representation to his later work, Representations, Targets,
and Attitudes. The major innovation of Representations, Targets, and Attitudes seems to
be his account of cognitive error, although there is also some acknowledgment and
discussion of the uniqueness problem in it not present in the former work. I will briefly
recapitulate the positive account of mental representation that Cummins has called
“interpretational semantics” here, concentrating especially on those elements of the

account that allow him to articulate the uniqueness problem.

1.1 Cummins’ Positive Account of Representation

1.1.1 The Picture Theory of Meaning

Cummins himself classifies his theory of mental representation as a version of the
picture theory of meaning. He believes that picture theories advocate three basic claims
about representation and mental representation in particular. These are:

(1) The most basic form of representation is the form of representation used in

mathematics—specifically, representation is a relation between two structures.
Supposing that St and 93 are both structures, 9! will represent (5 just in case 3! and (S

are isomorphic. Cummins makes what he calls the “radical” suggestion that the kind of

representation described here is not only the kind which is most appropriate to

mathematics and computer science, but that it is the only kind capable of accounting for
meaning.
(2) Only structures (or things which are mathematically equivalent to structures)
represent, since only structures can be related by isomorphism.
Cummins isn’t explicit about what he means by “structure.” However, his discussion of
structures and how they represent would suggest that he sees them simply as sets of
elements which are related to one another. The nature of the relationships between
elements is what can be characterized as the “structure of the set.” This description of
structures is sufficient for the purpose of understanding Cummins’ theory of
representation. However, I will give a more formal characterization of structures in
Chapter 3.4
(3) Mental representation is, likewise, representation which is grounded in
isomorphism.
This is the third and final claim that Cummins attributes to picture theories of mental

representation. In order to understand more about exactly how mental representation

works on this view, let’s turn to a more specific description of the theory Cummins calls

 

4 Notice that if only structures represent, then the picture theory may encounter difficulty with
explaining things which seem to acquire meaning conventionally. For example, a stop sign
(conventionally) means ‘stop’ because we have, as speakers of a language, decreed that it mean ‘stop.’ It
doesn’t seem to be in virtue of any isomorphism that stop signs acquire the kind of meaning that they
have. It is arguable that Cummins will need to handle this problem if interpretational semantics is going
to provide a comprehensive account of representation. But perhaps Cummins isn’t going for a
comprehensive account of representation after all. Cummins could contend (and I believe he should
contend) that conventional meanings are derivative or dependent on mental meanings in some way and
that interpretational semantics is really a theory about mental meanings.

 

8

interpretational semantics.

1.1.2 Isomorphism and Multiple Levels of Representation
The crux of Cummins’ representational theory is the claim that the representation
relation is characterized by an isomorphism between representing and represented

structures. He specifically describes the relation as follows:

The idea is that there is a mapping between the two structures such that (1) for every object in
the content structure C, there is exactly one corresponding object in the representing structure
R; (2) for every relation defined in C, there is exactly one corresponding relation defined in R;
and (3) whenever a relation defined in C holds of an n-tuple of objects in C, the corresponding
relation in R holds on the corresponding n-tuple of objects in R .5

Although a more formal definition of isomorphism than Cummins offers here will be given
later, it is usefiil to remark here that Cummins’ definition implies that isomorphic fimctions
are fimctions which (among other things) map a domain of elements into a range of
elements. A function maps one set of elements into another when the range of the ftmction
is a subset of the elements belonging to the set that is the object of the mapping.
Isomorphic fimctions are also one-to-one. Functions which are one-to-one map distinct
elements of the domain into distinct elements of the range. Most standard definitions of

isomorphism as given in abstract algebra and group theory include one addition condition

 

5 Cummins (1996), p. 96.

on isomorphic mappings, namely, that the mapping be onto.6 A mapping f between two
sets A and B is onto when: Vb E B, 3a 6 A such that fla) = b.

Although Cummins’ intuitive English rendition of the notion of isomorphism does not
seem to imply that it requires an onto mapping, I will understand isomorphism as requiring
that mappings be both one-to-one and onto.

Note that Cummins uses the phrase “content structure” to refer to the structure that
gets represented, while the phrase “representation structure” is used to refer to the
structure that is doing the representing. In addition to the content structure however, there
are other things which get represented in virtue of the isomorphism between R and C. In
particular: “If a Structure R represents a structure C, then:

1. An object in R can represent an object in C.

2. A relation in R can represent a relation in C.

3. A state of affairs in R—a relation holding of an n-tuple of objects—can represent a

state of affairs in C.”7
In other words (in a move that may sound a lot like conceptual role semantics), the
capacity of individual objects, relations, or states of affairs within a structure to represent
objects, relations, or states of affairs in another is derived from the relation which holds

between the structures themselves (i.e., the isomorphic relation)—not the other way

around.

 

6 See, for example, the following: Hungerford, Thomas. Graduate Texts in Mamematics:
Algebra. New York: Springer-Verlag, 1974. p. 30; Fraleigh, John B. A First Course in Abstract Algebra.
5'h ed. Reading: Addison-Wesley Publishing Company, 1992. p. 170; Jacobson, Nathan. Basic Algebra I.
2““ ed. New York: W.H. Freeman and Company, 1985. p. 37; Fletcher and Patty. Fpundations of Higper
Mathematics. 2"" ed. Boston: PWS-Kent Publishing Co., 1992. p. 170.

7 Cummins (1996), p. 96.

10

Note however, that R does not represent C because the objects, relations, and states of affairs in
R represent objects, relations, and states of affairs in C, but the other way around: things in R
represent things in C because R is isomorphic to C. According to [the Picture Theory of
Meaning], there is no such thing as an unstructured representation, except in the derived sense
just introduced: an unstructured element in a representing structure R may be said to represent
its counterpart in a represented structure C. We must be careful, then, not to think of the
objects, relations, and states of affairs in R as independent semantic constituents of R. [The
Picture Theory of Meaning] entails that every genuinely representational scheme is molecular
in that the meaningful constituents of a structure are meaningful only in the context of some
structure or other.8

Although this does begin to sound like conceptual role semantics, Curmnins firmly
disassociates himself fiom such a view and I believe that he is correct in so doing. After
all, the objects, relations, and states of affairs which depend upon their place in the
representational structure for obtaining the content they have, do not, in so doing, depend
on their role in a conceptual network. The relations and objects which collectively
constitute the representational structure need to be conceived of as nothing more than a
formalism, and so the claim that their semantic content is derived fiom the higher-level
features of that formalism needs to be conceived of as nothing more than a statement
about their formal relations to one another and to the structure they help compose.
Cummins thinks it necessary to abandon conceptual role semantics primarily because it
uses cognitive capacities to explain representation, where he believes one of the rmjor

functions of a successfiil theory of mental representation is to explain cognitive capacities.9

 

3 Cummins (1996), p. 97.

9 Cummins (1996), p. 90. Perhaps it is not obvious, despite Cummins’ program, how this view
is appreciably different from a kind of conceptual role semantics. Indeed, whether or not it really is seems
to depend on how loosely you construe the notion of conceptual role. If you argue that a mental state is
defined by its conceptual role just in case the relations it bears to other mental states (or would bear to
them, if placed in the right circumstance) are what primarily or even exclusively determine its content,
and that nothing much more is required for conceptual role semantics, then Cummin’s view certainly
qualifies. On the other hand, if you think that there is something more to conceptual role semantics than
just a commitment to meaning holism, as do I, then Cummins’ interpretational semantics may or may not

11

1.2 Cummins’ Theory of Error

What is unusual about Cumrnins’ approach to accounting for mental representation,
besides its explicit endorsement of picture theory, is that it begins with an account of how
such representations could be in error. The idea is that competing theories of mental
representation have often performed well when it comes to providing an explanation of
how cognitive entities represent, but that their very success in this endeavor has prohibited
them fi'om generating a robust explanation of representational error. Cummins admits that
his approach may amount to little more than “squeezing the balloon in a different place”

but claims that the new “bulge” that results may be an interesting one.10

1.2.1 An Example of What Can Go Wrong: Fodor’s Causal Theory

An example of how a successful account of representational content can compromise
one’s ability to provide a successful positive account of representational error may be
useful in order to illustrate the dificulty that Curmnins confionts.

Jerry Fodor has defended a theory of mental representation which has it that the
content of a mental representation is, by and large, the cause of that representation. He
begins with two assumptions: (1) what Cummins calls the “Representational Theory of
Intentionality” or the view that the content of mental states is derived from the contents of

their constituent representational states,11 and (2) that mental states are language-like

 

be a version of this view.

’° Cummins (1996), p. 5

11 Note that this approach is the opposite of Cummins’ own holistic strategy.

12

symbols, or symbols in what Fodor refers to as the “language of thought.”12 Since for
Fodor, mental representations are just meaningfiil symbols in the language of thought,
explaining mental representation amounts to explaining how terms in the language of
thought (sometimes called “Mentalese” by Fodor) can have semantic content. The answer
Fodor gives, in a nutshell, is that token symbols of Mentalese denote their causes, while
symboltypes denote properties whose instantiations reliably cause their tokenings.

The problem that arises for Fodor, and for similar causal accounts of mental
representation, is that not all mental representations of X (hereafter denoted
| X I) are caused by XS.13 In other words, sometimes we misrepresent the world around
us, and the idea is that a good theory of representation ought to be able to explain failures
like this one as well as it can explain the successful cases of representation.

Closely connected to the problems causal theories have with accounting for
representational error is what is known as the “disjunction problem.” It has been noted
that anytime it appears that a non-X has caused the tokening of | X | in some subject, it
may be possible to claim that in fact, the content of the subject’s mental representation
was not I X | at all, but I X V Y | (where Y is the misperceived object responsible for the
tokening of the subject’s representation). For example, suppose I mistake an Oak tree for
an Elm. Since I Elm l was really caused by the Oak tree outside my window, perhaps my
mental representation ought not be thought of as merely an I Ehn I but rather, as an

I Ehn V Oak | (or, surely even worse, as a | Elm V the Oak tree outside my window | ). If

 

‘2 Fodor, Jerry. Psychosemantics.. Cambridge: The MIT Press, 1987.

‘3 Nor do all Xs reliably cause tokenings of | X l.

13

we think of representations as having disjunctive content of this sort, then the contents of
representations will always match their causes exactly (or, should they appear not to, this
is just as much a reason for thinking we should add another disjunct to our description of
the representation’s content as it is for thinking that a misrepresentation occurred).

This result is problematic for at least a couple of reasons. First and foremost, it seems
to eliminate the phenomenon of misrepresentation rather than explain it. It seems obvious
that misrepresentations do occur and therefore that any theory which is incapable of
explaining them without explaining them away is a theory which does not match up well
with the way we believe the world (and our mental representations in particular) to be. In
addition, the notion of disjunctive content is epistemologically suspect. If disputes about
the content of a representation can always be settled by appending another disjunct to the
representation’s description—and it would seem that nothing in this view as we have
characterized it so far prohibits this—then there will never be any way of establishing the
correctness or incorrectness of claims about content. If Jim swears that the content of
Ella’s representation was | Elm | and I swear that it was I Elephant | for example, then
what, in this view, prevents us fiom compromising to the effect that Ella’s representation
was really about | Elm V Elephant I?

But Fodor has a response to the disjunction problem. Indeed, like Cummins, Fodor
starts out by trying to solve the problem of misrepresentation (and the disjunction problem
in particular), but in the process, forms for himself a fiill-blown theory of content. F odor’s
account is known as “asymmetric dependence theory.” I’ll briefly explain it, its

advantages, and its drawbacks before moving on to consider the advantages of Cummins’

14

approach.
Fodor argues that any given mental representation is consistently correlated with the

instantiation of certain “psychophysical Properties”

or properties which reliably—even
lawfillly—impinge on our senses whenever we come into contact with an object
possessing them. Because psychophysical properties are causally responsible for the
tokening of specific symbols in us, the causal theory of representation, together with an
account of just how much exposure we have to have to such properties and in what
manner, etc. (a task F odor leaves to the science of “psychophysics”), “... provides a
plausible sufficient condition for certain symbols to express certain properties ...”.”

If it is an object’s psychophysical properties which ultimately causes tokenings of
mental symbols in us, then we may have a way ofexplaining howl X |s could be caused by
non-l X |s after all. Suppose that my I Elm | representation is provoked by the impingement
of some set of psychophysical properties instantiated in the Oak outside of my window
(e. g., leafyness, brownish-colored tree trunk shapes, the smell of sap, the look of
branches). It is arguable that the Oak outside my window can only cause I Elm |s in me
because it shares certain psychophysical properties with Ehns—that is, because it looks
like an Elm. Indeed, if Oaks didn’t look like Elms then I would not ever produce the

symbol | Elm | in the presence of an Oak. For example, I would not produce the symbol

| Elephant | in the presence of an Oak since the two do not share a sufficient number of

 

'4 Fodor, Jerry. Psychosemantics. Chapter 4: “Meaning and the World Order.” Cambridge: The
MIT Press, 1987. p. 120.

‘5 Fodor (1987), p. 113.

15

psychophysical properties. This may seem like a simple, even trivial observation. But note
that it explains what about my tokening of the symbol | Elm | on this particular occasion
was a mistake (i.e., because it was caused by a non-Elm) while at the same time allowing
that the content of my representation was in fact I Elm | in a way that does not
compromise the causal theorist’s positive account of representation. In particular, my
tokening of | Elm | in the presence of an Oak is made possible only because of the
existence of an “asymmetric dependency relation” between Elms and Oaks. And in fact, if
the psychophysical properties had by Elms never produced the mental representation
| Elm | in me, then presumably those same psychophysical properties instantiated in Oaks
wouldn’t produce the relevant representation. To put it plainly: if Ehns didn’t cause
| Elm |s then Oaks wouldn’t cause them either. After all, this is just what it means to say
that such psychophysical properties covary lawfitlly with the production of mental symbols
in us. Fodor’s account of mental representation is therefore still a causal account, but one
which allows for principled misrepresentation without the price of disjunctive contents.
Perhaps the biggest difficulty with asymmetric dependence is that its avoidance of the
disjunction problem seems to come at the price of getting rid of both disjuncts in favor of
more proximal stimuli. After all, if what makes Oaks asymmetrically dependent on Elms is
the fact that they produce a “look” like that of an Ehn (which, albeit they wouldn’t
produce were it not that Elms produced the same “look”), then why not make some
description of that “look” the content of the mental representation rather than either the
disjunction of the two distal objects or the more foundational of the two objects in the

dependency relation? Fodor anticipates this criticism. He makes the same point during the

16

course of considering how we form theoretical concepts such as that of “proton,” the
psychophysical properties of which might include things like the “look” of a photographic
plate, for example.

We’ve got something into the belief box for which instantiations of proton are causally
responsible; but it’s the wrong thing. It’s not a token of the concept PROTON; rather, it’s a
token of some (probably complex) psychophysical concept, some concept whose tokening is
lawfully connected with the look of the photographic plate Something needs to be done
about this. Here’s where the cheating starts.l

Fodor attempts to circumvent this problem by adding a restriction to the account of
asymmetric dependence given above. In particular, he now requires that the
psychophysical properties which are responsible for the reliable tokening of mental
symbols in us are in fact causally connected to the presence of the objects which the
symbol denotes. So, in the case of | Elm ls, | Elm | will represent Elms just in case the
psychophysical properties which cause the tokening of | Elm l are themselves caused by
Elms. The requirement of a causal connection between the set of psychophysical
properties which provokes the mental representation and the object which instantiates the
set of psychophysical properties reestablishes a reliable link between object and mental
representation. However, in so doing, it seems to compromise the main advantage of
Fodor’s theory of error. For suppose, as we have been doing, that the psychophysical
properties causally responsible for my tokening of | Elm | can themselves be caused by
both Elms and Oaks. Which of these objects is the cause of my representation? Insofar as
both produce the psychophysical properties which are causally responsible for my

representation, either one ought to be an acceptable answer. It really is of no consequence

 

‘6 Fodor (1987), p. 120.

17

that in any given case, only one actually did cause these psychophysical properties to be
instantiated. For one thing, one would have absolutely no way of knowing this in cases
where the set of psychophysical properties produced by either candidate were sufficiemly
similar. But more importantly, it isn’t clear what we’d gain (apart from salvaging Fodor’s
version of causal theory) by reconnecting the more proximal stimuli with a distal object
and subsequemly assigning content to it. In fact, I think there is a strong case to be made
for the claim that there are a number of advantages to avoiding this move and to
deemphasizing the role of objects in fixing content in favor of the (specifically structural)
properties which they exhibit. A closer consideration of Cummins’ interpretational

semantics will begin to lay the groundwork for this argument.

1.2.2 How Cummins Handles Error

The causal theorist’s dilemma is not unique. It is common to encounter the problem
of being unable to account for how representations acquire their content, while at the
same time providing room for the possibility of representational error. Cummins will try to
avoid the dilemma by making what he views as a crucial, but commonly ignored

distinction between the content of a representation and its target.

1.2.2.1 Targets vs. Contents: An Example
While programming an application designed to average grades a few years ago, I
encountered a troubling problem whose solution seemed to elude me. The program was

designed to compute the average of 5 grades. It worked by prompting the user to enter a

18

grade, storing the grade in a data structure called an array, and prompting for the next
grade until the array was full. When the array was full, the application added the values
stored in it together and divided the sum by the size of the array (i.e., the number of
elements it contained). Although whole arrays may be treated as the objects of
computations in a program, they are usually used to store values which are themselves the
objects of computations, as was the case with my application. For this reason, the values
stored in an array must be “indexed” or given a reference, so that these values may be
retrieved by the program when they are needed. If the number of values to be stored in an
array is known in advance, an array of that size can be created to store the values. The
number corresponding to the size of the array is then a representation of the number of
values stored in it and can be used accordingly. Knowing that arrays behaved this way, I

created the following algorithm for my grading application:

19

 

: Create a data object called SizeOfArray = 5.
: Create an array of size SizeOfArray and call it ArraijGrades.
: Create a data object for temporary storage of the current grade called C urGrade.

: Create a data object which keeps track of how many grades have been entered called
NumGrades. Set NumGrades to an initial value of O to indicate that no grades
have been entered.

ingsUIAMNl-t

O

9: Create a data object called Sum which keeps track of the sum of the entered grades.
10: Set Sum to an initial value of 0.

l 1:

12: while NumGrades < or = SizeOfArray

l3: {

l4: Prompt the user for the next CurGrade.

15: Store C urGrade in the NumGrade index of the array.
16: Sum = Sum + CurGrade
17: Update NumGrade //add one to the value of NumGrade
18: }
19:
20: Create a data object called Average to store the average of the grades entered.
21: Average = Sum / NumGrade
22: Display Average

 

 

 

Figure l-A: Grading Algorithm

Unfortunately, this grading algorithm didn’t work very well. It seemed to produce lower
averages than it should have produced and it prompted me for six grades rather than five.
To see why the algorithm produced this error, study the trace of the program for the
following set of 5 grades: {67, 98, 87, 56, 71}. Note that this trace begins with a
representation of the values of the data objects after at least one cycle through the “while”

loop (see line 12). The value of Average is given at the end of the trace.

20

SizeOfA rray = 5
ArrayOfGrades =

.71

NumGrades = 1

C urGrade = 67

Sum = 67

Is NumGrades < or = SizeOfArray? Yes

 

 

 

 

 

 

 

 

Size OfA rray = 5
ArrayOfGrades =

 

 

 

 

 

 

 

67 98

 

NumGrades = 2

CurGrade = 98

Sum = 165

Is NumGrades < or = SizeOfArray? Yes

 

Size OfA rray = 5
ArrayOfGrades =

 

67 98 87

 

 

 

 

 

 

 

NumGrades = 3

CurGrade = 87

Sum = 252

Is NumGrades < or = SizeOfArray? Yes

 

21

Size OfA rray = 5
ArrayOfGrades =

 

67 98 87 56

 

 

 

 

 

 

 

NumGrades = 4

C urGrade = 56

Sum = 308

Is NumGrades < or = SizeOfArray? Yes

 

S izeOfA rray = 5
ArrayO/Grades =

 

67 98 87 56 71

 

 

 

 

 

 

 

NumGrades = 5

C urGrade = 71

Sum = 379

Is NumGrades < or = SizeOfArray? Yes

 

S izeOfA rray = 5
ArrayOfGrades =

 

 

 

 

 

 

 

67 98 87 56 71

 

NumGrades = 6

C urGrade = ?

Sum = 37 9

Is NumGrades < or = SizeOfArray? No (terminate loop and calculate Average)

Average = 379/6 = 63.2

Table l-A: Trace of Average Application

The problem here, as the trace demonstrates, is that the condition governing the

number of times the application will traverse the loop (i.e., NumGrades < or =

22

SizeOfArray—line 12) allows the application to traverse the loop one too many times. ‘7
As a result, the value of NumGrades becomes 6 and fails to represent what we wanted it
to represent, namely, the number of grades to be averaged by the application.

This example is useful for illustrating Cummins’ conception of misrepresentation. For
him, a cognitive system 2 is in error when “... there is a mismatch between the state of
affairs 2 needs to represent when it tokens [a representation r] and the state of affairs
[r] actually represents ....”"’ The state of affairs that 2 “needs” to represent is called its
target while the state of affairs that 2 actually represents is called the content of 2’s
representation. ‘9 In our example, the system 2 which computes the grading algorithm
“needs” to represent the actual average of the five grades entered. The average of these
grades is the target of a tokening of A verage. Nonetheless, what Average represents in
fact (i.e., the actual representational content of A verage) is the sum of those grades
divided by 6. The tokening of the representation Average is in error, “...when the target of
tokening it on that occasion fails to satisfi» its content”—that is, when the target of the

token and the representational content of the token are not the same.20 Curmnins writes:

 

‘7 Actually, this algorithm would probably cause the computer to crash if implemented in some
common programming languages, since it would attempt to insert the value of CurGrade in the last loop
traversal into a non-existent index of the array, but we will ignore this complication for the meantime,
since the example can be used to illustrate another kind of error closer to misrepresentation. Also note that
pointing to the loop condition as the source of the error is a little bit misleading since it is only one of at
least two possible sources of the algorithm’s failure. Had we indexed the array starting at 1 instead of 0
(see the initialization of NumGrades—line 6), this error would not have occurred.

‘8 Cummins (1996), p. 6.

19 In what sense a system can “need” to represent one thing rather than another is something
which needs explaining. I’ll show how Cummins attempts to naturalize this notion.

2° Cummins (1996), p. 6.

23

It is precisely the independence of targets from contents that makes error possible. If the
content of a representation determined its target, or if targets determined contents, there could
be no mismatch between target and content, hence no error. Error lives in the gap between
target and content, a gap that exists only if targets and contents can vary independently. It is
precisely the failure to allow for these two factors that has made misrepresentation the Achilles
heel of current theories of representation."

The distinction between targets and contents is a nice distinction particularly when
attending to representation and error as they are understood in computer science. For
computer scientists, a program can perform incorrectly when one or more of the data
structures it utilizes fails to represent what the programmer intended it to represent.
Moreover, it is possible to think of this failure in performance as consisting in a
relationship between the actual representational content of a data structure and its
intended representational content, or target. After all, the algorithm described above
would perform correctly if the target of the representation Average was indeed the result
of dividing Sum by the number 6.22 It is because the target of the Average is Sum divided
by 5 that we say the algorithm performs incorrectly or is in error.23

A distinction similar to the one between content and target can be found in

 

2‘ Cummins (1996), p. 7.

22 Note that the fact that the representation has been referred to as an “Average” would not
prohibit it (Tom having the target Sum/6 in this algorithm since referring to the data structure in this
manner is merely the programmer’s way of reminding herself of what Average represents and not a
condition which controls its actual content.

23 This kind of error may seem more like an error in the performance of the system, rather than
an error in representation. In fact, Cummins solution to the problem of error is designed to explain
performance error, not representational error. I will say more about this presently, but for now, note that
Cummins himself is aware of the distinction between performance and representational error, makes the
observation elsewhere (Cummins (1996), p. 99) that his account is more adequate as an explanation of the
former, and is perfectly comfortable with the limitations which accompany this observation (Cummins,
(1996), p. 102).

24

discussions of perceptual error. In the case of perception, we can distinguish the content
of the perception fiom the object being perceived. In fact, it seems arguable that, as with
representation, the very possibility of perceptual error turns on just such a distinction. To
revisit an earlier example, when l misperceive an Oak as an Elm on a dark night, the
content of my perception is I Ehn I, but the object that is actually being perceived (read:
the target of my perception) is an Oak tree. Indeed, there does seem to be a distinction
between perceptual content and perceived object that is closely related to the distinction
which Cummins attempts to articulate between representational content and target.

This observation raises a number of interesting issues for Cummins’ discussion of the
distinction between contents and targets. For example, how do we distinguish the content
of a representation and its target? Or alternatively, how are targets specified? In the Elm-
Oak example, it seems arguable that the target of a representation is equivalent to its
cause while the content of a representation may have nothing to do with the cause at all,
or may be related only asymmetrically to it. But unlike Fodor, Cummins does not avail
himself of a causal account of target fixation so he will have to propose an alternative
method for making the distinction. In addition, whatever theory of target fixation we
adopt must imply or—at the least—be compatible with a completely naturalized
conception of “target.” And finally, we have intimated (two paragraphs back) that
Cummins’ notion of target is more useful for explaining performance error than for
explaining representational error. What reasons are there for accepting an explanation of
performance error as adequate for a theory of error? Do we really need a notion of

representational error, or is performance error good enough?

25

1.2.2.2 Performance Error vs. Representational Error

Cummins acknowledges that when representations are misused (as in the case where
a representation of Sum/6 is used to signify Average), error certainly occurs, but not
representational error. For example, a map of the Upper Peninsula of Michigan can
contribute to error if the map’s user wrongly takes it to be a map of Florida, but the error
is not representational—the map represents the UP. accurately. Instead, the kind of error
involved is a kind of “performance error”: error which is the result of the misapplication of
representations which accurately signify states of affairs external to the representing
system.24 Indeed, Cummins argues that we should “...resist any formulation that forces us
to infer misrepresentation from a mere failure to perform, since misrepresentation is
compatible with—sometimes essential to—successful performance, and failed

Performance is compatible with accurate representation?”

But if misrepresentation and
failure to perform (i.e., failure to apply representations correctly so that the system
performs in the way expected) are not the same, then Cummins’ account of error is not an
account of how misrepresentation is possible, but rather, of how cognitive systems fail to
perform when correct representations are misapplied. Cummins thinks that this kind of

account of error is good enough. That is, any case where before it seemed we needed an

account of representational error will now be explained by citing a mismatch of

 

2‘ Even though Cummins terms such representations “accurate,” the notion of an “accurate”
representation is somewhat out of place in an account which does not allow for representational
inaccuracies. It would probably be better (and just as effective) to say, “... the misapplication of
representations which map isomorphically to states of affairs external to the representing system.”

25 Cummins (1996), p. 99.

26

representational content with intended target. In short, performance error will do the job
previously done by the concept of representational error. To be clear, Cummins himself
does repeatedly describe his as a theory of representational error. However, as we shall
presently see, it seems clear this his account is perfectly compatible with the view that the
contents of one’s representations are always “correct”—that is, that they will never fail to
correspond with their content. The account simply explains the case in which a “correct”
representation is misapplied.

In the end, it may make little sense to call a representation “correct” under these
circumstances at all. More likely, representations are not properly thought of as correct or
incorrect in the context of an account like Cummins’—instead, there simply is or is not a
correspondence between some potentially representing structure26 and another, potentially
represented object. When there is such a correspondence, representation occurs. When
there isn’t representations aren’t “incorrect,” rather, they just aren’t there at all.

There may be good reasons for allowing performance error to do the work that
misrepresentation is normally thought to do. Let’s examine the way in which Curmnins
makes use of targets a little more closely in order to understand what some of these

reasons might be.

 

26 If Cummins is right and a structure represents in virtue of bearing an isomorphic relation to
another structure, then any given structure ought to actually represent any number of other structures. I
use the phrases “potentially representing structure” and “potentially represented structure” here to refer to
the possibility that some mental structure represents some particular external Structure. In other words,
while all mental structures will be isomorphic to some structure in the world and are thereby actually
representing structures, few if any of them will bear this relation to everything. It may need to be
established that a mental structure is in fact a representation of some specific external structure in which
we are presently interested.

27

1.2.2.3 How Are Targets Fixed and Can They Be Naturalized?

Following Cummins, I have been speaking of the target of a representation as that
which the system “needs” to represent in order to avoid performance error. As pointed out
above, this is a decidedly unnaturalistic way of speaking about target fixation since it
seems to imply that identifying error requires access to the system’s intended goals. Later
in his book, Cummins attempts to explain what he means by what a system “needs” to
represent more naturalistically. Cummins claims that, “the target of tokening [a
representation] r is what [a cognitive system] 27 expects to find when r is accessed.”27
What the system “needs” or “expects” is not a matter of the system’s intended goals, nor
is it a matter of the intended goals of the system’s programmer. Rather, what the system
“expects” to find is “... a matter of [the system’s] architecture or design”28 The idea is
that representing systems, and cognitive systems in particular, incorporate mechanisms
whose fimction it is to represent objects or sets of objects in the external world.29
Cummins cites the example of a visual module, whose function it may be to represent
sizes, shapes, and edges. Targets are determined, he claims, by the function of the
mechanism in question together with the current state of the world. Mechanisms which

have the kind of representational function described here are called intenders.

 

27 Cummins (1996), p.18.
2" Cummins (1996), p.18.
29 Cummins (1996), p. 8.

28

The conception of target fixation I want, then, is the conception that the target of tokening r
is what I} expects to find when r is accessed. Not literally ‘expects’ of course; the idea is that
27 incorporates a design assumption to the effect that a representation generated by a certain
intender will be a representation of t. This is the conception that falls naturally out of
thinking of intenders as mechanisms for binding programming variables. Just as programs are
designed around assumptions about what will be accessed when a given variable is evaluated,
cognitive systems are designed around assumptions about what will be represented by various
intenders. Specifying a system’s design or functional architecture involves specifying what
intenders are possible, and hence, on the current conception, what targets are possible. The
targets that are possible for a given system are thus fixed by its functional architecture.”

But is the concept of “firnction” as it is utilized here a naturalizable concept? And if so,
will fimction together with the “current state of the worl ” be enough to explain why, for
example, the Average program failed? More important, will it be enough to describe its
failure as representational error rather than as performance error and is such an
explanation even necessary?

If it is true that targets are determined, in part, by what fimctions the mechanisms in
cognitive systems have, and if a system’s functions are derivative of its “architecture or
design,” then we may be on our way to a more naturalistic account of target fixation.
However, there seems to me to be a significant difl‘erence between saying that the function
of a representing mechanism is determined by a system’s architecture and saying that it is
determined by a system’s design. The difference lies in the distinction between two distinct
and independent senses of the term “firnction.” When Curmnins claims that some
representing mechanisms in a system (namely, intenders) have “functions” which specify
their targets, he might mean that they actually compute formal mathematical functions.

Although it is somewhat misleading to say in this circumstance that the mechanism “has”

 

3" Cummins (1996), p. 18.

29

the function in question, we will suppose that the relationship the mechanism bears to the
function it computes (in virtue of the fact that it computes it) is enough to make sense of
this terminology. On the other hand, Cummins might mean that mechanisms acquire
“functions” in a more teleological sense of the term. Perhaps systems subject to
evolutionary development, for example, acquire mechanisms whose fiinction is to promote
survival. Let us signify that we use the term “fimction” in the former sense by writing it as
“function,” while signifying its use in the latter sense with the notation “function,”

Now, if a system’s mechanisms have ftmctions1 and they can be used to determine the
targets of the system’s representations, then target fixation does seem to be a matter of a
system’s architecture. This would be consistent with a fully naturalistic account of target
fixation, since one has only to describe the formal properties of the system in order to
understand its firnctionsl. The problem with understanding the function of a mechanism
this way however, is that it cannot provide for a distinction between targets and
contents——a distinction which Cummins needs if he is to retain his account of cognitive
error. To see why this is so, consider the grading algorithm described above: Conceive of
the algorithm as a simple system whose data structures are functioning mechanisms. In
particular, consider the mechanism, Average. If we want to know the target of A verage,
then we need to know what its function, is (i.e., its formal description as implied by the
system’s architecture). To obtain this, we examine the algorithm and find that Average
filnctions1 to calculate the Sum divided by 6. When the program is executed, moreover,
the content of A verage is also the Sum/6—this is what Average in fact represents. The

problem with conceiving of an intender’s fimction as a fimction in the forrml mathematical

30

sense, as this example illustrates, is that this is the same strategy employed, at least on
Cummins’ view, for obtaining the representational content of an intender. Hence, doing so
will not allow for a distinction between target and content.

Suppose, instead, that it is the function2 of a mechanism which determines its target.
If the content of the representing mechanism is determined by its firnction,, but the target
of the mechanism is determined by its fiinctionz, then we can reestablish the distinction
between target and content Curmnins believes necessary for a successful account of error.
However, unless the origins of functions2 are explained without an appeal to the intended
goals of the system and/or without appeal to the goals of the system’s designer (a difficult
task in the case of the grading algorithm, to say the least), this approach threatens the
project of providing a fully naturalistic account of representational error, and by
association, of representation itself.

Accounts which attempt to provide a naturalized account of a system’s functions are
readily available.31 And although Cummins’ notion of “function” turns out to be
something close to fimctionz, he is not satisfied Simply to help hirnselfto one of them by
way of offering up a solution to the dilemma illustrated above. As he sees it, there are at
least two reasons why a typical rendering of the notion of “fimction” (as understood
especially in contexts such as functional and adaptational role theory) will not sufiice for
the problem of describing how targets are fixed. First, Currmiins contends that most

understandings of fimction have it that only event types can have functions—that is, once

 

3‘ See, for example Ruth Millikan’s “Biosemantics” in Journal of Philosophy, Vol. 86, No. 6,
1989.

31

the function of a thing or event has been determined, all things or events of that general
type must have the same function. The notion of fimction is not, as ordinarily understood,

something which can vary from token to token:

...available theories of functions apply to events only as instances of types: ‘the function of x is
f is defined in terms of what x does Normally (Millikan 1984) or typically or ideally, or what x
does whenever some specified condition holds. But it is essential to the current project [i.e., the
project of explaining target fixation in a way that is consistent with the positive theory of
interpretational semantics] that different tokenings of the type “tokenings-of-r” can have
different representational functions, for r can be applied to different targets on different
occasions. Not every tokening of| elm I has as its function representing elms, for, if this were
the case, there could be no error. Representational error arises when an I elm | is applied to
something that isn’t an elm, for example, when the function of a tokening of l elm | is to
represent a beech or the number 9 or the proposition that roses are red. So we need a theory of
functions that allows for the function of tokening r to differ from occasion to occasion.32

In other words, since targets can vary fi'om token to token, either we cannot make target
and fimction synonymous, or we must find an alternative notion of fimction which allows
us to identify functions at the token-level. Second, Cummins believes that most “over-the-
counter” analyses of fimction equate successful representation with correct functioning.
But, as we have already seen, Cummins does not equate successfirl representation with a
content-target match. Therefore, if the appropriate target of a representation is going to
get determined by the function of that representation, we must have a notion of function

which does not immediately imply a kind of success.

...we need an analysis of fimctions ...[which] allows for the fact that representational success is
neither necessary nor sufficient for the nonrepresentational success of representational systems.
Target Fixation will be incorrect if we expand it via a theory of functions that yields the

 

3’ Cummins (1996), pp. 113-14.

32

implication that cognitive processes require representational correctness to function properly.
Representational error can be quite compatible with success.33

Cummins circumvents the first of these two difficulties by proposing that we give up
the idea that representations themselves ever have the function of representing a particular
target. Instead, Cummins argues that substructures of the representing system called
mechanisms have the fimction of representing one or another particular target, and that
representations inherit this function when they are tokened by the relevant mechanism.
Mechanisms which fimction like this are referred to elsewhere by Cummins as “intenders”
(see above) and when they function specifically to represent a target t he calls them “t-
intenders.” Since mechanisms having very different functions might token the same type of
representation, the representation in question can have one function/target when tokened
by one mechanism and a very different function/target when tokened by another. Similarly,
the same mechanism might token different types of representations, leading to a case in
which different types of representations have the same fimction/target.

This raises the question of how to understand a mechanism as having the function of
representing some target or other. Cummins notes that if we are to satisfy his second

criterion for the analysis of function, then we must not understand it in a way that equates

 

33 Cummins (1996), p. 114. Cummins argues elsewhere at length for the point that
representational error can be compatible with successful performance. (See pp. 27ft; 44-47, 50, 99, 116-
118.) For example, on p. 27 (footnote) he states: “Less accurate representations are often tolerable because
they are less costly to compute. Misrepresenting a crow as a hawk is a less serious error for a field mouse
than misrepresenting a hawk as a crow. Given that recognition must occur quickly, a fast but inaccurate
system may be better than a slower more accurate one. Since crows greatly outnumber hawks, a fast
system that generates many false-positive hawk indentifications but no false negatives is less accurate but
more effective than a slower system that generates fewer false positives while still avoiding false
negatives.”

33

the notion of function with anything ensuring the correctness of the representation on the

part of the system which incorporates the mechanism.

Our problem is this: how can it be the function, or a function, of a mechanism to
represent t [the target] if neither it nor its ancestors have actually succeeded in accurately
representing it? The key to understanding this issue is to realize that it is the (or a) function of
[a mechanism] N to produce representations oft if it is the representational relation—the
degree of fit—between t and the tokens N produces that underlies N’s contribution to the
system that contains it. To get a feel for this, we may imagine the situation in which a map M
is completely accurate for ml, but used by [a system] 2 to get around the somewhat different
m2. To understand 2’s success and failures in negotiating m2, and Ms contributions to these,
we have to know how accurately M represents m2, since 2’s errors are measured relative to
m2, M’s target on the occasions in question. Hence E’s capacity to negotiate m2 as well as it
does cannot be understood in terms of its capacity to perfectly represent m1, even though any
representation that represents ml perfectly will in fact represent m2 accurately enough (and in
just the right way) to explain 2’s performance in m2. For while it is true and lawlike that 2
performs as it does in m2 because it perfectly represents m1, this holds only because of the
relation between m1 and m2. To assess 2’s performance in m2, we need somehow to assess
E’s representation of m2. We can do this by assessing E’s representation of m1, but only if we
are already in a position to assess how well m1 represents m2. Since M and ml are isomorphic
by hypothesis, assessing the representational relation between m1 and m2 is equivalent to
assessing the representational relation between M and m2.“

What Cummins provides here is a case in which the system 2 exploits a representation in
the service of negotiating some real-world problem—we might imagine that navigating the
city of East Lansing is the problem which 2 wishes to handle. The representation M which
E exploits bears a relationship to the city of East Lansing (m2), but it just so happens that
it bears an even better (read: more accurate to the point of isomorphism) relation to the
city of East Splansing (m1). Cummins’ account of target fixation requires that East
Lansing is the target of 2’s representation on this particular occasion, because it is the city
of East Lansing that 23’s representation functions to navigate. But the problem is this:
How can East Lansing be the target of M when M doesn’t accurately represent East

Lansing, and in fact, does accurately represent East Splansing? Why is the target of M

 

3" Cummins (1996), p. 117.

34

East Lansing (m2) and not East Splansing (m1)? Cummins argues that m2 can be the
target of M in virtue of the fact that there is some degree of fit between 2’s representation
and the actual city of East Lansing that informs 2’s behavior. In Cunnnin’s terms, it is the
existence of some “fit” between representation and represented thing which “underlies [the
mechanism which produced the representation] N’s contribution to the system that
contains it.”35

This move sounds a lot like F odor’s asymmetrical dependence to me, although I can’t
be certain how Cummins actually makes use of the similarity between m1 and m2 the way
it seems obvious that F odor does. Perhaps one way of describing Cummins’ move is that
he makes ml (in virtue of the fact that is bears the closer (isomorphic) relation to M) the
primary player in a F odor-like asymmetric dependency relation. Something like, “M can
only fiinction to assist in the navigation of East Lansing (m2) because m2 looks like East
Splansing (ml).” Translation: M can only have m2 as a target because this target looks like
the content of M. If Cummins is attempting to make a move which employs a notion
similar to asymmetric dependence, then his characterization of target fixation is subject to
criticisms like those made against F odor. I leave it to the reader to determine this.

Let’s step back to consider what this way of understanding target fixation
accomplishes and what it does not. We have already seen that the introduction of the
notion of a mechanism or “t-intender” makes it possible for Cummins to avoid
understanding functions as pertaining only to event and thing types. This latest move

would also seem to lay the groundwork for the possibility of maintaining a distinction

 

’5 Cummins (1996), p. 117.

35

between accurate/successfiil representation and the proper fimctioning of a system’s
mechanisms, resulting in the conclusion that mechanisms can have functions/targets which
do not require (totally) successfirl/accurate representation. For while m2 is the target in
this example, it is not the most accurate (read: most closely mapped to M) of the available
objects which stand in relation to M. So it seems that Cummins has met both of the
conditions he set out to meet when seeking to fix targets via the identification of a
system’s proper function.

Notice that what still cannot be distinguished are the fimctions/targets of a system’s
mechanisms and what promotes the successes and/or goals of the system as a whole.
Cummins has shown that “accurate” representation may not always be required for what
maximizes a system’s benefits, but this point necessarily involves the retention of the
notion of what is beneficial to the system, and it remains unclear how this can be
understood naturalistically. In Chapter 6, I’ll revisit this issue in more detail. For now, it
will have to be sufficient to note that there may be questions which a theory seeking to
naturalize representation cannot in principle address naturalistically—which is to say that
it cannot address these questions at all.

Cummins has made a significant contribution to the project of providing a naturalized
and highly formalized account of mental representation. In addition to reviving what many
thought to be a discarded form of representation theory namely, picture theory, Curmnins
has updated the view and shown how it can be used in the service of providing a solution
to one of the most challenging problems of representation theory: the problem of error.

However, the review of his work provided here has demonstrated, I believe, that there are

36

still some significant issues to be addressed if a modern form of the picture theory of
representation is to be taken seriously. Among these issues is the problem of uniqueness,
which, as we shall see, influences everything from the fixation of representational content
to the identification of the targets of our representations. Therefore in the remainder of
this chapter, I will attempt to set out in greater detail the particular challenges which the

uniqueness problem raises.

1.3 The Uniqueness Problem

Perhaps the biggest advantage of interpretational sermntics rests in the sheer
simplicity of its account of mental representation. A representation represents in virtue of
bearing a relation to some object or state of affairs which shares its structure—plain and
simple. But what makes interpretational semantics so attractive as an account, may also be
the source of its greatest weakness. This is because the simplicity of the account may
allow that mental structures represent all kinds of things—indeed, more than one thing at a
time. If the content of a mental representation is just an external object with which it
shares a structure, which external object should constitute the content of a mental
representation when it is isomorphic to more than one? Shouldn’t mental representations
map onto external objects uniquely if they possess the content that they do exclusively in
virtue of this mapping?

It is important to recognize that this is not the same problem that we have been
recently examining when considering the manner in which targets are fixed. The problem

of target fixation is a problem about which things a system intends or functions to

37

represent in any given instance—not about which things a system actually does represent.
When we examined the problem of target fixation, we assumed, for the moment, that there
was no problem about which things the system actually did represent. We start out by
assuming that this much is clear: 2 possesses a representation r with the content C. The
problem of targets is the problem of figuring out whether it was C which 2 was supposed
to represent on this occasion or not. In contrast, the problem of uniqueness is the problem
of knowing what the content of r is in the first place.

There are a number of ways in which a “uniqueness” problem can arise for
interpretational semantics and for picture theories of representation in general. I have
provided diagrams of three varieties of the problem below. Only one of these versions of
the problem is addressed explicitly by Cummins in Representations, Targets, and
Attitudes. The others are acknowledged by other authors in other contexts such as
measurement theory and abstract algebra. For the moment, I will primarily address the
threat to picture-theory-style semantics which is posed by Curmnins’ own version of the
problem and will comment only briefly on what I consider the “standar ” version of the
problem. Ultimately, I will argue that some of these versions of non-uniqueness are not
legitimate concerns for the interpretational semanticist.

Cummins recognizes that there is a “uniqueness problem” for representational
content when it is accounted for in the way he (and picture theory in general) propose to

account for it. He describes the problem as follows:

...isomorphisms, where they exist at all, are not unique. A structure R may be said to represent
another structure C wherever R is isomorphic to C. In general, however, there are bound to be
many isomorphisms, that is, many one-to-one structure-preserving mappings, between two

38

structures if there are any. Hence, when R is isomorphic to C, there are many different ways in
which each structure can represent the other.”

C, c

., C
.\
f, f\\
\\
\

'3
3

 

 

 

 

 

C
-/f, it
,L/rf, Figure l-C:
’"l ‘ Non-Uniqueness—Standard
L r“'/ -f3
R
Figure l-B:
Non-Uniqueness— R5
C ummins

Figure l-D:
Non-Uniqueness—Measurement

 

36 Cummins (1996), p. 97. Note that it is odd for Cummins to speak of isomorphisms “where they
exist at all” since isomorphisms are actually quite common. For example, for any given structure, an
isomorphism can be established between its elements and the set of natural numbers (more on the threat
this type of isomorphism poses to a Cummins-like account in Chapter 5). Cummins seems to be focused
on a very specific type of uniqueness problem wherein if any two objects can be said to be isomorphic in
virtue of the existence of some function mapping the elements of one onto the elements of the other, then
it seems that not only that firnction, but an indefinite number of other functions ought to exist which map
the two objects to one another. In other words, for Cummins, the problem seems to be that there is more
than one mapping between the same two objects. A more serious form of the problem arises when more
than one external object can be mapped to the same mental structure (see Figure l-C). This would occur
whenever there is an isomorphic relationship between the mental structure and more than one external
object. Although Cummins’ interpretational semantics is threatened by this sort of non-uniqueness too, he
only addresses the former variety.

39

For Cummins, the uniqueness problem arises because it is possible for there to be
several isomorphic functions which constitute a mapping between two structures: R, a
representing structure and C, a represented one. His version of the problem might
therefore be represented graphically as shown in Figure l-B (where each fl is an
isomorphism of R onto C).

To illustrate this brand of non-uniqueness, Curmnins imagines a mechanical car
capable of navigating a maze called the “Autobot.” which is. The Autobot is crafted with
cog wheels on the rear axle which pull a card parallel to the ground through the car as it

moves. In the card is a jagged slit. A pin on the tie rods controls the steering of the car as

 

TheCard

    
 

Thecogmelsonthereararde
marwllthecaruthromhmecar.

car enters a turn

Figure l-E: Autobot
4O

it passes through the slit (see Figure l-E).37 The idea is that each left hand groove in the
card represents a left hand turn in the maze, while right hand grooves represent right hand
turns. Supposing that the card is designed in such a way as to represent the turns of the
maze, the car should navigate the maze successfully when its motion is governed by the
card. Of course, the card represents the maze if and only if there is an isomorphic
relationship between the slit in the card and the path through the maze. But there is such a
relationship—let us refer to this isomorphism as f,.

The uniqueness problem arises when it is observed that f,, which promotes a
successfirl navigation of the maze by the Autobot, is not the only isomorphism which
exists between card and maze. Imagine that the card was inserted into the Autobot upside
down such that now left hand grooves in the card are mapped to right hand turns in the
maze and right hand grooves in the card are mapped to left hand turns in the maze. This
too is an isomorphism between the slit in the card and the path of the maze—let us call
this isomorphic relationship f2. Now we have a version of non-uniqueness which matches

Cummins’ description of the problem (see Figure 1-F).

 

37 reprinted from Cummins (1996), p. 95 with permission fiom The MIT Press.

41

 

 

 

Card
Figure 1-F:
Non-Uniqueness in the
Autobot

Obviously, f, promotes the successful navigation of the maze by the Autobot whereas
f2 does not. However, f, is certainly isomorphic to the correct path (just as f, was) since it
maps the slit in the card to its mirror image. Since both firnctions establish an isomorphism
between the slit in the card and the path through the maze, but only one promotes its
successful navigation, it may seem that both firnctions could not serve to specify the
representational content of the card. In fact, it seems (if the goal is ultimately to represent
the maze) that f, produces an accurate representation, while f, promotes representational
error.38 This may make it seem like we need a criterion for choosing f, over f, rather than
be satisfied with the mere existence of an isomorphic relationship between the card and the
maze. Indeed, Cummins believes that it is tempting to see the only solution to this
problem as consisting in the development of a criterion for choosing between competing

isomorphic mappings.

 

38 Or perhaps it is simply performance error that f; promotes—that is, behavior which is
inconsistent with the overall goals of the system or of the system’s programmer. This is the conclusion
you would reach if you accept Cummins’ strict distinction between targets and contents.

42

Of course, when considering which of these mappings identifies the target of the
card’s configuration—the one which maps left turns in the card to left turns in the maze,
or the one that maps the card’s left turns to the maze’s right turns—we do have to make a
choice. Since both mappings do not promote the Autobot’s successfirl navigation of the
maze equally well, and since we can presume for the sake of argument that successfirl
maze-navigation is the goal of the system, one mapping must fail to match the
representational content of the card with its target. But Curmnins argues tlmt no such
choice is required in order to know the content of the card’s configuration. There are any
number of things which are represented by the slit in the card. What explains the ability of
the Autobot to exploit its representations nonetheless is that the target is among them.39

Cummins’ strategy for overcoming problems of non-uniqueness is not, therefore, to
show how to eliminate all but one of those mappings which provide a representation with
the content that it has. Rather, Curmnins’ strategy is to Show that non-uniqueness (of the
variety described in Figure l-B anyway), does nothing to inhibit the explanatory power of
interpretational semantics, which turns out to be more concerned with how well a system’s
representations match up with their targets than with whether there might be other things

that the system’s representations could have represented equally well.

If your map is not properly oriented, you won’t get to your destination, but that is not the map ’s
fault. The correct information is there; you just do not or cannot exploit it.... Grounding
representation in isomorphism entails, as we have seen, that representational content is never
unique. This non-uniqueness, however, does not undermine the explanatory value of
representation in any way, for the explanatory work is done by the match (or mismatch)
between content and target. The fact that a representation correctly represents many things
other than the target has no tendency to devalue the important fact that it does (or does not)

 

’9 Cummins (1996), p. 99.

43

represent the current target, for it is the match or mismatch (or rather the degree of match)
between target and content that bears an interesting explanatory relation to performance. ‘0

But how the distinction between target and content comes into play here may reveal a
difficulty in Cunnnins’ account. The Autobot illustrates a situation in which a single
representational structure and a single content structure are related by more than one
isomorphic mapping relation. According to interpretational semantics, nothing more than
the existence of an isomorphism relating R to C is necessary for saying that C is the
content of R. This would make it difficult to see how a mismatch between the content of R
and its target could ever arise. In this scenario, there is only one content structure that can
serve as the target. And there is nothing in the interpretational senmnticist’s account of
content fixation which would imply that the content of R ever changes either. It seems to
me that, the only way to conceptualize mismatches between target and content in the
interpretational semanticist’s view is to allow that either (a) there are several content
structures available to which a single representation can be mapped, or (b) that there are

several representational structures which are related to a single content structure. Given

 

4° Cummins (1996), 99 & 102. It may seem that Cummins is alluding to something more like the
standard version of the uniqueness problem (Figure l-C) when he says: “The fact that a representation
correctly represents many things other than the target has no tendency to devalue the important fact that it
does (or does not) represent the current target, for it is the match or mismatch (or rather the degree of
match) between target and content that bears and interesting explanatory relation to performance”
[emphasis mine]. However, I think this is not what Cummins had in mind here. First, one cannot read
Cummins’ notion of “correctly represents” as equivalent to “has as its representational content” (which is
how the comment above must be read if it is to be interpreted as alluding to the standard problem of
uniqueness as I’ve characterized it) since the content of a representation is not technically correct or
incorrect on Cummins’ view—only the relationship between it and the target of the representation can
give rise to error (and hence to correctness). Second, Cummins’ notion of “correct representation” is one
which only partly appeals to the notion of representational content. Representational correctness, for
Cummins, occurs when content and target are consistent. But reading the passage above with this in mind
makes the claim that “a representation correctly represents many things other than the target”
nonsensical. Therefore, I think it best to assume that Cummins meant to describe the same manifestation
of the problem he has been addressing all along.

44

that this is so, it is curious that Cummins does not spend more time on the sorts of
uniqueness problems illustrated in Figures 1-C and 1-D respectively.

Cummins’ version of the uniqueness problem cannot be “solv ” by interpretational
semantics if the criterion for a solution is to be able to specify some way of picking one
mapping, responsible for giving R content, over all of the others. The reason that no such
criterion can be specified by interpretational semantics is that it treats isomorphism as a
sufficient condition for representational content. What this implies is that the
interpretational semanticist cannot even represent distinct mappings between C and R let
alone choose among them. The uniqueness problem, as characterized by Cummins, is
probably intractable for interpretational semantics."l

But there are reasons to take heart despite this. I will argue that if a meta-system 2
cannot represent the difference between two mappings of R onto C, then, relative to 2,
there is no difference between them.42 And as E acquires the capacity to represent
differences between C and R, the picture starts to look more like the sort of uniqueness
described in Figure l-C, for example, than like Curmnins’ brand. In particular, as E
acquires more information about the external world, C can been seen as part of a larger,
more complex structure than the one of which E was previously cognizant and which R
accurately represented.

Curmnins worries less about the effect of the varieties of non-uniqueness on content

 

41 Once again, this is so on the condition that your criterion for a solution is that some method be
specified for choosing between the competing mappings. Perhaps it seems obvious that this is the only
satisfactory way of responding to the uniqueness problem, but ultimately I will argue that it isn’t.

42 More on this in Chapter 5.

45

fixation than he should, but perhaps this is understandable given the de-emphasized role of
representational content in explaining system behavior according to his theory. More
important than fixing content for Cummins’ view is the ability to fix targets, and
ultimately, to fix them for the purpose of comparing them with the contents of a system’s
representations. But note that we will be unable to identify target-content mismatches if
we are unable to specify the content of representations with equal acuity. In other words,
if we cannot determine which external structure provides R with the content that it has in
the first place, then we will not be in a position to compare the content of R with an
appropriately chosen target. Even if Curmnins is correct, therefore, in putting the
explanatory burden on the relationship between targets and contents, rather than on
representational content alone, this does not exempt one from the need for solving the
uniqueness problem for representational content.

There is, I believe, a solution to the varieties of uniqueness problem for
representational content and Cummins has most of the tools he needs to construct this
solution. Most of the remainder of this dissertation is devoted to explicating and defending
the solution I’ll propose. Before concluding my summary of Cummins’ own views
however, I’d like to say a final word about targets and the possible disadvantages of

relying on them so heavily in an account of error and representation.

1.4 Two Problems About Targets
As I see it, there are at least two problems with Cummins’ use of targets (in addition

to those we’ve already considered in detail above):

46

1. Degrees of Accuracy in Representation: Most people think that representation can
occur in degrees. In fact, Cummins hirnselfhas insisted that a desideratum of a theory of
representation is that representation admit of degrees43 and mentions similar requirements
on representation in this work. Nonetheless, it’s hard to see how Cummins’ use of targets
will generate a theory of error which, among other things, accounts for representation by
degrees. This is true in part because targets don’t figure in to a legitimate theory of
representational error but rather, account for performance error understood as what
occurs when correct representations are misapplied. And as for representational content,
since representations have content in virtue of being isomorphic with what they represent,
there is a sense in which all representations are perfectly accurate, or better: a sense in

which the notions of accuracy and inaccuracy simply don’t apply.

2. Degrees of Completeness in Representation: Since representational correctness is
achieved “... when the content of a representation is the same as the target of its
application” targets will have to be structures too, for Curmnins [emphasis mine].“ If not,
then there is no sense of “same as” that will account for the match up between target and
content. However, if the representation relation consists in a strict isomorphism, then, to
have a perfect match, for example, between the city, which is the target of my
representation, and the imp of the city, there must be an isomorphism between the city

and the map. But this is surely requiring the unreasormble. Cummins is aware of this

 

‘3 Cummins (1989) and Cummins (1996), pp. 107 & 108.

4" Cummins (1996), p. 109.

47

difficulty when he writes:

...we speak loosely when we say, for example, that a map represents Tucson. Common sense
recognizes this looseness by acknowledging such facts as that a topographical map and a street
map of Tucson, while both maps of Tucson, nevertheless map different things. A street map
abstracts away from many features of the city, including contours and altitudes, while a
topographical map abstracts away from many features, including the layout of the streets. All
forms of representation, except for particle-for-particle duplication, are abstract in this sense:
they capture certain structures and are silent about others [emphasis mine]."’

Let me begin with the second problem. Although I am comfortable with the notion
that all representation is abstract, I believe it an undesirable consequence of what is
contained above that all representation short of “particle for particle duplication” is
representation only in a “loose” sense. I also believe this to be an entirely unnecessary
consequence of a picture theory of representation. As we consider the uniqueness problem
in its various manifestations more closely, I will present evidence that the representation
relation should be thought of as homomorphic rather than as isomorphic. That is, instead
of requiring that every element and relation in the content structure be mapped into by the
function which connects it with the representing structure (instead of requiring that the
representation relation be an onto nmpping), I will require only that some subset of the
elements and relations in the content structure be mapped into by the representation
relation. Cummins does not explicitly rule out the possibility of having mappings which are
not onto in his intuitive definition of isomorphism expressed earlier in this dissertation. But
insofar as “particle for particle duplication” suggests an onto relation between structures,

Curmnins does seem to imply that anything short of “loose” representation requires the

 

4’ Cummins (1996), p. 109.

48

stronger, onto mapping. Conceiving of the representation relation as homomorphic seems
particularly well-motivated now. If the representation relation is weakened in this way,
then, speaking intuitively, representation can remain abstract without applying “only
loosely” to anything short of duplication of the object represented. In my view, only some
features of the city need be mapped into by features of the map in order to achieve the
relationship of representation. This seems to be much more natural than relegating all but

particle-for-particle duplication to “loose” representation.

 

Figure l-G: Isomorphism ‘Modulo’ A
Substructure
Making this move may help to account for degrees of representation (the first
problem mentioned above), and can do so without the use of targets. To see how, suppose
that R , and R 2 are both representational structures in a cognitive system/structure R and
that f1 and f2 are homomorphisms of R, and R, into C. Suppose further that c1 and c2 are

substructures of C and that R1 and R2 are isomorphic to c1 and c2 respectively. We may say

49

that R, is an inferior representation of C compared with R2 just in case the domain of c1 is
less than than the domain of c,. The intuitive idea is, of course, that if R2 is more complete,
i.e., if fewer elements need to be excluded from C in order to achieve an isomorphic
mapping with some R 2 when compared with R,, then R2 is a better representation of C that

R, is (see Figure l-G above)!“S

 

4" This approach may also have the advantage of providing an account of what makes
representations be representations of the same or different types; or equivalently, of explaining the ability
of cognitive systems to select from among several different ways of typing objects. When a system
classifies objects as belonging to the same type we “mod out” the features or substructures of the
representational structures which are the objects of our typification. Alternatively, when the system
classifies these objects as belonging to different types, the relationship between them remains a
homomorphism.

50

CHAPTER 2: The Picture Theory of Meaning

Before moving on to consider the uniqueness problem and its possible solutions in
more detail, I’d like to pause to provide the reader with a more detailed introduction to
the view of mental representation known as the picture theory in order to compare this
view with interpretational sermntics and to determine the capacity of interpretational
semantics for responding to traditional difficulties with the view. Accordingly, this chapter
is organized into three primary parts: First, I discuss the basic tenets of a typical version of
the picture theory of mental representation; next, I discuss how interpretational semantics
may be seen as a refinement of this view; and lastly, I consider in detail some of the more
troubling objections raised against a picture theory of representation and investigate

whether or not interpretational semantics can surmount them.

2.1 An Introduction to the Picture Theory of Representation

Perhaps the best known proponent of the picture theory of meaning was Ludwig
Wittgenstein, despite the fact that in his later work, Wittgenstein largely rejected the view.
Other philosophers, such as Aristotle, and some of the Empiricists also put forth versions
of the view.47 It is arguable that Leibniz held a version of the picture theory where his
theory of perception is concerned, insofar as he held that one thing represents or expresses

another when “there is a constant and fixed relation between what can be said of one and

 

47 Compare for example, the empiricist John Locke’s views on primary qualities and how they
resemble their objects.

51

what can be said of another?”8 In general, philosophers who held a picture theory of
representation claimed that representations have the content that they have in virtue of a
resemblance between the representation and the object represented. Aristotle took the
notion of resemblance quite literally—for him, representations contained the form of the
things that they represented. In addition, ancient philosophers such as Epicurus and
Empedocles, believed that eidola or “images” of external objects, impinged on the senses
to create our perceptions of them. Epicurus allowed that perception could be firrther
influenced by judgment, occasionally leading to perceptions of things such as unicorns and
centaurs, but maintained that all perception can be traced to the reception of eidola.49

Later advocates of the picture theory, such as Wittgenstein, did not require that
representations resemble objects in the sense of “looking like” those objects. Rather,
representations, for Wittgenstein, could depict objects and/or states of affairs “... provided
there were as many distinguishable elements within the [representation] as within the
situation it represents, so that the [representation] possessed the appropriate pictorial form
to be isomorphic to [the object or] state of affairs?”0 With Wittgenstein, therefore, the

picture theory of meaning took a formal tum—features shared between the representation

 

‘8 fi'om “letter to Amauld, G. 2:112/L 339" referenced in Simmons, Alison. “Changing the
Cartesian Mind.” in The Philosoplfll Review. January 2001. Simmons writes: “In other words, [for
Leibniz] representation involves an isomorphism between res repraesentans and res repraesentata.
Resemblance is the paradigm case, but other forms of isomorphism will do; planar projective drawings
represent solids, maps represent cities, musical notation represents a musical composition and so on.”
(Simmons (2001).

 

49 Copleston, Frederick, SJ. A History of Philosophy Vol. 1, Greece and Rome. New York:
Doubleday, 1993. pp. 402 & 403.

5° Logue, James. printed in The Oxford Companion to Philosophy. ed. Ted Honderich. New
York: Oxford University Press, 1995. p. 681.

52

and the thing represented are described more formally (i.e., in terms of isomorphism) than
is possible when one invokes the notion of a visual image or picture to account for
pictorial similarity. In addition, describing pictorial similarity in terms of isomorphism
made it possible to give an account for the representational capacities of things such as
propositions, which are presumably not pictorial images of the things they represent at all.
Should a proposition be analyzable into elements which correspond to the state of affairs it
describes, that proposition would thereby be said to represent that state of affairs, in virtue

of the formal resemblance which the proposition bears to it.

2.2 Interpretational Semantics and the Picture Theory of Representation

Cummins’ interpretational semantics provides a fully forrml interpretation of the
notion of “pictorial similarity” and hence, is a forrmlized version of a picture theory of
representation. Like Wittgenstein, Cummins also depicts the representation relation as an
isomorphic relation between representation and thing represented. In his view,
representations are mathematical structures and have all of the associated characteristics
of such structures. In Chapter 1 we saw that whenever a representational structure R
represents a “content” structure C:

1. An object in R can represent an object in C.

2. A relation in R can represent a relation in C.

3. A state of affairs in R—a relation holding of an n-tuple of objects—can represent a
state of affairs in C.’1

 

51 Cummins (1996), p. 96

53

Specifically, R and C are related isomorphically in virtue of the existence of a special kind
of function called an isomorphism fi'om R onto C. Although we will want to say more
about the notion of isomorphism later, for now, let us define an isomorphic firnction as
follows: A fimction h is an isomorphism mapping a structure fit onto a second structure (5
just in case:

(a) for each n-place predicate symbol P and each n-tuple <a1, ..., an> of elements of

91, <a1, ..., an> E P3 ifl‘ <h(a,), ..., h(a,,)> E P‘ (where P‘ should be read as

“the predicate P in structure i”).

(b) for each n-place function symbol f and each such n-tuple,

h(f"(a,, ..., a,)) =j‘(h(a,), ..., h(a,,)) (where f should be read as “the function f in the

structure i).

(c) h is one-to-one and onto.52

The proponent of interpretational semantics claims that whenever such a fimction
exists between two structures, the two structures are similar or resemble one another,
where, like Wittgenstein, we no longer mean to imply by this that the two look similar.
2.3 Common Criticisms of Picture Theory and Responses from the Perspective of
Interpretational Semantics

Although which criticisms are most effective against a picture theory of meaning
generally depends, in part, on whether the theory has undergone the formalization
characteristic of Wittgenstein’s view and interpretational semantics, some criticisms are

classical objections to picture theory in any of its manifestations. I summarize versions of

some of these criticisms here and suggest strategies for responding to them from the

 

52 Enderton (1972), pp. 89 & 90. Enderton does not require that isomorphisms are onto
mappings. I append this condition to bring the definition of isomorphism into accord with the way it is
understood in contexts such as group theory and abstract algebra.

54

perspective of an interpretational semanticist. Finally, I consider a criticism of picture
theory (raised indirectly by both Dennett and F odor in their discussions of the picture
theory of representation) which I believe to be related to what I have been calling the

“standar ” uniqueness problem for interpretational semantics.

2.3.1 The Problem of Abstraction Via Daniel Dennett

Daniel Dennett makes his case against the picture theory of representation in an
article entitled, “The Nature of Images and the Introspective Trap?”3 His use of the term
‘image’ may be understood as highly analogous to our use of the term ‘picture’ thus far,
insofar as images for Dennett have representational capacities in virtue of the relationship
they bear to the thing of which they are an image. However, as we shall see, Dennett’s
characterization of picture theory is more restrictive than the view embraced by a
Wittgensteinian-type of picture theory. Specifically, Dennett portrays picture theory as a
view which requires that pictures share monadic properties with what they
represent—cg, the picture must have some color, shape, smell, taste, etc. in common
with the represented object.

An image is a representation of something, but what sets it aside fi'om other representations is
that an image represents something else always in virtue of having at least one quality or
characteristic of shape, form, or colour in common with what it represents. Images can be in
two or three dimensions , can be manufactured or natural, permanent or fleeting, but they must
resemble what they represent and not merely represent it by playing a role—symbolic,
conventional, or functional—in some system.”

Dennett’s chief concern is whether there are mental representations that represent in virtue

 

53 Dennett, Daniel. “The Nature of Images and the Introspective Trap.” in Readings in
Philosophy of Psychology. Vol. 1]. ed. Ned Block. Cambridge: Harvard University Press, 1981.

5" Dennett (1981), p. 129.

55

of being pictures (i.e., whether there are mental representations which are pictorial). His
strategy is to Show that similarity in general (whether structural or monadic) cannot
ground representation at all (and therefore, that pictures cannot be representations). Since
interpretational semantics just is the attempt to Show that the representation relation is
grounded in a kind of similarity, namely structural similarity, Dennett’s criticisms are
applicable here.

Dennett thinks that mental representations have at least one characteristic which is
essentially unlike the characteristics of pictures,” arguing that if mental representations
and pictures do not share all of the same characteristics, then mental representations
cannot be pictorial. Specifically, Dennett thinks that mental representations can be
abstract in a way that pictures cannot. This is true, he claims, because unlike pictorial
representations of a thing, the mind, and in particular, the imagination, can represent while
simply leaving out some of a thing’s characteristics fiom consideration. For example, a
picture of a man with a wooden leg could not refiain from mentioning the detail that the
man either has or fails to have a hat. But on the other hand, the imagination can conjure
up a representation of the man with the wooden leg without taking information regarding
his hat into consideration—that piece of information is simply not part of the mental
representation. In conclusion, argues Dennett, since mental representations, formed in the
imagination, can have the quality of being detail-poor, but pictures cannot have this

quality, mental representations cannot be pictures.

 

55 Note that I (and most authors) concentrate on pictures whose properties are detectable through
vision. However, presumably there are auditory and tactile “pictures” as well.

56

We can, and usually do, imagine things without going into great detail. If I imagine a tall man

with a wooden leg, I need not also have imagined him as having hair of a certain colour,

dressed in any particular clothes, having or not having a hat. If, on the other hand, I were to

draw a picture of this man, I would have to go into details. I can make the picture fuzzy, or in

silhouette, but unless something positive is drawn in where the hat should be, obscuring that

area, the man in the picture must either have a hat on or not. As [J.M.] Shorter points out, my

not going into details about hair colour in my imagining does not mean that his hair is colored

‘vague’ in my imagining; his hair is simply not ‘mentioned’ in my imagining at all."5

In addition to persuading Dennett that mental representations carmot be the same as
pictorial images, this argument convinces him that a mental representation is more like a
description than like a picture. For like mental representations descriptions can either be
detail—poor or detail-rich. I can describe Sephra with or without mentioning that she is a
cat; I can describe the man with the wooden leg with or without mentioning that he has a
hat. Pictures of Sephra however, would address the information about the type of animal
she is necessarily (either by revealing that she is a cat or representing her in such a way
that she is not), and so insofar as mental representations of her need not do so, Dennett
believes that they are distinct fi'om pictures and closer to descriptions.

It is arguable that pictures are not always as (necessarily) revealing of the details as
Dennett suggests here. For example, suppose that I want to represent Sephra not only as a
cat, but as a Siamese cat. Since Sephra’s Siameseness is something primarily detectable by
her coloring (at least, if your only source of information is a picture), then a monochrome

picture of Sephra might not convey the information in question, nor would a silhouetted

picture of her. Even her shape in silhouette would not reveal the breed of cat she is, since

 

5‘ Dennett (1981), p. 130.

57

Russian Blues, for example, look strikingly similar to Siamese in silhouette.” Dennett’s
example really only demonstrates that pictures convey all of the details about an object
that can be conveyed in virtue of things like its shape (and maybe its size, although this is
doubtfirl). But this doesn’t show that pictures are incapable of abstraction per se—only
that they do not abstract away from certain kinds of features. And this is certainly true of
mental representations (if not of descriptions) as well,’8 for it is doubtfirl that I can have a
representation of the man with the wooden leg unless I have a representation of certain of
his (perhaps more fundamental) characteristics

There is an additional argument Dennett raises for believing that abstraction is a
persistent problem for the view that mental representations are pictorial (and for believing
that they are more appropriately thought of as descriptive). The following is his famous

“striped tiger” example:

Consider the Tiger and his stripes. I can dream, imagine or see a striped tiger, but must the
tiger] experience have a particular number of stripes? If seeing or imagining is having a
mental image, then the image of the tiger must—obeying the rules of images in
general—reveal a definite number of stripes showing, and one should be able to pin this down

 

57 This is particularly true of “stick figures” and other line drawings. This example (Figure 2-A)
probably conveys that the represented thing is some sort of cat, but it certainly doesn’t convey all of the
available details.

Figure 2-A:
‘Sephra Approaching

the Food’

58 Figure 2-A also illustrates how pictures can be descriptive without filling in all of the details.
This figure might serve the same purpose as its more descriptive caption.

58

with such questions as ‘more than ten?’, ‘less than twenty?’. If however, seeing or imagining
has a descriptional character, the questions need have no definite answer. Unlike a snapshot of
a tiger, a description of a tiger need not go into the number of stripes at all; ‘numerous stripes’
may be all the description says.59

I believe that this argument is similar to the first and makes the same mistaken
assumptions about the capability of pictures to abstract. Dennett’s criticisms here seem to
be based on the assumption that pictures must have features isomorphic to the features of
that which they are images of. It is not diflicult to see why Dennett would make this
assumption—indeed, I have characterized both Wittgenstein’s version of picture theory
and interpretational semantics in a way that would suggest it. It is the assumption which is
behind his contention that images cannot abstract away from even some of the features of
their object—all of the stripes of the tiger must be represented in the image of the tiger,
for example. This is the grounds for rejecting images as good candidates for mental
representation. But on closer examination, this assumption is curious, to say the least. Not
only is it obvious that images do not depict, as a rule, every feature of the thing of which
they are an image, but assuming so would be inconsistent with Dennett’s own initial
definition of “image” as something which must have “... at least one quality or
characteristic of shape, form, or colour in common with what it represents?”0 In any case,
it is Dennett’s initial definition of “image” that is more analogous to the notion of pictorial
similarity I will ultimately defend. I will argue that, contrary to Cummins, the
representation relation is better characterized as a homomorphic relation, rather than as an

isomorphic one. If resemblance does not require an onto relation between representation

 

’9 Dennett (1981), p. 130.
60
Dennett (1981), p. 129.

59

and represented object, then, contrary to Dennett’s view, the picture theorist need not
portray the representing “image” as one which shares every feature with the object it
represents. If some features can be left out, then perhaps the door is opened to the
possibility of accommodating abstraction in representation after all.

Conceptualizing the representation relation as one which does not require an onto
mapping between structures, namely, as a homomorphism, may prove to have the
advantage of making abstract representations easier to conceptualize within the context of
a picture theory of representation. But Dennett’s arguments raise another difficulty for
picture theory which we have not yet acknowledged or responded to. Although it now
seems clear that pictures can represent only some of the properties of their objects,
Dennett’s examples also raise the possibility that pictures can, and possibly must, have
properties that are not meant to correspond to anything in the represented object. For
example, in Figure 2-A above, the color of the lines used to draw the stick-image is blue,
but we would not ordinarily imagine that the color of the lines is a representation of any
property of Sephra. For interpretational sermntics, wherein the representation relation is
an isomorphism, this presents a special challenge. On this view, R is a representation of C
if and only if R and C are isomorphic structures (and are therefore, among other things,
one-to-one); hence, it would seem that there can be no element in C which is not mapped
into by some corresponding element in R and therefore no element in R which does not
represent some aspect of C.

Strictly speaking, something in C must get mapped into by each element of R, else we

cannot characterize the representation relation as a proper function. But we can imagine

60

that in cases such as the one given above, elements of R map into 0, providing the
intuitive grounds for saying that some elements of our representations do not correspond
to “anything” in the external world. But such a mapping would not be an isomorphism,
since it would not be a one-to-one mapping (presumably there would be more than one
thing mapped into 6). However, this would qualify as a homomorphic mapping of R into
C.

So the move to characterize representation as homomorphism may turn out to be
usefirl in formulating a response to this criticism as well since it drops the requirement that
the mapping is one-to-one. But before firrther examining how this is so, let us take a look

at a similar criticism of picture theory described by Jerry Fodor.

2.3.2 The Problem of Abstraction Via Jerry Fodor

Jerry Fodor raises similar problems concerning the ability of pictures to represent
abstractly and concerning the non-representatiorml features of pictures in an article entitled
“Imagistic Representation.”"1 F odor’s notion of a “picture is similar to Dennett’s although
he is less precise about the conditions under which pictures resemble their objects. For
him, pictures are simply representations that refer by resemb ° .62

Like Dermett, Fodor sees the problem of accounting for mental representation as a
problem of choosing between two possible alternatives: we will either get a theory of

representation which is pictorial, or we will get one which is discursive (what Dennett has

 

6‘ Fodor, Jerry. “Imagistic Representation.” in Readings in Philosophy of Psychology. Vol. II. ed.
Ned Block. Cambridge: Harvard University Press, 1981. pp. 135-148.

62 Fodor (1981), p. 135.

61

called “descriptive”). F odor reconceptualizes this dilemnm as one between “iconical” and
“symbolic” theories of representation and will ultimately favor the latter over the former."3
Fodor associates iconical theories of representation (theories wherein the reference of
icons is mediated by resemblance) with Jerome Bruner.“ His own view is that the
reference of symbols is mediated by “convention or something”65 but that regardless,
representations are firndamentally symbols rather than pictures.

Also like Dennett, Fodor’s strategy is to cast doubt on iconical theories of
representation by showing that icons are not the kinds of things that can genuinely
represent. A good theory of representation, F odor claims, will make representations
capable of denoting states of affairs, by which he means facts of the matter regarding

relationships between objects, persons, and events. Icons, he claims, can never do this.

...it makes a sort of sense to imagine a representational system in which the counterparts of
words resemble what they refer to, [but] it makes no sense at all to imagine a representational
system in which the counterparts of sentences do.66

 

63 Both Dennett and Fodor seem to assume these are the only two alternatives available to
theorists of mental representation. Both use this assumption to support a theory of discursive
representation by rejecting the plausibility of the picture theory. However, if a theory like structural or
interpretational semantics is right, then there may be a third alternative. In particular, structural semantics
will be revealed to have many of the properties of both symbolic and discursive theories of representation,
leading one to speculate on whether the alternatives considered by Fodor and Dennett may form a false
dilemma.

6" See Bruner, J.S. “On Perceptual Readiness,” in Psychological Review. Vol. 64, 1957. pp. 123-
152.
Bruner, J.S. , Goodnow, JJ. and Austin, G.A.. “A Study of Thinking.” New York: Paperback
Wiley Science Editions, 1962.
Brunet, J.S., Olver, RR. and Greenfield, P.M. “Studies in Cognitive Growth,” Wiley, New
York: 1966.

‘5 Fodor (1981), 136.

6‘ Fodor (1981), 136-37.

62

Since we know that words and sentences represent, the idea is that their representational
capacities must be explained in terms of pictorial similarity (if an iconical theory of
representation is correct). Iconical representation makes words into icons (i.e., things with
representational capacity), but it cannot, thinks Fodor, do the same for sentences. But if
icons cannot represent what sentences represent, then, representation (and specifically, the

representational capacity of sentences) cannot be iconical.‘57

In Iconic English, words resemble what they refer to but sentences don 't resemble what makes
them true [e.g., states of affairs]. Thus, suppose that, in Iconic English, the word ‘John’ is
replaced by a picture of John and the word ‘green’ is replaced by a green patch. Then the
sentence ‘John is green’ comes out as (say) a picture of John followed by a green picture. But
that doesn’t look like being green; it doesn’t look much like anything. Iconic English provides
a construal of the notion of a representational system in which (what corresponds to) words are

 

67 Fodor is primarily concerned about the capacity of sentences to represent, and though I do not
believe that he provides us with reasons to doubt that icons can represent abstractly here, there may be
something to his more general point: that sentences are harder to think of as icons than are individual
words. The problem with words (vs. sentences) and their meanings arises in the context of questions about
how language acquires meaning whereas interpretational semantics, and ultimately structural semantics,
are views about the representational capacities of mental states. I don’t spend a lot of time talking about
how language acquires meaning, although I think that the account would have to involve something like
the notion that linguistic terms acquire meaning conventionally. Specifically, we might say that people
associate the contents of their mental states with words by convention, and that sentences have meaning in
virtue of expressing a relation between mental states with representational content. Since both
interpretational and structural semantics have it that the contents of mental states and networks of mental
states is given by the holistic role which they occupy within a larger representational network, the upshot
of this is that linguistic meaning is likewise given holistically. That is, insofar as individual mental states
have content in virtue of their role in the representational network, individual words, whose meanings
derive from the content of individual mental states, have meaning in virtue of their role in sentences. This
is a decidedly different view about language fi'om Fodor’s in that F odor views language as compositional:
The meaning of a sentence is a function of (is “composed” from) the meanings of its words. Alternatively,
Randall Dipert has expressed views on the holistic nature of language which would accord well with the
position of structural semantics. In an article entitled, “The Mathematical Structure of the World: The
World As Graph,” (The Journal of Philosophy Vol. XCIV, No. 7, July 1997), Dipert writes, “It is a long-
standing custom to speak of a word or phrase as having meaning; this everyday approach, of treating
words as having semantic and syntactic (monadic) properties, has been aided and abetted by theories of
language since the Middle Ages. As is often briefly noted but rarely developed, however, this is quite
misleading: no series of marks, or of sounds, really—intrinsically—has a meaning. A word has a meaning
only in the bosom of a language.”

63

icons, but it provides no construal of the notion of a representational system in which (what
corresponds to) sentences are.68

Since thoughts are sentence-like (according to Fodor) and since sentences cannot be
iconical, thoughts cannot be iconical either.69 But this is just to say that mental
representation is not iconical.

Of course, F odor’s guess about how the statement ‘John is green’ would be conveyed
iconically is probably not the best guess one could make. Intuitively, it seems a much
clearer conveyance of the information could be accomplished with a picture of John
colored all-green. But this does not avoid the central point of Fodor’s objection. Fodor
admits that if an iconical theory was to seriously approach the representation of states of
affairs, it would probably proceed more in accordance with my guess: He acknowledges,
for example, that a picture of the sentence ‘John is fat’ would probably picture John with a
bulging tummy.70 But there is still a problem: Even if my intention is to represent John’s
fatness with the image, the image certainly represents many other facts about John having
nothing to do with this. How are we to know which of these ways of depicting John the
picture represents? For example, if the picture of John with a bulging turmny is also a

picture of John standing, then does the picture in question represent the thought: “John is

 

‘8 Fodor (I981), 136.
69

Fodor (1981), 136.
70

Fodor (1981), 136.

64

fat,” or does it represent the thought “John is standing?” F odor contends that theories of

iconical representation have no way of determining this.71

...symbols really are abstract. A picture of fat John is also a picture of tall John. But the
sentence ‘John is fat’ abstracts from all of John’s properties but one: It is true if he’s fat and
only if he is. Similarly, a picture of a fat man corresponds in the same way (i.e., by
resemblance) to a world where men are fat and a world where men are pregnant. But ‘John is
fat’ abstracts fiom the fact that fat men do look the way that pregnant men would look; it is
true in a world where John is fat and false in any other world.72

Fodor puts the problem like this: “The trouble is precisely that icons are insufficiently
abstract to be the vehicles of truth”73 So, like Dennett, F odor believes that pictures lack
at least one of the characteristics necessary for grounding representational content,
namely, the ability to represent abstractly. But Fodor’s emphasis in this example may be
slightly different from Dennett’s. Dennett’s claim was that pictures cannot fail to represent
all of the properties of the objects of which they are pictures, and therefore cannot be
capable of abstraction. I argued that this position was inaccurate by showing examples of
pictures which do convey only some of an object’s properties. Fodor acknowledges the
ability of pictures to convey only some of an object’s properties, but seems to think that
icons still fall short of being “sufliciently abstract.” To see why, suppose that we have a
picture of Jolm and that this picture in fact conveys only some of his characteristics. In

particular, the picture is one which portrays John standing up, colored green, and with a

 

7‘ Of course, one could always ask me what I intended to represent with the image, but as soon as
one remembers that we are ultimately interested in claiming that mental representations are images,
invoking intentions threatens the project of providing a fully naturalistic account of mental
representations.

7’ Fodor (1981), 138.

’3 Fodor (1981), 137.

65

sightly bulging tummy. Fodor’s claim is that when we are presented with a picture of a
green, standing John with a bulging tummy, which of these features is currently being
represented by us is not clear, and the picture can do nothing to make it any clearer. On
the other hand, if someone were to utter the sentence, ‘John is fat’ it would be clear fiom
this that this person represents those characteristics of John’s which contribute to his
fatness, and not necessarily those which may indicate his color, his stance, etc. Since
sentences can “abstract away” from John’s other properties in this way, while pictures
cannot, F odor maintains that representations must be more like sentences than like
pictures.

In addition, like Dennett, Fodor’s arguments show that there will probably be
properties of pictures which are not ordinarily understood to be representational at all, and
therefore do not necessarily “resemble” anything about the objects they depict. For
example, one may find that there are variations in the darkness of the lines used to create
the picture of John, the result of unevenly applied pressure on the artist’s pencil. We
would not ordinarily think of such variations as representations of any particular property
of John’s though they are most certainly properties of the representation itself.

There are at least two responses which can be made to Fodor’s criticisms concerning
the ability of pictures to represent abstractly. First, it is arguable that although pictures do
not always pick out only one of the available properties of a subject and fi'equently
(perhaps always) contain properties which are non-representational, the same can be said
of sentences. If this is so, then Fodor has not provided us with a convincing reason to

prefer descriptive theories of representation to iconical theories of representation. Second,

66

there may be clear criteria for judging that a picture represents but one of a wide variety of
possible targets, despite F odor’s worries that there are no such criteria. To see why, it is
necessary to be more specific about how “pictures” ought to be conceptualized in the

context of a picture theory of representation. I’ll consider both of these points in turn.

2.3.2.1 Non-Representational Properties of Pictures and Sentences

F odor’s case requires that both of the following are true: (1) sentences do a better job
than icons do of abstracting away fiom all of a subject’s properties except for those the
sentence represents, and (2) sentences, unlike pictures, do not have properties which are
non-representational. I believe that both of these statements are false. Let’s consider (2)
first.

It is important to point out that it is easy to get into trouble if one takes it for granted
that there are properties of pictures which are not supposed to be representational. Any
claim that a property of some icon is not one which represents is a claim which must be
couched in an antecedemly accepted theory of representational content. It wouldn’t be
surprising therefore if advocates of a non-iconical theory of representation were able to
identify properties of icons that didn’t seem to represent anything. But for the sake of
argument, let us suppose that icons can have representational content, but that they may
nonetheless have specific properties which do not represent anything in the object they
depict. Since interpretational semantics and Wittgensteinian picture theory both require
that representations be isomorphic to the structures which give them content, they are

hard-pressed to explain such a scenario. Isomorphism requires that the representing

67

structure and represented structure are one-to-one, and this prohibits the possibility of
having properties in the representation which do not correspond to anything in the object
represented.

Understanding the representation relation as a homomorphism may provide a way out
of this quandary. Since homomorphism drops the one-to-one requirement, there is nothing
preventing a situation in which a number of elements in the representational structure get
mapped to o in the represented structure.“ This is one way to explain what we intuitively
describe as a case where some features of a representation don’t correspond to any
features of a structure in the external world. We say nonetheless that we have a
representation of the external structure in such cases. On the other hand, as we shall see,
dropping the one—to-one and onto requirements on the representation relation may raise
other difficulties for this account of representational content. Among the most significant
of these is an exacerbation of the uniqueness problem, which we will consider in detail
presently.

Interestingly, it may not be necessary to weaken the representation relation in order
to respond effectively to Fodor’s criticism. Instead, advocates of interpretational semantics
could argue that non-representational properties of pictures are always monadic, and that
though such properties always have to be mapped to momdic properties of the
represented object, these monadic properties don’t have to be the same. For example, if

the picture of John is drawn in green, the isomorphism does not have to map this property

 

74 a will always be an element of the represented structure, since a is a member of all sets.

68

to an actual instance of the color green, or to any color at all even if it is necessary that
this feature is mapped to some feature of John.

So in summary, it looks as if pictures may be able to contain non-representational
properties without compromising the possibility tint iconical theories of representation are
true. But regardless, the existence of non-representational properties of icons does not
distinguish them fiom sentences, since sentences can also have properties which have
nothing to do with their representational content. For example, the proposition, ‘John is
green’ can be tokened in pencil on a piece of paper, can be spoken, can be stored as
magnetic media on a hard disk, or can be emblazoned on the back of a vintage jacket.
Each of these manifestations of the proposition has properties which in no way represent
aspects of the proposition itself. As a result, it is hard to see why the possible presence of
non-representational properties in pictures would be cause to compare them disfavorably

with sentences.

2.3.2.2 Can Discursive/Symbolic Devices Uniquely Specify A Subject’s
Properties?

Next we need to consider whether sentences do a better job than icons do of
abstracting away from all of a subject’s properties except for those the sentence represents
(Fodor’s assumption (1) from above).

F odor argues that pictures usually portray a number of the properties possessed by

their subjects, even if not all of them, and there is no way to know which of these is the

69

one of which the picture is a representation.75 Since sentences presumably do better at
uniquely specifying the content of a representation, representations are probably more like
sentences than pictures. But do sentences really pick out the things they represent as
uniquely as Fodor suggests? Probably not. Consider the following sentences:

(a) ‘Bob is larger than Tom.’

(b) ‘Mary is swift.’

(0) ‘Cleveland is not close to Sydney.’
In the first case, if we suppose that Bob is both heftier and taller than Tom, then (a) does
not pick out one or the other of these properties uniquely. The sentence could function to
specify that Bob is heftier, taller, or even both heftier and taller, but nothing about its
structure alone will serve to determine which Sentence (b) is ambiguous due to the
ambiguity of the adjective it contains. Assume that Mary is in fact both intelligent and
fleet-of-foot. Perhaps (b) is supposed to represent the fact that Mary is smart, but we have
no way of knowing whether this was the intention, or whether the fact that Mary runs fast
was the target of the expression. Finally, (c) contains two nouns, ‘Cleveland’ and ‘Sydney’
whose references might change depending on how one takes the ambiguous phrase ‘is not
close to.’ For example, if we take the phrase to represent a judgment about distance, then
both nouns would likely name locations, specifically cities. But if we interpret the phrase

as one which describes the status of a personal relationship, then ‘Cleveland’ and ‘Sydney’

might well name people. Therefore, not only does (c) fail to specify uniquely which

 

75 That is, short of asking someone what they intended to represent with the picture, which is not
an option here, since we want an explanation of the representational capacities of pictures which does not
appeal to the intentions of a “designer” or “interpreter.”

70

properties of an object are represented, it does not even uniquely pick out the objects
themselves.

Of course, the sentences I have used here all have one thing in common: They are
ambiguous in one way or another and could be disarnbiguated if more information was
obtained from the speaker. But note that we cannot ask the speaker about his or her
intentions in uttering these statements for the same reason that we could not inquire about
the intended target of a pictorial representation. If pictures or sentences are going to be
the kinds of structures which possess representational content primitively, then they
cannot possess it in virtue of having been interpreted or designed to do so. Whatever the
mechanism by which our mental structures have representational content, it carmot be one
which relies on the more primitive capacities of an outside designer if we are to achieve a
firlly naturalistic account of mental representation.

F odor might agree that these sentences are not, by themselves, capable of revealing,
say, whether it is Bob’s relative tallness or his hefiiness that we represent. However, he
might argue that statements are, in general, capable of such abstraction, whereas pictures,
in general, are not. In other words, while (a) does not achieve the kind of abstraction

F odor wants representations to be capable of achieving, perhaps (a') does:

(a') Bob is larger than Tom because he is taller than Tom.

It is probably true that most sentences, even if perhaps not sufficiently descriptive to

represent a unique property of their subjects, can be made to be descriptive enough for

71

this purpose. But ] want to argue that the same can be said for pictures. In order to
understand why this is so, let’s take a closer look at how both F odor and Dennett
understand the notion of “picture” for picture theory semantics and how this
understanding differs fiom Cummins’ and my own notion of “picture” for interpretational

and structural semantics.

2.3.2.3 Can Iconical Devices Uniquely Specify A Subject’s Properties?

It seems as if both Dennett and Fodor want to require that in order for pictures to be
used as representations, they must share at least one monadic property (e.g., a color, a
shape, a taste, a smell: in general, a look) with what they represent. Neither Dennett nor
F odor explicitly state this to be a criterion of picture theory as far as I know, but the
examples they use, and specifically, the arguments which they launch against the ability of
pictures to represent abstractly, suggest that this criterion is implicit in their views. In
particular, by arguing that pictures cannot specify which of an object’s properties they are
representations of, Fodor seems to ignore some of the non-monadic properties of pictures
which might be useful for exactly this purpose. Take the following case, illustrated in

Figure 2-B.

Figure 2-B: Man
Climbing/Descending
A Mountain

72

It cannot be clear fi'om the depiction in Figure 2-B whether the figure pictured is
climbing the incline or descending (backwards) along it. Presumably the picture represents
one or the other and, by hypothesis, it shares certain monadic properties with the scenario
which it represents. For in either case, there is an incline at (roughly) the angle described
by the picture, and a figure who (very roughly) resembles in shape the figure pictured
here. Despite these shared properties however, there is no way to determine which of two
scenarios (climbing or descending) the picture represents.

Interestingly, interpretational and structural semantics do not require that a picture
share monadic properties with the thing which it represents. Many representations will
share such properties, but pictures and the things they represent are not required to share
them on these views."5 Instead, for structural and interpretational senmntics, pictures are
structures characterized in terms of their relational properties. Whether or not a given
structure represents a given object (read: is a picture of a given object) has to do with both
the formal relationship which it bears to that object and with the relationship which it bears
to a larger network of representations. This ends up allowing pictures a kind of flexibility
not apparent in either Dennett’s or Fodor’s description and criticism of picture theory
semantics. To see how this works in the case of the example illustrated by Figure 2-B let

us imagine a slightly altered version of the figure, such as that shown in Figure 2-C.

 

76 There may be one exception to this: homomorphism implies preservation of reflexivity, and
reflexivity is arguably monadic. But even if this is true, it in no way compromises the possibility that
monadic properties of pictures are mapped to different monadic properties of the represented object,
implying that relational properties are what matter for object individuation.

73

,/
’\/ /’l
.l/’
/

l

l

/’ l, ./
l I!
/ ,t _,,/
I /’
(1’
//

Figure 2-C: Man
Climbing A Mountain

While the picture given in Figure 2-C may have characteristics which fail to “look like”
those present in the actual state of affairs depicted, it is nonetheless a representation of
that state of affairs, just in case the two have a similar structure. In Cummins’ case,
establishing structural similarity amounts to showing tlmt the representation is
isomorphically related to the state of affairs it represents. Although I do not suggest that
the climber represented here has a set of arrows at his back in reality, what I do suggest is
that the arrows restrict the available interpretations of the picture’s meaning in virtue of
their placement on the diagram in relation to the climber and the incline, and that the
relationship between the climber and incline that they thereby express is one that is in fact

mirrored in reality.77 Thankfully, this is all that picture theory semantics needs to require

 

77 Note that the arrows themselves are symbols which have meaning. Probably most of us take
them to indicate the direction of the arrow’s pointed side. How these symbols come to have the meaning
that they do undoubtedly has a similar explanation to that given for the meanings of symbols in a
language. On the other hand, if you see the arrows as more akin to pictures than to descriptors, then it is
arguable that Figure 2-C does no better at specifying a unique interpretation of the man’s activity than
Figure 2-B does. This is because the arrows, qua pictures, do not necessarily represent a direction of
movement uniquely. Although this is a legitimate criticism of the example used here, it does very little
beyond reestablishing the relevance of the primary problem which structural semantics must address,
namely, the uniqueness problem.

74

of representations, and thus it seems that pictures may well be capable of representing
abstractly after all.

Of course, showing that some pictures are capable of representing abstractly does
nothing to avoid the fact that there are pictures with representational content (such as that
given in Figure 2-B) which do not clearly represent only one or the other of two very
different objects or states of affairs. F odor is wrong to think that this problem is restricted
only to pictorial representations, but is correct to point out the problematic nature of
pictures which are not related to a unique object. If all that is required for representation is
that the representation is homomorphically mapped to the thing which it represents, as I
shall ultimately argue, then Figure 2-B represents a climbing figure as well as it represents
a descending one. Taken by itself, there is really no criterion for choosing which of these
two distinct states of affairs the picture ought to denote. But this problem is, by now, a

familiar problem—it is the problem of uniqueness.

2.3.3 The Problem of Uniqueness for Picture Theory

Having attempted to address the allegation that picture theory cannot support
representations which are sufficiently abstract, let us examine Fodor’s criticism more
carefully as it applies to the uniqueness problem for interpretational semantics and picture
theory in general. Very generally, this problem arises as the result of the fact that pictures
can be interpreted in a variety of different ways. As we have seen, the state of affairs
described by the sentence, ‘John is fat’ could be represented pictorially, but the

representing picture would simultaneously represent many other aspects of John, making

75

its content indeterminate. This problem is easy to conceptualize as a kind of uniqueness
problem and in particular, as what I have called the “standard” sort of uniqueness problem.
In Figure 2-D, each C, is a possible interpretation of the pictorial image doing the

representing (R). Eachf, is a homomorphism fi'om R into C,.

 

Figure 2-D: Standard Non-Uniqueness
Via Fodor’s Critique

Although Fodor’s criticism raises the standard uniqueness problem for picture theory,
it is consistent with Cummins’ version of the problem as well. To illustrate how both
problems can arise for pictorial representations, consider Dennett’s rendition of the

dilemma associated with Joseph Jastrow’s infamous duck-rabbit (see Figure 2-E):78

Figure 2-E:
The Duck-Rabbit

 

78 Joseph Jastrow 1900 “Duck-Rabbit Drawing.” in Henry Gleitman, Psychology New York: W.
W. Norton and Co., 1981.

76

What can possibly be the difference between seeing it [the duck-rabbit] first one way and then
the other? The image (on the paper or the retina) does not change but there can be more than
one description of that image. To be aware of it first as a rabbit and then as a duck can be just a
matter of the content of the signals crossing the awareness line, and this in turn could depend
on some weighting effect occurring in the course of afferent analysis. One says at the personal
level ‘First I was aware of it as a rabbit, and then as a duck,’ but if the question is asked ‘What
is the difference between the two experiences?’, one can only answer at this level by repeating
one’s original remark. To get other more enlightening answers to the question one must resort
to the sub-personal level, and here the answer will invoke no images beyond the unchanging
image on the retina.79

The idea is this: Let the duck-rabbit picture be R (see Figure 2-F). Then there are two
content structures, C, and C, such that, under one homomorphic mapping, R has content

C, (the duck) and under the other, R has content C, (the rabbit).

C1: duck (1,: rabbit
6 6
Q
R
Figure 2-F: Standard Non-Uniqueness Via
The Duck-Rabbit

This is the standard sort of uniqueness problem wherein both content structures seem to
serve as equally good candidates for specifying the content of R. On the other hand, if we

think of the mental image which we have when viewing Figure 2-F as R and the picture

 

7’ Dennett (1981), p. 131.

77

itself as the object of our representation, C, then we have something which looks more

like Cummins’ version of the uniqueness problem (see Figure 2-G).

(3679
“~- .

I;

fix!

 

 

R (our mental image)

Figure 2-G: Cummins’ Non-Uniqueness
Via The Duck-Rabbit

Since Fodor’s articulation of the uniqueness problem for picture theory can be construed
as similar to (indeed, perhaps even identical to) the corresponding problems facing
interpretational semantics, a more in-depth look at these problems and the responses
available to advocates of interpretational semantics is in order. In the next chapter, I
characterize non-uniqueness more formally by exploring how it arises in the context of
measurement theory. This more formal characterization of non-uniqueness will put us in a
better position to formulate solutions to the problem, and will also reveal at least one
additional context in which problems with uniqueness can arise. I’ll also consider the
proposal to characterize the representation relation as a homomorphism rather than an
isomorphism more carefirlly. This may reveal additional options for responding to the

uniqueness problem as it has been characterized so far.

78

CHAPTER 3: Representation As Homomorphism
And The Problem of Uniqueness

Until now, I have only briefly alluded to the need for a revision in the way that
Cummins characterizes the representation relation: the isomorphic relation he claims holds
between representations and what they represent. However, from the preceding discussion
and in what follows, it should be clear that the uniqueness problem is not unique to
interpretational semantics, and that where it arises in other contexts, part of the approach
to dealing with the problem involves, ironically, a weakening of the relation between the
structures that are not uniquely mapped. I’ll begin this chapter with a more detailed look
at the nature of the representation relation as Cummins characterizes it. This I will follow
with a detailed proposal for a weakening of the representation relation and will argue in
particular that the relation is a homomorphism rather than an isomorphism. I’ll discuss
some of the advantages that this revision would have over Cummins’ original view and
will attempt to say how the move to a homomorphic representation relation influences our
understanding of how mental representation works, and most importantly, how it can
contnbute to a solution to various manifestations of the uniqueness problem. Ultirmtely,
the theory that emerges to account for mental representation is a variety of picture theory
which is, nonetheless, different enough fiom Cummins’ own view, that I refer to it as

“structural semantics.”

79

3.1 Cummins’ Isomorphic Mapping Relation

Cummins characterizes mental representation as consisting of a mapping between
structured features of a cognitive system (e.g., a brain or computer) and things in the
external world. Specifically, a structure of the cognitive system is a representation just in
case a mapping, and in particular, an isomorphic mapping exists between it and some
structure in the external world. Since the account of representation proposed by Cummins
requires only that isomorphisms of this sort exist and not that we or any other agent are
aware of them, this theory of the nature of representation is a non-question begging one
(that is to say, we do not require that an agent who already possesses representations
identify or construct the isomorphism which is in turn to account for the presence of
representation in cognitive systems originally).

A closer look at the precise nature of the relation Cummins uses to account for the
representational capacities of cognitive systems will reveal why a fiiendly amendment to
his account may be in order.

Cummins does not, to my knowledge, describe the notion of isomorphism with much
precision in any of his works on interpretational semantics, but does say that it should be
understood the same way in the context of mental representation as it is in mathematics.
Following this lead, I will endeavor to describe a view of isomorphic relations for the
reader which is more precise than that offered in the previous chapters so that their role in
the explication of mental representation for Cummins, and for me, can be understood more

clearly. I will characterize isomorphic relations, at least initially, as relations that hold

80

between relational systems, where a relational system80 is “a finite sequence of the form 8
= (A, R ,, ..., R,,), such that A is a nonempty set of elements called the domain of the
system 8, and R ,, ..., R,, are relations on A.”81 Relational systems may be of a certain type.

The following are examples of relational systems which differ in type:

S, = (A ,, R,> where A, is the set ofall cats and R, is the binary relation of
being cuter than, such that for any a, b E A ,, aR,b iff a is
cuter than b.

S, = (A,, R ,) where A, is the set of all people and R, is the quaternary

relation of one pair of siblings being more alike than
another. That is, for any a, b, c, and d E A ,, abR,cd iffa and
b are siblings, c and d are siblings and a and b are more

alike than c and d.82
In general, we may define the type of a relational system as follows: “If s = (m,, ..., m") is
an n-termed sequence of positive integers, then a relational system 8 = (A, R ,, ..., R,,) is of
type s if for each i = 1, ..., n the relation R, is an m,-ary relation.”83 Note that this implies

that 8, in the example above is of type (2) while 8, is of type (4). A relational system 3,

 

8° Tarski, A. Contributions to the Theory of Models, 1, H. Indagationes Mpthematica_e_. 1954, pp.
5 72-588. and P. Suppes and J. L. Zinnes, "Basic Measurement Theory", in R.D.Luce, R.R. Bush, and E.
Galanter (eds), Handbook of Mathematical Psychology, Vol. I, New York: Wiley, 1963. pp. 1-76.

8‘ The concept of a relational system is analogous to the concept of a structure (Enderton (1972))
for a formal language.

82 Of course, the relations that form a relational system are really sets of ordered n-tuples. In the
first case, for example, R , is a set of ordered pairs (of cats). The phrase “is cuter than” is the intuitive
English rendition of this relation. I use the intuitive formulation here because I will ultimately want to
consider the role which relational systems play in representation, and the more intuitive rendition of the
relation will help make that role easier to understand.

83 Suppes, Patrick and J. L. Zinnes, "Basic Measurement Theory.” in R.D.Luce, R.R. Bush, and
E. Galanter, eds. Handbook of Mathematical Psychology. Vol. I, New York: Wiley, 1963. p. 5.

81

= (A 3, R ,. R,) where R, and R, are both binary relations would be of type (2, 2).“ Two
relational systems are similar if they are of the same type.85

We are now in a position to state a definition of isomorphism for relational systems.
Let us start with the particular case of relational systems of type (2). Suppose that 8 and
St are type (2) relational systems such that 8 = (A, R) and at = (B, S). Now, 8 and 9!
are isomorphic if there is a one-to-one fimction f from A onto B such that for every 0 and

binA:

aRb ifffla) = S(/(b)).

In general, let 8 and at be similar relational systems, such that a = (A, R,, R,,) and at

= (B, S,, ..., S"). Then, 53 is isomorphic to fit if there is a one-one function f fiomA onto B

such that, for each i = 1, ..., n and for each sequence (a ,, ..., am.)86 of elements of A,

Ri(al.9 "'9 amt) if? Si Mal), °-°9f(ami))87

The objective, of course, is to use the notion of isomorphism between relational

systems to account for a cognitive system’s ability to represent the external world. As we

 

84 Suppes et al. (1963), p. 5.

85 Note, knowing when two relational systems are of the same type is important because it
fonnalizes part of the notion of “same structure” that is employed in the claim that a cognitive relational
system is a representation of a sn'ucture in the external world in virtue of the fact that the two systems
have the same structure.

86 Please read a," as “a sub or sub 1”.

87 Suppes’ and Zinnes’ definition of isomorphism also assumes the restriction that mappings be
both one-to-one and onto.

82

have seen, Cummins wants to say that a given representation has the content that it does
(and indeed, is a representation) in virtue of the existence of an isomorphic mapping
between it and the thing it presumably represents. Suppose, to take a concrete example,
that we want to explain how some cognitive system rudirnentarily represents its
neighborhood. There is a relational system 8, = (A, R ,, ..., R,,) such that A, the domain of
8,, contains streets88 and (R ,, ..., R,,) is a set of relations between streets (for example,
streets may be related in the following common ways: “is longer than,” “intersects,” “is to
the east of,” “dead-ends into,” etc.). Together the domain of 3, and the relations on that
domain constitute a neighborhood. Now, let St, = (B, S ,, ..., S,,) be a relational system
instantiated by some cognitive system P. P instantiates 91, in the sense that the domain, B
of St, consists of a set of data structures (implemented in P) which are ordered by the set
ofrelations (S,, S,,). Ifit is true that there is an isomorphism between 8, and St, in the
sense described above, then, on the present view, Si, is a representation of the
neighborhood of P (since P’s actual physical neighborhood just is the relational system
531)-

Of note is that the elements of the domain in this example may be complex
entities that, viewed fiom a lower level, are themseleves relational systems—systems
which could, from that lower level, be representations for the cognitive system. For

example, in the present case, streets may be—and probably are—relational systems with

 

88 I consider the representation of the system’s neighborhood to be a relatively un-complex case
(in that it does not involve a lot of subsystems (like streets) of the main relational systems under
discussion which might themselves be thought of as representable), but I do not intend to suggest that it is
the simplest case of representation. What the “simplest” case of representation is might be an interesting
question to pursue in this context.

83

their own elements and relations, capable of being independemly represented by the
system. In general, this example demonstrates that there is an ambiguity, thus far
unexplored, in what constitutes the “structures” being mapped in the representation
relation. In some cases, the representation relation maps structures that consist of what are
intuitively simple elements and relations on those elements. But other times what are
mapped are structures whose elements might normally be seen as complex entities
themselves. Indeed, the elements could even be properties and relations. One way of
understanding the ambiguity described here is to think of there being different levels of
representation; sometimes we are representing things at a higher level of abstraction,
sometimes at a lower level. One place one ofien gets such changes in level is in moving
from one scientific field to another (e. g., ecology often treats species as individual
elements where a lower level in biology would treat them as complex interacting systems.

Therefore, it is important to recognize that we must be able to specify what elements
belong to the domain of a relational system, and what relations are defined over these
elements (i.e., what ordered n-tuples belong to the relational systems in question) before it
is possible to determine whether two (or more) systems meet the conditions required for
them to be isomorphic to one another, or alternatively, what two (or more) “systems” we
are even dealing with in the first place.

As a concrete example, consider the construction of a database table for storing user
information. Suppose that I have a database housed on a mainfi'ame computer which can
only store information using scalar data types. A table containing the user’s unique id, his

first name, last name, and phone number has been created in the system which contains

84

several million records. Though the company I work for has historically been interested
only in the home phone number of each user, it would now like to begin collecting and
discriminating between the user’s home, work and cell phone numbers. Though I would
like to be able to store all three numbers in the “phone” field of the users table (see Figure
3-B), the limitation on the type of data storage the mainframe supports will not allow me
to treat it as an array of values. As a result, I decide to create a new table called
“phone_numbers” which contains the three types of number, plus an integer value
corresponding to the value of the “phone” field in the original “users” table. Where before
this field was a discrete representation of the home phone number of individual users, now
it serves as a pointer to an array of numbers, each related in virtue of belonging to the

same individual.

 

 

 

 

 

 

 

 

 

 

 

users
id
first_name
last_name ,. _ .,
phone a phoneflmbers
home_phone
V work _phone
ce||_phone

 

 

 

Figure 3-A: Mainframe Database Tables

Examples like these, illustrating the value of locating representations at different
“levels” of amlysis, suggest that Cummins’ requirement that features of the cognitive
system be isomorphic to features of the external world is a significant restriction on the
representation relation. This is true because of the requirement that, to be isomorphic, two

relational systems must be related by a function which is one-to-one. Since one-to-one,

85

onto functions must map distinct elements of the domain into distinct elements of the
range, it is difficult to see how isomorphisms could support mappings involving “higher-
order” representations of the sort described here. Specifically, if we take the “phone” field
of a particular users table entry to represent the entire table of phone numbers for that
record, then it seems that we have a mapping which resembles something more like the

following:

, * * .homejhane
phone Awaruhone.
\mlljhone

Figure 3—B: Failure of One-to-One

Although it is possible to think of the set {home _phone, work _phone, cell _phone} as a
single object: phone_numbers, thereby preserving a one-to-one, onto mapping between
phone and phone_numbers, this understanding of the napping relations obscures the
complex structure of the phone_numbers construct, and is not, in my view, the most
natural way of viewing the relation between phone and the phone_numbers table. IfI am
correct, then we need a construal of the representation relation which does not require
that the relation is a one-to-one, onto mapping between structures. This condition is met
by the notion of homomorphism.

In addition to allowing only one-to-one mappings between relational structures,
isomorphism, in virtue of the onto requirement, guarantees that the sets of elements
composing each relational system are equivalent in cardinality. Despite these requirements,

it seems obvious that the relational system which constitutes the cognitive system’s

86

representation and the relational system which exists in the external world may not always
have the same number of elements in their domains. Speaking intuitively, if the cognitive
system’s representation is one that does not pick out each and every one of the features of
the thing which it represents, then no isomorphic mapping between the two structures is
possible.89 Likewise, if the cognitive system’s mental structure has features not mirrored in
the object it represents or possesses more than one mental structure that can be mapped to
this object, then no isomorphism obtains between them in this case either. It seems likely
that most relationships between mental and external relational systems will fall under one
of these two cases, leading one to conclude that a better description of the representation
relation is that it is a homomorphic relation between relational systems. We understand
homomorphic relations in exactly the same way that we understand the relation of
isomorphism, with the exception that for homomorphic relations we drop the requirements
that f be one-to-one and onto. Notice that even cases wherein the mental and external
relational systems do possess domains of equal cardinality will meet the conditions
necessary for representation, since, under this formulation, it is implied that every
isomorphism is a homomorphism, but not the converse.

It makes sense to ask therefore, whether we might be able to be satisfied with the
weaker relation of homomorphism in accounting for the nature of mental representation
given that it does not require that the fimction mapping the two relational systems be one-

to-one and onto. In the next section, I’ll endeavor to show how the account of

 

89 I use the term “features” here to denote elements of both the representational relational system
and the external relational system.

87

representation is affected by the move to a homomorphic relation, what advantages the
move may present, and what difficulties, if any, it may raise.
3.2 Understanding the Representation Relation as a Homomorphism: Structural
Semantics

3.2.1 How Homomorphism Accounts for Paradigm Cases of Representation

The way in which the notion of homomorphism replaces Cummins’ use of
isomorphism is really quite simple. Instead of saying that the representation relation
consists in an isomorphic mapping f between the relational system of the cognizer and the
relational system which constitutes representable features of the external world, we now
state that only a homomorphic mapping is required. This amounts, in part, to allowing that
mappings occur between individuals in one relational system and the subsystems of other
relational systems. We define a subsystem of a relational system 8 as “a relational system
obtained from 8 by taking a domain that is a subset of the domain of 8 and restricting all
relations of 8 to this subset.”90

This may be a more significant alteration of the relation f than it initially seems to be.
Part of the advantage of requiring an isomorphic relation between the network of
representations in the cognitive system and representable things in the external world may
have been that no one needed to decide which features of the cognitive system (the range)

should belong to the relationship. It is as simple as this: What features exist in the

cognitive system are those that belong to the range of the function mapping the two

 

9° Suppes et al. (1963), p. 10.

88

systems together.91 Setting it up this way seems to help one to avoid the problem of
requiring an agent (who would have to already possess representational capacities) to
identify which features are most important to the representation relation (and hence, which
features belong in the range)——a problem because it would make the account of
representational content dependent on a more primitive representational capacity. Since an
isomorphic relation has it that all elements belonging to each of the relatiorml structures
get mapped to one another, no choices must be made regarding which elements in the
range ought to be mapped into and therefore, no agent capable of such choice is
required.92 A homomorphic mapping, on the other hand, will be constructed in such a way
that only some elements of the cognitive relational system get mapped into by the system
realized in the external world. Hence, it may seem that a decision about which elements
these are will be required.

This is not an unfamiliar difficulty: A version of it was raised in connection with
Fodor’s criticism of picture theory semantics, when he argued that pictures, uner
descriptions, do not abstract away from any features of the things they represent.
Typically, a picture immediately depicts more about the represented object than our
thought about it does, and may say nothing about which of the features it depicts are the

ones we care about. Since mental representations are frequently more targeted than this,

 

91 Or alternatively, what features of a represented object exist are those that belong to the range
of the relational system which is mapped into by the cognitive system. Above I am speaking at the level of
the entire network of represented objects rather than at the level of individual represented things.

92 It may be necessary, however, to determine which elements there are. This problem is closely
related to the problem of specifying the “level” of analysis for a particular relational system.

89

that is, since they do pick out the one thing we are focused on in any given situation,
Fodor thinks that pictures are not the kinds of things that can be mental representations.

It may seem that Curnmins’ isomorphism fares better in this regard, but this
assmnption is mistaken. While it is true that characterization of the representation relation
as homomorphic seems to leave one without a clear criterion for identifying which of the
features of represented objects ought to belong in the range of the mapping in any given
case, characterization of it as isomorphic eliminates the need for this decision at a fairly
heavy price.93 For if all properties belonging to an object must be mirrored in the
representing structure, then the we reopen ourselves to Dennett’s arguments against
picture theory. Recall that Dennett, too, was concerned with the ability of pictures to
support abstract representation. His view was that they would inevitably portray all of a
subject’s features, making abstraction from one or more of those features impossible.
Dennett’s view makes sense if the representing structure must in fact be isomorphic to the
structure it represents. But without this requirement, 1 have argued that it is difficult to see
why pictures cannot do as good or even better a job at portraying only some of a subject’s
characteristics as descriptions can.

It seems, therefore, that at least one advantage of viewing the representation relation
as homomorphic is that it enables pictures, understood now as relational systems, to pick
out only some of the properties of external objects. What remains to be explained is how

one ought to determine which elements of each relational system are the ones belonging to

 

93 Actually, conceiving of the relation as isomorphic doesn’t eliminate this need at all, although
it may reduce it somewhat. See below for a discussion of why this is so.

90

the homomorphic relation that constitutes our representation. But this is, as the reader has
probably already noticed, simply another way of articulating the uniqueness problem for
picture theory semantics. Which elements of the cognitive structure are mapped into
elements in the external structure depends on how the elements of each relational system
are defined and which relations on the elements in these systems are preserved. In other
words, figuring out which elements are the ones relevant to a given representation
amounts to deciding which homomorphic function, out of many homomorphic functions
between the two relational systems, constitutes the representation.

Here, in dealing with Fodor’s critique, is where it may appear that Cummins’
isomorphic relation performs better, but this too is a mistaken view. Even though
conceiving of the representation relation as isomorphic does eliminate mappings between
relational systems which are not of equal cardinality, and hence the need for choosing
among fimctions which map the domain of the representational relational system into
different elements of the cognitive relational system, it does not prevent the possibility of
obtaining several isomorphic mappings between two relational systems of equal
cardinality. In other words, it does not prohibit versions of non-uniqueness such as those
seen in connection with Cummins’ Autobot, wherein it was possible to specify one
fimction mapping each of the turns in the card to each of the turns in the maze, and
another mapping each and every one of the turns in the card to the inverse of each and
every turn in the maze.

Since both homomorphism and isomorphism give rise to versions of the uniqueness

problem, but only homomorphism is consistent with the idea that representations do not

91

consistently pick out all of the elements of those objects which they represent, it seems
that, in this respect at least, understanding the representation relation as homomorphic is
the preferable choice.

3.2.2 How Homomorphism Accounts for Representation Under Degraded

Conditions

As we have seen, there is an ambiguity in this account of the representation relation
about whether the relational system instantiated by a cognizer and mapped to the external
world is an individual representation or an entire network of representations. This
ambiguity arises in both Cummins’ account and in mine. Although I believe that for a
picture theory of representation to overcome the uniqueness problem, the focus must be
on a mapping between a wider cognitive network and a wider network of things it
represents, part of this picture definitely involves the presence of individual mappings
between nodes of the cognitive network and individual objects in the external world. I
propose that these individual mappings also be thought of as homomorphic. Doing so will
allow us to account for the ability of a cognitive system to represent objects successfully
with only partial information about their features (something we believe cognitive systems
have the ability to do).

One can graphically represent a mapping between a cognitive system and an external

world structure as shown in Figure 3-C.

92

 

 

 

 

 

U
I 1} r L In If: 4
ga—
g/A
' 8::
u a g” E

Figure 3-C: A Representational Network

Let each g, be an element of the cognitive system that, once mapped to some element
G, of the external world, constitutes an individual representation for that system. Let each
I mapping g, into C, be a homomorphism. I is also a homomorphism, but one which maps
the entire network of gs (u) to the entire network of Gs (U). In this model, u and U are
also relational systems.

In general, each g, may be thought of as a relational system itself. Suppose in
particular, that g, is a representation I have of my cat Sephra in virtue of a homomorphic
mapping between it and Sephra herself (0,). By hypothesis, the relational system g, maps
into the relational system 0,. In other words, not every feature Sephra has (i.e., every
element in the range of I") will be captured by my representation of her. For example, if
Sephra were to have an above average size liver organ, my representation of her would

not likely map into that feature. Our intuition is that, despite such shortcomings in the

93

completeness of our representational picture with respect to many of the things which
occupy the external world, we can nonetheless aptly describe ourselves as possessing a
legitimate representation of such things. Moreover, by characterizing the representation
relation as homomorphic, we allow for the possibility that we can acquire additional
information about external objects which was not previously available to us, with the
result that representations which were “incomplete” depictions of some particular object at
one time, can become more “complete” at a later time. If an ultrasound of Sephra’s liver
was shown to me, for example, then additional elements of g, would become part of the
range of the function mapping g, into 6,.“ In this way, we can see how conceiving of the
representation relation homomorphically allows for the involvement of mental
representations in learning and knowledge acquisition in general.

Of course, it will always be possible to specify an isomorphic mapping between a
subset of the elements in the domain of the cognitive relational system to a proper subset
of the elements composing the domain of the real-world external relational system. Given
that this is the case, it may seem that there is not a lot of motivation to conceive of the
representation relation as homomorphic rather than isomorphic. After all, in any given
case, what we do represent “completely,” we represent isomorphically. But requiring the
representation relation to be homomorphic makes sense if we think of what is represented
as being the real world “out there” in all of its complexity, assuming that this complexity is

usually not represented by cognizers to the fiillest extent (and perhaps never is). Relative

 

9" Or, equivalently, we would simply specify a new function whose domain and range now
included the relevant elements.

94

to this assumption, it makes sense to imagine that our representations always fall short, in
the direction of the weaker relation of homomorphism as opposed to full isomorphism.
Indeed, it may make sense to assume that even our representations of complex

substructures of the external world typically fall short in this way.

3.2.3 How Homomorphism Accounts for Substitutivity of Identity Failure
Recall that in my introduction to the problem of mental representation, I stated that
any good theory of representation ought to provide a plausible explanation for a peculiar
characteristic of mental representations, namely, that when representations refer to the
same external object they cannot necessarily be substituted for one another without
altering the content of the thought in which they play a part. There, we used the following
example:
(s) I want to be a Jedi like my father.
(s) I want to be a Jedi like Darth Vader.
where s is a thought possessed by the character of Luke Skywalker in the film Star Wars
and s’ is the result of substituting ‘Darth Vader’ for ‘my father’ where it appears in s. In
the context of the film, both ‘my father’ and ‘Darth Vader’ are descriptions of two
representations which can be mapped to the same external relational structure, namely, the
Jedi knight, Anakin Skywalker.
Since interpretational semntics has it that a cognitive relational structure represents
an external relational structure just in case there is an isomorphic relationship between the

two structures, it is certainly arguable that so long as Luke Skywalker has a representation

95

of his father, he also has a representation of Darth Vader, and visa versa. That is, since
Luke’s father and Darth Vader are the same person (read: are one and the same external
relational structure), then so long as Luke’s possesses a representation of his father, he
must also possess a representation of Darth Vader. Indeed, given that representations
must be isomorphic to the external relational structures which provide them with the
content they have, we may say that Luke representation of his father and his
representation of Darth Vader are one and the same.”

This is a problematic result for interpretational and structural semantics because it
seems to leave no way of distinguishing between thoughts such as those expressed in s and
s’ above. But clearly there is a difi‘erence of content between these thoughts, for one (s)
accurately expresses Luke’s desire, while the other (s ’) does not.

If the representation relation is homomorphic, we can argue that failure of
substitutivity of identity arises when a subject’s cognitive relational system 3t contains
two different representational substructures R, and R2 which are both homomorphic to a
single external substructure C. R, and R2 are distinct substructures bearing distinct
relationships to the other substructures in 9! despite the fact that both are
homomorphically related to the same external relational structure. For example, in Figure

3-D below, though both R, and R2 are semantically related to Luke’s representation of a

 

95 I base this claim on the assumption that there would be nothing structurally distinctive about
the individual representations, and that differences in their monadic properties are not relevant to
distinguishing them. However, it may be that if ‘my father’ and ‘Darth Vader’ are treated as substructures
of a larger representational scheme, there would be ways of distinguishing between them. Specifically,
they might be distinguished in virtue of the diflerent relationships they bear to other substructures of the
cognitive relational system. I’ll address this possibility presently.

96

Jedi knight, only one is related to his representation of Obi Wan Kenobi’s disciple. In the

external world, Anakin Skywalker is defined by all of these relations.96

strong in the Dark Side of the Force

  
   
 
  
  

brother of Owen Lars K Jedi knight

evil Imperial lord
disciple of Obi Wan Kenobi

good with a light saber

father of Princess Leah
f. I.

Rica R255!

brother of Owen Lars o\ ,’ Jedi knight strong in the Dark Side of the Force

 
    
   
 

1, Jedi knight
disciple of Obi Wan Kenobi

evil Imperial lord
f1 and f; are homomorphisms from

R, and R2 into C respectively 9006 with a light saber

father of Princess Leah

Figure 3-D: Substitutivity of Identity Failure

 

96 One response available to both interpretational and structural semantics is to argue that the
contents of representations is not best specified by considering how individual representations are mapped
to the external world, but rather, by understanding the contents of representations holistically. In
particular, one could argue that the failure of substitutivity of identity arises when a subject’s cognitive
relational system 91 contains two different representational substructures R, and R2 which are both
isomorphic (and therefore homomorphic) to a single external substructure C. Ordinarily, we would not be
able to say that R, and R2 are “different” substructures under these circumstances, and relative to the
external structure to which they are mapped, they are not. But relative to the entire cognitive relational
system 91 of which they are a part, they are distinct substructures. They are different in virtue of being
related to other substructures of 3! in distinct ways.

97

The reason, therefore, that “Darth Vader” (R2) and “my father” (R,) can't be treated as
identicals in the context of Luke Skywalker's representational scheme, is that they are not
identicals. They nonetheless have something important in common, which is that both are
homomorphically related to the same external “real-world” structure.97

Characterizing the representation relation as a homomorphism brings with it a number
of advantages, such as providing part of the explanation for how representations can be
abstract, accounting for the failure of substitutivity of identity in mental contexts, and
allowing for the possibility that cognitive systems can represent objects more or less
completely, opening the door for a kind of representation-driven explamtion of learning
about the external world. There are other advantages to “structural semantics,” my name
for the view that representation is a homomorphic relation between relational systems. In
particular, we shall ultimately see how structural semantics can address issues of non-
uniqueness in representation. Before considering possible ways of responding to the
problem of uniqueness however, there is still one more version of the problem to consider.
This manifestation of the uniqueness problem clearly arises in the context of measurement
theory and is described by Suppes and Zirmes in their work entitled “Basic Measurement

”98

Theory.

 

97 Notice that this explanation accounts for both the “de-re” and “de-dicto” readings of s ’. There
is a sense in which Luke does want to be a Jedi knight like Darth Vader, insofar as he wants to be a Jedi
knight like his father, and Darth Vader is his father. This is the “transparent” or de re reading of s ’. The
de re reading of s’ is justified by the fact that an actual relationship obtains between the elements of R,
and those of R2: namely, both sets of elements are mapped to elements of the some external substructure.
The de dicta or “opaque” reading of s ’, is justified by the observation that nonetheless, R, and R2 are not
the same structure. The fact they stand in different relations to the other substructures composing 9! is
enough to establish the non-necessity of treating the two as identicals within the context of 31.

’3 Suppes et al. (1963), pp. l-76.

98

3.3 A Final Form of Non-Uniqueness

3.3.1 Suppes’ and Zinnes’ Project

In “Basic Measurement Theory” Suppes and Zinnes propose to describe a
“fundamental” theory of basic measurement. By offering a theory of fundamental
measurement, Suppes and Zinnes mean to provide a mathematical interpretation of many
of the concepts common to measurement without appealing to alternative concepts
already in place in the theory. This is the sense in which they account for “fundamental”
(as opposed to “derived”) concepts of measurement. Suppes and Zinnes identify two basic
problems in the theory of measurement which parallel some of the issues I have discussed
above in the context of mental representation. These are “The Representation Problem”
and “The Uniqueness Problem.” Although there are some divergences between what
Suppes and Zinnes attempt to explain in the field of measurement theory and what I have
discussed in the context of a theory of mental representation, there are also some very

striking parallels which make their view worth examining.

3.3.2 The Problem of Representation in Measurement Theory

Suppes and Zinnes write: “The first problem for a theory of measurement is to show
how various features of [an] arithmetic of numbers may be applied in a variety of empirical
situations. This is done by showing that certain aspects of the arithmetic of numbers have
the same structure as the empirical situation investigated?” This statement captures, if

only in rudimentary form, the problem referred to as the problem of representation for

 

99 Suppes et al. (1963), p. 4.

99

measurement theory. More precisely, the problem of representation is the problem of,
“characteriz[ing] the formal properties of the empirical operations and relations used in the
[measurement] procedure and show[ing] that they are isomorphic to appropriately chosen
numerical operations and relations.”100 Suppes and Zinnes point out that this problem is
equivalent to proving what it called a “numerical representation theorem.”101
The authors employ the notion of a relational system in their efforts to provide a
numerical representation theorem for certain procedures of measurement. It is their notion
of isomorphism (and homomorphism) between relational systems that I employed
above.102 But before we can understand the kind of numerical representation theorem the
authors provide, we must understand the specific kinds of relational systems they believe
are most relevant to a theory of measurement. Our intuitive notion of measurement is that
it is a process through which sets of numbers (and relations on those numbers) or some
other appropriate formal system (such as a formal language, or a natural language) come
to represent items in the external world and the relationships between those items. For
example, we measure the height of a person by associating a number with the physical

distance corresponding to the spatial interval between that person’s vertical extremities.

We measure the ratio of one temperature difference to another by assigning numbers to

 

'00 Suppes et al. (1963), p. 4.
‘01 Suppes et al. (1963), p. 4.

'02 Although the following account of Suppes’ and Zinnes’ solution to the problem of
representation in a theory of measurement is articulated in terms of an isomorphic relation between
relational systems, they readily acknowledge the usefulness of considering the relation a homomorphic one
when the function mapping one system onto another is not one-to-one (see Suppes et.al. (1963), pp. 7 &
14-16).

100

each value in the two pairs of values, determining what the relationship is between the two
numbers associated with the first pair, and comparing that relationship to the one that
holds between the two numbers associated with the second pair. Suppes and Zinnes seek
to formalize our intuitive notion of measurement, in part, by conceiving of the items in the
external world and the relations between them as a relational system and in particular, as
what they call an empirical relational system. In short, an empirical relational system is
“... a relational system whose domain is a set of identifiable entities, such as weights,
persons, attitude statements, or sounds,” while the formal system which is used to
represent the empirical relational system is a numerical relational system. "’3 A numerical
relational system is defined by Suppes and Zinnes as “... a relatiorml system (A, R ,, ..., R,,)
whose domain A is a set of real numbers.”‘°"
Keeping in mind the concepts of numerical and empirical relational systems, the
problem of representation can be stated in still another, and final way. Suppes and Zinnes
write: “The first fimdamental problem of measurement may be cast as the problem of

showing that any empirical relational system that purports to measure a given property

of the elements in the domain of the system is isomorphic (or possibly homomorphic) to

 

’03 Suppes et al. (1963), p. 7.

10" Suppes et al. (1963), p. 7. Suppes and Zinnes note that although this definition of “numerical
relational system” places no restrictions on the relations which help to compose the numerical system, in
practice the relations will be limited to relations which obtain between numbers (Suppes eta]. (1963), p.
7). For example, you would not expect to encounter a relation like “is the sibling of’ in a numerical
relational system, despite the fact that the definition does not officially disallow it. I believe that the
analogue to a numerical relational system in representation theory will not need this practical restriction:
In Chapter 5, I’ll consider the reasons why in more detail.

10]

an appropriately chosen numerical relational system.”"’5 The problem of representation
for a theory of measurement is, therefore, quite similar to the problem of representation
for a theory of cognition. In the former case, it is in virtue of an isomorphism (or possibly
a homomorphism) between a numerical relational system and an empirical relational
system that the measurement has meaning; that is to say, the members of the numerical
system’s domain and relations between them represent what they do. In the case of mental
representations, it is in virtue of the existence of an isomorphism (or perhaps, a
homomorphism) f between the cognitive relational system and empirical relational systems
that the elements in the domain of the cognitive relational system (representations) as well
as the relations between themw" have the content they have.

Demonstrating the existence of the sort of isomorphism (or homomorphism)
described above amounts to proving a numerical representation theorem for the
measurement procedure. Some comments are in order about the significance of providing

a numerical representation theorem however. The authors argue that,

The representation problem is not solved if the isomorphism is established between a given
empirical system and a numerical system employing unnatural or ‘pathological’ relations. In
fact, if the empirical system is finite or denumerable (i.e., has a finite or denumerable domain),
some numerical system can always be found that is isomorphic to it. It is of no great
consequence therefore merely to exhibit some numerical system that is isomorphic to an
empirical system. It is of value, however, to exhibit a numerical system that is not only
isomorphic to an empirical system but employs certain simple and familiar relations as well. A
complete or precise categorization of the intuitively desirable relations is unfortunately
somewhat elusive, so for this reason, the statement of the representation problem refers to an

 

"’5 Suppes et al.(l963), p. 7.

106 These relations could be statements in Mentalese (Fodor) or perhaps simply relations between
subsets of representations which compose the domain of the cognitive system—relations which are
mirrored in the actual world.

102

“appropriately chosen” numerical system.‘07
Although much needs to be said regarding the notion of an “appropriately chosen”
numerical system, especially if Suppes and Zinnes are to avoid a question-begging account
of representation, the main import of this comment is that it raises the issue of whether
there are unique isomorphic mappings between numerical and empirical relational

systems. "’3

As in the context of mental representation, since the existence of an
isomorphic (or homomorphic) mapping between numerical system and empirical system is
what provides meaning to the process of assigning a measurement to empirical quantities,
it is important that some way of dealing with non-unique mappings be identified. In what
follows, I will characterize the formal problem which Suppes and Zirmes refer to as the

“problem of uniqueness” more carefully.

3.3.3 The Problem of Uniqueness in Measurement Theory
I nformally, the uniqueness problem arises in the theory of measurement when it is

realized that there can be more than one numerical relational system mapped into by a

 

‘07 Suppes et.al.(l963), p. 8.

'08 My guess is that, for Suppes and Zinnes, an example of a numerical system employing

“unnatural or pathological relations” would be a system using the relation K, defined as {(2, 17), (26, 5)}.
We would then have that K(2, 17) is true and that K(26, 5) is true and that K is false for all other pairs of
numbers (n, n,). This is “pathological” because it has no intuitive or heuristic mathematical value. It’s
pathological in the same way that the set consisting of {the empty box of apple juice on my table, Times
Square, Bill Clinton} is—there is just no clear purpose to its specification. Contrast this with the
“healthy” relation corresponding to “less than”: {(1, 2), (2, 3), (3, 4), (4, 5), (I, 5) } or the healthy set
of square numbers: {1, 4, 9, 16, 25, ...}. Of course, it is not enough to simply give examples of such
relations—it seems as though we need a more formal way of guaranteeing that such numerical systems are
somehow ruled out as candidates for mappings with empirical relational systems. One consequence of
failing to establish the grounds for their elimination (in the context of a theory of mental representation)
would be acceptance of the claim that absurd statements (or more generally, absurd cognitive states of
affairs) such as these have representational content.

103

single empirical relational system. This is perhaps the most significant dissimilarity
between the problems Suppes and Zinnes consider in the context of measurement theory
and the problems I consider in connection with interpretational and structural semantics.
The standard uniqueness problem for interpretational semantics is the problem of the
existence of more than one empirical relational system for a single cognitive relational
system. And in Cummins’ version the problem is that more than one mapping relation
exists between a single empirical relational system and a single cognitive relational system.
Suppes’ and Zinnes’ uniqueness problem appears to be yet another variant on the basic
difliculty of non-uniqueness: In fact, if we take the analog of the cognitive representing
system to be the numerical system, it is an inversion of the standard version of the problem

(see Figure 3-E).

 

R5

Figure 3-E: Non-Uniqueness in
Measurement Theory

In the case of mental representation, non-unique mappings are a problem because the
elements of the cognitive relational system get the representational content they have in

virtue of a mapping between the cognitive system and an empirical system. If there is more

104

than one empirical system to which the cognitive system can be mapped, the
representational content of the elements of the cognitive system is at best ambiguous, and
at worst, unspecifiable (because the existence of non-unique mappings might render the
whole account of representation untenable). In the case of fundamental measurement
theory, elements in the numerical relational system (and statements constructable in that
system) will represent the same things (in the empirical relational system to which they are
mapped) as elements and statements in competing numerical relatioml systems. The two
situations are different in that in the former case, we find single elements in the cognitive
system representing diflerent things in the external world whereas in the latter case we
find multiple elements from competing numerical systems representing the same things in
the external world.

There are a number of additional contexts in which Suppes’ and Zirmes’ version of
non-uniqueness can be identified. In semi-recent years for example, there was a move fi'om
analog representation of sound (via the old-style LP and/or cassette tape) to digital sound
representation via the modern compact disc recording. Analog systems use a continuum of
values to represent any real-world structure. Armlo g recordings of sound are theoretically
quite accurate, since there is no principled limitation on how finely grained the
representation of the sound being recorded can be. However, how much of the auditory
information can be captured in an analog recording depends heavily on the media used to
capture and store the information. In contrast, digital representations of sound can be
much better because they assign pre-set values to physical properties in such a way that

perturbations in these properties rarely cause error. Both models can be used to represent

105

one and the same set of sounds however. The first record I ever owned was Bruce
Springstein’s “Born in the USA.” These days I listen to Bruce on a CD. But there was
but one studio session in which Springstein recorded the sounds I hear on both the analog
and the digital medium.

Computer scientists interested in the problem of representing knowledge in artificial
systems encounter a Suppes-and-Zinnes form of non-uniqueness as well. For them, a
single body of data can be represented using at least four very common methods. The first
is via the standard relational database, wherein a set of relationships between elements in a
relational system is symbolized by placing these elements in one or more structured tables

(see Table 3-A).

 

 

 

 

 

 

 

 

 

 

users_id First Name Last Name Address State/ Occupation
Country

3456 Jerry Fodor 56 Cambridge MA . philosopher

9679 Socrates Smith 76 Athens Pkwy Greece philosopher

 

Table 3-A: Relational Database Table

An alternative method often used is one which employs the notion of inherited knowledge.
This method can capture more about the data it represents and the relationships between
the chunks of information fi'om which the data is formed. Specifically, inheritance models

or property inheritance allows individual elements or classes of elements to inherit

106

attributes and values from more general classes in which they are included.109 The notion
of a semantic network as it is understood in cognitive science is a good example of an
inheritance model for knowledge representation.

Two additional forms of knowledge representation, inferential knowledge and
procedural knowledge are also commonly used for the purpose of knowledge
representation by computer scientists. Inferential knowledge is very similar to inheritance
models but implements the hill set of axioms and inference rules for first-order predicate
logic in order to describe the relationships between elements in the system it represents.
As a result, inferential knowledge systems often capture more of a system’s structure than
do either of the former kinds of representing systems. Procedural knowledge systems are
probably the most complex of all of these methods in that they too capture a system’s
structure through the use of inference rules, but can, in addition to this, perform certain
actions based on the information they possess. Each and every one of these methods can
be used to describe one and the same knowledge base and therefore collectively constitute
a good example of the sort of non-uniqueness Suppes and Zinnes identify in the context of
a theory of measurement.

In choosing which of these models is the best for representing data in computer
science, or which method is preferable for audio recording, a number of factors such as
efficiency, purpose, ease of implementation, and space play a decisive role. Non-unique

ways of representing one and the same external relational structure do not typically give

 

'09 Rich, Elaine & Kevin Knight. Artificial Intelligence. 2"d ed. New York: McGraw Hill, lnc.,
l99l.p.110.

107

rise to serious philosophical concern. However, as illustrated by the examples above, the
choice of which representational scheme to use is often informed by practical
considerations, and does not necessarily present any deep philosophical problems about
meaning. As a result, I will focus primarily on the two sorts of non-uniqueness which I
have described as the “standar ” and “Cummins” versions of non-uniqueness and in
general, will only attempt to defend structural semantics against versions of the problem
which do present deep problems about meaning and representational content. In the next
chapter, I will consider how the notion of translational indeterminacy, as explicated by
W.V.O. Quine, presents one such threat to the structural semanticist’s account of
representational content. This examination of non-uniqueness, manifested as a form of
translational indeterminacy, will begin to reveal some ways in which to respond to the
various forms of the problem we have seen thus far, and will consider some of the

potential difficulties that these responses may inspire.

108

CHAPTER 4: Withstanding Quine ’s Criticisms of Picture Theory

A proposal for a picture theory of meaning cannot be complete without some
consideration of William Van Orman Quine’s devastating critique of traditional theories of
meaning. In some sense, interpretational and structural semantics are unique and
unprecedented approaches to the problem of accounting for meaning and mental
representation. However, as has already been demonstrated, both theories bear striking
similarities to the picture theory of meaning, and these similarities make interpretational
and structural semantics likely objects of criticism—in particular, the relationship of these
theories to the picture theory inspires charges not unlike those raised by Quine against

what he calls the “museum” view of meaningfulness.

4.1 The “Museum View” of Meaning

In “Ontological Relativity,” as in most of his works, Quine is interested primarily in
the meanings of terms in a language. Throughout this chapter and for reasons that I will
make clearer presently, I will view Quine’s comments concerning the meaningfulness of
terms as directly relevant to considerations of content for mental representations, and
hence Quine’s assertions regarding how terms acquire their meanings as applicable to

questions about how representations acquire content.

109

Quine describes the “museum” view:

[It] is the myth of a museum in which the exhibits are meanings and the words are labels. To
switch languages is to change the labels.”°

According to the museum view, there are three kinds of things that are relevant to an
explanation of meanings and our knowledge of meanings.111 These are (1) the things in
the world to which terms with meaning refer (“referents”), (2) the meanings themselves,
and (3) the behavior through which we communicate the meanings of terms to others
(linguistic behavior might be the most common example of this).112 Quine’s primary
dispute with the museum theory concerns whether or not there ought to be a distinction
between (2) and (3). Although museum theory does distinguish these, Quine does not. For
the museum theory, meanings are in the head—they are mental states which our behavior
(e. g., linguistic behavior) describes, conveys, or reproduces in others.113 Meanings are
“private” in the sense that they exist in the minds of individual persons as mental entities.

Words have meaning in virtue of being associated with these private mental states—i.e., in

 

llO

Quine, William Van Orman. Ontological Relativity and Other Essays. New York: Columbia
University Press, 1969. p. 27.

111 The “museum” view as Quine characterizes it is really applicable to a whole class of theories
about how language acquires meaning. It shares many features with the class of picture theories of
representation, although it is a somewhat narrower class as it is characterized by Quine. Quine refers to
this class of theories with the term “museum” both because of the analogy with labels on exhibits and, I
believe, in order to convey his idea that this view is outdated.

”2 Although this description of the museum view makes it a view about how words get their
meaning, the analogue of this three-part description in the general case would be (1) the things in the
world to which representations refer (the referents of representations), (2) the content of representations,
and (3) the behavior through which we communicate the content of our representations to others
(linguistic behavior might also be the most common example of this).

“3 Commonly, the museum view is associated with the idea that meanings are mental states in
the mind of the speaker. Although Quine’s discussion of the museum view is mostly consistent with that
trend, he does acknowledge that the meanings of terms can be, in this view, “...Platonic ideas or even
denoted concrete objects” (Quine (1969), p. 27).

110

virtue of labeling them—and are usefiil only insofar as they manage to reproduce in others
the same mental states (i.e., the same meanings) as those I possessed when writing or
speaking them. In this way, words are ‘public” whereas meanings are not. In addition, the
museum view is an “atomistic” theory of meaning, in that it implies that terms (and
representations generally) possess meaning independently of the linguistic and/or
conceptual scheme into which they fit. In order to establish the meaning of the term,
‘dog,’ I need only establish that (1) it is the term I utilize to label a specific mental state;
namely, a mental state which constitutes my concept of dog, and (2) it produces in others
the same mental state it does in me when so utilized. I do not need to take account of how
the mental state the term labels is related to other mental states I may possess. In this
sense, terms are utilized atomistically—they are simply labels—to be associated directly
with a mental state or set of mental states without regard for how these states are related

to others (see Figure 4-A).

term labels mental state
mental state ' refers to actual object

     

(dog,

Figure 4-A: Terms As Labels for Mental States

Finally since, on the museum view, terms are simply labels for mental states which

refer to concrete objects, it is a consequence of this view that meanings can be gotten right

111

or wrong—that there is a fact of the matter concerning any given term’s meaning. To be
correct in one’s understanding of a term, one must imagine that the term applies to some
mental state. Whether one is correct in his/her application of the term to a mental state
must be judged relative to the convention governing the use of the terms in one’s own
native language (this is part (2) above of the two part “museum view” of meaning).
However, whether or not a mental state means what we say it means is a matter of
whether there is a relationship between that mental state and the object it presumably
means (part (1) of the view described above). Whether such a relationship exists is, in the
museum view, distinguishable fi'om the behavioral conventions of a culture. It is the
second part of the relationship, depicted in Figure 4-A above, that must be “gotten right,”
in this view, in order to translate terms accurately.

In contrast to both structural semantics and the museum view of meaning, Quine
holds that terms do not admit of fixed, determinate meanings.“4 Although it may make
sense to say that one has a meaning wrong (say, if everyone who responds to his usage of
the term behaves in strange and unanticipated ways), a term’s meaning is never, can never
be determined solely by its relationship to a concept or referent. Rather, meanings are bad
by a subject in virtue of that subject’s “dispositions to overt behavior.””’ I understand, I
possess the meaning of ‘chair’ if I am disposed to behave in such and such a way (for

example, if I am disposed to sit rather than stand on the object in question when it is

 

”4 At least, not in any absolute sense. Meanings may be fixed, for Quine, only relative to a

background theory.

”5 Quine (1969), p. 27.

112

placed at a table, if I am disposed to answer “Yes” when someone points at such an object
and inquires, “Chair?”, etc.). Although I may be in a certain mental state when (and maybe
even only when) I see a chair, knowing the meaning of ‘chair’ does not amount to
possessing a mental representation, or to being in any kind of mental state for that matter.
Rather, knowing the meaning of ‘chair’ is to be disposed to certain kinds of behaviors.
And there is nothing more to the meaning of ‘chair’ than knowing its meaning. For Quine,

there are no “private” meanings.

4.2 Quine’s Translational Indeterminacy

4.2.1 Meaning and Translational Indeterminacy

Central to understanding the motivation for Quine’s rejection of the museum theory
of meaning in general, and the notion that a term’s referent bears a relationship to its
meaning in particular, is the concept of translational indeterminacy. Quine has argued that
possessing the meaning of a word amounts to getting acquainted with the behaviors of
others who use the word and mimicking those behaviors in such a way as to obtain social

sanction for one’s performance. He describes this process in detail in Ontological

Relativity:

The word refers, in the paradigm case, to some visible object. The learner has now not only to
learn the word phonetically, by hearing it from another speaker; he also has to see the object;
and in addition to this, in order to capture the relevance of the object to the word, he has to see
that the speaker also sees the object. Dewey summed up the point thus: ‘The characteristic
theory about 3’5 understanding of A ’s sounds is that he responds to things from the standpoint
of A’ (178). Each of us, as he learns his language, is a student of his neighbor’s behavior; and

113

conversely, insofar as his tries are approved or corrected, he is a subject of his neighbor’s
behavioral study. ‘ '6

Because my knowledge of meanings depends, in this sense, on observations of my
neighbor’s behaviors, and because such behaviors must then be interpreted by me as, for
example, confirming my correct usage of the term in question, I can never be certain that
my understanding of a term is consonant with my neighbor’s. There will always be, in
other words, a degree of indeterminacy in my attempts to translate my neighbor’s words
into my own. This is part of what Quine refers to as the “indeterminacy of translation.”
Note however, that the indeterminacy of translation is not merely an epistemological
problem for Quine—that is, it is not the case that only our ability to know that the referent
of our’s and our neighbor’s term is the same is threatened here (although this is, in fact,
threatened). Rather, it is the very existence of a uniquely correct translation that is
problematized by translational indeterminacy. Any attempt to pick out a unique referent
for a term will fail, argues Quine, because the behavior of native speakers in the
community in which the term is utilized will always be interpretable as consistent with any
number of distinct referents. Indeed, Quine argues tlmt it will not even be possible to
determine whether the referent is a concrete object or an abstract universal based on the
native speaker’s behavior alone.117 Moreover, this difficulty is not limited to the would-be
translator of an unfamiliar language. The dificulty of associating words with unique

referents applies to the process of learning a language as well—indeed, even one’s own

 

“6 Quine (1969), p. 28.

”7 Quine, W.V.O. Word and Object. Cambridge: MIT Press, 1960. sect. 12 pp. 51-57 and Quine
(1969), pp. 30-37.

114

native language.

Quine’s views about translational indeterminacy implicitly invoke a distinction
between what are merely epistemological obstacles to overcoming indeterminacy, and
what obstacles are ontological. Quine has identified other contexts in which
epistemological indeterminacy may be found, that is, contexts in which our claims “go
beyond” the available evidence. For example, one form of epistemological indeterminacy
arises, according to Quine, when we realize that our scientific theories are
“underdetermined” by the available empirical evidence. A scientific theory is
underdeterrnined when there is no experience we can produce which would count as
evidence for it and against an alternative incompatible theory. As in the case of translation,
this sort of scenario is problematic because it seems that both options cannot be correct,
and yet, we have no method of choosing between them. But this may be where the
similarity between translational indeterminacy and the underdetermination of theory by
fact ends. In an article entitled “Translation, Physics, and Facts of the Matter,”118 Roger
Gibson argues that it is Quine’s view that the underdetermination of theory by evidence
presents an epistemological dilemma since our only way of knowing which is the best
choice is by consulting the evidence, but the evidence is equally consistent with both
options. However, Gibson argues that Quine does not conclude, fiom this, that there is no
fact of the matter regarding which of the two competing theories is the correct one. In the

case of the underdetermination of theories by fact, Quine believes there is a right or wrong

 

”8 Gibson, Roger. “Translation, Physics, and Facts of the Matter.” in The Philosophy of WV.

Quine. eds. L.E. Hahn and RA. Schilpp. LaSalle: Open Court, 1986.

115

answer to the question of which of the two competing theories is in fact true. In contrast,
Quine holds that translational indeterminacy is more than just an epistemological problem
about evidence. According to Gibson, Quine holds that meanings are inherently

indeterminate.

In effect there is a fact of the matter to the question of which of two physical theories, both

of which are consistent with all possible observations, is the correct one but there is no fact
of the matter to the question of which of two translation manuals, both of which are consistent
with the speech dispostions of all parties concerned, is the correct one.”°

Quine does seem to suggest exactly this in me Pursuit of Truth. when he writes:

There is evident parallel between the empirical underdetermination of global science and the
indeterminacy of translation. In both cases the totality of possible evidence is insufficient to
clinch the system uniquely. But the indeterminacy of translation is additional to the other. If
we settle upon one of the empirically equivalent systems of the world, however arbitrarily, we
still have within it the indeterminacy of translation.120

Gibson correctly suggests that Quine’s views about the special status of translational
indeterminacy are heavily influenced by his commitments to both physicalism and
empiricism. A closer examination of translational indeterminacy will reveal why Quine
thinks that it is more than just an epistemological dilemma. In addition, understanding how
Quine’s physicalism and empiricism motivate his convictions concerning the indeterminacy
of translation is essential to appreciating how his perspective differs from that of structural
semantics and how this perspective may pose a threat to any theory that attempts to fix the
meanings of terms by associating them with their referents. Indeed, if well founded,
Quine’s views are particularly troublesome to a theory such as structural semantics. As we

have seen, structural semantics has it that mental states mean or represent what they map

 

”9 Gibson (1986), p. 140.

12° Quine, W.V.O. The Pursuit of Trgh, Cambridge: Harvard University Press, 1990. p. 101.

116

into via homomorphic functions. Objects thus mapped can quite correctly be called the
“referents” of these mental states. While Quine’s focus is on language, and not on mental
states, his arguments concerning the difficulties of uniquely fixing the referents of terms
ring familiar as a kind of analogue to the problem of uniqueness. If it can be argued that
there is no fact of the matter as to what my mental state refers to, then presumably there is
likewise no fact of the matter concerning what my mental state represents. We will
certainly need to examine whether the obstacles inherent in fixing the translation of a term,
of linking it uniquely to another term in another language, and therefore of linking either
to a unique referent, are obstacles which face the attempt to map mental states uniquely to

referents in the external world via homomorphism.

4.2.2 Translational Indeterminacy and Non-Uniqueness

In order to appreciate the relationship of translational indeterminacy to the problems
of uniqueness, let us first examine some specific examples of the phenomenon of
translational indeterminacy in the works of Quine. Quine provides this example in

“Ontological Relativity”:

To see what such indeterminacy would be like, suppose there were an expression in a remote
language that could be translated into English equally defensibly in either of two ways, unlike
in meaning in English. I am not speaking of ambiguity within the native language. I am
supposing that one and the same native use of the expression can be given either of the English
translations, each being accommodated by compensating adjustments in the translation of other
words. Suppose both translations, along with these accommodations in each case, accord
equally well with all observable behavior on the part of speakers of the remote language and
speakers of English. Suppose they accord perfectly not only with behavior actually observed,
but with all dispositions to behavior on the part of all the speakers concerned. On these
assumptions it would be forever impossible to know of one of these translations that it was the
right one, and the other wrong. Still, if the museum myth were true, there would be a right and
wrong of the matter; it is just that we would never know, not having access to the museum. See

117

language naturalistically [and meaning behavioristically] on the other hand, and you have to
see the notion of likeness of meaning in such a case simply as nonsensem

Let me represent pictorially the difficulty Quine is describing here. Let n be a term in
the native language, e, and e2 terms in English. The functions f, and f2 are translation
functions which facilitate a mapping from n onto e, and hem n onto e2 respectively (see

Figure 4-B).

e, g
\{g f,

\\n/

Figure 4-B:
Translational Indeterminacy

Given this picture, we can easily see how the meaning of n in English is
indeterminate. It might either mean e, or e2. Observing the behavior of English speakers
when n is translated as e, rather than as e2 will not resolve the indeterminacy, for by
hypothesis, either usage will allow a speaker to behave in ways consistent with other
English speakers’ expectations. Indeed, even in cases where e, and e2 are semantically

related to122 other terms in the language, Quine imagines that n can be mapped to both in

 

1" Quine (1969), pp. 29-30.

122 By “semantically related to” here I mean only that e, and e, bear relations to other terms in
the language. In particular, you may think of e, and e2 as elements in the domain of a relational system.
The relations which order the domain may be characterized, in this example, by speakers’ dispositions.
For example, if e, is the term ‘mammal,’ it may be related to the term ‘dog’ as long as the speaker is
disposed to behave in ways that would suggest his or her belief in the proposition, “Dogs are mammals.”

118

such a way that either mapping would “accord equally well with all observable behavior
on the part of speakers of the remote language and speakers of English.” All that is
required is that the translations of the semantically related terms be adjusted as well. This

more complicated, but more likely possibility is illustrated below, in Figures 4-C and 4-D.

 

 

 

Figure 4-C: n Translated As c, Figure 4-D: n Translated As e,

In these figures, two semantic networks are portrayed. In one (Figure 4-C) the term
n in the native language is mapped to the term e, in English. Other terms of the native
language are mapped to other terms of English and in both languages, individual terms

bear relations to others. The idea is that it should be possible to map n into e2 instead of e,

 

When terms are semantically related in this way (i.e., related in virtue of their meanings or related in
virtue of the speaker’s disposition to behave as though they referred to objects which bear a relation to one
another) it seems harder to imagine that either might be one of many possible translations for a native
term. This is because the translation of n as ‘dog’ would carry with it the implication that ns are mammals
and an alternative translation for it might not preserve the truth of this implication. Quine’s example is
meant to show how a persistent indeterminacy is, nonetheless, possible under these circumstances.

119

without anyone noticing so long as accommodating adjustments are made in the mappings
of the other native terms as well. This possibility is illustrated in Figure 4-D where n is
now mapped to e2 instead of e, while preserving the structure of the networks.123

Although the homomorphisms f, and f2 (see Figure 4-B) are homomophisms between
terms in one language and terms in another, and not between mental states and entities in
the world, the scenario Quine describes nonetheless bears a striking resemblance to the

uniqueness problem for structural semantics. To see why, let us examine two concrete

cases of translational indeterminacy.

4.2.2.1 Two Cases of Indeterminacy
Quine’s concrete example of the sort of indeterminacy he has been describing
involves the French language construction ‘ne...rien.’ The term ‘rien’ can, by itself, be

translated correctly either as “anything” or as “nothing” in English.

 

‘23 The claim that the structure of the graph is preserved between Figures 4-C and 4-D is based
on the assumption that there are only two properties of this graph which constitute its structure and hence,
only two which need to be preserved. The first of these is the degree of each of the graph’s nodes (a
node’s degree is determined by counting the number of nodes to which it is directly connected). The
second is the specific set of nodes to which each node is connected. The reader may wish to verify that
Figure 4-D maintains 5 nodes of degree 4 and 4 nodes of degree 2, just as Figure 4-C did and that
(excepting n, e, and e, themselves) each node in Figure 4-D is connected to the same nodes it was
connected to in Figure 4-C, despite the fact that most nodes are mapped into by different native terms in
each case. Although providing this example is made easier by the deliberate symmetry of the graphs,
Quine’s indeterminacy could be illustrated with non-symmetric graphs as well, even if one allows that
additional kinds of properties (such as weightedness, directedness, etc.) ought to be counted as relevant to
the graph’s structure. An interesting and relevant question pertaining to the problem of uniqueness
concerns whether larger, less symmetric, and/or more complex graphs would make it systematically more
difficult to obtain alternative, structure-preserving mappings of this sort. Randall Dipert discusses this
issue as well as what types of properties are relevant to the judgment that two or more graphs are
structurally equivalent in an article entitled “The Mathematical Structure of the World: The World as
Graph.” The Journal of Philosophy Vol. XCIV, No. 7, July 1997.

120

‘nothin g'

'anym mgl
In er trier-ll II"! e' fie n'

Figure 4-E: Translational Indeterminacy for ‘ne rien’

 

What is required is that the ‘ne’ is translated accordingly as either “not” or as a redundant
phrase. In the one case, ‘rien’ is rmpped to “anything,” in the other, to “nothing.” Figure
4-E illustrates the relevant relationships. Quine calls this example “disappointing” because

he believes it is easy to argue that the indeterminacy it describes is circumventable:

You can object that I have merely cut the French units too small. You can believe the
mentalistic myth of the meaning museum and still grant that “rien” of itself has no meaning,
being no whole label; it is part of “ne...rien,” which has its meaning as whole.”‘

Quine anticipates the proposal that the reason you can’t get a determinate translation
of ‘rien’ into English is because ‘rien,’ by itself, has no (unique) meaning. But it is
arguable that ‘rien’ is part of a larger linguistic unit and that it is that unit which gets
mapped uniquely onto the terms of English—it is that unit which has unique meaning. The
terms ‘ne’ and ‘rien’ only look as though they have non-unique meanings when they are

considered apart from the unit to which they belong (as shown in Figure 4-F).

 

'2‘ Quine (1969), p. 30.

121

‘not' ‘no' ‘anvthing' ‘nothing'

\/

‘ne' ‘rierl'
Figure 4-F: ‘ne rien’ Treated Atomistically

But if treated as part of an indivisible unit, the indeterminacy disappears (see Figure 4-G).

'nothing‘

 

'ne rien'

Figure 4-G: ‘ne rien’ Treated Holistically

Attempting to resolve the uniqueness problem through an appeal to holism is an
attractive option to the structural semanticist. Since both representations and external
objects are thought of as complex relational systems in this view, it seems intuitively true
to think that the more complex the cognitive relational system we are considering, the
harder it will be to specify fimctions which map it to more than one external object and
visa versa.

Part of a resolution to the uniqueness problem would thereby be achieved by

conceiving of each representation (here analogous to terms in a language) as possessing

122

content in virtue of its role in a larger unit (specifically, a larger relational system). It is the
mapping of the sum total of our representations to a network of external objects that
provides individual representations with the content that they have. If we adopt a holistic
strategy, and consider first how the entire representational relational system best maps
onto the external world, taken in its entirety, then the result will be that certain constraints
are placed on the relationships between individual substructures of each relational system
which were not there when we considered them outside of the context of the more
complete relational system of which they are a part.

The result is that uniqueness of representational content is achieved, for all practical
purposes on this view, by finding a homomorphic relationship between one’s entire
network of representations and the objects in the world. Ultimately, to achieve uniqueness
for individual representations will require one to acknowledge that the contents of each of
these representations are derived from am homomorphic relation between an external
relational structure and the network of representations one possesses.

But in order for this tactic to resolve translational indeterminacy altogether, it would
have to be the case that every instance of indeterminacy could be handled by showing that
the indeterminate elements are part of a larger unit and do not themselves have
independent meaning. If not, then there will be some cases of indeterminacy that are
intractable, and analogously, some cases of representational non-uniqueness which remain
unresolved by the “holistic” approach.

Quine believes that an appeal to holism is not efi‘ective for resolving indeterminacy,

 

‘25 ‘the’ if the mapping is in fact unique.

123

and gives his famous ‘gavagai’ example as evidence. When the native points to a rabbit
and utters ‘gavagai,’ we will not be able to tell, he argues, whether the native term maps
to the English term ‘rabbit’ or to the English phrase ‘undetached rabbit part.’ This is
because the native’s pointing action is simultaneously a pointing toward a rabbit, and a
pointing toward an undetached rabbit part. ‘26 Which is the intended referent of the native
when he utters ‘gavagai?’ So far, this example is similar to the one involving ‘ne rien’:
This artificial [gavagai] example shares the structure of the trivial earlier example ‘ne rien.’
We were able to translate ‘rien as ‘anything’ or as ‘nothing,’ thanks to a compensatory
adjustment in the handling of ‘ne.’ And I suggest that we can translate ‘gavagai’ as ‘rabbit’ or
‘undetached rabbit part’ or ‘rabbit stage,’ thanks to compensatory adjustments in the
translation of accompanying native locutions. Other adjustments still might accommodate
translation of ‘gavagai’ as ‘rabbithood,’ or in further ways. I find this plausible because of the
broadly structural and contextual character of any considerations that could guide us to native
translations of the English cluster of interrelated devices of individuation. There seem bound to

be systematically very different choices, all of which do justice to all dispositions to verbal
behavior on the part of all concerned.”'”

To understand why the ‘gavagai’ example is different, and specifically, why, in Quine’s
judgment, the appeal to a more holistic context is ineffective for it, we must understand
something about Quine’s views concerning the nature of language, and how, like his views
on translational indeterminacy in particular, they are informed by his deep commitment to
physicalism and empiricism. We need also to understand what stande would generally

have to be met, in Quine’s view, in order for translational indeterminacy to be

 

'26 In fact, the situation is even worse, as Quine points out in Word and Object. The native’s

ostensive act would also be compatible with the translation of ‘gavagai’ as “all and sundry undetached
parts of rabbits” or as “a single term naming the fusion of all rabbits” or still further as, “a singular
term naming a recurring universal, rabbithood.” Hence not only is the particular referent of ‘gavagai’ at
issue, we cannot even be sure of whether the term itself is a general or singular one, or of whether it refers
to a concrete or abstract object. Quine, W.V.O., Word and Objegt, Cambridge: MIT Press, 1960. p. 52.

127 Quine (1969), p. 34.

124

surmountable—or put in slightly different terms—for there to be a “fact of the matter”

regarding the translation of terms.

4.2.2.2 Quine on Language and Meaning

As we have already seen, Quine believes there is a fact of the matter where scientific
theories are concerned and believes this despite the underdetermination of theories by
empirical evidence. We have also seen that Quine contends there is no fact of the matter
concerning matters of translational indeterminacy. If we are to understand why Quine
believes that underdetermination is surmountable, but indeterminacy is not, then clearly we
will need to appreciate his notion of a “fact of the matter.” Understanding this will provide
a deeper look at what motivates Quine’s belief that translational indeterminacy is
intractable in the ‘gavagai’ case, and ultimately will more clearly identify the obstacles
translational indeterminacy poses to a theory of representation such as structural
semantics.

According to Gibson, many of Quine’s commentators have gotten his notion of a
“fact of the matter” all wrong. In particular, Noam Chomsky, Richard Rorty, and Dagfinn
Follesdal have all assumed that Quine has a epistemological or methodological
understanding of the notion, such that facts of the matter are those things for which there
is a rational procedure for reaching agreement about what to assert (Rorty), things for
which we have ample empirical evidence (Chomsky), or things for which we have ample
evidence and which meet the standards of simplicity in addition to other extra-evidential

concerns (Follesdal). Still others (Bruce Aune) have assumed that Quine’s notion of a

125

“fact of the matter” is more transcendental, implying that there are facts which may be
beyond all evidence.128 None of these captures Quine’s own view about facts of the

matter, according to Gibson. Instead, he contends,

Quine’s understanding of this term is decidedly naturalistic and physicalistic. When Quine
says that there is a fact of the matter to physics and no fact of the matter to translation, he is
talking about physical facts, and he is talking from within an already accepted naturalistic-

physicalistic theory. ‘29

So facts of the matter, where they exist, must be understood as physical facts. This is to be
distinguished fiom the evidence we have for what is true about the objects in our
ontology, as surely as epistemology is to be distinguished fi'om ontology. According to
Gibson’s Quine,

despite the fact that it is meaningful to say of alternative theories that they are equally

warranted by the same sensory evidence, it makes no sense to say that they are equally true. It

makes sense to say they are equally warranted, because we are speaking from within the same

(physicalisitic) theory of evidence: given all the (possible) evidence, this ontology is warranted,

that ontology is warranted, and so on. However, it makes no sense to say they are equally true,

because we are not speaking from within the same theory of objects (i.e., ontology). The trouble
isn’t with ‘equally’, it is with ‘true’.130

Why can’t the same be said for translational indeterminacy? That is, why can’t it be
said that though one or another translation is equally warranted by the same sensory
evidence, and in particular by observations of the same behavioral dispositions on the part
of speakers, there is yet but one correct translation of a term? Taking the ‘gavagai’ case as

our example, it would go something like this: “Although the available empirical evidence

 

‘28 Aune, Bruce. “Quine on Translation and Reference.” Philosophical Studies. Vol. 24. April
1975. pp. 221-236.

‘2’ Gibson (1986), p. 143.
130 -
Gibson (1986), p. 152.

126

(that is, the totality of behavioral dispositions to stimuli and to others’ speech behavior)
may underdetermine which of one or more referents is the intended referent of ‘gavagai,’
there is nonetheless a physical fact of the matter concerning which referent is the correct
one.” For Quine, the reason that the translational indeterminacy for ‘gavagai’ is intractable
is rooted in his decidedly naturalistic views about language. If there were going to be facts
of the matter concerning language and concerning the synonymy of linguistic expressions
in particular, then these would have to be found in observable behavior.131 This follows
from what Quine believes language is. When we think about what distinguishes a language
from a mere collection of meaningless sounds and physical distortions or from an
unorganized scribbling on a medium, we recognize that it is a collection of linguistic
behaviors. In attempting to discover from what source the collections of terms in a
language derive their meaningfiilness, one cannot, if one is a staunch physicalist such as
Quine, point to things like abstractions, Ideas, or even non-physical mental states. The
behavior of a linguistic community around a term is a quantifiable, naturalistic alternative.
But by hypothesis, the behavioral dispositions of this conununity are insufficient for
distinguishing between the possible referents of the term. And since there is nothing else to
do the job which also meets Quine’s strict empiricist standard, no distinction can, even in
principle, be made. That there is no physical-behavioral fact of the matter concerning
which of the available translations is correct, means that there is no fact of the matter

concerning which is correct at all.

 

‘3‘ Gibson (1986), p. 144., Anne (1975), p. 224.

127

The current theory of the world is physicalistic—and with good reason, Quine thinks. And this
physicalistic world-view settles, for the present, the physical facts of the matter and thereby
what can be said to be true or false given these facts. Any putative meanings, therefore, that
fall between the cracks of the physical facts just aren’t meanings at all. And, further, since
there just aren’t any facts for such putative semantical statements to be about, it follows that
such statements are indeterminate, that is, they are neither true nor false. In this
straightforward, naturalistic-physicalistic sense, then, there is no fact of the matter to the
question of which of two manuals of translation is the right one.132

Quine thinks that “putative meanings fall between the cracks of the physical facts” quite
simply because the only physical facts available where language is concerned are facts
about behavior, and behavior does not always distinguish between the content of one term
and another. This conviction, that what counts in matters of meaning and translation is the
predictable, conditioned responses of speakers, is also expressed clearly in Chapter 1 of
Word and Obiect.

In Word and Object, Quine portrays a natural language as consisting of words and
sentences which, when uttered within a social context, evoke predictable and regular
behavioral responses fiom other speakers. This predictable response to linguistic stimuli
on the part of other speakers is what distinguishes mere noise fi'om meaningful linguistic
units of speech. Specifically, the meaning of a term is given in the disposition to a
behavioral response which that term elicits in others. Quine thinks that most of the time,
the kinds of linguistic units that consistently elicit the same socially sanctioned responses
are not individual words, but sentences. Take the English word “Red” as an example: For
Quine, when a subject applies this word to a specific stimulus, it is more natural to think of
the subject as having uttered a one-word sentence, and in particular, to have uttered a

shortened version of the sentence, “This is a red thing.” After all, it is the application of

 

"2 Gibson (1986), p. 153.

128

the term to a stimulus which is approved or disapproved by the subject’s linguistic peers,
and therefore it is the application of the term to a stimulus which has meaning. “Red,” qua
one-word sentence captures the intention to apply the adjective to a stimulus, and hence
captures what it is that is meeting with the community’s approval. “Red” qua isolated
term, captures no such application to a stimulus. As an utterance untethered to experience,
it leaves us with no basis for the approval or disapproval of its usage. And since linguistic
units derive their meaning from such social sanction of their usage, “Red” is best
understood as a one-word sentence, if it is to be understood at all.

The interesting and relevant thing to note here is that the way Quine divides up the
units of a language has everything to do with what elements of the language reliably
covary with conditioned responses on the part of the linguistic community. Specifically,
just in case a unit of speech can be mapped consistemly to a set of behavioral responses,
then it counts as a meaningful unit of speech. Understanding this is part of what is
necessary to appreciate why the ‘gavagai’ example is an intractable case of translational
indeterminacy for Quine, while the ‘ne rien’ case is not. In the latter case, it is arguable
that there is no consistent and identifiable behavioral response to either of the two units
‘ne’ or ‘rien’ that compose the French phrase when taken in isolation. That they are
therefore ambiguous in any translation attempt is far from problematic—on the contrary, it
is exactly what Quine’s view would predict. As in the case of “red,” it is more appropriate
to think of the “longer” unit, ‘ne rien’, as the unit of meaning, rather than to attempt to
translate the phrase compositionally. For it is ‘ne rien’ that covaries with specific,

predictable responses on the part of French speakers. When we do so, we find that

129

translation comes easily and the ambiguity drops away.

‘Gavagai,’ on the other hand, does, by hypothesis, elicit specific, predictable
responses on the part of the community and does so all by itself. Taken as a one-word
sentence implying the application of the term to a stimulus, it will elicit approval or
disapproval from other native speakers in predictable ways. The problem arises when we
recognize that these predictable, conditioned responses are consistent with a myriad of
translations for the word ‘gavagai’ into English. As we’ve seen, it might be translated as
‘rabbit’ or possibly as ‘undetached rabbit part’ among other options. The utterance,
‘Gavagai,’ taken as a one-word sentence applying to a stimulus, is compatible with either
translation, making determinate translation impossible.133

Why not narrow down the options, as in the ‘ne rien’ case, by considering the
relationship of ‘gavagai’ to other terms in the language in hopes that its relationship to

these terms might constrain its translation into English? Certainly Quine would not deny,

especially in light of his association with meaning holism)“ that fixing the meanings of

 

'33 The fact that it is even possible to imagine the term ‘gavagai’ having a whole host of different
referents might lead one to conclude that indeterminacy of translation, far from being a problem for a
view such as structural semantics, is quite consistent with the view. This is because translational
indeterminacy seems to provide a counterexample to the view that meanings are given exclusively by the
behavioristic trends of a linguistic community. If ‘gavagai’ fails to evoke behaviorially distinct responses
from speakers whether taken as referring to rabbits, Rabbithood, or undetached rabbit parts, then meaning
must not be a matter of behavioral responses after all. But this line of reasoning makes the mistake of
equating meaning with reference, something which Quine would not allow. The fact that ‘gavagai’ can be
taken to have a number of different referents, while still evoking the same behavioral responses fiom
speakers, would suggest that meaning and reference are not equivalent, so long as one accepts Quine’s
account of meaning.

13" It is certainly arguable that Quine is a meaning holist given his position in “Two Dogmas”
wherein he critiques the logical positivist’s view that sentences can be confirmed or disconfirmed
individually, without the benefit of considering their relationship to other claims in a scientific theory. In
this work, he writes, “But the dogma of reductionism has, in a subtler and more tenuous form, continued
to influence the thought of empiricists. The notion lingers that to each statement, or each synthetic

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some terms in a language constrains the available interpretations of those which remain.
And indeed, he does not deny this. If we consider all of the native sentence utterances
containing the word ‘gavagai,’ and require that they all be translated into true (or at least
plausible) English sentences, Quine’s claim is that there will be more than one
translation/interpretation (actually an infinite number) which will satisfy the whole corpus.
Interpretations of individual terms within a translation might constrain the interpretations
of other terms within that same translation, but there is presmnably no way of deciding
between translations on a language-wide level. In theory, doing so would require that one
be able to find a decisive reason for choosing one over a multitude of available mappings
between a term in the language and a specific stimulus. This initial assignment of term to
stimulus might then constrain the mappings of other terms to other stimuli, assuming that
something like meaning holism is true. But the gavagai example is meant to illustrate the
inherent difficulty in establishing even one such initial, constraining mapping between term
and stimulus. For apart from interpreting ‘gavagai’ relative to an already-established

translational framework, there is simply no behavioral criterion for assigning it to the set

 

statement, there is associated a unique range of possible sensory events such that the occurrence of any of
them would add to the likelihood of truth of the statement, and that there is associated also another unique
range of possible sensory events whose occurrence would detract from that likelihood. This notion is of
course implicit in the verification theory of meaning. The dogma of reductionism survives in the
supposition that each statement, taken in isolation fi'om its fellows, can admit of confirmation or
infirmation at all. My counter-suggestion is that our statements about the external world face the
tribunal of sense experience not individually but only as a corporate body.” (“Two Dogmas of
Empiricism” in The Theor_'y of Knowledge. ed. Louis P. Pojman. Belmont: Wadsworth, Inc., 1993. p.
404)

Quine’s view, that it is only our entire system of beliefs that can be tested against experience, and
not each individual belief taken in isolation, might be better described as a kind of “verification holism” or
“evidential holism.” However, since Quine seems to retain, to some extent, the equation of verification
and meaning here, it is not entirely implausible to view him as adhering to some version of meaning
holism as well.

131

of rabbits instead of to the set of undetached rabbit parts. Indeed, there is no such criterion
for assigning any arbitrary term to one stimulus rather than to another, and hence no way
of establishing constraints which would recommend one translation of the whole corpus
over another translation. What this seems to show, is that unique translations of terms are
indeed possible, but only relative to an arbitrarily chosen translational framework. One can
have unique translation of individual terms which is “internal” to this fiamework, but
cannot have unique translations which go beyond, or transcend such frameworks, and this

despite wholeheartedly embracing a holistic approach to meaning.“

4.2.2.3 Two Kinds of Non-Uniqueness

The ‘gavagai’ example bears some extended scrutiny, because it reveals much more
about the extent to which the indeterminacy of translation is a special case“ of the
problem of uniqueness for structural semantics. In efiect, the ‘gavagai’ case seeks to
recreate indeterminacy (and hence, non-uniqueness) in spite of the insistence that linguistic
units may sometimes derive their meanings from the role they play in a larger unit (i.e., in
spite of the “holistic” response to indeterminacy suggested above.

If Quine is correct in assuming that an appeal to holism fails to eliminate translational

indeterminacy in this case, then this bears on one of the responses which structural

 

‘35 Compare this to “internal realism,” the view that though we can talk about a reality
independent of us, this talk must always take place within a rational framework of beliefs and concepts:
There is no God's Eye view of the world.

'36 In particular, it covers the case of linguistic representation while the problem of uniqueness is
a problem for accounts of representation generally.

132

semantics has to offer to the uniqueness problem In particular, if the structural semanticist
intends to eliminate as many competing mappings as possible by arguing that it is the
mapping of entire networks of representations to entire networks of objects which confers
meaning, then any claim that indeterminacy persists in such a scenario, or that individual
representations in the network ought to possess meaning independently of it, is a threat to
her account of representation.

One way of interpreting the difiiculty that Quine’s arguments raise is to see them as
demonstrating the persistence of Cummins’ version of the tmiqueness problem. Recall that
the standard version of the problem arises when a single representation can be rmpped to
more than one external object. Each mapping between representation and external object

is facilitated by a distinct homomorphic relation.

\/

‘gavagar

     

Figure 4-H: Standard: Rabbit vs. Undetached-Rabbit-Part

133

Cummins version of the problem, in contrast, posits the existence of several homomorphic
mappings between a single representation and a single external object. Figure 4-H
illustrates the ‘gavagai’ case as an example of the standard uniqueness problem for

structural semantics.

 

 

 

'uovogll'

Figure 4-1: Cummins: Rabbit vs. Undetached-Rabbit-Part

It shows multiple mappings fiom the term ‘gavagai’ to objects in the world (the translation
functions f, and f2 map ‘gavagai’ to rabbits and undetached rabbit parts, respectively).
Figure 4-1 demonstrates the problem as an instance of Cununins’ brand of non-
uniqueness.

The standard version of the uniqueness problem illustrates a situation in which two or
more distinct structures are viable candidates for the translation/meaning of a single

term/representation. When the ‘ gavagai’ case is considered to be a problem about the

134

translation of a single term as either one of two distinct sets, rabbits or undetached rabbit
parts, it makes sense to see the case as an instance of the standard uniqueness problem.
However, when we recognize that the real importance of the ‘gavagai’ case lies in its
demonstration that a single language can be mapped in multiple, non-equivalent ways to a
single ontology, something like Cummins’ version of the uniqueness problem comes more
to mind. Ultimately, which version of the uniqueness problem you take the ‘gavagai’
example to illustrate depends on how you cut up the world. Quine’s deep physicalist
convictions lead him to conclude that the only patterns and structures that can serve as a
means of establishing constraints on the interpretation of linguistic units, and on the
establishment of content for representations generally, are physical, empirically accessible
structures and patterns. And the only kind of structure which meets the physical-empirical
standard according to Quine, is a structure formed out of the totality of behavioral

dispositions on the part of a specific community of speakers.

When with Dewey we turn thus toward a naturalistic view of language and a behavioral view
of meaning, what we give up is not just the museum figure of speech. We give up an assurance
of determinacy. Seen according to the museum myth, the words and sentences of a language
have their determinate meanings. To discover the meanings of the native’s words we may have
to observe his behavior, but still the meanings of the words are supposed to be determinate in
the native’s mind, his mental museum, even in cases where behavioral criteria are powerless to
discover them for us. When on the other hand we recognize with Dewey that ‘meaning is
primarily a property of behavior,’ we recognize that there are no meanings, nor likenesses nor
distinctions of meaning, beyond what are implicit in people ’s dispositions to overt behavior
[emphasis mine].'37

Since the totality of these behavioral responses would not, by hypothesis, distinguish

 

‘37 Quine W.V.O., “Ontological Relativity”, p. 29.

135

between rabbits, undetached rabbit parts, and rabbithood, there is no empirical basis for
distinguishing between these things at all—as far as meaning and representational content
are concerned anyway. This is why Quine’s translational indeterminacy is something more
akin to Cummins’ version of the uniqueness problem: There is but one empirical relational
system to which ‘gavagai’ can be mapped in several different ways, and no ontological
basis for characterizing the external world as being composed of more than one such
system in this particular example. Although, in Cununins’ case, these several mappings are
facilitated by the complexity of the relational system, the structure responsible for this
complexity is not something we can exploit in the service of uniquely specifying a
mapping, because it is not, again by hypothesis, something which makes any difference to
the behavioral responses of speakers using the representation. If we see translational
indeterminacy as a property of whole systems in this way, then it is arguable that Quine’s
critique shows the holistic approach to be effective only against non-uniqueness within
translational systems (read: non-uniqueness of the standard variety) but not against non-

uniqueness of the sort which Robert Cummins describes.

4.3 Where Does This Leave Structural Semantics?

By way of summary, it seems that a convincing case can be made for the intractability
of translational indeterminacy if we can grant Quine a few of his more basic assumptions.
In particular we must start out, with Quine, fi'om a decidedly physicalist perspective, and

adopt with him the further view that the only physical facts of the matter relevant to the

136

meaningfulness of terms are facts about the behavioral dispositions of speakers.138
Adopting this point of view is easier to do if one is already sympathetic to empiricism. A
physicalist who was not sympathetic in this way, might seek to identify other physical facts
about language which, perhaps, are not as readily apprehended by the senses as speech
behavior undoubtedly is.

In order to assess the effect of Quine’s critique on structural semantics, we must first
situate that critique relative to these assumptions, for while structural semantics shares
some of them, it does not share them all. And a departure from Quine on any of these
po ints—physicalism, empiricism, or behaviorism—might change the degree to which his
arguments concerning translational indeterminacy are compelling.

Regarding the commitment to physicalism: I have always thought of structural
semantics as a position which is sympathetic to physicalism and most of the examples I
have used of “structures” which could be exploited in the service of discovering mappings
between representations and the external world, have been physical structures, or, at least,
structures which could be realized in physical media. In point of fact, however, there is
nothing that has been said so far regarding structural sermntics’ account of how
representations have content, which would commit it exclusively to a physicalist
metaphysics. I see this as an advantage of structural semantics since it allows the view to
recommend itself to a range of metaphysical positions. However there may be an

advantage to this flexibility which we can only now fully appreciate in light of having

 

‘38 It’s also possible that a belief in the intractability of translational indeterminacy requires a

commitment to a strong version of verificationism, such that the meaning and evidence are one and the
same.

137

considered Quine’s views on translational indeterminacy. For if translational indeterminacy
is a consequence of the belief that the only properties of language which count in the
attempt to fix meanings are the physical properties of language, and in particular the
behavioral dispositions of speakers of the language, then translational indeterminacy does
not necessarily follow from a non-physicalist starting point. It certainly seems easier to
accept that there are no distinguishable physical-behavioral responses to rabbits when
compared with responses to undetached rabbit parts, than to accept that there are no
means of distinguishing them whatsoever.

Of course, there are, at least in my view, a significant number of more serious
problems which structural semantics might have to contend with if held in conjunction
with a non-physicalist metaphysics. So I have defended, and will continue to defend the
view under the assumption that some form of physicalism is true. What I do not believe,
and will not defend, is the implication of a behaviorist approach to meaning from a
physicalist metaphysic combined with an empiricist epistemology. Implicit in my
discussion of structural semantics until now has been the assumption that we are capable
of appreciating structural properties of networks of representations which go beyond just
the patterns of behavior exhibited by the people who possess them. Specifically, I’ve
talked about the formal properties of such structures and have assumed that these types of
properties are what individuate representations fiom one another and which ultimately
enable them to stand in relation to that which gives them the content that they have. Even
assuming physicalism, translational indeterminacy is only a threat to the structural

semanticist’s account of meaning if it can be shown that, in addition to there being no way

138

of using behavior to distinguish between two or more proposed referents of ‘gavagai,’
there is no appreciable physical difference whatsoever between them such that one would
recommend itself above the other as the most appropriate translation for the term. In the
next chapter, I will examine whether or not such a case can be made against structural
semantics.

It is worth noting that there is starting to emerge a very significant difference between
the specific kind of view that translational indeterminacy threatens (namely, a view about
the meaningfulness of terms) and the view that is represented in structural semantics. I
have been assuming that Quine’s arguments for the indeterminacy of translation carry over
to mental representations, thereby posing a new argument for the problem of uniqueness.
But this assumption may be false. For it is possrble that the structural semanticist could
admit all of what Quine assumes, even that there are no significant properties beyond
speakers’ behavioral properties to consider when learning the meanings of terms in a
language, and still maintain her affirmed account of representational content. This is
because structural semantics is a theory about the representational content of mental states
and not a theory about the meaningfulness of terms in a language. Perhaps, insofar as
natural languages are uniquely social phenomena, the properties of natural languages
(including the meanings of terms in such languages) are primarily characterized in terms of
dispositions toward behavior on the parts of speakers. And if so, perhaps translational
indeterminacy is an interesting phenomenon which Quine has identified as inherent in
natural language. This does not necessarily imply that the same is true for mental states.

Once again, as a physicalist, I would urge that mental states are physical (possibly

139

physical-functional) states of the brain. As such, there ought to be a whole host of
properties which can be discovered about them and which go beyond just the behavioral
evidence for their existence. In my view, mental states are arguably more like the objects
of a scientific theory than like the terms in a language and, as we have seen, even Quine
would admit to there being physical facts of the matter concerning what is true of the
objects of a scientific theory. For structural semantics to maintain viability as a theory of
mental representation in the face of translational indeterminacy via Quine, we have only to
contend, minimally, that there is a fact of the matter as to which mental states are
structurally similar to which external, physical states. Next, let’s consider whether or not

this minimal condition can be met by the theory of structural semantics.

140

CHAPTER 5: Formal and Metaphysical Criticisms of Structural Semantics

Much of the preceding discussion has relied on the assumption that the notion of
“similar structure” can be successfully understood in terms of homomorphic relations
between cognitive and external relational systems. Though Quine’s discussion of
translational indeterminacy reveals a potential obstacle to the acceptance of this
assumption, I’ve argued that a departure from any of Quine’s basic starting assumptions
could do much toward removing the force behind the obstacles his version of non-
uniqueness presents to structural semantics. Thus far, I’ve also portrayed the main positive
response to non-uniqueness problems as involving the following sort of move, which, for

convenience, I will refer to as “the holism solution”:

The Holism Solution: Attempt to eliminate as many alternative mappings from a
representation to objects in the external world as is possible by locating the
representation within a network of other representations, themselves mapped to a
network of objects. Which is the appropriate mapping for individual representations
will then be determined by deciding which of the original competitors is compatible
with the mapping relation that holds for the entire representational network.

In this chapter, I’ll consider some reasons for thinking that a solution to the
uniqueness problem will not be this easy. Already, Quine has made a convincing case for
the claim that there is a persistent version of the uniqueness problem, namely Cummins’
version of the problem, which threatens the project of fixing unique referents for
representations, and thus, for structural semantics, threatens the possibility of assigning

unique content to them. Furthermore, he has argued effectively that the holism solution

does not adequately address this version of non-uniqueness. Now we must consider

141

whether there is a way for indeterminacy to persist even where physical-behavioral
distinctions are not the only kinds of distinctions we allow between two or more content
structures. As background to this discussion, let’s first examine an interesting consequence

of the Lowenheim-Skolem theorem.

5.1 The Liiwcnheim-Skolem Theorem
The Liiwenheim-Skolem theorem states:

LS: Let I‘ be a satisfiable set of formulas in a language of cardinality ref” Then
I‘ is satisfiable in some structure of cardinality s x. ”0

The theorem says that there can be a model (an interpretation) for the sentences in P
which is lower or equal in cardinality to that of the language fiom which I‘ is constructed.
Since by the “cardinality of a model” we simply mean the size of that model’s domain, we

can restate the deenheim-Skolem theorem as follows:

Let I‘ be a satisfiable set of formulas in a language of cardinality it. Then I‘ has a model whose
domain is of a cardinality less than or equal to x. 4'

This theorem, in conjunction with the compactness theorem, can be used to prove many
interesting results. This version of the theorem, usually referred to as the downward
Lowenheim-Skolem theorem (DLS), has a counterpart (the upward Lowenheim-Skolem

theorem or ULS) which provides that there exist models for P which are higher in

 

139 By “a language of cardinality it” we mean, a language containing at least it terms
(expressions which can be interpreted as naming an object).

”0 Enderton, Herbert B. A Mathematical Introduction to Logic. New York: Academic Press,
1972.p.14l.

141 This is what Quine refers to as “the strong version” of the theorem in “Ontological

Relativity.” Quine (1969), p. 60.

142

cardinality than the language fiom which P was constructed. Although it is arguable that
both versions of the problem present a threat to the uniqueness of models, Quine describes
an argument which relies primarily on the DLS result, as we shall presently see.

Let us consider one result of the DLS. Let 1" be a satisfiable set of formulas in a
language L of cardinality No, the cardinality of the natural numbers. One may think of F as
some set of true sentences of L, in particular, as some theory constructed in L. Suppose
the intended interpretation of this theory has an uncountable domain, e. g., the real
numbers.‘42 The downward version of the L-S theorem tells us that there will be an
alternative model that has a domain of cardinality s No.

This is a remarkable consequence because, given that the domain of any acceptable
theory is, in effect, reducible to a denumerable ontology (via the DLS), it suggests that any
domain “... is reducible in turn to an ontology specifically of natural numbers, simply by
taking the enumeration as the proxy function, if the enumeration is explicitly at hand.”143
In effect Quine is saying that, no matter what the intended model of a theory, this theory’s
ontology is mappable, via a proxy function, to the natural numbers (or some subset of
them), and hence, that its ontology is reducible to the natural numbers or one of its
subsets.

What effect, if any, does this chain of reasoning have for the theory of structural

semantics? Conceive of the “theo ’ in Quine’s ar nt as uivalent to the network of
gume e(l

 

“2 A set is countable ifit is either finite or countably infinite. A set s is countably infinite
provided that S and the set of natural numbers, N have the same cardinality. A set S is uncountable if it is
not countable.

“3 Quine (1969), p. 59.

143

interpreted representations possessed by some entity (e. g., a person) with representational
capacity. Suppose this network has only No elements (as is plausible for any finite being).
This theory (call it (D) is interpreted in virtue of a homomorphism to a network of objects
(and the relations between them) in the actual world—such objects are the intended model
of the theory. However, given the DLS together with Quine’s point about proxy functions,
we know that (I) my also have, as a model, the natural numbers and the relations between
them. DLS provides that a model equivalent in cardinality to the natural numbers exists.
Quine’s proxy function provides a mapping fiom the elements of this model to the natural
numbers. The result is that there are at least‘“ two models for (I), that is, two
interpretations of it. Since (I) is just a network of interpreted representations, another way
of putting this result is that DLS together with Quine’s proxy fiinction implies persistent

non-unique interpretations of the same representational structure (see Figure S-A below).

N actual objects and
relations between them

(I)
Figure 5-A: The Pythagorean Problem

 

”4 Actually, there will be an infinite number of competing interpretations, because if a theory
can be mapped to the natural numbers, then it can be mapped to the evens, the odds, etc.

144

But that is not all. The prospect of having more than one possible model for a
representational network is not a new one. DLS perhaps allows a more formal illustration
of the problem than has been offered until now, but the problem itselfis simply what I
have been addressing all along. What DLS shows that is new, is that among the competing
interpretations of (I) are models which would make the content of representations in (D
nothing more than numbers and relationships between numbers, when what we’d wanted
was a representation of the objects and relationships in the actual world with which we are
familiar. Quine recognizes this result in “Ontological Relativity” and thinks it is a kind of
Pythagoreanism regarding representational content that ought to be avoided, if possible.
Notice that the problem of Pythagoreanism arises regardless of whether or not one
embraces an ontology of external structures which are manifest only through behavioral
response. Even if it is true that allowing for a richer ontology of structures will eliminate
many of the competing translations of a given term that a more spartan ontology would
retain, the DLS shows that there will, nonetheless, always be more than one model for any
given representational structure. The assumption behind the holism solution is that where
two or more models exists for a single representation, it will always be possible to
discover finer structural detail in the external world which will provide the justification for
considering one competitor a better model than the other. What DLS shows is that this
assumption is false: Even if the holism solution may eliminate competing models up to a
point, there will always be at least one structure which can model a representation as well

as any other, and that is the model provided by the ordered set of natural numbers.

145

5.2 The Pythagorean Problem

So how can competing mappings be eliminated in spite of the DLS result? The
answer is that they cannot be eliminated and happily, do not need to be. But before
explaining why this is so, let us take a closer look at how the reduction described in the
DLS would proceed.”5

The downward Lowenheim-Skolem theorem shows that “all but a denumerable part
of a [theory’s] ontology can be dropped and not be missed.”“’" If we begin with a theory
whose universe of discourse is non-denumerable, then we can imagine that the reduction

proceeds, in theory, as follows:

[We begin by] partitioning the universe into denumerably many equivalence classes of
indiscriminable objects, such that all but one member of each equivalence class can be dropped
as superfluous; and one would then guess that where the axiom of choice enters the proof is in
picking a survivor from each equivalence class. If this were so, then with the help of Hilbert’s
selector notation we could indeed express a proxy function.”7
Figure 5-B represents a procedure for the construction of a proxy function. On the first
line we havethe set ofallrealnurnbersxsuchthatl s x s 2, 2 s x s 3, andso on. This is
the non-denumerably infinite set of real numbers between 1 and 2, 2 and 3, etc. If a

survivor is picked fiom each subset, then we are left with the set of natural numbers N.

 

”5 Note that this “proof’ is not intended to be a proof of DLS or of the method by which an
appropriate proxy function is constructed either. The DLS only shows that a model exists which is
equivalent in cardinality to the natural numbers. It does not provide the proxy function which maps the
elements of this model to those of the intended model or any other specific model. This illustration is
simply one example of how such a proxy function might work.

‘46 Quine (1969), p. 60,

147 Quine (1969), p. 60. Note that the proxy function reduces a homomorphic function between
relational systems to that of isomorphism, where the latter is a special case of the former.

146

{ [1r-u): [2]."): [3,.u),." } Original Non-denumerable
Ontology

Equivalence Classes

 

New, Denumerable Ontology

Figure 5-B: Construction of a Proxy Function for DLS

There doesn't seem to be much point in disputing the truth of the Lowenheim-Skolem
theorem or of the possibility for constructing proxy functions like the one Quine describes.

So our available options seem to be these:

(I) accept the consequences of Pythagoreanism outlined above or,

(2) add an additional constraint to the structural semanticist's account of

representation. That is, insist that there is something more to the representation

relation besides homomorphism and make sure that the extra condition doesn't allow
representations to be mapped to the natural numbers but only to the universe of
objects you really want to be your model.

Of these options, (2) seems to compromise the main advantage of structural
semantics—namely, the beauty and simplicity of the account. Structural semantics makes
no appeal to causality, evolutionary history, or behavior in order to explain
representational content as do most all other theories. Indeed, there are also well known

difficulties with all of these alternatives which are, in my opinion, much more daunting

than those associated with structural semantics. Moreover, to appeal to one of these

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alternatives now would make the account of mental representation I am defending here ad
hoc.

This leaves us with option (1), which, on the face of it, sounds unacceptable too. But
is it really? What happens if we accept that a homomorphic account of representational
content will always leave at least one competitor (and in fact, an infinite number of them)
to the mapping between representations and objects in the world, namely, a mapping
between representations and the natural numbers or some subset of them? What seems to
happen is that objects and numbers become ontologically indistinguishable relative to the
theory. For example, it seems that there would be no difference between a representation
whose content is actual dogs and a representation say, of the number 4.

Remember that, according to structural semantics, the content of a representation
can't be determined by looking at the way it individually maps to an element of the
external world, but instead is determined first by considering how the entire
representational system maps to the external world, and deriving the contents of individual
representations fi'om this. And we have seen that even when the representational and
external networks are considered holistically, we are still committed to Pythagoreanism.
But it is a very unusual sort of Pythagoreanism to which we’re committed if it is a form of
Pythagoreanism at all: What constitutes a mapping between two structures is not that they
contain certain kinds of objects or the same number of objects or really anything to do
with the objects themselves. Rather, what constitutes a mapping is the relationships that

the objects (or numbers, as the case may be) bear to one another (and of course, whether

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the representations to be interpreted enter into the equivalent"8 relationships). That is,
whether or not a network of representations can be mapped to another structure depends
on whether or not that whole structure serves as a model for the whole representational
network.

Now here is the crux of the point: If any model consisting of the natural numbers is a
model for a network of representations, then it would have to be the case that the natural
numbers were related to one another in this model in precisely the “same way” that the
representations in the network are related, otherwise it would not be a model. But this is
all that was ever required of a model consisting of objects!” Hence, though there may be
two potential models for the representational network, both “say” the same thing about
the content of representations in that network. What provides the representations in the
network with content is exactly the same in both models.

Paul Benacerraf makes a similar point in “What Numbers Could Not Be,” an article

that probes the question of what kinds of things (e.g., sets, objects, relations, etc.)

 

”8 By “equivalent relationships” here, I do not mean to suggest that the relations which order the
elements in the two structures mapped to the representational network must be the same relationships.
Rather, I mean to say that whatever relations order the two structures, they must be equivalent in respect
of what makes them models for the representational network. Hence “Structure A and structure B are
equivalent” should be read as shorthand for “A and B are equivalent in respect of which makes them both
models for a representation R.”

”9 It is possible to argue that this begs the question. After all, couldn’t somebody (e.g. Fodor) say

that for the content or meaning of language or thought, more is required, namely that it be the right set of
actual objects, not just some homomorphic model? Even if what is claimed here is correct when applied to
models, when applied to content or meaning in general, does it not beg the question? I guess it does beg
the question, strictly speaking. But only if taken in isolation from the other arguments contained in this
defense of structural semantics. I see this argument as addressing a problem which is internal to structural
semantics. For even if one believed the theory wholeheartedly, you would still be confronted with the
problem of avoiding Pythagoreanism.

149

numbers might be.‘50 Benacerraf claims that, given a set of sufficient conditions for
explaining the concept of number, we can yet arrive at interpretations of the concept of
“number” which are contrary to one another. In effect, Benacerraf illustrates a version of
the uniqueness problem regarding the content of “number.” To take a particular number as
an example, suppose that the dispute is over the matter of the proper interpretation of ‘3.’

Figure 5-C illustrates some of the possible ways in which ‘3’ might be interpreted:

[[[IJJJ [IIIILII’ICJJJ

C 3 I
Figure 5-C: Interpretation of Numbers
as ‘Objects’

Both interpretations suppose that the number ‘3’ should be understood as the successor
under some relation R of the number 2. The problem is that in one case (right) the
successor under R of 2 is understood as the set consisting of 2 and all the members of 2,
while in the other (left) the successor of 2 is simply [2], the unit set of 2—the set whose
only member is 2.151 Which interpretation is correct?

The curious thing is that there can be such a high degree of commence regarding

what (relational) properties the numbers have (e. g., that each is the successor under R of

 

‘50 Benacerraf, Paul. “What Numbers Could Not Be” in The Pmlosoppical Review. Volume 74,
Issue 1. January, 1965. pp. 47-73.

151 Benacerraf (1965), p. 54.

150

the number 2), and yet so much disagreement over what kinds of objects (e.g., which set)
the numbers are. And indeed, Benacerraf thinks that the problem arises from the
assumption that numbers must be characterized as some kind of set-defined object at all.
Ntunbers, on Benacerraf’s view, ought to be characterized much as representations are
interpreted on the structural semanticist’s view: they are individuated exclusively on the

basis of their role in an ordered (relational) system.

Any object can play the role of 3; that is, any object can be the third element in some
progression. What is peculiar to 3 is that it defines that role—not by being a paradigm of any
object which plays it, but by representing the relation that any third member of a progression
bears to the rest of the progression.

So what matters, really, is not any condition on the objects (that is, on the set) but rather a
condition on the relation under which they form a progression. To put the point
differently—and this is the crux of the matter—that any recursive sequence whatever would do
suggests that what is important is not the individuality of each element but the structure which
they jointly exhibit. This is an extremely striking feature. One would be lead to expect fiom
this fact alone that the question of whether a particular “object”—for example, [Hem—would
do as a replacement for the number 3 would be pointless in the extreme, as indeed it is.
“Objects” do not do the job of numbers singly; the whole system performs the job or nothing
does. I therefore argue that numbers could not be objects at all; for there is no more reason
to identify any individual number with any one particular object than with any other (not
already known to be a number)”2

While Benacerraf reaches the conclusion that numbers could not be objects, I say
(above) that numbers and objects are arguably more or less indistinguishable. These may
sound like very different points of view but they are really not. Both positions embrace an
ontology in which relations are wlmt are real. Bermcerraf sees the notion of “object” as

one which suggests singularity and isolation and therefore discards the idea that numbers,

 

‘52 Benacerraf (1965), pp. 69 & 70.

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qua essentially relational entities, can ever be “objects.” But I see “object” as just another
way of labeling a cluster of relations. This is why I don’t mind when Quine argues,
effectively, that one might just as soon interpret some appropriately structured
representation as ‘3’ rather than as a dog. Assuming that both really are models for this
representation in respect of the relational properties which they instantiate, it doesn’t
really matter to me what name they are given. And indeed, it seems to me that so long as
the same relations are being modeled in each case, there is no sensible way in which to

understand this as a case of non-uniqueness. Benacerraf writes:

Why so many interpretations of number theory are possible without any being uniquely singled
out becomes obvious: there is no unique set of objects that are the numbers. Number theory is
the elaboration of the properties of all structures of the order type of the numbers. The number
words do not have single referents. Furthermore, the reason identification of numbers with
objects works wholesale but fails utterly object by object is the fact that the theory is elaborating
an abstract structure and not the properties of independent individuals, any one of which could
be characterized without reference to its relations to the rest.153

This seems to me to be the right argument but I would put the conchrsion in a slightly
different way: Indeed, why so many interpretations of a single representation are possible
without any being uniquely singled out is that there is no unique object that is the
representation’s referent. Contrary to Benacerraf, I would argue that representations do
have single referents nonetheless. Their referents are the sets of relations which each
competing model instantiates.

Though Quine may be right in demonstrating a persistent variety of non-

uniqueness—one which the holism solution is of no use to eliminate—it seems that the

 

‘53 Benacerraf (1965), pp. 70-71.

152

sort of non-uniqueness we are left with is completely irrelevant to the task of fixing
representational content, if it is even fit to be described as a version of “non-uniqueness”
at all. Those characteristics which distinguish alternative external structures from one
another are not the characteristics in respect of which both constitute models for a
particular representation. As a result, the fact that there may be more tlun one kind of
object that can realize a model for a particular representation does not imply that the
representation has more than one model.

You may be thinking that an approach such as this resolves the problem DLS posed
for uniqueness of models in interpretational semantics, but that it still leaves us with
something too close to a Pythagorean view. After all, what is a “Pythagorean view of
content” if not the view that numbers and objects can be used to “say” the same things
about the content of a representation? 1 would argue, however, that saying this no more
makes one a Pythagorean than does accepting the principle of the identity of
indiscernibles. For someone who accepts this principle, things do not differ unless their
properties differ. Hence, the only real difference between objects and numbers must reside
in a difference in their properties, including relations.

To articulate this position in still another, final way: If a set of numbers and a set of
objects are both models for a system of representations, then there can be no difference in
their properties (or at least, no difference in the properties which are represented by the
system). So it seems just as legitimate to call what gives representations their content
“objects” on this scenario as it does to call them “numbers.” Indeed, it must either work

out that in our ordinary language (where it seems we can distinguish objects from

153

numbers) we can also express some property of one that is not a property of the other, in
which case they could not both be equally good models, or else we are wrong in thinking
we can distinguish the two in ordinary language. In the first case, both objects and
numbers would not turn out to be equally good models of the representational network we
seek to explain, with the result that the uniqueness problem for that network never arises.
In the latter case, where both options turn out to model the representational network
equally well, the models are unique in respect of that which makes them models and hence
do not provide conflicting answers to the question of what content representations in the
network may possess.

Of course, insofar as being a Pythagorean means that one believes numbers are what
really exist, this is anything but a Pythagorean View. For in this view, numbers are not
what really exist and give content to representations, relations are."" It's just that
“numbers” can assume these relations as easily as anything else. If this view is nonetheless
akind onythagoreanisminthereader’s opinion,thenit isakindwhichthe structural

semanticist ought not to regret embracing.”

 

15" This may seem like a curious claim to make given that, in mathematics, relations are just sets

of ordered pairs. If we think of relations as understood in mathematics, then it seems like there must exist
“elements” or “individuals” which belong to ordered pairs in order to make sense of the concept of
relations. However, we can acknowledge that the notion of “relation” requires that there are “elements”
which form sets of ordered pairs while still thinking of these elements as relational systems themselves
rather than as discrete individuals. (See Chapter 3 for a more inedepth discussion of “levels” of
representation.) This way of thinking about relations may ultimately only push back the question of which
n-tuples of elements constitute the relations on representational and external relational systems to a lower
level of analysis. If so, then it may make sense to ask whether an alternative model of relations might
suffice for the structural semanticist’s account of representation (cf., Randall Dipert’s “The Mathematical
Structure of the World: The World As Graph.” and graph theory).

155 This sort of “pythagoreanism” is already finding its way into mainstream thinking. Consider
the following case, reported by The New York Times in early March of 2001: Professor David Touretzky, a
computer scientist at Carnegie-Mellon, sponsors a website containing representation of software written to

154

5.3 The Role of A Background Theory Revisited

Finally, it would be nice to know what bearing the preceding discussion may have on
the earlier question, raised in connection with Quine, of the degree to which a background
theory is required in order to make sense of talk of the relations between theories and
between theories and their models.

In Chapter 4, we saw that Quine’s views concerning translational indeterminacy
are compatible with the view that terms can have fixed reference, so long as one
understands that the reference of such terms is relative to a fiamework. In the ‘gavagai’
case, the “homework” or “background theory,” as Quine calls it, is the native language,
the terms of which are translated/interpreted as the terms of English. For structural
semantics, the background theory consists of a network of mental representations, a
cognitive relational structure which is mapped to an external relational structure, thereby
interpreting it. According to Quine, it makes sense to speak of the interpretation of a term

or representation only relative to this background theory: Outside of any overarching

 

allow users of the Linux operating system to play DVDs on their home computers. In 1999, when the
software was initially developed, no software was available for viewing DVDs on Linux machines.
Creating the software required developers to crack the encryption scheme protecting the discs. The new
software, called DeCSS was published on sites like Professor Touretzky’s as part of the open-source
software movement, causing it to run afoul of the Digital Millenium Copyright Act, a law passed in 1998
that makes it illegal to offer a way to gain unauthorized access to a copyrighted work protected by
encryption. In February of 2001, the Motion Picture Associate of America contacted Touretzky’s ISP in
attempt to convince them to terminate his website on the grounds that it was illegal. In response,
Touretzky invited open source programmers from around the world to participate in a “contest” to develop
shorter and shorter versions of the code that would break the MPAA’s encryption scheme. He then
published the results on his website. One submission came from a programmer and mathematician in
Finland, who contributed a submission which hides the DeCSS code in the digits of a very long prime
number. The MPAA is contending that this number is illegal under the Digital Millenium Copyright Act.
(Source: New York Times Online, March 2001,
http://www.nytimes.com/2001/03/30/technology/30CYBERLAW.html). It seems clear, therefore, that
from the perspective of the MPAA, there is no difference between the code which breaks their encryption
scheme and this particular prime number.

155

theory, the notion of an interpretation is nonsensical, since there is no criterion for
assigning one interpretation to a term in the theory over any other.

It certainly seems that at least one result of the preceding discussion is the realization
that if we can distinguish (i.e., describe the differences between) two competing
structures, C, and C 2, both represented by R, then we must be doing so in a “background”
representational structure which is more “complete” (or richer) than R. Were this not the
case, then the tmiqueness problem would not arise in the first place. For in order to see C,
and C 2 as competitors, we must have some way of representing their differences. The
uniqueness problem arises because R, by itself, represents both C, and C2 equally well, but
unless our representation scheme is richer than R, there would be no way to know that C,
and C 2 were different objects, and therefore no way to articulate any “problem” about
which one of them R should be taken to represent."6

Having said this, it should be clear that at any point in time, our most complete R (our
most complete representation of the structure of the external world) Ins only one model,
C that we can describe. The progress of our knowledge consists in discovering that our
most complete R is in fact not complete, and in introducing new representations into R
that bring it back into harmony with the structure of the external world. In this very
specific respect, whether or not a unique model is available for any given representation is
something which must be decided fi'om within the context of a richer background theory.

In particular, the background theory fixes our assumptions about what sorts of things

 

'56 A similar observation arose in Chapter 1 in connection with Cummins’ theory of error and the

possibility of distinguishing between alternative targets.

156

(e. g., relations, objects, behavioral dispositions, etc.) are legitimate members of our
ontology. From there we can do our best to establish which representations are mappable
onto which elements of this ontology. Perhaps another way of putting Quine’s point, the
point that a background theory is required for the project of fixing content, is to say that a
theory of content cannot be defended independently of a set of ontological commitments.
Although Quine’s ontological commitments and those implied by structural semantics
are very different, it seems that, in the end, his views are not entirely difl'erent from my
own. Quine would seem to agree that there can be no basis for distinguishing between
competing external structures (here analogous to the referents of terms in his discussion)
so long as they do not differ in respect of that which makes them models for the
representational structure: This is really just another way of putting the idea behind
translational indeterminacy. For Quine, what characteristics are allowed to be considered
in determining a thing’s “structure” are undoubtedly very different, and indeed, more
limited, than those characteristics considered by the structural semanticist. However,
beyond this difference in point of view, Quine and the structural semanticist have more in

common that it originally seemed.

157

CHAPTER 6: The Problem of Error and Indirect Representation

Having attempted to provide an account of how successful representation works and
to defend this account against some of its more serious obstacles, finally it is necessary to
consider how one should explain representational error. Typically, explaining how
representations err has been a daunting task for theories of representation. Philosophers
often find that having spent considerable time perfecting an account of how representation
works when it works, there is no room left for representational failures. But clearly we do
make mistakes which we describe as “misrepresentations,” consequently, a good theory of
representation ought to be able to account for this.

In addition to misrepresentation, a related phenomenon requiring explanation is the
representation of objects that do not exist in the external world. This too has been a
challenge for most theories of representation, but it presents a special challenge for the
theory of structural semantics. Given that structural semantics grounds representational
content on a mapping relationship between properties in the representing entity and
properties of the thing which is represented, it might seem that the theory is hard-pressed
to explain how “things” that don’t even exist are to be represented. After all, there will
presumably be no entities which can stand in the relation that is supposed to characterize

representation in this view.

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6.1 The Problem of Misrepresentation

Since structural semantics has it that representations have the content that they do in
virtue of the existence of a homomorphic mapping between the representation and some
relational structure in the external world, there is a real sense in which no representation is
ever “inaccurate.” Insofar as any external relational structure is a model for a structure in
the cognitive system, the structure in the cognitive system can be said to represent that
external relational structure. There is really no way out of this for structural semantics
given its account of what representation is. So if there is no sense in which a
representation R fails to be a representation of C when R is mapped to C
homomorphically, then how can the concept of misrepresentation even be well defined in a
structural semanticist’s theory, let alone explained?

The answer is that for structural semantics, misrepresentation cannot be defined
simply as a failure to represent. And accordingly, freedom from error is not simply a
matter of achieving successful representation. This claim should sound familiar. Curmnins
argues precisely this in his own discussion of error and interpretational semantics, as we
saw in Clmpter 1. But if there is something more to error and accuracy than the failure and
success of representation respectively, then what is it?

You’ll recall that Cummins thinks that the answer to this question can be found in
making a distinction between what a representation ought to represent, namely, its target,
and what it actually does represent: the content of the representation. For Curmnins, and
for structural semantics, the content of a representation is never in error. This follows

fi'om the fact that the content of a representation is, in both views, given exclusively in

159

virtue of the presence of a mapping relation between representation and external object. If
such a mapping relation obtains between representation and external object, then the
content of the representation is the external object and, in general, there is nothing more to
determining the content of representations. There is no sense in which a representation can
have content which is “incorrect.” Representations either have the content that they do, or
they fail to have it. Therefore, rather than equate misrepresentation with a failure to
represent, Cummins claims that error arises whenever the content and “target” of a
representation are not the same.

For Cummins, the “target” of a representation is something which can only make
sense relative to a particular representational system with a particular architecture.
According to him, the target of a representation r is what a system 2 which tokens r
“expects” to find when it accesses r.157 Substructures of 2 called “mechanisms” or
“intenders” function to produce tokenings of representations, which in turn inherit their
functions or “targets” from the function of the mechanism that tokens them. Mechanisms
have the functions they have in virtue of having the structure they do, that is, in virtue of
“incorporating a design assumption” inherent in the architecture of the system."8 As a way
of illustrating part of what Cummins means when he says that a mechanism “incorporates
a design assumption,” consider the phenomenon of “reverse engineering” as applied to

software programs. Reverse engineering projects rely on the assumption that it is possible

 

‘57 Cummins (1996), p. 18. Recall that Cummins explains the notion of what a system “expects”
to find by stating that it is equivalent to what the system, or more specifically, the system’s mechanisms,
function to represent.

"3 Cummins (1996), p. 18.

160

to discover the function or purpose of a piece of software by reasoning “backwards” from
the code in which the program is implemented. Other projects involve “decompiling”
machine code in order to obtain the original source code in which the program was
designed. In the first case, if enough inforrmtion is known about how the program
functions in relationship to its environment (usually a specific operating system) and its
input, one can use this, together with what is known about the program’s own structure,
to determine what the program does. In many cases, there is only one, unique, behavior
the program exhibits when executed in a specific environment with specific input. This
behavior is arguably the “function” of the program insofar as it produces output or
behavior which systemtically covaries with specific input, together with the program’s
own internal architecture. In Chapter 1, we saw that Cummins wants to argue that
mechanisms of 2 possess firnctions in exactly this sense of the term—in the sense that
their physical architecture determines their function, and ultimately, the function of the

representations which they serve to token.

Just as programs are designed around assumptions about what will be accessed when a given
variable is evaluated, cognitive systems are designed around assumptions about what will be
represented by various intenders. Specifying a system’s design or functional architecture
involves specifying what intenders are possible, and hence, on the current conception, what
targets are possible. The targets that are possible for a given system are thus fixed by its
functional architectural”

One advantage of Cummins’ conception of function is that, unlike adaptational role

theory, for example, it does not imply that every type of representation has the same

 

'59 Cummins (1996), p. 18.

161

function/target. In his view, mechanisms or “intenders” possess fimctions implied by the
structure or architecture of the system. Representations inherit their fiinctions from the
mechanisms which token them, so, for example, if a mechanism with frmction f, should
produce a representation | elm | at time t ,, and a mechanism with function f, should
produce | elm | at time t2, then the result is that I elm I has function f, at t, and functionf, at
1,. This is a feature of Cummins’ account which is all but essential if proper and improper
exploitation of function is going to underwrite an account of representational error in the
way he has envisioned. For Cummins, error occurs when there is a misnmtch between
target and content. The content of a representation is always “accurate” in Cummins’
view—that is, a representation always has the content that it does in virtue of an
isomorphic mapping between it and the structure it represents. In the absence of such a
mapping the representation isn’t “incorrect”——it simply doesn’t represent at all.“50 This
implies that for error to occur (i.e., for there to be a variance between target and content)
it is the target of the representation which has to change. But since the target of a
representation just is its function, the function of a representation must be allowed to vary
from token to token. Cummins therefore needs a notion of “function” which is compatible
with this requirement.

Another advantage Cmnmins seeks for his notion of “function” is that correct
functioning will not end up being understood as equivalent to, or as implying, “successful”
representation. It is now easy to see why Cummins can say that it is mistake to equate

correct representation with fieedom from error, or alternatively, to equate error with

 

‘60 and therefore isn’t really a representation.

162

incorrect representation. If 23 tokens | elm l on an occasion when the target of the tokening
was beech, then there is error borne of a content-target mismatch, although there is
nothing whatsoever the matter with the content of | elm I. It represents elrns just as surely
as it would have had the target and content of the representation been the same. Likewise,
if 23 tokens | elm | on an occasion when the target of the tokening was an elm, it is the
consistency between target and content that is responsible for this success and not the
content of the representation, which remains constant in both cases.

In Chapter 1, I argued that the biggest challenge to Cummins’ strategy for accounting
for error lay in his efforts to separate the notion of fiinction fi'om representational success
while maintaining that mechanisms’ functions can be understood in a way that does not
rely on antecedently meaningful concepts involving the goals or intentions of a system or
its designer. There, I distinguished between two senses of the term “fimction.” One of
them, which I termed function,, is the sense of function one employs when one intends to
talk about the behavior a system exhibits in virtue of facts about its physical architecture.
In contrast, fimction2 is the term reserved for talk of behavior consistent with the
maximization of a system’s purposes or goals. For a given system, its functions, and
functions2 may be, but are not necessarily, the same.

I want to argue that, in order to distinguish successfully the content of a
representation fi'om its target, Cummins needs a conception of function which is closer to
that of fiinctionz. However, I believe Curmnins does not (and possibly carmot) provide an
account of fiinction2 while still maintaining a strict distinction between function and

representational success.

163

In Chapter 1, I argued that a notion of function which is close to function, cannot do
the work of distinguishing target and content. In that chapter, I described a grading
algorithm which, while designed to compute the average of a set of five grades and set the
data object Average equal to this value, nonetheless prompted the user for six grades and
set the value of A verage to a lower quantity tlmn was expected. In Curmnins’ terms,
though the target of A verage was the average of a set of five grades, its content was the
average of six.

Putting aside what we know about the grading algorithm from its use in Chapter I,
suppose that this piece of code was offered to us for the purpose of reverse engineering it.
If we want to know the target of the data object Average, then we need to know what its
function is. In the absence of any access to the program’s designer, our best way (and
arguably our only way) of determining this is to examine the architecture of the algorithm.
We examine the algorithm and find that its structure, combined with the input we provide
to it on request functions, to calculate the Sum divided by 6 and to assign this value to
Average. This provides us with the content of A verage but, by hypothesis, not with its
target. In fact, it is hard to see how examination of the architecture or function, of the
grading algorithm will ever reveal what the true target of A verage had been. The
firnctions, of systems seem good only for revealing the content of the representations they
employ. This leaves firnction2 as the only alternative for understanding target in terms of
function. But I believe that Cummins cannot describe a sense of function2 that does not
involve an appeal to representational success, contrary to what he claims he must do.

As a matter of fact, Curmnins does acknowledge that correct functioning might be

164

understood in terms of the types of performances that promote the system’s survival and
fitness.161 A system’s mechanisms might arguably have the function of producing
representations of edges, given that edge—detection may be an important skill to possess
for the purpose of avoiding falls and collisions, for example. Representations tokened by
these mechanisms would then have edges as their targets, but might have anything
whatsoever as their actual contents. That is to say, in Cummins’ view, it should be
possible to identify a representation as having a specific target, even if no representation

has ever been tokened whose content matches this target!

...selectionist theories assume that something must actually perform its function successfully in
order to contribute to its replication. Whatever you may think of this assumption in other
contexts, it won’t do in the current one, for it amounts to the assumption that something is a t-
intender only if it sometimes successfully represents t, and that is just the assumption we must
avoid for the very good reason that it is not true.‘62

If this is the case, then performative success, understood by Cummins as the
convergence of target and content, might never be observed in the system. But if
observance of successful performance is a prerequisite to the identification of
targets/functions (as I have argued it is), then we cannot use the concept of a “target” to

evaluate successful performance.163 To do so is to be involved in a kind of vicious

 

16' Cummins, 1996, p. 115.

“2 Cummins, 1996, p. 115.

163 In addition, so long as targets/functions are considered to be those things which would
facilitate the “right” performance of the system 2, it would seem that for E to be in error, it must be the
case that the content of one of 2’s representations r, is not consistent with that which facilitates right
performance. This would seem to imply that Cummins cannot allow for errors which actually promote
right performance. This is a consequence which he clearly wants to avoid, since he believes that there are

165

circularity.

This leaves us to wonder if there might be an alternative way of identifying targets,
namely, one which does not tie the notion of a target so closely to the system’s correct
performance, or one which maintains that connection, but can evaluate what it means for a
system to perform correctly without appealing to the notion of a target antecedently. I
believe there is an alternative, but accepting it comes at a price.

If ‘rnisrepresentation” is not going to be understood as incorrect representation (i.e.,
as having a representation, the content of which is somehow “wrong”), then I believe that
the only alternative is to understand it as a species of performance error. In addition,
conceptualizing performance error may indeed require the identification of a target,
application, or “planned use” for representations, which is violated whenever
misrepresentation occurs, as Cummins has argued. But contrary to Cummins, I do not
believe that it will be possible to identify the targets of representations exclusively through
an appeal to a system’s physical architecture. In fact, it is possible that the only way of
identifying targets is to allow their identification to be informed by considerations of value.
The notion that one object is better understood as the target of a representation than
another is a judgment which seems to invariably require a normative component, even
when the notions of “target” and “function” are explained, for example, in terms of the
evolutionary fitness of systems. That a system errs when it represents the world in a way

that is inconsistent with its long term survival or that it is successfirl when it represents the

 

a number of cases in which error can promote better performance in a system (see Cummins, 1996, pp.
27ff, 44-47, 50, 99, 116-118).

166

world in such a way to promote survival, is a judgment that implicitly places a higher
priority on survival than extinction and on fitness above unsound health. To think that
these ends should not be valued may sound ridiculous, and indeed, I am not arguing here
that they ought not to be. Rather, the point I am making is that there is a non-naturalizable
component to the judgment that would classify representations which do not promote
these ends as “errors.” The most that a naturalized theory of representation can do is
explain how representations come to have the content that they do. It cannot explain why
some representations are more appropriate in particular circumstances than others, or why
some promote successful perfornmnce in a system and others don’t. These explanations
simply fall beyond the scope of a naturalized theory of representational content.

Once the intended application of a representation is well-defined, perhaps, with the
help of an appeal to a set of widely accepted value-based assumptions, we can avail
ourselves of a number of theories concerning how and why representations which do not
match the intended target are activated instead of representations which do match the
target. Cognitive scientists who believe that the mind is composed of representational
networks, or “semantic networks” point to several reasons why one portion of the
network may be more strongly activated than another. Among them is the idea that some
elements of the semantic network are more strongly associated with each other than
others. The strength of the association between two or more elements of a semantic

network is referred to as the “weight” of that association.

Various factors are thought to influence the weights (Smith and Media 1981): for instance, the
probability that the feature is true of an instance of the concept, the degree to which the feature

167

uniquely distinguishes the concept from other concepts, and the past usefulness or frequency of
the feature in perception and reasoning.164

In addition to perceptual influences, there is evidence that some “top-down” processing

may be involved in the activation patterns of semantic networks:

Semantic network models of concepts also predict priming and context effects. Conceptual
information processing will be facilitated if a prior context has activated the pathways that are
required for the task. Loftus and Cole (1974) gave subjects a superordinate category name and
a feature and asked them to produce an instance of the category possessing the feature. For
example, when given vehicle and red, the subject might respond fire engine. In some cases the
category was presented before the feature, and in other cases the feature was presented before
the category. Subjects were able to produce instances faster when the category was presented
first; for example, they produced fire engine faster in response to vehicle followed by red than
in response to red followed by vehicle.1

If top-down processing really does influence patterns of activation, this would also
imply that subjects could be influenced to activate representations which are not the
closest available structural matches to the stimulus with which they are presented. If there
is a way to conceptualize the stimulus as the target in this example, this would be a case in
which the target and content of the subject’s representation were inconsistent.

There seems to be good evidence that something like Curmnins’ account of error is
correct. In particular, it seems that there are good arguments for concluding that error is
best understood as what happens when there is a mismatch between the target of one’s
representation and its content. The main weakness in Curmnins’ account is that it tries to
do too much. A naturalized theory of representational content cannot account for

essentially normative aspects of mental representation. But nor does it need to account for

 

164 Cognitive Science: An Introduction. 2”d ed. eds. Neil Stillings, et al. London: MIT Press,
1995.p.90.

“5 Stillings, et al. (1995), p. 91.

168

all of this. Naturalized theories of content ought to explain representation in such a way
that allows for the possibility of error, and this can be a challenging task in and of itself.
However, they do not carry the burden of providing a theory of error, insofar as error is

not naturalizable.

6.2 The Problem of Representing Non-Existent Objects

Next we turn to the problem of how to account for the representation of objects that
do not exist. A lot of theories of representation have problems accounting for how we
represent objects that do not exist. For example, a theory of representation that claims
representations get their content from the objects that cause or invoke representations in
us, will need to make special efforts in order to explain how representations of unicorns
are caused—there are no unicorns to cause them. Similarly, since structural semantics has
it that the contents of representations are determined by what they are mapped to in the
external world, it will have trouble accounting for representations of things which are not
a part of that world. Nonetheless, since we clearly do have representations of objects
and/or states of affairs that do not exist externally, a good theory of representation ought

to be able to account for them.

6.2.1 The Axiom of Referring
Behind most of the problems associated with the representation of nonexistent things

is a principle which Avrum Stroll calls the “axiom of referring?“ In “Proper Names,

 

‘66 Stroll, Avrum. “Proper Names, Names, and Fictive Objects.” The Journal of Philosophy.
1998, pp. 522-34.

169

Names, and F ictive Objects” Stroll characterizes this principle as the view that reference
requires the existence of the referent. He is not the first to identify the axiom of
reference—as a matter of fact, a number of philosophers have accepted it as a starting

point for discussions of reference and naming, some expressions of which are given below:

Whatever is referred to must exist.167

it would not in general be correct to say that a statement was about Mr. X, or the so-and-so,
unless there were such a person or thing.168

He who does not acknowledge the nominatum cannot ascribe or deny a predicate to it.169

Stroll himself ultimately denies that the axiom of reference ought to be accepted,

claiming that it is “patently false.”

The irony of the situation is not merely that it is false, but that it is obviously so; for it is a
plain fact that we do use language to refer to nonexistent (including fictive) objects by name,
and to make true (or sometimes false) statements about such objects. I submit that we should
abandon the axiom of reference in any of its forms, since it is palpably false.170

Regardless of the merits of those who accept (or reject) the axiom of reference, it
seems that structural semantics is connnitted to some version of it. In particular, the
positive account of representation advanced by structural semanticists implies that
representation is impossible without something which is the object of the representation.

This follows fiom the fact that representation depends on the existence of a relation

 

‘67 Searle, John. Speech Acts. New York: Cambridge. 1969- P- 77.

‘68 Strawson, P.F. “On Referring,” in A. Flew, ed., Essayp in Conceptual Analfiis. New York:
Macmillan, 1960. p. 35.

'69 Frege, Gottlob. “On Sense and Nominatum,” in The flilosopby of Language. ed. A.P.
Martinich. New York : Oxford University Press, 1996.

”0 Stroll (1998), pp. 531-32.

170

between mental and external structures. When one half of the relation is missing,
representation cannot occur.

The upshot is that however representation of nonexistent objects is explained, it must
not imply that the axiom of reference is false. As a result, I prefer to adopt a rather
different terminology for this type of representation. Rather than call it “representation of
nonexistent things,” which seems to build in the rejection of the axiom of reference fiom
the onset, I will call this type of representation, “indirect representation” for reasons which
will become clear presently. Next, I will proceed to show how the structural semanticist
can construct an account of a class of “indirect” representations which is intuitively co-
extensive with those that we normally refer to as “representations of nonexistent things”
and how this account is consistent with what the theory has to say about representations in

general.

6.2.2 Indirect Representation

There are really two distinct issues that must be addressed in order to give a complete
account of indirect representation. First, structural semantics must be able to explain how
indirect representation is possible. That is, we require an explanation of what it is that
forms the other halfof the representation relation in cases where, intuitively, it seems that
nothing is available. As we shall see, there are at least three types of indirect
representation which will need to be described in order to fulfill the first of these
requirements. Second, it will be necessary to account for how it is that some indirect

representations are recognized by the subject doing the representing for what they

171

are—namely, as being about seemingly nonexistent objects—whereas others are not. This
is necessary because it is arguable that the content of a subject’s representation would
differ in each case even though the object which the representation involves remains the
same. Consider, for example, Virginia’s representation of Santa Claus: Speaking
intuitively, it seems that there must be a difference between the content of her
representation and the content of mine, who doubts his existence. If, in fact, the contents
of two such representations would be distinct, then we must explain how the theory of

structural semantics differentiates them.

6.2.2.1 The Classical Empiricist View

Before considering each type of indirect representation in detail, it is useful to note
some similarities between structural semantics and classical empiricism on the matter of
indirect representation. According to the classical empiricist view, every simple idea
formed by the mind corresponds to something in the external world. No idea can be
formed which is not either the result of direct experience or the mind’s operations on
direct experience. Concepts such as that of a unicorn, a Pegasus, or a ‘wirtuous horse” are
formed by combining simple ideas got directly fi'om contact with the external world as is

expressed in this passage from Htune’s An Enquiry Concerning Human Understanding.

Nothing, at first view, may seem more unbounded than the thought of man, which not only
escapes all human power and authority, but is not even restrained within the limits of nature
and reality. To form monsters, and join incongruous shapes and appearances, costs the
imagination no more trouble than to conceive the most natural and familiar objects. And while
the body is confined to one planet, along which it creeps with pain and difficulty; the thought
can in an instant transport us into the most distant regions of the universe; or even beyond the
universe, into the unbounded chaos, where nature is supposed to lie in total confusion. What

172

never was seen, or heard of, may yet be conceived; nor is any thing beyond the power of
thought, except what implies an absolute contradiction.

But though our thought seems to possess this unbounded liberty, we shall find, upon a
nearer examination, that it is really confined within very narrow limits, and that all this
creative power of the mind amounts to no more than the faculty of compounding, transposing,
augmenting, or diminishing the materials afforded us by the senses and experience. When we
think of a golden mountain, we only join two consistent ideas, gold and mountain, with which
we were formerly acquainted. A virtuous horse we can conceive; because, from our own
feeling, we can conceive virtue; and this we may unite to the figure and shape of a horse,
which is an animal familiar to us. In short, all the materials of thinking are derived either from
out outward or inward sentiment: the mixture and composition of these belongs alone to the
mind and will.171

Similarly Locke held that nothing contributes to knowledge save experience and

reflections on experience:

Let us then suppose the Mind to be, as we say, white Paper, void of all Characters, without any
Ideas; How comes it to be furnished? Whence comes it by that vast store, which the busy and
boundless Fancy of Man has painted on it, with an almost endless variety? Whence has it all
the materials of Reason and Knowledge? To this I answer, in one word, From Experience: In
that, all our Knowledge is founded; and from that it ultimately derives it self. Our Observation
employ’d either about external, sensible Objects; or about the internal Operations of our
Minds, perceived and reflected on by ourselves, is that, which supplies our Understandings
with all the materials of thinking. These two are the Fountains of Knowledge, from whence all
the Ideas we have, or can naturally have, do spring.172

If we apply the empiricist approach to the problem of explaining indirect
representation for structural semantics, we find that even though representations have
content in virtue of sharing structure with what is represented, what is represented need

not always be a single, unified object.173 Rather, the thing represented may simply be a set

 

171 Hume, David. An Enguigy Concerning Human Understanding. Section 11: Of the Origin of
Ideas, § 4 & 5. ed. Antony Flew. Chicago: Open Court Press, 2000. pp. 64 & 65.

”2 Locke, John. An Essay Concerning Human Understanding. Book 11, Chapter I: “OfIdeas in
general, and their Original.” Oxford: Clarendon Press, 1975. lines 15—26, p. 104.

173 Note that structural semantics differs from the classical empiricist view on the matter of

whether all representations must come flow the objects of experience. More will be said about this later,
but for now, note that the similarity between these views lies chiefly in the tendency of both to allow for

173

of properties which are instantiated by a number of different external objects. The objects
through which these properties are instantiated need not bear the relations to one another
that would make them form a more complex “object” in the typical sense of the term.”"
So, a representation of a unicorn could be the result of mapping some mental structure
onto selective properties of horses and horns, for example. And in general, when a thing
exists, one would argue there is a co-instantiation of its relevant properties (suppose they

are p ,, p2, and p3), and no co-instantiation of these properties when a thing doesn’t exist.

 

Representation of Representation of
non-existent object existing object

Figure 6-A: Representation of Non-Existent Objects

Happily, neither scenario prohibits the presence of homomorphic mappings between

mental structures and {p ,, p2, p3}. For such mappings are possible notwithstanding the

 

representations to possess structure which maps to more than one external object.

174 Perhaps the relations instantiated properties must bear to one another in order to form
“objects” are relations such as “share the same proximal space,” “exist in the same time,” etc. I leave this
question to the metaphysicians, and shall speak here simply of “co-instantiated” properties, leaving this
notion undefined.

174

possibility that p ,, p,, and p, are not instantiated in the same object; that is to say: Such
mappings are possible notwithstanding the possibility that the mapping between
representational and external relational systems is not one-to-one. See Figure 6-A.

At first, this way of explaining representation of non-existent objects may seem like a
weakening of the structural semanticist’s account of the representation relation. After all,
representations are supposed to share a structure with the things they represent. In fact,
that shared structure is at the crux of what makes mental structures “representations” at
all. Ifl represent a unicorn, then presumably I possess a mental structure which contains
elements corresponding roughly to a horn and to a horse, among other things. Moreover,
these elements certainly do stand in some relation to one another which indicates that the
horn belongs on the horse and not on some other object I might also be capable of
representing. How can I explain the representational content of this mental structure by
pointing to properties that are not co-instantiated by a single object? Clearly this is a case,
it will be said, where some of the structure had by mental representation is missing in the
elements of the external world to which it is mapped.

It is true tint some of the representations we have would seem to work better if
mapped to an object which possessed most of the structure present in the representation,
but it is important to realize that the structural semanticist’s account of representation is
one grounded in a homomorphic relation, not an isomorphic one. Since homomorphism
also drops the onto requirement, if is not necessary that every element of an external

relational structure be mapped into by its representation. The fact tint some aspects of a

175

mental structure are not replicated exactly in the external world does not prohibit one
from nonetheless identifying the mental structure as representational.

This may nonetheless seem like a hollow technical point. Surely most of what we do
successfully represent, we represent in virtue of shared structural similarities that approach
isomorphism. So even if the structural semanticist can provide a theoretical defense of less
“strict” forms of representation, we may feel that in many cases we are owed an account
of representation of non-existent objects where there is more shared structure than the
theory minimally requires.”’ I believe structural semantics can provide this more robust
account. Next, let us examine in detail some difl‘erent types of indirect representation in

order to see how.

6.2.2.2 Representation of Fictions

In cases such as the representation of tmicorns, it is arguable that our representations
do closely mirror the structure of certain objects in the external world. Pictures,
sculptures, and other renditions of unicorns are likely to instantiate many of the properties
related to one another in our representation. Of course, someone had to have a
representation of a unicorn without the presence of these artifacts, else they would not
exist themselves. But this does not in any way obfirscate the claim that the content of our

representations of unicorns is usually a function of the relation between them and any

 

”5 Note also that until now, the failure to achieve a 1-1 mapping between representation and
represented object has always stemmed fi'om the fact that there have been fewer elements in the
representing structure than in the represented one. In this case however, the failure stems from the
opposite circumstance.

176

number of drawings, paintings and statues of the creatures. Remember, structural
semantics is not a theory about why we have representations or about what caused the first
mental representation of some object or other. Rather, it is a view about how
representations have the content that they do. And this question is answered equally well
by structural semantics whether the representation in question is caused by a real unicorn,
or by a picture of one.

I will call representations which map to artifacts such as stories, pictures, myths, and
works of fiction in general, representations of fictions. There is a significant body of work
available which deals with representations of this sort. I’ll draw on some of that work in
what follows in order to illustrate how fictional representation can be transparent to the
subject.

6.2.2.3 Representation of Non-Physical Objects, Contradictory Objects,

Hallucinations and Ideas

Not all representations which seem to involve nonexistent objects can be explained by
pointing to an image or other representation of the object concerned. For example, there
are no pictures of the present king of France, nor of MacBeth’s dagger.176 Similarly, it is
certainly possible for us to have ideas which have not been committed to record, or to

develop mathematical structures which currently cannot be. m We have also to consider

 

'76 Although there will presumably often be linguistic representations of these objects: for
example, ‘the present king of France.’

177 I am thinking here of a 1,000,000,000-sided polygon or Gabriel’s hom—the parabolic conical
object with infinite surface area but finite volume.

177

representations of abstractions such as the general preferences and feelings of a culture or
of predicted and hoped for events. What will structural semantics have to say about these?

Some of the these examples can be handled by the classical enrpiricist view, if they do
not qualify as representations of fictions. For example, though there may in fact be no
present king of France, it is possible for us to form the representation just in case its
constituent parts may be found externally. Similarly, if I hallucinate a dagger, it arguable
that a representation is activated in me which maps quite naturally to external objects
(namely, real daggers). What makes it a hallucination is that presumably there is nothing
which is responsible for the activation of this representation. But as we have seen already,
structural semantics is a theory about the content of representations, not a theory about
the mechanism through which representations are activated. ”8

What about contradictory objects, such as Gabriel’s Horn? We can represent
Gabriel’s Horn as a parabolic conical structure which has an infinite surface area but a
finite volume. If one thinks of Gabriel’s Horn as an enormous paint pot, the idea is that the
Horn cannot contain enough paint to paint its own surface, no matter how thin a coat of
paint is used (See Figure 6-B). Surely such an object carmot exist in reality. But we

nonethelem seem to possess a representation of it.

 

”8 Hallucinations of things never experienced by the subject, or that cannot be broken down into

constituent parts which have themselves been represented by the subject may present a problem for
structural semantics. But it is not obvious that such hallucinations are possible.

178

 

 

 

Figure 6-B: Gabriel’s Horn

I would argue that what this case shows is not that we have representations of things
which do not exist in reality and therefore which defy structural semantics’ account of
representation, but rather, that the “external” structures to which our representations map,
do not always have to be physical structures. Gabriel’s Horn is paradoxical only as a
physical object—not as a mathematical object. There is nothing about the formal
description of this solid which in any way implies a mathematical inconsistency. As a
result, what we often represent when we represent Gabriel’s Horn is not a physical object,
but a mathematical one.I79

Admittedly, this may seem to require a commitment to the view that mathematical

structures “exist” externally to the mind—if not, then despite all, we still have a case in

 

”9 It is possible to represent Gabriel’s Horn imperfectly—as in Figure 6-B, which is a picture
suggestive of the actual structure of the Horn, but unreflective of all of its actual properties. This is not
really an example of mental representation though, as the picture is what is doing the representing here,
not the mind. Having said that. it is noteworthy that the representational capacity of Figure 6-B is
consistent with the structural semanticist’s theory of representational content. For although the picture is
not isomorphic to the mathematical structure, it does possess properties which can be mapped
homomorphically to it.

179

which the mind represents something which does not seem to exist in reality. While I don’t
want to deny that mathematical structures, ideas, or other sorts of abstraction might exist
in reality, neither do I want to commit myself to this view here. But there is really no need
to do so. For there is nothing in the structural semanticist’s account that prevents
structures formed in the mind from being the objects of representations
themselves—nothing that equates existence in reality with existence outside of the mind.
Presumably not all mental structures will be representational structures. Why not hold that
these can be the objects of representation? For that matter, it seems that there is nothing in
the structural semanticist’s account which would prevent other representational structures
from being the objects of representation. Indeed, something like this would most likely

form the basis of a structural semanticist’s account of introspection.

6.2.3 Meta-Mental Operators

In summary, what is usually referred to as representation of nonexistent objects is
really a kind of indirect representation of existing objects. Some representations are really
representations of pictures, myths or stories which themselves purport to describe
something “out there.” Others are composed of mental substructures, the elements of
which individually pick out external objects, but which are collectively associated in the
mind alone. Still others may be representations of other mental structures.

It is an advantage of structural semantics that it can explain different types of
representation (e. g., both ordinary representation and indirect representation) using

essentially the sanre explanatory model, but it ought not to leave us with no way of

180

distinguishing between them. It seems like common sense to assume that I can sometimes
represent the difference between an imaginary dagger and an actual one, and if so, there
must be something in the account of representation to explain what distinguishes the two
cases. However, considering only what has been said so far there would be fundamentally
no difference in content between “direct ” representations and representations of
seemingly nonexistent objects (“indirect ” representations), according to structural
semantics. To illustrate, consider the following two cases:

Case I: P has a representation R of a non-existent thing and knows that what he/she

represents doesn’t exist.

Case I]: P has a representation R of a non-existent thing and doesn’t know that what

he/ she represents doesn’t exist.

Note that for Case 11, P is not aware, and therefore does not represent that R is about
something with properties which are not co-instantiated, which are instantiated in a
picture, myth or story, or which are exemplified in some other mental structure. In other
words, P does not know, and therefore does not represent that R is an indirect
representation. This makes Case 11 the easier of the two to handle. For even though there
surely is a difference between direct representation and indirect representation, in this case
P does not know, and therefore does not represent the difference between the two. A
theory of representation does not have to explain a difference where no difference (in
representation) exists.

Case I, however, is more challenging. In this case, P knows that what he/she

represents is represented indirectly. Since all knowledge involves representation, P must

181

therefore represent that R is an indirect representation. This aspect of P’s representation
is not explained by the classic empiricist view concerning how the mind forms complex
ideas for example. For that view explains only how representations of properties which are
not in fact co-instantiated might come about, not how subjects might also represent the
fact that these properties are not co-instantiated. Similarly, we have not said how a
subject can represent that his/her representation of Pegasus is really relative to the
creature’s mythological description or how he/she can know that a representation is about
another of his/her own mental structures. Representing these facts seems to require
another level of discourse—a way of talking about representations and the relationships
between them. Representation such as this seems to suggest the need for a metalanguage.

Rod Bertolet suggests that when we knowingly represent seemingly nonexistent,
predicted, or fictional objects, people, and events, we do so with the use of a kind of
meta-mental operator.180 For example, consider the following statements which seem to
involve reference to nonexistent entities:

(1) Pegasus is a winged horse.

(2) Santa Claus has a white beard.

According to Bertolet, there are circumstances in which these statements are
semantically equivalent to the following:

(1') [According to Greek mythology] Pegasus was a winged horse.
(2') [According to legend] Santa Claus has a white beard.

 

18° Bertolet, Rod. “Reference, Fiction, and Fictions.” Smthese. Vol. 60, 1984. pp. 413-37.

182

Specifically, it is when the subject knows that his or her reference to Pegasus or Santa
Claus is indirect that the claims in (1') and (2') are semantically equivalent to those in (1)
and (2). When the subject does not realize that this is the case, the claims are not
necessarily equivalent. I say, “not necessarily” here because a subject who is not aware
that (1) is false, may nonetheless believe (1'). That is, a subject may believe that according
to myth, Pegasus was a winged horse, but may also believe that it is true in reality that
Pegasus was a winged horse. The bracketed portion of the statement is the meta-mental
operator. The job of the meta-mental operator in this case is to relativize the statements in
(1) and (2) to a certain context. The presence or absence of a meta-mental operator is the
difference between representations which are recognized by the subject as indirect, and
those which are not.

As an additional example, consider the following statement:

(3) The earthquake will be here in three hours.

This statement can be translated into:

(3') [According to the prediction] the earthquake will be here in three hours.

In this case, it is arguable that the meta-mental operator is doing more than it did in
the previous example. Here, the operator, when present, seems to relativize the truth of
the statement to a prediction. When absent, the sentence implies a belief that there will be

an earthquake, rather than merely the belief that there is a standing prediction. In this case,

183

the meta-mental operator alters the semantic content of the statement in a way not seen in

the first example.

But then, the objection continues, suppose that no earthquake comes: won’t (3), and what the
speaker says by uttering it, be false? And won’t it be false because no earthquake comes? And
doesn’t this show that with predictions, unlike perhaps the Pegasus myth, we are dealing not
just with what the prediction is, but with its accuracy as well—whether the world is, or comes
to be, as it claims? Doesn’t the parallel then break down, since whether what one says by
uttering ‘Pegasus is a winged horse’ is true depends only on what the story says, whereas
whether what one says by uttering ‘The earthquake will be here in three hours’ is true is not
settled by the content of the prediction alone? 81

The problem seems to be that (3) carries with it a certain endorsement of the
prediction which (3') does not. But if this is correct, then (3') and (3) are semantically
equivalent only in cases where the speaker suspends belief concerning whether or not the
prediction is correct. Hence, we ought to apply the meta-mental operator only in such
cases. Where the subject transparently believes the prediction to be true, the meta-mental
operator is not applied. Once again, this suggests that the role of the meta-mental operator
is to distinguish those cases in which the subject knows that his/her representations are
indirect fiom those in which the subject has no such knowledge.

Meta-mental operators should be thought of as firnctions which operate on the
representational structures to which they are applied. In general, they will take
representations as arguments, and return representations. Meta-mental operators which
take a representation r and return a result which describes r as indirect are the kinds of
operators which pronounce negatively or neutrally on their input. They are therefore the

kinds of operators which are needed for describing Case I above. Case 11 can be thought

 

'8' ibid., p. 428.

184

of as the “default” kind of case and can be described with operators of either of the two
remaining types—those which pronounce positively on their input, and those which are

neutral (see Figure 6-C).

n(r) = R
x(r) 1: R"

 

ml! men

p(r) pronounces positively on r
Mr) pronounces negetlvely on r
x(r) 13 neutral on r

Figure 6-C: Meta-Mental Operators As Functions

Although the notion of a meta-mental operator does introduce another level of
representational complexity to the structural semanticist’s account, it fits in nicely with the
overall strategy of the view. Using the basic idea that representation is a matter of
homomorphic mappings between structures as a starting point, structural semantics, with
the use of meta-mental operators, can account for the mechanics of indirect representation

while explaining how it is different from representation in the standard case.

185

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