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dissertation entitled

Quine's Criterion of Onotological Reduction

presented by

Dai Y. Yun

has been accepted towards fulfillment
of the requirements for

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QUINE'S CRITERION OF
ONTOLOGICAL REDUCTION

By

Dai Young Yun

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Philosophy

1997

ABSTRACT

QUINE'S CRITERION OF
ONT OLOGICAL REDUCTION

By

Dai Young Yun

In W. V. Quine's philosophy, there is an apparent dilemma or
conflict between his commitment to physicalism and his
encouragement of pure set-theoretic ontology through an ontological
reduction. My strategy to (dis)solve the dilemma is, first, to explicate
Quine's particular point of view about the two crucial notions of 'ontic
reduction' and 'ontic commitment. In particular, I emphasize the
requirement of a proxy function in Quine's conception of 'ontic
reduction', and contrast it with an adequacy condition of the
typical/standard conception of ontic reduction—namely, explicit
definability of the predicates of the reduced theory in terms of the
predicates of the reducing theory. The proxy function criterion for an
adequate ontic reduction, however, implies as its corollary a
structuralist view of ontology which in turn leads to 'ontic relativity'.
For Quine, however, this structuralism is not ontology per se but the
epistemology of ontology; while 'ontology' and 'truth' are theory-
immanent concepts. Thus, I conclude that Quine's position with
respect to the dilemma is identified with what he terms
naturalism—namely, "it is within science itself that reality is to be

identified and described," which is physicalism.

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ACKNOWLEDGMENTS

I am obliged to my thesis director, Professor Joseph F. Hanna,
for his meticulous guidance during the writing of this dissertation. I
am deeply grateful for his patience and encouragement, and endless
discussions. I am also grateful to the committee consisting of
Professors Albert Cafagna, Fred Gifford, and Richard Hall for their
helpful insights and critical comments.

A special thank must go to the late Professor Herbert Hendry,
my original thesis advisor. Despite his untimely death, his assistance
in the preparation of this thesis was invaluable. I am also thankful to
Professor Steve Esquith, Graduate Chair, and to Ms. Loriane Hodack,
Departmental Secretary, for their administrative help.

I am forever indebted to HJ and Min for their unconditional
support through the "eternity" of past years. They have simply
become sine qua non of my life and work. I am also indebted to my
extended family members who from distant places have generously

given me a helping hand, moral and otherwise.

Inspirational and edifying writing is admirable,
but the place for it is the novel, the poem, the
sermon, or the literary essay. Philosophers in the
professional sense have no peculiar fitness for it.
Neither have they any peculiar fitness for helping
to get society on an even keel, though we should
all do what we can. What just might fill these
perpetually crying needs is wisdom: sophia yes.

philosophia not necessarily. (Quine, 1981)

TABLE OF CONTENTS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. Introduction 1
1.1 The Dilemma 1

1.2 Formal Analysis and the Method of Explication ............... 2

1.3 'Ontic Reduction' and 'Ontic Commitment' ........................... 9

1.4 ASynopsis 31

. Frege's Logicism 34
2.1 Logicism 34

2.2 Frege's Definition of 'Number' 36

2.3 Deducibility and Truth-Preservation 45

2.4 Frege's Platonism 52

2.5 Adequacy Conditions 56

. Ascent to 'Truth' 59
3.1 Material Adequacy and Formal Correctness .................... 59

3.2 Tarski's Definition of 'Truth' and 'Satisfaction' ............... 65

3.3 Interdependence of Truth and Existence .......................... 69

3.4 Model-Theoretic Interpretation 79

. Ontological Relativity 92
4.1 Predicament of Frege's System 92

4.2 Multiplicity of Reduction or Model 97

4.3 Ontological Relativity 106

. Proxy Function 119
5.1 Ontic Decision 119
5.2 The Pythagorean Challenge 123

5.3 Quine's Criterion of Proxy Function 131

5.4 Ontology Naturalized 151
References 164

 

vi

Introduction

In this thesis, I address a single question that raises a possible
dilemma or incoherence in W. V. Quine's philosophy. My objective in
this dissertation is, first, to give a correct analysis of the issues
associated with the question, then to articulate some adequate and
comprehensive answers to the question, building upon Quine's own
views. Besides addressing this apparent dilemma (section 1.1),
however, this "introductory" chapter is also to provide a broad
background or framework for the main discussion of later chapters,
which will help to clarify the particular point of view developed in
Quine's various works. It consists first in a preliminary clarification
of the method of analysis and some key notions that are involved in
the main discussion (sections 1.2 and 1.3, respectively): secondly, a
synopsis of the chapters to follow and how they fit together,
including a brief statement anticipating the conclusion of my thesis

(section 1.4).

1.1 The Dilemma

My question concerns the following passages in Quine's
writings. Quine expresses, on the one hand, "unswerving belief in
external things—people, nerve endings, sticks, stones."1 This belief,

according to Quine's own criterion of ontological commitment,

 

1 W. V. Quine, "Things and Their Place in Theories," Theories and Things,
Harvard U., 1981, p.21.

commits him to an ontology of physical objects. On the other hand,
Quine encourages an ontological reduction. If "we make the drastic
ontological move," says Quine, "all physical objects go by the

board—atoms, particles, all—leaving only pure sets."2 For

By identifying each space-time point with a quadruple of real
or complex numbers according to an arbitrary system of
coordinates, we can explain the space-time regions as sets of
quadruples of numbers. The numbers themselves can be
constructed within set theory in known ways, and indeed in
pure set theory; that is, set theory with no individuals as
ground elements, set theory devoid of concrete objects. The
brave new ontology is, in short, the purely abstract ontology of
pure set theory, pure mathematics.3

These passages, and similar ones elsewhere in Quine's work,4 seem to
give rise to a dilemma; there is an apparent incompatibility between
his two philosophical positions, physicalism and pure set-theoretic
ontology. By 'physicalism' is meant the ontology of physical objects

plus sets of them, since sets are a must in any ontology.5

1.2 Formal Analysis and the Method of Explication
As is typical of attempts to resolve dilemmas that have arisen
in the analytic philosophical tradition, in this dissertation I am

seeking a correct understanding of some important notions or

 

2 W. V. Quine, "Facts of the Matter," in R. Shahan & C. Swoyer (eds.), Essays on
the Philosophy of W. V. Quine, U. of Oklahoma, 1979, p.165.

3 lbid., p.164.

4 See, for instance, "Whither Physical Objects?" in Boston Studies in the
Philosophy of Science, 39 (1976), pp.497-504; "Ontological Reduction and the
World of Numbers" in his The Ways of Paradox and Other Essays, Harvard U.,
1976, pp.212-220.

5 See W. V. Quine, "Wither Classes," Word and Object, MIT, 1960, pp.266-270;
"Scope and Language of Science," The Ways of Paradox and Other Essays,
Harvard U., 1976, p.244.

concepts that are involved in the present dilemma. In particular, my
goal is to provide a philosophically adequate analysis for the two
crucial notions of 'ontic commitment' of a theory and 'ontic reduction'
between theories. The analysis in question is of the general type that
Carnap and other philosophers (including Quine) have called an
explication or rational reconstruction, purporting to provide a
cognitive clarification for a familiar but, to some extent, defective
concept. The task of formal analysis (of explication or rational
reconstruction), thus, is not one of defining a new concept but rather
of redefining an old one. For instance, Tarski states in reference to
his "semantical" or formal analysis of the concept of truth that "The
desired definition does not aim to specify the meaning of a familiar
word used to denote a novel notion; on the contrary, it aims to catch
hold of the actual meaning of an old notion."6 It can generally be
agreed, therefore, that the method of explication consists in replacing
an old concept (the explicandum), used in a more or less vague way
either in everyday language or an earlier stage of scientific language,
with a new and precise notion (the explicans) which will be couched
in a more systematic language.

Such an explication does not need to establish a synonymy that
holds between the explicandum and the explicans. As a matter of
fact, the pursuit of a synonymy claim in connection with explication
could engender the so-called paradox of analysis, which I believe
usually goes as follows. Either we know the meaning of the

explicandum or we don‘t. But if the goal is synonymy and we already

 

6 A. Tarski, "The Semantic Conception of Truth," in L. Linsky (ed.), Semantics
and the Philosophy of Language, U. of Illinois, 1970, p.13.

know the meaning of the explicandum, then there is no need of
explication. While if we do not already know its meaning, then we
can never know if the explicans is adequate. It is, however, obvious
that the explicans must not be allowed to be unrelated in meaning to,
or semantically independent of, the explicandum. Thus, according to
Carnap, "it is not required that an explicatum [explicans] have, as
nearly as possible, the same meaning as the explicandum; it should,
however, correspond to the explicandum in such a way that it can be
used instead of the latter."7 In this sense of 'correspondence' in
meaning, the function of explication somewhat resembles that of
explicit definition, where the definiens makes it possible to
substitute for the definiendum salva veritate in every context in
which the latter may occur.

However, there is an important difference between definition
and explication. Carnap's theory of reduction-sentences, developed
first in his "Testability and Meaning" (1936), may offer a convenient
access to the difference. Previously, the logical empiricists, including
Carnap of the Aufbau (1928), held that in order to be empirically
meaningful a scientific term must be explicitly definable on the basis
of terms in the observation vocabulary, which refer to directly
observable aspects either of immediate phenomenal experience or of
physical objects. But dispositional terms (or dispositions in the
material, ontological mode) posed a problem for this conception of

empirical meaningfulness. So Carnap proposed, by the time of

 

7 R. Carnap, Meaning and Necessity, U. of Chicago, 1956 (2nd ed.), p.7. It is
noteworthy that there is not a uniform usage in the contrasting terms of
explication that are analogous to the familiar terms 'definiendum' and
'definiens'. Carnap uses the pair of terms 'explicandum' and 'explicatum' while
Quine uses 'explicandum' and 'explicans‘.

"Testability and Meaning", that the meaning of a scientific term is
introducible not by means of explicit definitions alone, but rather by
a more general method called 'reduction'. It was argued that the
predicate 'soluble in water', for example. cannot be introduced by

what might appear to be the obvious and explicit definition
(E) Sx <—) (Vt)(Wxt —~) Dxt),

namely, 'x is soluble in water just in case if x is put in water at any
time t, then it dissolves'. For on this definition, a piece of iron which
was never immersed in water would incorrectly be declared soluble.
The problem is that if the antecedent Wxt is false for all times t, then
the conditional 'Wxt —9 Dxt' would be true and therefore 'Sx' would be
true no matter what x is.

One way of avoiding this difficulty that Carnap considered was
to construe the definiens of (E) not as the standard extensional or
truth-functional conditional, but as the counterfactual conditional
which reads 'if x were to be put in water, then x would dissolve'. At
first glance, this construal seems to reflect the actual meaning of
'soluble' in scientific practice more closely than the original reading.
But to state (E) as a counterfactual or subjunctive conditional is to
assert that for a given object x, D must be connected with W by
virtue of a general law expressing a modal or causal connection.
Thus, to attain a clear understanding of the subjunctive (B), it would
be necessary to explicate first the concept of a general law or
nomological sentence or the concept of causal necessity. However,
since Hume raised it in connection with the problem of induction, the

concept of causal necessity has proved highly resistant to analytic

efforts. In regard to this enduring situation Quine laments that "the
Humean predicament is the human predicament."8

An alternative analytic strategy that avoids the difficulty
associated with (E) can be obtained, according to Carnap, by means of

the following reduction sentence:

(R) (Vx)(Vt)[Wxt —) (Sx (—) Dxt)].

This sentence uses extensional connectives, but no longer exhibits the
undesirable aspect of (E), i.e., the so—called paradox of 'material
implication'. For it consists in determining the meaning of '8' only
partly—namely, for just those objects that meet condition W. For
those which do not meet condition W the meaning of 'S' is left
unspecified. If an object is not subject to the test condition W, then
the entire formula (R) is true of it, but this implies nothing as to
whether the object does or does not have the property S. Thus, (R)
provides a precise formulation of the "open-ended" character of the
role that dispositions like S play in science, by expressing merely a
partial specification of the meaning of '8' associated with operational
criteria pertaining to different contexts.9

A fundamental difference between definition and reduction as
exemplified by Carnap's treatment of reduction sentences, therefore,
is as follows. Explicit definitions provide a means of translating

sentences into equivalent sentences, i.e., they provide both necessary

 

8 W. V. Quine, "Epistemology Naturalized," Ontological Relativity and Other
Essays, Columbia U., 1969, p.72. For more about this thorny "philosophic"
problem, see Nelson Goodman's Fact, Fiction, and Forecast, Harvard U., 1955.

9 R. Carnap, "Testability and Meaning," Philosophy of Science 3 (1936),
pp.419-471; 4 (1937), pp.1-40. See, for a fuller discussion, section 7; and
especially, pp.440-441.

and sufficient conditions in terms of the given vocabulary for the
application of the definiendum. The definition sentence thus makes it
possible to eliminate the definiendum entirely from any sentence in
favor of its definiens salva veritate. Reduction sentences of Carnap's
liberalized kind, on the other hand, do not in general give
equivalences. They give implications, which do not preserve the
truth values of original sentences, so that the specifying sentences do
not permit elimination of the definiendum from all contexts in which
it may occur. Reduction sentences explain a new term, if only
partially, by specifying some sentences which are implied by
sentences containing the term, and other sentences which imply
sentences containing the term.

Notice, however, the dual manner in which the analysis of
meaning enters in the above example. On the one hand, Carnap is
concerned here with the reduction of the meaning of a quasi-
theoretical dispositional term to manifest observation terms. On the
other hand, Carnap is engaged in an analysis whose goal is to
explicate the concept of 'empirical meaningfulness'. Carnap
introduces reduction-sentences as an "alternative" explication of
'empirical meaningfulness' that does achieve the goal of transforming
the material question of the ontology of dispositions into a question
of direct empirical observation (the truth conditions of the manifest
predicates) together with a formal question (the truth conditions of
the conditional and the other truth-functional connectives), albeit at
the cost of having provided only a "partial interpretation" of
dispositions. Again, on Carnap's notion of explication, there is no

requirement that the formal explicans provide a translation whose

truth conditions match those of the informal explicandum. This
"conventional" aspect regarding the choice of explicans, given the
explicandum, allows Carnap to view explication as largely a matter of
pragmatic consequences. Should the word 'raven' (or 'fish'), for
example, be defined in such a way as to be applicable only to black
birds (or gill-breathing marine animals)? Or should the possibility of
a white bird (or dolphins) not be excluded? From Carnap's point of
view, one explication is preferable to another if it proves itself more
useful, e.g., more useful in the development of biology. He regards
our knowledge as a matter of progressive improvement of the
explicit conventions or what he calls 'meaning postulates' within the
language of science.

This Carnapian methodology of analysis, then, seems to be
largely shared by Quine, when he defends his "quantificational"
criterion of ontological commitment against the philosophers of

ordinary-language:

Now a philological preoccupation with the unphilosophical use
of words is exactly what is wanted for many valuable
investigations, but it passes over, as irrelevant, one important
aspect of philosophical analysis—the creative aspect, which is
involved in the progressive refinement of scientific language.
In this aspect of philosophical analysis any revision of
notational forms and usages which will simplify theory, any
which will facilitate computations, any which will eliminate a
philosophical perplexity, is freely adopted as long as all
statements of science can be translated into the revised idiom
without loss of content germane to the scientific enterprise.lo

 

10 W. V. Quine (1953), "Reifications and Universals,” From a Logical Point of
View, Harvard U., 1980 (revised ed.), p.106.

But explication or philosophical analysis has not always been seen in
this way. Russell praises Frege's analysis of 'cardinal number'
because it "leaves unchanged the truth values of all propositions in
which cardinal numbers occur."11 Such a truth-preserving analysis
requires explicit-definitions by equivalent sentences, and rejects
postulational implicit-definitions as "theft over honest toil."12 Russell
had in mind the preservation of truth values not just within the
general Fregean model, but in more applied contexts as well. That is,
Russell wants "our numbers not merely to verify mathematical
formula, but to apply in the right way to common objects . . . and our
use of numbers in arithmetic must conform to this knowledge."13 He
apparently thought that this additional condition singles out the
Frege-Russell logicistic definition of number as the proper explication
of number, elucidating or conforming best to the "customary
meaning" of talk about numbers. Even though, from the arithmetic
point of view, other equally adequate explications of number such as

Zermelo's or von Neumann's are readily available.

1.3 'Ontic Reduction' and 'Ontic Commitment'

Evidently, a concept or predicate may require explication for
two separable sorts of reasons, depending on the nature of the
"vagueness" contained in the concept. We may phrase these different
senses as external and internal vagueness, in that the defective

concept stands in need of analysis, respectively, either to fix the

 

11 B. Russell, "Logical Atomism," in R. Marsh (ed.), Logic and Knowledge,
Unwin and Hyman, 1956, p.327.

12 B. Russell (1919), Introduction to Mathematical Philosophy, Allen &
Unwin, 1960, p.71.

13 Ibid., p.9.

10

boundaries of its application or to characterize "paradigm cases" of its
correct usages.l4 An example of the former would be the

replacement of the taxonomist's term 'piscis' by the ordinary
predicate 'fish', where the main effect of the new predicate is to
draw a sharper boundary between fishes and nonfishes. In everyday
discourse, our customary intuitions alone are not adequate to
determine whether dolphins, like sharks, are fishes. But the gill-
breathing and egg-laying 'piscis‘ clearly excludes dolphins from its
extension, despite the fact that they are marine animals for their
whole lifetime, while it includes sharks among the fishes proper.
Frege's analysis of the concept 'number', on the other hand, seems to
be a clearcut case of the latter (internal) sort of "vagueness". For,
prior to Frege, mathematicians were simply not in a confident
position to say what sort of entity a numeral really refers to. They
could not pinpoint or characterize what should be taken as a typical
instance of number, although everybody seemed to know exactly
how to enumerate the complete series of the whole numbers.

In their presystematic usages, both notions of 'ontic reduction'
and ‘ontic commitment' appear to be internally vague. However,
there is a danger of subtle confusion here. On the one hand, it may be
that particular cases of ontic reduction or commitment involve
concepts that are internally vague—as in the example of numerals.

But, on the other hand, it may also be that the concepts 'ontic

 

14 J. F. Hanna, "An Explication of 'Explication'" in Philosophy of Science, vol.
35 (1968), p.35 and p.39. Kaplan said similarly that "Ordinary vagueness is
external—that is, it concerns the difficulty of deciding whether or not
something belongs to the designated class. . . Meanings are open not only with
respect to what is comprised under the term but also with respect to what
should be taken as a typical, standard, or idea! instance" (in A. Kaplan, Conduct
of Inquiry, San Francisco, 1950, p.67).

11

reduction' and 'ontic commitment' are themselves internally vague.
That is, it might be the case that we simply cannot point to a clearcut
case of ontic reduction or commitment. The latter issue or question,
which might be construed as an issue over second-order concepts, is
my primary concern here. Since I am after all looking for the
particular point of view of Quine, among other philosophical views, as
to the concepts in question (in order to solve or dissolve the
"ontological" dilemma mentioned above). Of the two second-order
concepts under discussion, the question regarding the notion of 'ontic
commitment‘ appears to be less controversial than that regarding
'ontic reduction'. For there seem to have been no serious conceptions
or analyses of 'ontic commitment‘ that are alternatives to Quine's
own criterion of ontic commitment, which has since its inception set
the standard regarding the philosophical concept in question. As
Church has pointed out, "Quine's criterion of 'ontological commitment'
is indeed the only existing proposal" for clarifying the logical issue
associated with the perennial ontological debate among philosophers,
e.g., debate over ‘universals'.15 Whereas there are several different
but individually persuasive conceptions of the notion of ontic
reduction, as will be seen later. So let me begin with 'ontic
commitment' first and proceed with the task of providing a
philosophical background for Quine's criterion rather than an
analysis proper.

To reach an adequate or precise characterization of 'ontic

commitment', let us ask first what would count as the evidence for

 

15 A. Church, "Symposium: Ontological Commitment," The Journal of
Philosophy, vol. 55 (1958), p.1008.

12

someone's ontic commitment to a certain object or objects of a certain
kind—cg. numbers, sticks, stones or, for that matter, Homeric gods.
An obvious way to manifest such a commitment to an alleged entity,
which is being referred to in someone's discourse or theory, is simply
to maintain that the alleged entity exists. For the discourse or theory
would be false if that object did not exist. However, it has become
pervasive in philosophic thought since the Critique of Pure Reason to
deny that 'existence' is a predicate. As Kant put it, "'Being' is
obviously not a real predicate; that is, it is not a concept of something
which could be added to the concept of a thing. . . Logically, it is
merely the copula of a judgment."16 After Russell's "On Denoting"
(1905), the viewpoint gained an even wider following. Thus, in his

Mathematical Logic, Quine observes that

Russell undertook to resolve the anomalies of existence by
admitting the word 'exists' only in connection with descriptions,
and explaining the whole context '(1x)(. . . x . . .) exists' as short
for '(3y)(Vx)(x = y (-—) . . . x . . .)'. This course supplies a strict
technical meaning for Kant's vague declaration that 'exist' is not
a predicate; namely, 'exists' is not grammatically combinable
with a variable to form a matrix 'y exists'. (p.151)

Russell's "strict technical" method, capable of explicating Kant's
"vague" slogan, was his theory of singular descriptions which Ramsey
lauded as a paradigm of philosophical analysis.” The theory is best
appreciated when seen in the context of certain epistemic as well as

semantic positions that Russell advocates. Russell, at the threshold of

 

“5 I. Kant, Critique of Pure Reason (1787, 2d ed.), tr. by N. Kemp Smith, p.504.

17 F. P. Ramsey, The Foundation of Mathematics and Other Logical Essays, ed.
by R. Braithwaite, Routledge & Kegan Paul, 1931, p.263n: "that paradigm of
philosophy, Russell's theory of descriptions."

13

his theory, made a distinction between knowledge by acquaintance
and knowledge by description. His idea was to distinguish objects "of
which we are directly aware, without the intermediary of any
process of inference,”8 from objects that are not directly known to
us, but can be described by means of what Russell calls denoting
phrases or descriptions—such as 'a man' or 'the philosopher who
drank hemlock'. The kinds of things with which we can be

acquainted are, according to Russell, either particulars (e.g. present
sense-data, memories, and introspection of one's mental activity) or
universals (e.g. whiteness, roundness or the relation north-of, the
awareness of which Russell calls 'concepts'). They exhaust the domain
of our epistemological acquaintances, so that familiar physical objects
and other minds are not among the objects of acquaintance and will
only be known rather by descriptions.

Russell also made a semantic distinction between a logically
proper name and a name merely in a grammatical sense. A logically
proper name, Russell claimed, "can only be applied to a particular
with which the speaker is acquainted"; and for any expression, "if it
were really a name, the question of existence could not arise, because
a name has got to name something or it is not a name."19 On the
other hand, most common nouns and proper names, like 'Socrates',
that we view grammatically as names are not really (logically
proper) names at all, but are mere abbreviations for truncated
descriptions. For instance, 'Socrates' cannot simply stand for a

historical figure that we can be presently acquainted with, but stands

 

18 B. Russell, The Problems of Philosophy, Oxford U., 1912, p.25.
19 B. Russell, "The Philosophy of Logical Atomism," in R. Marsh (ed.), Logic
and Knowledge, Unwin and Hyman, 1956: respectively, p.201 and p.243.

14

rather for the person 'who was a teacher of Plato', 'who was married
to a woman named Xantippe' and so on. Thus, for both Kant and
Russell, the grammatical and logical forms of language are
incongruent, and it is the logical rather than the grammatical
structure of language that reveals correctly what there is or what
exists.

Traditionally, philosophers believed in the denotative theory of
meaning according to which the meaning of a (categorical) word is its
reference or denotation, which corresponds to something in the
world. In particular, they believed that all subject-expressions in
subject-predicate statements must name or denote, since otherwise
the sentences in which they occur would be meaningless. On this
view, only if a word like 'unicorn' denotes or refers to something is it
even meaningful to deny of unicorns that they exist. Meinong, for
instance, said that "any particular thing that isn't real (Nichtseiendes)
must at least be capable of serving as the object for those judgments
which grasp its Nichtsein. . . In order to know that there is no round
square, I must make a judgment about the round square."2° In other
words, Meinong regards any grammatically correct denoting phrase
as standing for an object, contrary to what we mean by asserting the
nonexistence of unicorns or round squares (i.e. there is no such thing
as a unicorn or a round square). If pressed for further explanation,
he would maintain that existence is one thing and subsistence, or

being, is another. Russell, who at one time accepted Meinong's

 

20 A. Meinong, "Kinds of Being," in G. Iseminger (ed.) Logic and Philosophy,
Appleton-Century-Croft, 1968. p.123.

15

view,21 soon found it absurd and intolerable, because it would not
only offend our vivid sense of reality, but ”infringe the law of
contradiction . . . that the round square is round, and also not
round."22 It was to solve this problem that Russell devised his theory
of singular descriptions.

Russell's solution was, as noted, to say that most common nouns
and proper names are in fact concealed descriptions, and are not
'logically proper' names at all—which are alone suitable for
occupying the subject (i.e. argument) place in genuine subject-
predicate statements. The names to which Russell's theory directly
applies are, thus, descriptive names such as 'the author of Waverly'
or 'the present King of France'; and Russell's theory of singular
descriptions makes explicit their logical form. Russell analyzes such
descriptive phrases systematically as fragments of the whole
sentences in which they occur, that is, as so-called incomplete
symbols which acquire meaning only in context and have no
significance on their own account. The sentence 'The present King of
France is wise', for example, is to be explained as a whole as meaning
'Something is the present King of France and is wise, and nothing else

is the present King of France'. Or, more formally,

(EXXKX & [(VY)(KY —> Y = 70] & WX).

 

21 In his early philosophy of the Principles of Mathematics (1903), Russell
himself embraced a Meinongean view that "what does not exist must be
something, or it would be meaningless to deny its existence; and hence we
need the concept of being [or subsistence], as that which belongs even to the
non-existent." (p.450)

22 B. Russell, "On Denoting" in Logic and Knowledge, p.45.

16

asserting that the unique existent x, which is the present King of
France, is wise.

The virtue of this analysis is that a sentence containing a
definite description is to be translated into an equivalent set of
sentences in which no references by description occur. In
consequence, the burden of objective reference which had been put
upon a descriptive phrase is now replaced by logical words like
'something', 'nothing' or 'everything'——namely, by variables of
quantification. These words or bound variables are an essential part
of our language, and their meaningfulness, at least in context, is not
to be challenged. When a statement like 'the present King of France is
wise' is analyzed by Russell's theory, thus, it ceases to contain any
expression which even purports to name the alleged entity whose
being is in question, so that the meaningfulness of the statement no
longer can be thought to demand that there be such an entity. For
the variables of quantification do not purport to be names at all; they
refer rather to entities generally, with a systematic ambiguity
peculiar to themselves. The statement 'the present King of France is
wise' is palpably meaningful, i.e., being either true or false, despite
having nothing for the subject-expression to denote. Russell's theory
of descriptions, therefore, shows how we might meaningfully use the
seeming names without supposing there to be the entities allegedly
named.

Russell did not, however, explore further and generalize the
results so as to arrive at a criterion which could be put to use in
order to determine the ontological commitment of any given theory

or piece of discourse. What is important to note in this connection is

17

Russell's "eliminative" explication or reduction of commitment to
entities through proper names and descriptive phrases to
commitment to entities through variables. It is now worth noting
that Quine's "criterion" of ontological commitment derives directly
from Russell's strategy, but without espousing either of Russell's
doctrines concerning logically proper names or the epistemic notions
of acquaintance and description. Quine's epistemological outlook,
which is holistic, is quite opposite in character to Russell's
thoroughgoing atomism.23 Quine's criterion, however, can be viewed
as a corollary of Russell's doctrine that "I take the notion of the

variable as fundamental."24 Since Quine maintains that

We can very easily involve ourselves in ontological
commitments by saying, for example, that there is something
(bound variable) which red houses and sunsets have in
common; or that there is something which is a prime number
larger than a million. But this is essentially the only way we
can involve ourselves in ontological commitments: by our use
of bound variables. The use of alleged names is no criterion, for
we can repudiate their namehood [as Russell has shown] . . . To
be assumed as an entity is, purely and simply, to be reckoned
as the value of a variable.25

Enough for a preliminary investigation of Quine's conception of
the notion of ontic commitment. A fuller account of the conception
must wait until later chapters (especially Chapter 4). Let us now turn
to the notion of 'ontic reduction' and provide a brief analysis of it.

The subject of reduction holds a special place of philosophical

 

23 For this point, see especially Quine's "Epistemology Naturalized" in his
Ontological Relativity and Other Essays, Columbia U., 1969, pp.69-90.

24 B. Russell, "On Denoting" in Logic and Knowledge, p.42.

25 W. V. Quine, "On What There Is," From a Logical Point of View , Harvard U.,
1980 (revised ed.), pp.12-l3.

18

interest, especially in logic and methodology of science. For reduction
affords an important mode of scientific explanation. It raises
intriguing questions concerning the relation between the concepts of
the reducing theory and of the reduced theory. Its potential scope
reaches across the boundary lines of scientific disciplines, as diverse
as physics and chemistry, biology, psychology, and the social and
historical disciplines. Reductions between these diverse fields
investigate and illustrate the relations that obtain between the terms
and the laws of two theories when one of them is reduced to the
other. There have been, for example, "theoretical reductions”
associated with the 'Unity of Science' movement, where one is
concerned to reduce chemical and biological, and ultimately
sociological and psychological, theories to physics. A special and
much discussed case would be the reduction of the phenomenal gas
laws (Boyle's law) to Newton's laws by means of "bridge principles"
that connect the properties of pressure and temperature with the
average momentum and average kinetic energy of the molecules of a
gas. These intertheoretic reductions are not necessarily to be
construed as ontological reductions, even if there appears to be an
ontological dimension within each such reduction. Reducibility
between sciences, as the question is usually treated in the
methodology of science, is rather construed as linguistic or
conceptual only, in that, for instance, the chemico-phenomenal term
'gas' is reduced to the physicalistic term 'collection of molecules in
motion'.

The linguistic construal of reduction goes back at least to early

logical empiricist studies, especially to the works of Carnap and

19

Neurath, which hold that all significant issues in the philosophy of
science can be restated so as to concern exclusively the syntax of the
language of science. In particular, Carnap emphasized that "the
question of the unity of science is meant here as a problem of the
logic of science, not of ontology, . . . concerning the logical
relationships between the terms and the laws of the various
branches of science" with respect to reducibility.26 And he further
claimed that, although the unity of laws of science may be at present
an unattainable goal, already "there is unity of language in science,
viz., a common reduction basis for the terms of all branches of
science, this basis consisting of a very narrow and homogeneous class
of terms of the physical thing-language."27 The construal of the
notion of reduction that is linguistic in a wider sense has proved
remarkably fruitful in many recent and current studies in general
methodology of science—after all "the linguistic turn" is the defining
characteristic of 20th century 'analytic philosophy'. However, one
further reason for the fascination that the subject of reduction has
held for philosophers lies, I think, in the ontological roots of many
questions concerning reductions in science. There are questions such
as: Are mental states nothing else than brain states? Are living
organisms simply nothing else than swarms of electrons and other
subatomic particles? Are social phenomena no more than complex
physicochemical processes? Or is it the case, as the "doctrine of
emergence" would have it, that as we move from subatomic particles

to molecules, to living organisms, and to social and cultural

 

26 R. Carnap, "Logical Foundations of the Unity of Science," in H. Feigl &. W.
Sellars (eds.), Readings in Philosophical Analysis, Appleton-Crofts, 1949, p.413.
27 Ibid., p.422.

20

phenomena, we encounter at each stage various novel phenomena
which are irreducible and cannot be accounted for in terms of
anything that is to be found on the preceding levels?28 These
questions appear to concern genuinely ontological issues, namely, the
basic identity or difference of various kinds of empirical states and
processes which, prime facie, exhibit striking differences.

The point is that the notion of 'ontological reduction' is pre-
analytically ambiguous. In other words, one and the same "reduction"
in science can be viewed either as linguistic and conceptual reduction
only or as demonstratively ontological. Carnap recognized such
ambiguity as a potential problem connected with explication and
argued that in the process of "clarifying the explicandum" it was
often necessary to distinguish different senses of the preanalytic
notion. Thus, in his study of the foundation of 'probability', Carnap
distinguished between two concepts of probability, namely, logical
probability (eventually explicated in terms of the technical concepts
of state-descriptions and structure-descriptions associated with
regimented languages) and empirical probability (explicated by
Reichenbach and others as the limiting relative frequency of
outcomes in infinite random sequences of events of a particular

type).29 Likewise, different senses of the explicandum 'ontological

 

23 For instance, Davidson argues for 'Anomalous Monism' (also known as
'token physicalism') according to which there is no mental substance other
than physical states and processes, but there are irreducibly mental ways of
grouping physical states and events. Thus, he uses the term 'supervenience'
rather than the traditional usage 'causation' or 'causal nexus' to characterize
the relationship between the mental and the physical. See D. Davidson, "Mental
Events" in his Essays on Actions and Events, Oxford-Clarendon, 1980.

29 "The Two Concepts of probability," in H. Feigl & W. Sellars (eds.), Readings
in Philosophical Analysis, op. cit., pp.338-339. Also his Logical Foundations of
Probability, Chicago U., 1950: especially pp.168-75.

21

reduction' must be identified before the notion can properly and
fully be explicated as intended in the present work. There are, I
believe, at least three distinctive conceptions of or approaches to the
preanalytic notion of ontological reduction. We may call them Frege-
Russell's "platonistic" approach, Carnap's "linguistic" or
conventionalist approach, and Quine's "structure-relative" approach.
Naturally, these diverse conceptions of 'ontic reduction' adduce
different standards for and constraints upon the adequacy of a
purported ontological reduction.

Let me start with the platonistic or realist conception. In the
lectures on "The Philosophy of Logical Atomism," Russell defends "the
view that you can get down in theory, if not in practice, to ultimate
simples, out of which the world is built, and that those simples have
a kind of reality not belonging to anything else."30 We saw
previously that Russell's "ultimate simples" are identified with the
objects of our acquaintances, which are immediately knowable
without intermediate inferences. Then, Russell formulated as a
philosophical method the central motto of his logical atomism:
"Wherever possible, substitute constructions out of known entities
for inferences to unknown entities."31 One of the ontological
consequences of this reductive method is that, for him, the force of
ontic commitment goes only up to the point "that I am not denying
the existence of anything; but I am only refusing to affirm it."32 An
example of such Russellian reductions was already seen in our earlier

discussion of his explication or formal analysis of the English word

 

30 R. Marsh (ed.), Logic and Knowledge, op. cit., p.270.
31 lbid., "Logical Atomism," p.326.
32 lbid., "The Philosophy of Logical Atomism," p.273.

22

'the' by means of the first-order logical vocabulary of quantifiers and
variables. As a result we need not be committed to the existence of
physical things and other minds as ultimate features of reality; they
are what Russell calls "logical fictions", as contrasted with his "logical
atoms" (to which we have epistemic access by acquaintance) that are
the ultimate constituents of the world.

Frege's conception of 'ontological reduction' is on a par with or
even more extreme than Russell's view, with respect to the
assumption of ultimate reality. As is well known, Frege initiated and
exemplified more confidently than anybody else the method of
"formal analysis" as a philosophical method, by laying down in
painstaking detail his logicist view that arithmetic is reducible to
logic—or to logic and set theory one might better say. given the
distinction that is standard in the field nowadays. For Frege,
however, the logicistic reduction in question is not a reduction to be
construed as having a mere linguistic or theoretical purpose only, but
rather as having a demonstratively ontological purpose. Because he
holds that "the individual number shows itself for what it is, a self-
subsistent object" that is there to be discovered rather than
invented.33 Numbers exist, he believes, neither as psychological
objects nor as physical objects; but they do exist nonetheless in the
sense that "the mathematician cannot create things at will, any more
than the geographer can; he too can only discover what is there and
give it a name."34 This conception of 'ontological reduction', as

pursued in Frege's logicism, is definitely ontological in character, but

 

33 G. Frege, Foundations of Arithmetic (1884, u. by J. L. Austin), 1950, New
York, Philosophical Library, p.68.
34 lbid., p.108.

23

not obviously reductive at all. It is rather discovery or demonstration
that the numbers always have been classes (logical objects) or sets,
but we mistakingly took them to be something else or to be no-thing
at all that can be distinguished from classes. Thus, in a sense, Frege-
Russell's conception of 'ontic reduction' is rather that of ontic
clarification. This understanding of 'reduction' resembles more or less
the reductive claims of "scientific realism" that, for instance, light is
in fact electromagnetic waves, temperature is in fact mean molecular
kinetic energy, and mental-states are in fact brain-states and so
forth.

For Carnap as the champion of "linguistic" reduction, on the
other hand, the Frege-Russell assumption-of or commitment-to
ultimate existence simply does not make sense. Carnap's so-called
'principle of tolerance' comes into play here. Consider two theories
that are mutually reducible to each other, say, the theory of ordered
pairs of real numbers and plane (two-dimensional) Euclidean
geometry. One theory is ontically committed to geometric points, and
the other to pairs of real numbers. In any given theoretic
construction representing the world, only one of these reductions, at
best, is likely to be imbued with ontological significance. Which
should it be? (In the Aufbau, Carnap finds the physical and the
phenomenal to be mutually reducible, and faces this problem in an

acute form.)35 Finding no logical reason that could be given in favor

 

35 Of course, for Carnap, there is "epistemic primacy" for the direction of
reducibility in question, even though no such ordering is available for the
ontic reduction. Because he believes ontological questions are simply
meaningless. His "epistemic primacy" prefers the reduction of the physical
into the phenomenal. See his Aufbau .

24

of either one of two such metaphysical presuppositions, Carnap

eventually declared that

In logic there are no morals. Everyone is at liberty to build up
his own logic, i.e. his own form of language, as he wishes. All
that is required of him is that, if he wishes to discuss it, he
must state his methods clearly, and give syntactical rules
instead of philosophical arguments.”6

This tolerant attitude illustrates the linguistic status of 'ontic
commitment' for Carnap, as opposed to the Frege-Russell realist

position. For, as he later expressed this point of view,

If someone wishes to speak in his language about a new kind of
entities, he has to introduce a system of new ways of speaking,
subject to new rules; we shall call this procedure the
construction of a framework for the new entities in question.
And now we must distinguish two kinds of question of
existence: first, questions of the existence of certain entities of
the new kind within the framework; we call them internal
questions; and second, questions concerning the existence or
reality of the framework itself, called external questions.37

Of course, for Carnap, the internal questions are ones that can
occur only from within the specified contexts of some linguistic
framework designed specifically for the purpose of talking about
such entities; hence, answers for them are meaningfully pursued and
empirically tested (in science) in the manner specified also by the
rules of the framework. Whereas the external questions occur

outside the context of any particular framework, and will thus be

 

36 R. Carnap, Logical Syntax of Language, (translated by Amethe Smeaton),
Routledge & Kegan Paul, 1937, p.52

37 R. Carnap, "Empiricism, Semantics, and Ontology," in L. Linsky (ed.),
Semantics and the Philosophy of Language , U. of Illinois, 1970, pp.209-210.

25

formulated without reference to any particular language-framework
and thereby without reference to its purported ontology or reality.
They are meaningless questions for Carnap, for their answers will be
neither true nor false. Hence, for Carnap, ontological questions of an
absolute kind such as 'Are there numbers?‘ or 'Are there properties?’
or 'Are there bodies?‘ are meaningless, prior to our choice of a
linguistic framework, where words like 'number', 'property' and
'body' function as what he calls Allwo'rter or category words. Their
truth values are not verifiable (and, consequently, their claims not
meaningful) outside the framework of number-language, property-
language or thing-language, respectively. And within the framework
the external questions say nothing about the actual data of
experience, for they are analytic sentences deducible from our
"conventional" choice of the linguistic framework itself. Such
sentences become what Carnap calls 'meaning postulates' in such a
linguistic framework or theory. Sentences such as 'There are bodies'
or 'There are numbers' or 'There are properties' are trivially true,
either logically true or analytically true, because they are trivially
deducible either from the structure of the language itself or from the
meaning-postulates by fiat.

In consequence Carnap takes the paradigm of reduction to
involve. fundamentally, linguistic or conceptual translation only.
Specifically, Carnap states, "an object is said to be reducible to others,
if all statements about it can be translated into statements which

speak only about these other objects."38 In fact, in Carnap's

 

38 R. Carnap, The Logical Structure of the World (tr. by R. George), Berkeley,
1967, p.60.

26

conception of ontic commitment, "An alleged statement of the reality
of the framework of entities is a pseudo-statement without cognitive
content . . . cannot be judged as being either true or false . . . can only
be judged as being more or less expedient, fruitful, conducive to the
aim for which the language is intended.”9 Therefore, all that is
required for the construction of an object out of other objects or the
reduction of an object to other objects, according to Carnap, is a
transformation-rule specifying how a statement about the
constructed object can be translated into statements that refer only
to the previously given objects. This conception of 'ontological
reduction', then, obviously aims at conceptual economy with respect
to the 'Unity of Science'—especially, unity of language in science,
rather than economy in ontology or ontic commitments.

Quine rejects Carnap's dichotomy between internal and
external questions of ontology because of its intimate connection
with the semantic distinction between synthetic and analytic
statements—namely, a statement is analytic, or concerns "external"
question, when it is true by virtue of meanings and independently of
fact. Regarding the latter distinction, however, Quine comments that
"that there is such a distinction to be drawn at all is an unempirical
dogma of empiricists, a metaphysical article of faith.”0 The reason
Quine gives for rejecting the distinction derives from the doctrine of
'epistemic holism' (often associated with Duhem's thesis), according

to which "our statement about the external world face the tribunal of

 

39 R. Carnap, "Empiricism, Semantics, and Ontology," in L. Linsky (ed.), op.
cit., p.219.

40 W. V. Quine, "Two Dogmas of Empiricism," From a Logical Point of View,
Harvard U., 1980 (revised ed.), p.37.

27

sense experience not individually but only as a corporate body."“'1 In
our theories about the world, individual sentences cannot be
conclusively verified or conclusively falsified by the evidence of our
senses. For such sentences occur as part of a more general theory,
and because of this we have a choice about where to alter the theory
when things go wrong at the observational level. For instance, in
testing a scientific hypothesis, we are never forced by recalcitrant
experiences to alter the hypothesis in question, but can make a
drastic change elsewhere in the theory, perhaps even revising the
laws of logic, thereby preserving the hypothesis. Indeed,
Heisenberg's indeterminacy principle has been held to be a reason to
reject the law of excluded middle, because it provides a means of
simplifying quantum physics. Conversely, no sentence in our theory
is completely immune from revision. This breaks, then, the tenability
of the synthetic-analytic distinction, for an analytic statement has
been construed by Carnap or others as one that is confirmed no
matter what. If there is no proper distinction drawn between
analytic and synthetic, then there remains no basis at all for the
dichotomy that Carnap urges between "external" (ontological or
metaphysical) and "internal" (empirical or factual) statements of
existence.

And the untenability of the external-internal dichotomy leads

to Quine's claim that "ontological questions then end up on a par with

 

41 lbid., p.41. The Duhem thesis, thus, is the thesis of "underdetermination" of
a theory by evidence.

28

questions of natural science.”2 This point becomes more forceful

when Quine proclaims later in his Word and Object that

The quest of a simplest, clearest overall pattern of canonical
notation is not to be distinguished from a quest of ultimate
categories, a limning of the most general traits of reality. Nor
let it be retorted that such constructions are conventional
affairs not dictated by reality, for may not the same be said of
a physical theory? True, such is the nature of reality that one
physical theory will get us around better than another; but
similarly for canonical notations.43

Oddly, this is the result of stressing rather than downplaying the
importance of Carnap's linguistic doctrine. By extending Carnap's
doctrine to scientific and theoretical questions in general, not
confining it only to the philosophical questions as Carnap did, Quine
manages to reach what he calls the method of "semantic ascent". It is
the semantic shift from talking in certain terms to talking about
them, regardless of any subject matter. In essence it is a general
method of reduction or explication "from the material (inhaltlich)
mode into the formal mode, to invoke an old terminology of
Carnap's."44 In particular, in dealing with the ontological question of
what a theory's commitments to objects consists in, this semantically
ascending method of explication is able to provide what has been
known as Quine's criterion of ontological commitment—'to be is to be

the value of a variable'. For the criterion is simply a semantic test of

 

42 W. V. Quine, "On Carnap's Views on Ontology," The Ways of Paradox and
Other Essays, Harvard U., 1976, p.211.

43 w. v. Quine, Word and Object, MIT, 1960, p.161.

44 lbid., p.271.

29

what a theory says there is, not of what there is. What there is is
what a true theory says there is.

As a result, Quine cannot accept Carnap's linguistic conception
of 'ontic reduction' according to which the ontological
(re)interpretation of a language or theory can be carried out in a
fully determinate and independent manner, depending only on our

"arbitrary' choice of the language or framework in which the entities

needed are posited by fiat. This is a mistaken view from Quine's
standpoint. Because Quine's holistic attitude in science in general,
together with his method of semantic ascent (which would shift the
intertheoretic reduction-talk to a metalanguage or background

language), makes him observe:

Although signs introduced by definition are formally arbitrary,
more than such arbitrary notational convention is involved in
questions of definability; otherwise any expression might be
said to be definable on the basis of any expressions whatever.
When we speak of definability, or of finding a definition for a
given sign, we have in mind some traditional usage of the sign
antecedent to the definition in question . . . For such conformity
it is necessary and sufficient that every context of the sign
which was true and every context which was false under
traditional usage be construed by the definition as an
abbreviation of some other statement which is correspondingly
true or false under the established meanings of its signs.”

In other words, reduction is meaningful only if there is a further
language or a set of rules in which both the reduced and reducing
theories are related by the reductive definition in question. But if
there is any ontological significance to the sort of linguistic

construction Carnap proposes for the purpose of ontic reduction, it is

 

45 W. V. Quine, "Truth by Convention," in his Ways of Paradox, pp.78-79.

30

not because the construction serves as a set of reductive rules or
definitions for reducing or eliminating old entities in favor of new
objects, but rather because the construction serves as a means of
initiating talk of new entities entirely unrelated to old objects. Such
an ontic proposal may be regarded as an ontological introduction
rather than reduction, based on its own "undefined" meaning-
postulates which are given merely by linguistic decree. Quine argues,
however, that "what makes sense is not what the objects of a theory
are, absolutely speaking, but how one theory of objects is
interpretable or reinterpretable in another."46 For whenever we
undertake a theory-building by specifying the values of its variables
or extensions of its predicates we do so only from the vantage point
of some background theory or language; otherwise the theory would
be inexplicable or meaningless. (This notion of 'background theory'
will play an important role when we formulate later in a precise
form Quine's conception of 'ontic reduction', alternative to the so-
called "standard/typical" conception of 'ontic reduction'.)

Thus, Quine has urged that the translation-rules employed for
the purpose of ontic reduction between theories must be reflected in
the resources of a background theory where the reduction in
question proceeds. The sort of translation or (re)interpretation
wanted for ontic reduction can be achieved only by means of a
paraphrase of the original or reduced theory in terms of the
antecedently familiar terms of the background theory, in this case,

the reducing theory. This is so because "We are finding no clear

 

46 W. V. Quine, "Ontological Relativity," Ontological Relativity and Other
Essays, Columbia U., 1969, p.50.

31

difference between specifying a universe of discourse [for a given
theory]. . . and reducing that universe into some other."47 In such an
ontic reduction or specification, however, all that is required is the
internal-structure, not the objects themselves, of the reduced theory
to be mirrored somewhere in the vocabulary and sentential-
structure of the background theory or the reducing theory. Within
the accepted terms of our most inclusive background theory,
whatever that may be, however, we may indeed introduce genuinely
new types of entities in order to strengthen and simplify the overall
theory of the world—namely, the positing of the ultimate reality,
pending our ontic decision. Such entities are not reduced to or
equated with any other objects, not determined by any explicit rules
or definitions; but characterized only by the way the laws governing

them are linked with the rest of the theory itself.

1.4 A Synopsis

So far we've engaged in a preliminary examination of the
central concepts of 'ontic reduction' and 'ontic commitment'. It was
seen that we could identify at least three different conceptions of
'ontic reduction‘, reflecting different attitudes toward 'ontic
commitment'—namely, the platonistic, linguistic, and "theory
immanent" views. What has been said so far about the two concepts,
however, is somewhat inaccurate and vague. A precise
understanding or explication of them is required, prior to our being
ready to solve or dissolve the dilemma mentioned earlier. Just how

such explication should proceed cannot be said without successfully

 

47 lbid., p.43.

 

32

finishing the required study; but we know something about it from
the historical examples of ontic reduction and commitment, such as
Frege's logicism.

In the next chapter, we take as our paradigm of ontological
reduction and commitment Frege's platonistic logicism, according to
which arithmetic (and the subject matter of arithmetic) is reducible
to set theory. In showing what it is to reduce one domain of objects
(numbers) to another (sets or classes), we shall be able to illustrate a
set of conditions for an adequate ontological reduction. Presumably,
one theory is reducible to another just in case the first can be
translated or interpreted in the second, in such a way that the
translation preserves predicate structure and "theoremhood"—
namely, 'definability' and 'derivability'. In Chapter 3, we generalize
these conditions of adequacy for 'ontic reduction' in a precise form
provided by the "analytic technique" Tarski employs in his 'semantic
conception' of truth. Eventually, we will identify Tarski's notion of
interpretability of one theory in another as the formal representation
of our "typical/standard" adequacy conditions of 'ontic reduction'.

In Chapter 4, we consider some consequences of adopting
Tarski's 'interpretability'—in particular, the 'ontological relativity'
resulting from the multiple interpretability of a given theory. This
will lead us to reject the notions of 'ontological commitment'
associated with Frege-Russell's external realism or Carnap's linguistic
conventionalism. The multiplicity in question will be presented
through the point of view of Quine's famous thesis of 'indeterminacy
of translation', with set theoretic examples. As a consequence, we

need an alternative approach to 'ontological reduction' as well, since

33

different conceptions of 'ontic commitment' require different
accounts of 'ontic reduction'. In Chapter 5, we shall see some
inadequacies of Tarski's ‘interpretability' as the formal
representation of adequate ontic reduction. For the requirement of
biconditionals in his notion of 'possible definition' is too strong to
count as adequate many preanalytically acceptable reductions. Then,
I will present Quine's "particular" criterion or view on 'ontological
reduction'— namely, the requirement of a proxy function between
the reduced and reducing theories. But the criterion does not require
any "explicit" intertheoretic definability. A corollary of the proxy-
function ontic reduction is the "structuralist" view of ontology. That
is, what is essential to a theory is not its objects but its structure, and
such a structuralist ontology would give us a clue to a solution or

dissolution of the dilemma addressed in the beginning.

2. Frege's Logicism

In this chapter we will discuss Frege's logicistic programme
that involves not only a paradigm case of ontic reduction but also a
particular form of ontic commitment which has been called
'platonism'. Frege attempts an ontological reduction of arithmetic to
logic (and set theory) because he is deeply committed to a platonistic
ontology which holds that numbers are sets or classes. What is
important for our present concern, however, is not the logicistic
paradigm itself, but the relation between the account of reduction
per se and questions of ontology. Accordingly, by presenting Frege's
conviction in platonism and his logicist method of 'reduction', we
shall be able to illustrate a "standard/typical" set of conditions for
the adequacy of an ontic reduction—namely, definability of concepts

and derivability of truths.

2.1 Logicism

To study logicism as a paradigm of 'ontological reduction', we
are interested not in detailed execution of the project—such as
elaborated in Russell and Whitehead's Principia Mathematica or in
Frege's Grundgesetze der Arithmatik, but rather in the underlying
concepts or principles of the reduction in question. Logicism may
take many forms. The simplest account of it would be that arithmetic
can be translated, in principle, into logic. A more articulated account,

often regarded as the standard account, of what a logicist reduction is

34

35

may be found in the preface to Russell's The Principle of

mathematics:

[Logicism is the claim that] all pure mathematics deals
exclusively with concepts definable in terms of a very small
number of fundamental logical concepts, and that all its
propositions are deducible from a very small number of
fundamental logical principles. (p.xv)

We may characterize, thus, the underlying principles of logicism as
the principles of definability of arithmetic concepts in terms of
logical vocabulary, and deducibility of arithmetic theorems by means
of logical truths. We shall take definability first, and discuss
deducibility later.

It was Frege, however, in his Foundations of Arithmetic who
was particularly reflective on the philosophical foundations of the
logicist program, whereas Russell seemed to have more interest in
actually doing the reductive work. So let us discuss Frege in detail,
since we are more interested in the conceptual basis of logicism than
in its execution. Already before Frege, mathematicians in their
investigations of the interdependence of various mathematical
concepts had shown, though often without being able to provide
precise definitions, that all the concepts of arithmetic are reducible to
the natural numbers. Thus, in order to establish his logicist thesis,
Frege had only to give a definition of natural numbers in terms of
purely logical concepts, and to show in addition the logicist
translation of 'mathematical induction' in terms of logical inferences.
It is instructive to notice that Frege at the outset of his inquiry lays

down his methodological principles: "(i) always to separate sharply

36

the psychological from the logical, the subjective from the objective;
(ii) never to ask for the meaning of a word in isolation, but only in
the context of a proposition; (iii) never to lose sight of the distinction

between concept and object."1

2.2 Frege's Definition of 'Number'

Frege began his attempt to define number with the question
"How, then, are numbers to be given to us?"2 Since he believed that
numbers are not accessible by sense-perception or by Kantian
intuition, he answered that we have no choice but to look at the way
the number words are used in our judgments or statements. For
Frege, thus, to define number is to capture the appropriate content of
a statement in which a number word occurs. This way of introducing
numbers may be called a "contextual definition" of number. In
addition, he boldly claims that the content of a statement of number
concerns "concepts" rather than external things, contrary to what
most of Frege's predecessors had long believed. His bold claim is not

without justification. To prove his claim, Frege pointed out that

a statement of number contains an assertion about a concept.
This is perhaps clearest with the number 0. If I say "Venus has
0 moons", there simply does not exist any moon or
agglomeration of moons for anything to be asserted of; but
what happens is that a property is assigned to the concept
"moon of Venus", namely that of including nothing under it.
(Foundations of Arithmetic, p.59)

 

1 G. Frege, Foundations of Arithmetic (1884, tr. by J. L. Austin), 1950, New
York, Philosophical Library, p.x. Romanenumerals are added.

2 lbid., p.73. Further parenthetical page reference to Frege will be to this
work, unless noted otherwise.

37

To understand this, we must apprehend that Frege's term 'concept‘ is
a technical word of his philosophical semantics and stands in
opposition to the term 'object'. Frege's concepts are not subjective
ideas, nor they are identical with predicates. They are not predicates;
rather, predicates refer to them so that a predicate truly
characterizes an object just in case that object "falls under" or
"belongs to" the concept to which the predicate refers. Nor are
Fregean concepts subjective, because an object falls under a concept
whether or not anyone asserts that it does. For instance, the fact that
the Earth has one moon would be true even if no one ever thought or
said so. To have the number one, according to Frege, is a property not
of an idea or a word but of the concept moon of the Earth. In thus
ascribing numbers to concepts, Frege could guarantee both the
objectivity and the linguistic independence of numerical truths. As a
consequence, Frege was able to propose the thesis that statements of
number contain assertions about concepts, not about external things.3

Frege then goes on to suggest the following definition for the
sense of a complete statement involving a number, e.g. 'Jupiter has
four moons' or 'The number of Jupiter's moons is four'. In the
notation of modern logic, where F is an arbitrary concept or

predicate, his "contextual definition" may be expressed

'0 belongs to F' if and only if '(Vx)-Fx'

 

3 Frege also uses this doctrine to criticize the ontological argument for the
existence of God, to the effect that existence, like number, is a property of
concepts, not of external thing—i.e. God. See ibid., p.65.

38

that is, the number 0 belongs to a concept if no object falls under

that concept. Similarly, we would have

'1 belongs to F' iff '(3x)[Fx & (Vy)(Fy -) x = y)]'
'2 belongs to F' iff '(3x)(3y){Fx & Fy & (Vz)[Fz —-> (z = x v z = y)]}'

'n-t-l belongs to F' iff
'(3x)[Fx & it belongs to the concept "F(z) & (2 at x)"]'

In essence, what is said in these definitions is that the number m
belongs to a concept P if, and only if, there are exactly m objects
falling under that concept or, simply, there are exactly m F's. The
purpose of this scheme was to eliminate some occurrences of
"numerals" and expressions of the form "the number of F's", so that
numbers can be defined in the contexts using only logical symbols,
especially in the context of 'there are exactly m F's'.

Frege noticed, however, that these "logical" formulas of
number—concepts did not constitute an adequate definition of
number. Because, if useful at all, they are not strong enough to
enable us to prove that the number of F's is the same as the number
of G's. Nor do these formulas explain sentences of the form 'the
number of F's is x' with 'x' a variable, that is, with an indeterminate
number. Moreover, in such logical paraphrases of number
expressions, there seems to be no hint of how to decide whether any
given number, like 2, is the same or different from any other object,
say JULIUS CAESAR; in fact, whether JULIUS CAESAR is a number or
not. In short, Frege's preliminary attempt at the definition of number
did not help to prove "numerical identity". But we know that no

number-theoretic definition without numerical identity is adequate

39

for arithmetic, since identities are the most typical statements of
arithmetic.

The diagnosis of the difficulty with the above definitional
scheme is that it treats numbers not as singular terms, but merely as
elements of complex predicates or of numerical quantifiers—namely,
'there are n -things such that'. For number theory, however, we need
an account that introduces numerals as singular terms, in that each
numeral has its unique reference to be identifiable and reidentifiable
in various contexts. That would be possible only if numbers are
treated as objects which in turn make numerical identity
possible—say, 'no entity without identity'. That numbers are objects,
as a matter of fact, is another indispensable thesis for Frege's
logicistic programme. Frege's technical term 'object' used here also
belongs to his philosophical semantics, and it is being used as
standing in opposition to his term 'concept' previously mentioned.
For Frege, an object is the type of thing that can be referred to only
by a singular term, i.e., by names (including sentential-names) and
definite descriptions; whereas a concept is the type of thing that can
be referred to only by a general term or predicate. The relationship
between them, again, may be best characterized by saying that an
object falls under a concept if and only if its value for that object as
argument is true.4

Frege, to prove that numbers are objects, suggested that we
look at their identity conditions for a clue, which would capture the

sense of the statement that the number which belongs to the concept

 

4 For more about the distinction, see Frege's "On Concept and Object" and "On
Sense and Reference" in P. T. Geach & M. Black (tr. & ed.), Translations From
the Philosophical Writings of Gottlob Frege, Philosophical Library, 1960.

40

F is identical with the number which belongs to the concept G. He
adopted as his definition of "general" identity Leibniz's principle of
the substitutivity of identicals—namely, given a true statement of
identity, one of its two terms may be substituted for the other in any
true statement salva veritate. A bit more is desired, however,
because Frege wishes to be able to say that, for instance, the number
of Jupiter's moons is four but not JULIUS CAESAR. What Frege wants,
in addition to Leibniz's principle, is a "particular" identity that is
suitable specifically for numerical contexts. He argues that the notion
of one-to-one correspondence is capable of playing that "particular"

role, in such a way that

the number that belongs to the concept F is identical with the
number that belongs to the concept G if and only if the F's and
the G's can be put into one-to-one correspondence.

For Frege it is obvious that numbers measure the size of collections,

and two collections are of the same size if and only if their members
have a one-to-one correlation. And to show that the notion of one-to-
one correspondence does not presuppose that of number, Frege gives

the following example:

If a waiter wishes to be certain of laying exactly as many
knives on a table as plates, he has no need to count either of
them; all he has to do is to lay immediately to the right of
every plate a knife, taking care that every knife on the table
lies immediately to the right of a plate. (pp.81-82)

In fact, the notion of such correlation can easily be defined by a 1-1

correspondence relation, that is, by the two sentences

41

'(Vx)(Vy)(Vz)((ny & sz) —> y = z)' and '(Vx)(Vy)(Vz)((Ryx & sz) —) y
= z)' combined, where 'R' represents a matching relation.

Yet this criterion of one-to-one correspondence by itself cannot
properly address whether numbers are logical objects. For there
seems to be no such one-to-one correlation between the objects in
logic and the objects in arithmetic. In logic, we have only two logical
objects, the two truth values, even though we have plenty of
concepts. Whereas there are infinitely many numerical objects in
arithmetic. So it is only natural that Frege was led to look at concepts,
and somehow to try to connect them with logical objects, so as to
generate a one-to-one relationship between numbers and
"conceptual" objects. But we have a problem that identity only holds
between objects, and is not available for concepts. Fortunately, there
is a traditional criterion for an analogous notion of identity for
concepts—namely, every concept F has a class "x(Fx) as its extension.
And if we let the classes "x(Fx) and "x(Gx) represent the logical
objects associated with the concepts F and G, the one-to-one

correspondence in question may be formulated as
"x(Fx) = "x(Gx) <-—) (Vx)(Fx H Gx),

which is the famous axiom V of the logicist system in Frege's

Grundgesetze der Arithmetik.5 Thus, Frege was able to say that

The number which belongs to the concept F is the extension of
the concept "equal to the concept F". (pp.79-80)

 

5 G. Frege, The Basic Laws of Arithmetic (originally 2 vols., 1893 & 1903), tr. &
ed. by M. Furth, Berkeley, 1964. See, for a class as the extension of a concept,
sections 34 and 55; and for Frege's axiom V, see section 20. Also very useful is
Furth's expository "introduction".

42

(Frege uses interchangeably the phrases 'is equal to' and 'is in one-
to—one correlation with'——see sections 68 and 70.)

So each positive integer k is identifiable with a particular class,
which is the extension of a concept F in case the F's are exactly as
many as k or one-to-one correlated with k. And if Fregean concepts
are construed as classes themselves, (as was the case in Frege's later
work Grundgesetze but not in the Grundlagen), then the number five,
for instance, can be identified as the class of all classes that have five
elements. In general, therefore, we may conclude that Frege
succeeded in defining numbers in terms of classes or the extensions
of concepts, which he regarded as logical objects.

However, a definition of number is hardly complete unless it is
possible to individuate each number in relation with each other. For,
otherwise, there may be a logicist definition of individual numbers
but no logicist arithmetic in general, in the sense that numbers are
useful not only for measuring the size of collections, but also for
counting in a series of succession or progression. Such a definition of
"number-series", then, will give us the logicist notion of 'x is a
number' where 'x' ambiguously refers to an indeterminate number.
Thus, Frege next tries to individuate each number in the number
series in terms of zero and the successor function. A good start to this
effect may be seen in the definition of the dyadic term 'ancestor' on
the basis of 'parent'. To simplify the situation let us count a person as
one of his own ancestors, thereby counting as a person's ancestors
not only his parents, grandparents, and so on, but also the person

himself. Frege showed us by using second order logic how to get an

43

appropriate definition for 'x is an ancestor of y'.6 To understand his
procedure, let us call a property "F-hereditary" just in case, for every

x and y, if x has F and is a parent of y, then y has F: in symbols,
F-hereditary iff (Vx)(Vy)[(Fx & x parent y) —> Fy]

Hereditary properties are, in other words, those properties that are
passed from parent to child. So a necessary condition for x to be an
ancestor of y will be that y must have every hereditary property
that x has. Suppose now that y does have every hereditary property
which x has. Since 'having x as an ancestor' is hereditary and x has
this property, then, y must also have it. That is to say, 'x is an

ancestor of y' may be defined as
(VF){[Fx & (Vz)(Vw)((Fz & 2 parent w) —9 Fw)] -> Fy},

which gives us a definition in second order logic of 'ancestor' in terms
only of 'parent' and some purely logical symbols.

Frege's definition can also be formulated in terms of classes,
thereby avoiding the use of second order logic, by stating that x is an
ancestor of y if and only if x belongs to every class that contains y

and all parents of members. To symbolize, it would be:
(Vu){[y e u & (Vz)(Vw)((w e u & Pzw) —> z e u)] -—> x e u}

where 'Pzw' means '2 is a parent of w'.

 

6 G. Frege (1884), sec 81. See also his Begriffsschrift (1879) in From Frege to
Gddel (ed. by J. van Heijenoort), Harvard U., 1967, secs 23-26.

44

Frege's construction of the "ancestral" relation admits of many
applications besides this genealogical one. In particular, it can be
applied to define the class of natural numbers in terms of 0 and the
successor function. Let 'S(x)' abbreviate 'the successor of x’. As the
first number, then, 0 may be taken as "x(x at x) or the extension of
the concept 'nothing is identical with itself‘; then 1 is definable as the
successor of 0; and 2 as the successor of 1; and so on. That is, the
series of all individual numbers can be effectively given as 0, 8(0),
88(0), . . ., once we define S(x). Since to say that a collection of things,
say x, has y members is simply to say that x e y, the successor of 2,
8(2), will be the class of all those classes which, when deprived of a

member, become member of 2. In symbols,

S(z) = {y: (3x)(x e y & y-{x} e 2)}.

Here "-{x}" is the class of everything but x, so the conjunction y-{x} is
the class of all members of y but x. Thus, following Frege, if 0 is

definable as "x(x #3 x) or the extension of all empty concepts, then we

would have

'1' as "y(3x)(x e y & y-{x} e 0),
'2' as "y(3x)(x e y & y-{x} e 1).

Similarly we can define 3, 4, and so on, as far as we like. In general,
S(w) or w + 1 is definable as "y(3x)(x e y & y-{x} e w).

Thereupon, following the same technique, we can even explain
the expression that x is a number, for any object x, in such a way

that it will come out true when and only when the object x is either 0

45

or 1 or 2, etc. The ancestor relation provides a means of
accomplishing this. Just as 'ancestor of y' means y or parent of y or
parent of parent of y and so on, so 'number' means 0 or SO or 880
and so forth. That is to say, x is a number just in case 0 is "successor-

ancestor" of x. Thus, we can define 'x is a number 'as:

(Vz){[0 e z & (Vw)(w e z —> S(w) e 2)] -> x e 2}.

To be a number, then, is to belong to every class to which 0 belongs
and to which the successor of each member belongs. Recall that
previously we could not decide whether JULIUS CAESAR is a number
or not. We now can say that JULIUS CAESAR is not a number since
"it/he" does not belong to every class which is closed under zero and

the successor function.

2.3 Deducibility and Truth-Preservation

The second thesis of logicism is that all arithmetical statements
may be deduced from logic. This means that the translations of true
arithmetical sentences should remain as true logical sentences, which
are derivable from logical principles and the required definitions for
arithmetic expressions. So truths of arithmetic can be reduced-to or
derived-from truths of logic. For instance, the principle of
mathematical induction, a proof procedure central to number theory,
states that if 0 has a property F and a natural number has F only if
its successor does (i.e. F is "Successor"-hereditary), then every
natural number has F. To derive this in logic, let us assume that the
antecedent condition is true, i.e. F0 & (Vy)(Fy —> F(Sy)). and that x is

any natural number, say Nx. What we need is that (Vx)(Nx -> Fx). But

46

from Fregean second-order definition of "S-ancestral", we can easily

see that
(VF)(\7’x)[(F0 & F is S-hereditary) -—> (Nx —> Fx)].

So, 'Nx —> Fx' will follow by universal instantiation with respect to
any given hereditary property "F" and truth functional logic.
Similarly, other theorems of mathematics, such as '7 + 5 = 12', are
derivable in logic with the help of some appropriate definitions and,
if necessary, logical axioms.

However, the matter is somewhat more complicated than this
picture suggests. It appears clear to me that Frege in his
Grundgesetze, or Whitehead and Russell in their Principia
Mathematica, had in mind not semantic entailment of "truth" but
formal derivation in logic of "theorems" of axiomatized arithmetic. Of
course, for the earlier logicists like Frege and Russell (unlike more
careful later logicists like Carnap and Tarski), the distinction between
these would not have seemed so terribly significant. For, at the time
when Frege or Russell attempted their logicism, there were no
adequate metalogical studies yet developed in which the importance
of the connection between truth and theoremhood had been fully
appreciated. Nor did they have any doubt, while executing their
logicistic reductions, that they had in fact axiomatized set theory,
from which all arithmetical truths were said to be derivable. In the
latter respect, indeed, logicism has a methodological affinity with
formalism—the view that tends to identify truth with derivability or
theoremhood in some formal or axiomatic system. Truths, according

to formalism, can be known or obtainable if we can determine

47

theoremhood in a given system in which we do not worry about the
meaning of the primitive symbols. Logicism likewise proposed to
construct the logico-mathematical system in such a way that,
although the axioms and rules of inference are initially chosen with a
meaningful interpretation of the primitive symbols in mind, the
actual deductions are nevertheless carried through as formally as
possible, without recourse to any meanings or truth values.

As such, Frege and Russell never bothered to ask seriously
whether, in fact, all arithmetical truths were derivable in the
systems that they were concerned with. They did not ask the
question simply because they believed they were able to derive in
their logical or set-theoretical language every important arithmetic
theorem that had been known thus far. However, as Godel showed in
his famous Incompleteness Theorem (1931), the answer is negative,
in the sense that there is no first-order axiomatic system for any
reasonably strong portion of mathematics (meaning that it includes
arithmetic) in which every true statement of arithmetic is provable,
unless it is a system in which some false statement can be proved as
well. In other words, we cannot identify truth with provability or
derivability in any sufficiently powerful formal system. Therefore, if
we take the second logicist thesis as asserting that every true
mathematical statement can be derived within a first-order formal
system employing only set-theoretic expressions and definitions of
mathematical expressions in terms of them, then the thesis is simply
false.

As a result, if we are to save logicism as our paradigm of ontic

reduction at all, we must give up either the claim that all the

48

theorems of arithmetic are deducible from elementary logic; or the
claim that the second logicist thesis is that of formal derivability
within a first-order axiomatic system. The latter position may be
favored by formalists. For formalists tend to believe that if
mathematical statements are to be considered true or false at all, it
certainly is so not because mathematical truths correspond to the
facts, but because they are formally derivable from some "consistent"
set of axioms. Formalists in fact argue that the sole criterion for such
mathematical truth must be based on the consistency of the axiom

system. David Hilbert, the foremost representative of formalism,

asserted that

If arbitrary postulated axioms do not contradict each other
with their collective consequences, then they are true and the
things defined by means of the axioms exist. That, for me, is
the criterion of truth and existence.7

He even claimed that these axioms define the primitive terms, to the
extent that axioms are, after all, implicit-definitions in a way that the
meanings of primitive symbols are to be established merely by their
necessary relationships with each other in such definitions. He
entirely shifted the emphasis in mathematics from questions of truth
to questions of deductive relationships, thereby removing the stigma
attached to investigating axioms that do not describe any known
"reality", and opening the way to the creation of new mathematical

theories simply by laying down new axioms.

 

7 EH. W. Kluge (tr. & ed.), Gottlob Frege: On the Foundations of Geometry and
formal Theories of Arithmetic, New Haven, Yale U., 1972, p.12 ("Frege-Hilbert
Correspondence").

49

To the logicists' eyes, however, the formalist alternative is
strongly objectionable. For the most perplexing problems of the
axiomatic method of formalism concerns the status of axioms
themselves, (even leaving aside the question of a consistency proof,
to which Godel's Incompleteness Theorem represented a death blow).
These axioms cannot be proved within the theory which they
axiomatize. But then what justification do they receive? And what
force do proofs based upon them carry? Perhaps, formalists would
like to base their justification for a set of axioms upon their success
in application. But how does one measure the 'success'? How can we
say a mathematical system is successful or fruitful if it is not
ultimately to show that the system fits or explains some previously
recognized, though possibly unorganized, set of mathematical facts or
truths which unlike the formalist axioms are neither trivial nor
arbitrary? To these questions, the standard logicist answer would be
that to avoid circularity in proofs we must, of course, take some
truths as unproved; and logicist's claim that the source of these
truths must be sought outside the theory being axiomatized. In many
of his writings, Frege seemed to suggest that axioms must not only be
"true" but must also be "self-evident".3 This implies (1) that one
cannot select just any subset of the theorems of a theory as its
axioms, even if this set is sufficient for generating the others; (2) that
the existence of some mathematical entities and the truth of those
statements about such existents are ultimately needed prior to a
Hilbertian establishment of the consistency in question—i.e. Frege's

ontological commitment-to or conviction-of platonism. I shall discuss

 

8 For example, ibid., pp.6-21.

50

(2) in somewhat greater detail in the next section, while (1) will be
treated presently.

Frege never seems to have developed an epistemology for the
"self-evidence" of his logic or logical axioms, despite his whole
hearted convictions. However, he stated explicitly that logic could not
give a noncircular answer to this epistemic question. For he no doubt
believed in the universality of logic in all facets of our thinking,

saying that

The question why and with what right we acknowledge a law
of logic to be true, logic can only answer by reducing it to
another law of logic. Where that is not possible, logic can give
no answer.9

But it should be no surprise that those symbols and the resulting
axioms in a formal theory must bring some degree of ”obviousness”,
or at least some clarification and conformity, to the familiar practices

in our past usage of them. For as Quine remarks

Although signs introduced by definition are formally arbitrary,
more than such arbitrary notational convention is involved in
questions of definability; otherwise any expression might be
said to be definable on the basis of any expressions
whatever.10

Following a similar line of reasoning, Russell at some point
during his political imprisonment wrote an article ridiculing the so-

called formalistic method of postulating axioms, noting that "The

 

9 G. Frege (1893), p.xvii. I quoted it from M. Resnik, Frege and Philosophy of
Mathematics, Ithaca, Cornell U., 1980, p.175.

10 W. V. Quine, "Truth by Convention," in P. Benacerraf & H. Putnam (2nd
ed.), Philosophy of Mathematics, Cambridge U., 1983, p.331.

51

method of 'postulating' what we want has many advantages; they are
the same as the advantage of theft over honest toil. Let us leave
them to others and proceed with our honest toil."ll Russell's 'honest
toil', of course, consisted in explicit-definitions, such as illustrated by
Frege's definition of number (which is by the way the same as
Russell's own—save that Russell, unlike Frege, refuses to accept the
existence of classes).12 Indeed, our logicistic reduction of arithmetic
seems to be not possible at all without recourse first to some already
acknowledged arithmetic truths. For, as Wang argues, where (*)
means a translation of '7 + 5 = 12' in logical terms—"If we attempt to
give a proof of (*) . . . We are able to see that (*) is a theorem of logic
only because we are able to see that a corresponding arithmetic
propositions is true, not the other way around."13 This is also the case
for Russell, again, when he praises Frege's definition of 'cardinal
number' by saying that it "leaves unchanged the truth values of all
propositions in which cardinal numbers occur, and avoids the
inference to a set of entities called 'cardinal numbers'."14 It is clear

from these examples that logicists think of reduction as a relation

 

11 B. Russell (1919), Introduction to Mathematical Philosophy, Allen &
Unwin, 1960, p.71.

12 B. Russell (1918), "The Philosophy of Logical Atomism," Logic and
Knowledge (R. Marsh, ed.), p.273: "1 want to make clear that I am not denying
the existence of anything; I am only refusing to affirm it." This attitude is
what Russell took for the existence of classes, the position that is often called
his 'no-class' theory developed in his "Mathematical Logic as Based on the
Theory of Types" (in R. Marsh, ed.), and later extended in Principia
Mathematica.

13 Hao Wang, From Mathematics to Philosophy, New York, Humanities Press,
1974, p.235.

14 B. Russell (1924), "Logical Atomism," Logic and Knowledge (R. Marsh ed.),
Unwin & Hyman, 1956, p.327.

52

between two interpreted, but not axiomatized, theories, in which the

notion of truth plays a central role.

2.4 Frege's Platonism

Let us now turn to the question of whether and in what sense
numbers may be said to exist. Frege holds that numbers do exist,
though neither as psychological objects nor as physical object. They
exist nonetheless, according to Frege, as abstract and objective

entities:

we are driven to the conclusion that number is neither spatial
and physical, like MILL's piles of pebbles and gingersnaps, nor
yet subjective like ideas, but non-sensible and objective. (p.38)

This realm of numbers has obvious similarities to Plato's world of
Forms, and for that reason Frege's view of number has been called
platonism (realism) as opposed to formalism (nominalism) advocated
by Hilbert, and to intuitionism (conceptualism) propounded by
Brouwer and Heyting. Frege in fact further maintained that "the
individual number shows itself for what it is, a self-subsistent object"
(p.68) which is there to be discovered, rather than invented, in the
sense that "even the mathematician cannot create things at will, any
more than the geographer can; he too can only discover what is there
and give it a name." (p.108)

Discovering what is there and giving it a name, however, has a
special semantic dimension for Frege, for he cannot separate naming

from referring. This may be explained in accordance with Frege's

general semantics. For Frege, a sentence 'a is F', or Fa, is true if the

53

object denoted by 'a' falls under the predicate expressed by 'F’; false
if it does not. That 'a' refers to an object is a necessary condition for
the sentence 'a is F' to have a truth value. If 'a' lacks a reference, the
sentence may express a thought but it will lack a truth value. The
laws of logic for Frege, however, are laws of truth, not of thought.
They could deal with the relations of sentences only insofar as they
have truth values. They do not deal with relations between the
thoughts expressed by sentences. There is thus a presumption that
where the laws of logic apply, sentences must be considered as
having a determinate truth value. And this can only be the case if
each singular term, i.e. each name or definite description, is assumed
to have a reference. So all names in what Frege calls a "logically
perfect" language—a language to which the laws of logic apply and in
which correct reasoning can be exhibited—must denote objects. The
formal language of Frege's Grundgesetze was intended to be just one
such language. So was Russell and Whitehead's Principia
Mathematica. So, if numerals, or names of numbers, are to be defined
as abbreviations of complex singular terms of such a formal system,
(as we've seen above in defining a number in terms of classes), then
these numerals will be assigned a reference. In short, for Frege, to
give or define a name is to fix its reference.

However, all of this semantics, contrary to what Frege wishes,
seems rather to lead us to believe that Frege's numbers are not so
much full-fledged entities. Because in the end there is no epistemic
proof for the existence of a mind-independent domain of numbers.
At best, one might argue, what we have after all appears to be a

mere semantic presumption or stipulation that, only in order for a

54

"logically perfect" language to work, there ought to exist numbers as
the referents for the numeric expressions. At one point, indeed, Frege
himself suggests something very similar to that when he says "the
self-subsistence which I am claiming for numbers is not to be taken
to mean that a number word could still signify something when
removed from the context of a proposition, but only to preclude the
use of such words as predicate or attributes, which appreciably alters
their meaning." (p.71) And yet, I believe, there are some strong
reasons to insist that Frege was an ontological platonist. Two reasons
could be given, at least.

The first reason would be one which responds to the question
what kind of objects can be known to exist a priori, and which gives
us an epistemic justification for the existence of some abstract
entities. It is possible to give some sort of "transcendental" proof for
the existence of numbers, as Kant did for the possibility of our
experience. What, concerning Frege's "platonisitc" ontic commitment,
will satisfy this demand? The answer would be the necessary
existence of classes—namely, the apriori existence of the extensions
of concepts. Classes are things or objects that are given immediately
when the corresponding concepts are given. Even if a concept has no
application to any object, that concept still has an extension, i.e. the
empty class. The empty class, furthermore, is a necessarily existing
object, because the concept 'x #5 x' (which would generate the empty
class) is a logical concept. But then the concept 'being identical with
the empty class' is a concept under which at least one object falls,
and its extension is not identical with the empty class. So, now, there

are two distinct but necessarily existing objects. Similarly we may

55

argue that there must be infinitely many such objects, that is,
infinitely many classes. As we've seen, numbers are classes for Frege
so that numbers necessarily exist.15

The other reason would be that we must take semantics
seriously from an ontological point of view, to the effect that
determinations of truth value for a given theory must depend upon
extralinguistic objects of its domain and their characteristics.
Epistemically, of course, platonism would seem to create problems
because we stand in no causal or empirical contact whatever with the
abstract objects which platonism is committed to. Godel objects to the
notion that only empirically accessible relation are epistemically

legitimate, by saying that

Classes and concepts may, however, also be conceived as real
objects . . . It seems to me that the assumption of such objects
is quite as legitimate as the assumption of physical bodies and
there is quite as much reason to believe in their existence.
They are in the same sense necessary to obtain a satisfactory
system of mathematics as physical bodies are necessary for a
satisfactory theory of our sense perception.16

He even appears to be postulating a special faculty of intuition,
"something like a perception,"17 which relates us to abstract objects
and enables us to apprehend abstract truths.

Intuition or not, however, we can consequently expect some

sentences to commit us to the existence of an object, and in particular

 

15 See M. Tiles, "Kant, Wittgenstein and the Limits of Logic," in l. Grattan-
Guinness 9ed.), History and Philosophy of Logic, Abacus Press, 1980, p.162.

15 K. deel, "Russell's Mathematical Logic," in Benacerraf & Putnam (2nd
ed.), Philosophy of Mathematics, Cambridge U., 1983, pp.456-457.

17 lbid., pp.483-484: "But, despite their remoteness from sense experience, we

do have something like a perception also of objects of set theory, as is seen
from the fact that the axioms force themselves upon us as being true."

56

some theories to commit us to collections of objects, in the sense that
semantic analysis may indicate that the sentences can be true only if
some object or objects exist. Quantification, especially existential
quantification, provides a vehicle for making such commitments
explicit within a theory. For instance, in set theory, we find the axiom
of infinity, asserting the existence of an infinite multitude of sets; on
a more basic level, we come upon the null set axiom which is also
existential in force. In number theory, we find theorems asserting
the existence of infinitely many primes. As such, Quine has held that
classical mathematics "is up to its neck in commitments to an
ontology of abstract entities."18 And this is the approach taken by
platonism which I believe Frege in fact has adopted; the view that
mathematics has a definite subject matter, that mathematical objects,
in some sense or other, exist independently of us, and that
mathematical statements are true or false insofar as they agree or

disagree with facts about these objects.

2.5 Adequacy Conditions

As we've seen, it is far from clear that formal-derivability or
theoremhood alone can establish the reduction of arithmetical truth
once we give up the claim to its universality or completeness with
respect to truth. Philos0phically, this seems an unsatisfactory
situation. The alternative, of course, is to surrender the claim that we

ought to interpret our deducibility as formal derivability in an

 

18 W. V. Quine, "On What There is," From a Logical Point of View, Harvard U.,
1980 (rev. ed.), p.13.

57

axiomatic system, and instead embrace the possibility of a "truth-
preserving" translation of arithmetic into logic. The consequence of
this move will be that the two logicist theses blend together, in that
arithmetical truths, when we eliminate the characteristically
arithmetical expressions or concepts in them in favor of the logical
definiens of these expressions, becomes logical truths—that is,
arithmetical truths are entailed by logical truths together with the
definitions. The essential point of this method is that the logicist does
not establish a "logicist" arithmetic by laying down axioms or
postulates; but, through explicit-definitions, he produces logical
constructions that have, by virtue of these definitions, the usual
properties of the "natural" arithmetic.

Let us say, therefore, that one theory is reducible to another
just in case the first can be translated into the second, in such a way
that the translation is to preserve predicate structure and truth
value. We may call the former requirement 'adequacy of definability'
and the latter 'adequacy of truth-preservation'. In other words, it is
required in an adequate reduction that the "old" predicates be
definable in the "new" theory in such a manner that truths go into
truths. This may be closely related to the notion of 'modeling', in the
sense that "We have provided a model of arithmetic in set theory
when we have provided a way of so reinterpreting (a theory being
already interpreted) arithmetical notations in set-theoretic terms as
to carry the truths of arithmetic into truths of set theory . . .
Modeling proves tantamount to reduction because all traits relevant

to application carry over into the model."19

 

19 W. V. Quine, Set theory and Its Logic, Harvard U., 1969 (rev. ed.), p.135.

58

However, this alternative course would fare no better unless
we are to have a good account of how to translate between the
"interpreted" theories in accordance with the two adequacy
requirements or criteria (i.e. definability and truth-preservation),
especially in connection with the latter requirement. For there
appears to be a great difficulty in dealing with the notion of truth-
preservation in the reductive translation in question. The semantic
notions like 'truth', unlike the "clean" formal-derivability between
two axiomatized theories, often generate built-in paradoxes in a
language or theory, unless we seek a careful distinction of hierarchy
of languages in which they are used. Moreover, even if we assume
that we already have a general method of translating one theory into
another with respect to truth-preservation, there could still be a
question concerning our issue of 'ontic commitment': How do we
know that such a translation method has any ontological significance,
rather than involving mere linguistic stipulation?

For these questions, we turn to Tarski's 'referential semantics'
which provides us a formal method that fits nicely with both
demands—and fits well with Quine's criterion of 'ontological
commitment'. It gives an account of truth-preserving translation
between "interpreted" theories, in such a way that truth can be
explained in terms of 'reference' and 'satisfaction'. Then, we shall
further adopt Tarski's notion of interpretability of one theory in
another, through his method of 'possible definition', for the formal
characterization of our "typical/standard" adequacy conditions of
'ontic reduction'—that is, definability of predicates and preservation

of truth values.

3. Ascent to 'Truth'

In this chapter, we study Tarski's "formal methods" in his
referential semantics to explicate more precisely our notions of ontic
commitment and reduction. Especially his notions of 'satisfaction' and
'hierarchy of languages' will be closely examined, because it will
provide a precise explicans for the preanalytic notions of
'definability' of concepts and 'truth-preservation' in reduction of one
theory to another. With such 'formal semantics', we shall further
explore Tarski's notion of interpretability of one theory in another as
the formal representation of our "typical/standard" adequacy
conditions of 'ontic reduction'. As a result, we'll see that ontological
reduction allows us to translate a theory making some existence
implications or ontic commitments into another theory that may not
force the same commitments at all. In other words, ontological
reduction gives us a way of showing that (undesirable) ontological

commitments can be avoided.

3.1 Material Adequacy and Formal Correctness

According to Tarski, "The task of laying the foundation of
scientific semantics" for avoiding semantic paradoxes requires what
he calls 'materially adequate' and 'formally correct' definitions of all

semantic concepts or terms.1 A materially adequate definition is one

 

1 A. Tarski, "The Establishment of Scientific Semantics," in Logic, Semantics,
Metamathematics (tr. by J. Woodger), Oxford U. Press. 1956, p.402.

59

60

that successfully captures the intuitive and customary sense of the
term to be defined, so as to determine its extension correctly and
uniquely. A formally correct definition, on the other hand, fixes such
a "material" definition precisely by employing in the definition only
terms that are explicitly specified and admit of no vagueness or
ambiguity on their own account.

In particular, for Tarski, a materially adequate definition of
'truth' must be one that "[does] justice to the intuitions which adhere
to the classical Aristotelian conception of truth. . . To say of what is
that it is not, or of what is not that it is, is false, while to say of what
is that it is, or of what is not that it is not, is true."2 He wants us to
see "the so-called classical conception of truth ('true—corresponding
with reality') in contrast, for example, with the utilitarian conception
('true—in a certain respect useful')."3 Donald Davidson in his "In
Defense of Convention T" elucidates this classical or Aristotelian

intuition quite well:

Let someone say, 'There are a million stars out tonight' and
another reply, 'That's true', then nothing could be plainer than
that what the first said is true if and only if what the other has
said is true. This familiar effect is due to the reciprocity of two
devices, one a way of referring to expressions (work done here
by the demonstrative 'that'), the other the concept of truth. The
first device takes us from talk of the world to talk of language;
the second brings us back again.4

 

2 A. Tarski, "A Semantic Conception of Truth," in L. Linsky (ed.), Semantics
and the Philosophy of Language, U. of Illinois, 1970, pp.14-15.

3 A. Tarski, "The Concept of Truth in Formalized Languages," in Logic,
Semantics, Metamathematics, p.153.

4 D. Davidson, "In Defense of Convention T," Inquiries into Truth &
Interpretation, Oxford U., 1984, p.65.

61

The "intuitive" connection between language and reality, thus,

might take the form of the following schema
(T) X is true-in-L if and only if p

where we replace a (quotation) name, say X, by the sentence "There
are a million stars out tonight" that will be a logical subject for the
predicate 'is true-in-English'; and replace p by the sentence itself.
Indeed, this schema (T) is the criterion which Tarski adopted for his
material adequacy of a theory of truth. In his classic paper "The
Concept of Truth in Formalized Languages," Tarski proposes that any
acceptable definition of truth should have as consequences all
instances of (T), thereby uniquely determining the extension of the
term 'true' for the given theory. That amounts to saying that the
concept of truth has a unique extension, for, supposing two different
interpretations of our predicate 'true-in-L' each of which is
compatible with (T)-—say, 'true1-in-L' and 'truez-in-L', we would

have
X is truel-in-L if and only if X is truez-in-L,

no matter what actual sentence of L we substitute for X. Thus, the
schema (T) leaves no ambiguity as to the totality of true sentences
for a given theory, i.e., truth1-in-L and truthz-in-L are coextensive.
However, some formal constraints need to be imposed on the
schema if it is to provide what Tarski calls a 'formally correct'
definition of truth. First, the truth predicate should not be of the

form 'X is true' which is language-transcendent, but should be of the

62

form 'X is true-in-L' where L refers to a specific language. This
requirement of language-immanence will be obvious if we consider
an instance of the (T) schema such as 'Der Schnee ist weiss' is true-
in-German iff snow is white, where 'Der Schnee ist weiss' is a good
English word, i.e., a quotation name. The same expression which is a
true sentence in one language can be false or meaningless in another.
Another more important constraint is that a sentence asserting that
some sentence S is a true sentence of some language L cannot itself
be a sentence of the language L, but must belong to a metalanguage
M in which the sentences of L are not used but are mentioned and
discussed. Otherwise, it leads to an antinomy or contradiction. Take

for example a self-referential sentence that says of itself:
(F) (F) is not true.

If it is granted that (F) is allowable or meaningfully assertable in L,

we would have an instance of (T) which says

'(F) is not true' is true iff (F) is not true.

Since '(F) is not true' is itself the very sentence (F), we would have

by the substitutivity of identity that
(F) is true iff (F) is not true,

which is a contradiction and will thereby nullify the requirement of
material adequacy—namely, an exact truth-extension for any given
theory. Needless to say, a theory in which a contradiction is detected

will serve no useful purpose, for it will make everything true by

63

forfeiting the distinction between 'true' and 'false'. Such a theory is
no theory at all, and fails to capture the intuitive concept of truth.
Some may object to the use of a self-referential sentence, like
(F), in the classical bivalent logic to which Tarski adheres for a theory
of truth. For instance, Skyrms has maintained that "The principle of
substitutivity of identity, like the rest of classical logic, was
developed to handle bivalent sentences, and it is not adequate to the
strain of paradoxical sentences like the Liar."5 For he believes that
sentences like (F) fail to satisfy bivalence, and are neither true nor
false so that they are simply not subject to the classical logic of
identity. In other words, Skyrms identifies the derivative force of
semantic antinomies with nonbivalence of self-referential sentences,
and suggests either depriving the language L of self-referential
sentences or restricting the range of applicability of some laws of
classical logic. For many logicians and philosophers, however, neither
of the choices will provide a good solution. It would appear, on the
one hand, that the sacrifice of the logic of bivalence and of
substitutivity with respect to identity is "too high a price to pay even
to get rid of the Epimenides paradox."6 On the other hand, as to the
self-referentiality, Tarski observes that although there lingers a
sense of suspicion in the substitution for the (T) schema with self-
referential sentences like (F), nonetheless, "no rational ground can be

given why such substitutions should be forbidden in principle."7 In

 

5 B. Skyrms, "Notes on Quantification and Self-Reference," in R. Martin (ed.),
Paradox of the Liar, Ridgeview Pub., 1970, p.69. See also Skyrms' "Intensional
Aspects of Semantical Self-Reference," in R. Martin (ed.), Recent Essays on
Truth and the Liar Paradox, Yale U., 1970, pp.119-131.

6 F. B. Fitch, "Comments and Suggestion," ibid., p.75.

7 "The Concept of Truth in Formalized Languages," in Logic, Semantics,
Metamathematics, p.158.

64

fact, Godel's profound discovery of incompleteness of arithmetic, for
example, was entirely based on his innocuous formulation of
"syntactic" self-referentiality by which arithmetic expressions can be
made to refer to arithmetic expressions themselves.

As such, the distinction between object— and meta-language is
crucial to Tarski's requirement of 'formal correctness' for an
adequate definition of truth. For, only if there exists a hierarchy of
languages, similar to the hierarchy of types invoked in the theory of
types to avoid "set-theoretic" paradoxes, can we avoid "semantic"
antinomies and be able to define consistently the notion of truth. Just
as a variable of higher-order in the type hierarchy is needed to avoid
the "impredicative" definition associated with class-existence, an
essentially higher logical structure of a metalanguage M is needed in
order to curb the assertability of instances of (T) within the object
language L itself. According to Tarski, M must contain names for
expressions in L and descriptions of the logical structure of such
expressions. If this requirement is satisfied, then we may be able to
define the truth predicate in terms of the given metalinguistic
apparatus, so that X is true if and only if p, where X is a name of
some L-sentence and p is a translation into M of that very L-
sentence. M could include L as a part of itself, however, in which case
the sentences of L themselves would be their own metalinguistic
translations into M. Thus, the minimum requirement for the
hierarchy of languages would be that all the expressions of the
metalanguage except the 'truth' predicate can in principle be
expressed in the object language; but the predicate 'true' must belong

essentially to the metalanguage.

65

In addition to the point so far, there is this issue that the
problem of finding a satisfactory explication for the concept of truth
(explicandum) has an exact meaning (explicans) only for languages of
exactly specifiable structure. Tarski says, "The problem of the
definition of truth obtains a precise meaning and can be solved in a
rigorous way only for those languages whose structure has been
exactly specified."8 It is this conviction which led Tarski and others
to approach problems of definition and explication by way of the so-
called 'formal languages'. It seems obvious that the meaning of any
term is relative to the language in which it occurs. Consequently the
problem of clarifying or explicating the meaning of any term will be
a vague problem to the extent that the structure of the language in
which it occurs is not precisely and completely specified. Some
philosophers have objected to this aspect of Tarski's "formal method"
on the ground that it avoids philosophical issues. It seems to me,
however, that the utility of formal languages for the explication and
clarification of concepts is sufficiently proven by the work of modern
logicians who have used such languages to achieve clarifications of
the fundamental concepts of the deductive sciences and of their
terminology—in particular, for the concept of truth, by Tarski's
definition itself. So let us see Tarski's formal method for an adequate

definition of 'truth'.

 

3 "A Semantic Conception of Truth," in L. Linsky (ed.), Semantics and the
Philosophy of Language. p.19.

66

3.2 Tarski's Definition of 'Truth' and 'Satisfaction'

The schema (T) is not itself a definition of truth, but a mere
schema for such a definition in the metalanguage. Nevertheless, this
schema might be thought to provide an obvious way of giving such a
definition, in that a conjunction of all instances of the schema, each
being a partial definition, would constitute a complete definition.
Tarski maintains, however, that it is not possible to give such a
conjunctive definition, for the number of sentences of a language
may be infinite. Neither can we turn the (T) schema into a definition
of truth by universal quantification, with a quotation-name variable

'p', as in
(Vp)('p' is true-in-L <—> p),

which would apparently satisfy the requirement of material
adequacy, since all instances of the (T) schema are instances of it.
This quantified version of (T) is unacceptable because, as Quine
argues elsewhere, quotation is a 'referentially opaque' context, so
that the result of quantifying into quotation marks is meaningless.9
For, from the standpoint of logical analysis, the quotation-expression
is an indivisible unit, analogous to a proper name, such that "each
whole quotation must be regarded as a single word or sign, whose

parts count for no more than serifs or syllables."lo So the universal

 

9 See W. V. Quine, "Three Grades of Modal Involvement" in his Ways of
Paradox, 1976, Harvard U., p.161: "a context is referentially opaque if it can
render a referential occurrence [in that context] nonreferential. Quotation is
the reverentially opaque context par excellence."

10 W. V. Quine, Mathematical Logic, Harvard U., 1981 (revised ed.), p.26.

67

generalization of the (T) schema cannot be regarded as a valid form
of definition.

Tarski therefore considers the alternative strategy of a
recursive definition, for all instances of (T), that could first capture
truth for each and all simple sentences of (T), then explain truth for
compound sentences as a function of those simple ones. Even this
alternative faces a serious obstacle, though, in that for some
languages, such as Boolean algebra, the elementary formulas (e.g. 'a
c B' or 'x < y') are not themselves sentences in the proper sense but
sentential-functions or open-sentences which by themselves are
neither true nor false. Actual closed sentences in such a language can
be formed only by the binding of variables in quantification, e.g.

'(3x)(Vy)(x > y)’. Tarski nevertheless claims that

The possibility suggests itself, however, of introducing a more
general concept which is applicable to any sentential function,
can be recursively defined, and, when applied to sentences,
leads us directly to the concept of truth. These requirements
are met by the notion of the satisfaction of a given sentential
function by given objects.ll

'Satisfaction' purports to be a relation between an individual
sentential-function (or open-sentence) and the particular objects to
which the function applies. An account of truth in terms of
satisfaction, thus, represents a semantic analysis of the truth of a
statement as a function of the references of its component terms, and
raises concerns about ontological reduction and commitment in a way

which an unanalyzed notion of truth does not. For the concept of

 

11 A. Tarski, "The Concept of Truth in Formalized Languages," in Logic,
Semantics, Metamathematics, p.189.

68

satisfaction is essentially a semantic notion defined on the relation
between language and extralinguistic objects.

In proceeding to define 'satisfaction', Tarski finds it necessary
first to introduce into the metalanguage the notion of an infinite
sequence. Satisfaction, then, can be construed as a relation between
an open sentence and an infinite sequences of objects. An open
sentence with n free-variables, F(x1 . . . X“), is to be satisfied by the
sequence of objects < 01 . . . On, 0M1 . . . > just in case the open
sentence is satisfied by the first 11 members of the sequence, and
subsequent members may be ignored. The infinite length of
sequences is needed merely to ensure that there will always be large
enough "tuples" of objects to correlate with the variables in a
sentential function of any length. Satisfaction in this scheme, thus, is
a relation between open sentences with n free-variables and an
ordered n-tuple of objects. For instance, the open sentence 'x is a city'
is satisfied by London, Paris, or Rome; 'x is a teacher of y' is satisfied
by the ordered pair <Plato, Aristotle>; 'x bought y for z' is satisfied by
the ordered triple <John, roses, Mary>; and so on. Now, in the spirit of

convention (T), this 'satisfaction' can be schematized in such a way

that
(S) Va, a satisfies the open sentence x if and only if p,

where 'a' is an infinite sequence of objects, and 'x' is to be replaced
by the name of an open sentence while 'p' is replaced by the open
sentence itself with any number of free variables. From this schema

(S) we can obtain every individual instance of the materially

adequate use of satisfaction, such as "For every ct (object), a satisfies

69

'x is white' if and only if or is white"; "For every ct (ordered pair of
classes), on satisfies 'x1 c; x2' if and only if 011 is included in a2"; and so
forth. That is, satisfaction for each elementary open sentence of the
given object language can be directly defined under this schema,
although no general definition for (S) is possible for the same reason
that no general definition for (T) was possible.

A formally correct definition of satisfaction, then, is as follows.
First, we may suppose our object language to consist, as its syntax, of
variables (x1, x2, x3 . . .), predicate letters (F, G, . . .), sentence
connectives (-, &), and quantifier (3 . . .). The formulas and sentences
of the object language are to be the set of expressions which includes
each n-place predicate followed by n free variables (i.e. 'Fx1 . . . xn');
and which is closed under the usual laws governing the logical
symbols {-, &, 3}. Next, for our metalanguage, let a and [3 range over
sequences of objects, let 0t; denote the i-th object in any sequence a;
and let A and B be the metalinguistic translations of the open

sentences of the given object language. Satisfaction, then, is formally

defined as:
(1) For all sequences a and i (O S i S n), a satisfies 'Fx1 . . . xn' iff
(a1, a2, . . . an) of a, in that order, stand in the relation or attribute F.

(2) For all on and A, a satisfies '-A' iff on does not satisfy 'A'.

(3) For all a, A and B, at satisfies 'A & B' iff 0: satisfies A and on
satisfies B.

(4) For all on, A and i, on satisfies '(3xi)A' iff there is a sequence [3

such that cij = Bj for all j at i and [3 satisfies 'A'.

70

Clauses (1)-(4) represent Tarski's full recursive definition of
satisfaction, and all that remains for Tarski's definition of 'truth' is
for truth to be introduced in terms of satisfaction. It can be easily
appreciated that when all variables of an open-sentence (e.g. 'x is a
teacher of y') are bound by quantifiers, it will be a closed-sentence,
'(Vy)(3x)(x is a teacher of y)’, which is either true or false. In other
words, a closed sentence is a formula with no free variables and will
therefore be satisfied either by all sequences of objects or by none,

accordingly as it is true or false. Therefore,
(5) For all or and A, A is true if and only if it is satisfied by 01.12

It would now be a simple task to recursively define 'truth' of all the
remaining compound closed-sentences of the object language, for it
will exactly resemble the formulations (2) and (3) of the satisfaction

definition above.

3.3 Interdependence of Truth and Existence

At this point, in connection with the issues of ontic commitment
and reduction, some features of Tarski's method reflected in his
definitions deserve our attention. First, Tarski's method imposes an
objectual reading of quantifiers in that '(3x)Fx' is interpreted as
'There exists at least one object that satisfies F' or simply 'At least
one object is F'. The objectual interpretation of quantification appeals
to the values of the variables, the objects over which the variables

range.; whereas the substitutional reading appeals to the linguistic

 

12 lbid., pp.189-197 for Tarski's original formulations of (1)-(5).

71

"substituends" for the variables, the linguistic expressions that can be
substituted for the variables. (Philosophers use 'substituends' instead
of 'values' in order to distinguish two different readings of
quantifiers.) A substitutional interpretation of quantification would
avoid, for a definition of truth, the need for the detour via
satisfaction. For it would permit truth of quantified sentences to be
defined directly in terms of their linguistic substitution instances, not
by way of their satisfaction by extralinguistic objects.

The choice between the different readings of quantification is
no small matter, but may rather have important philosophical
consequences. One such consequence would be the crucial role played
by the objectual reading in Quine's ontological views, especially in his
criterion of ontological commitment which, according to Romanos,
"owes much to the treatment truth receives at the hands of Alfred
Tarski."13 Quine's criterion—'to be is to be the value of a variable'—
purports to be a test or standard to clarify what kinds of objects a

theory says there are,

For to say that a given existential quantification presupposes
objects of a given kind is to say simply that the open sentence
which follows the quantifier is true of some objects of that kind
and none not of that kind.14

 

13 G. Romanos, Quine and Analytic Philosophy, MIT press, 1983, p.123. In
Romanos' view, "It was Tarski's numerous observations concerning the nature
of semantics, along with the explicit procedure he provided for defining
semantic concepts, that formed the basis for Quine's vigorous advocacy of the
theory of reference over the theory of meaning." (p.123)

14 W. V. Quine, "Notes on the Theory of Reference," From a Logical Point of
View, Harvard U., 1980 (revised ed.), p.131. Italics is mine.

72

Of course, this is none other than Tarski's notion of 'satisfaction'. So
one is able to tell the ontological commitment of a theory by
formulating it in classical predicate logic which would provide the
metalinguistic apparatus, and asking by the "objectual" reading of
quantifiers what objects are required to exist as values of its
variables for the (open) sentences of a theory to be true (or
satisfied).

It is important to note that this account makes the concepts of
truth and existence interdependent. Also important is to note that
the criterion applies only to interpreted theories whose ideology or
set of predicates is already understood perfectly. They are, in other
words, semantically presented theories, for which the meanings of
terms and the truth conditions of statements are supposed to be
given beforehand. What Quine's criterion does for such a theory is to
make explicit its existential status, or ontic commitments, by raising
it on what Quine calls the plane of 'semantic ascent'——-up to the
(meta)language of "objectual" quantification or predication.15 Such an
ontic explication, with respect to Tarski's definition of predicate-
satisfaction or predication, is more clearly in view, when Quine
explains:

Another way of saying what objects a theory requires is to say

that they are the objects that some of the predicates of the

theory have to be true of, in order for the theory to be true.

But this is the same as saying that they are the objects that

have to be values of the variables in order for the theory to be

true . . . Predication and quantification, indeed, are intimately
linked; for a predicate is simply an expression that yields a

 

15 See From a Logical Point of View , pp.16 ff; Word and Object, pp.260 ff; and
Philosophy of Logic, pp.10 ff.

73

sentence, an open sentence, when adjoined to one or more
quantifiable variables.16

Another important feature of Tarski's method which deserves
explicit mention is that, as Haack opines, "Tarski gives an absolute
rather than a model-theoretic definition; 'satisfies', and hence 'true',
is defined with respect to sequences of objects in the actual world,
not with respect to sequences of objects in a model or 'possible
world'."17 Obviously, the sentence 'Plato is a teacher of Aristotle' is
true, absolutely, but false in a model or possible world in which there
exists no Aristotle, or in which Aristotle may have been the teacher
of Plato. The significance attached to the difference between model-
theoretic and absolute definitions will depend, of course, upon one's
attitude toward the notion of truth relative to a model or possible
world. We are not talking here about the Kripkean 'possible-world'
semantics for a modal logic, though it is not entirely unrelated, but
rather talking at a much more superficial level about the possibility
of a theory's being interpreted differently from its original or
"intended" interpretation. A new interpretation for a given theory
can be accomplished usually by way of stipulating the domain and
assigning new values for the predicates of the theory. Truth, then,
becomes relative only to a specific interpretation or model the theory
is presented within.

The point is that if Tarski's notion of satisfaction is of an

"absolute" sort, not allowing model-specific interpretations, then it

 

“5 W. V. Quine, "Existence and Quantification," Ontological Relativity and
Other Essays, Columbia U., 1969, p.95.

17 S. Haack, Philosophy of Logics, Cambridge U., 1978, p.108.

74

must commit one to an "ultimate reality", upon which 'satisfaction'
and 'truth' will be duly defined. Moreover, if that is really the case,
Tarski's definitions are on a par with the Frege-Russell conception of
'ontic reduction' as a mere ontological clarification or discovery, and
therefore are incompatible with both Carnap's and Quine's notions of
ontological reduction which together might be referred to as an
'internal realist' view as contrasted with Frege's 'external realism'.
Tarski's "absolute" definitions would not allow any ontic reductions
but rather only one "ultimate" reduction or discovery—whatever
ontology it is being committed to, for there would be no room left for
different interpretations of the world. However, it seems to me that
ontological reduction is possible only if we admit that ontology is
relative to how we specify it in virtue of our theories or conjectures
of the world. Otherwise, all reductions would turn out to be false; in
fact the history of philosophy as a whole will be false, for in it we can
find plenty of such reductive efforts. Thus, if 'ontological reduction' is
a philosophically valid notion at all, which I believe it is, then,
Tarski's definition of satisfaction, or anyone else's for that matter,
cannot be given this naive or "absolute" interpretation. Instead, any
such theory of satisfaction or of denotation in general, therefore, will
be "model-theoretic". So a little investigation will tell whether this is
the case.

Popper advocates Tarski's "absolute" theory of truth when he
develops an account of the role of truth as a regulative ideal of
scientific inquiry, that is, the notion of verisimilitude. And in doing so

he is by no means hesitant to equate Tarski's absolute theory of

75

truth with the correspondence theory of truth. For in his Conjectures

and Refutations he says that Tarski has

rehabilitated the correspondence theory of absolute or
objective truth . . . vindicated the free use of the intuitive idea
of truth as correspondence to the facts.18

He further indicates that his reason for equating Tarski's theory with
the correspondence theory comes from Tarski's own insistence on the
need for a metalanguage, such that in the metalanguage one can both
refer to expressions of the object language and say what the object
language says. It is as if Popper regards the left-hand side of each
instance of the (T) schema, e.g. 'Snow is white' is true if and only if
snow is white, as referring to the language, and the right-hand side
to the facts or extralinguistic entities. So it is as though Popper

interprets Tarski as saying
'Snow is white' is true if and only if snow is in fact white.

This Popperian interpretation of Tarski may have some
supporters. According to Dummett, Tarski's view countenances only
bivalent theories of truth which is what the material adequacy
condition requires, and such bivalent theories would carry
commitments to (external) realism. For if sentences are to be
regarded as determinately either true or false, Dummett argues, then

"there must be something in virtue of which" they are true or false.19

 

13 K. P0pper, Conjectures and Refutations, London, Routledge & Kegan Paul,
1963, p.224.

19 M. A. E. Dummett, "What is a Theory of Meaning? (11)" in G. Evans & J.
McDowell (eds.), Truth and Meaning, Oxford U., 1976, p.89.

76

In other words, "In making such an assumption, we are adopting a
realistic attitude" that the world is determinately constituted, and
that the truth conditions of sentences about such a deterministic
world may transcend our capacity to recognize whether or not we
can come to obtain the conditions—namely, either true or else false
absolutely.20 In fact, on Dummett's view, any theory such as
Davidson's which holds that the meaning of sentences is to be
specified in terms of truth-conditions implies realism concerning the
subject matter of those sentences. Thus, from Dummett's standpoint,
it could be maintained that in his definitions Tarski is committed to
realism, which would impose a "regulative" function on our
knowledge acquisition as Popper claimed—a regulative function
which underlies the correspondence theory of truth.

Suffice it to say that if Dummett is right, Tarski is committed to
realism, even if only marginally; and if he is wrong, it remains
unclear in what sense Tarski's definition or theory of truth can be
said to be "absolute" as Haack has described it. Hartly Field, however,
went a step further than Dummett and argued that Tarski is
committed not only to realism but to physicalism in particular. As his
reason for saying so, he cites Tarski's own stipulation that truth must
be defined without appeal to semantic primitives, because such an
appeal would make it "difficult to bring [Tarski's] method into
harmony with the postulates of the unity of science and of

physicalism (since the concepts of semantics would be neither logical

 

20 lbid., p.93.

77

nor physical concepts)."21 Tarski indeed wished to make semantics a
respectable study from the point of view of science, and since
'physicalism' is the only appropriate basis for the notion of unified
science, it would follow that Tarski took himself, in defining 'truth', to
be providing a characterization of truth of a physicalistic sort. Having
identified Tarski's position as physicalistic, Field argues further that
Tarski did not really succeed in reducing semantics to
physicalistically acceptable primitives. For, in Field's belief, "Our
accounts of primitive reference and of truth are not to be thought of
as something that could be given by philosophical reflection prior to
scientific information—on the contrary . . . without such a [scientific]
account our conceptual scheme breaks down from the inside."22 In
saying so, what Field seems to have in mind is a more empirical
account of primitive reference as opposed to the account rendered in
Tarski's analysis of satisfaction, in which there is no mention of
"empirical" qualifications. As a more promising recent step toward
the so-called physicalistic reduction of truth and denotation, Field
cites Kripke's development of causal theories of denotation.23
However, Tarski is himself somewhat equivocal or ambivalent
in referring to his theory as a correspondence theory, let alone a
physicalistic one. For, on the one hand, he asserts at the outset of his
definition of truth that the material adequacy for a theory of truth

should be in agreement with "the so-called classical conception of

 

21 See H. Field, "Tarski's Theory of Truth," Journal of Philosophy, vol. 69
(1972), pp.356-357: and his quotation comes from Tarski's Logic, Semantics,
Metamathematics, p.406.

22 H. Field, "Tarski's Theory of Truth," p.373.

23 S. Kripke, Naming and Necessity, Harvard U., 1979.

78

truth ('true—corresponding with reality')."24 On the other hand, he
rejects any need to appeal to such a correspondence, and warns his

critics of the danger of misunderstanding his position:

It has been claimed that—due to the fact that a sentence like
"snow is white" is taken to be semantically true if snow is in
fact white—logic finds itself involved in a most uncritical
realism . . . First, I should ask [the critic] to drop the words "in
fact" which do not occur in the original formulation . . . For
these words convey the impression that the semantic
conception of truth is intended to establish the conditions
under which we are warranted in asserting any given sentence,
and in particular any empirical sentence . . . In fact, the
semantic definition of truth implies nothing regarding the
conditions under which a sentence like (1):

(1) Snow is white

can be asserted. It implies only that, whenever we assert or
reject this sentence, we must be ready to assert or reject the
correlated sentence (2):

(2) the sentence "snow is white" is true.25

This is in effect saying that Tarski's theory is not a correspondence
theory at all, as Popper has assumed. Tarski emphasized instead that
his definition of truth is completely neutral and compatible with
various rival views of truth—"rival" in the sense of supplying
different criteria for the ascription of truth, such as empiricism or
rationalism, and realism or idealism—and is, therefore, free and
independent from any particular philosophical standpoint or

ontological commitment.

 

24 A. Tarski, "The Concept of Truth in Formalized Languages," in Logic,
Semantics, Metamathematics, p.153.

25 A. Tarski, "The Semantic Conception of Truth," in L. Linsky (ed.) Semantics
and the Philosophy of Language, p.33.

79

In his Quine and Analytic Philosophy, Romanos exploits Tarski's
semantic conception of truth to show that Tarski's method does not
make any special assumption about the objects involved, and is to
that extent a "neutral" theory of truth. He has us consider the open
sentence 'x is a prime number' as expressed by Tarski's satisfaction

schema (S), namely,
Va, a satisfies 'x is a prime number' iff a is a prime number,

(for the sake of convenience, a sequence a is simply identified with
the first element of the sequence itself). Romanos then argues that
this formula is ontologically neutral. For it holds regardless of
whether we choose to construe the objects of arithmetical discourse
(here a, the reference for x) as natural numbers or as sets or as mere
linguistic expressions. This formula, with respect to the open
sentence 'x is a prime number', fixes the satisfaction relation without
making any "truth-value" difference in the purported semantic
distinction between such diverse references; and thereby,
conversely, without making any "ontological" distinction as to
whether the open sentence 'x is a prime number' is itself expressing
a genuine number-theoretic property or a corresponding property of

set-theory or of protosyntax. Thus, Romanos is able to claim that

When defining satisfaction for the expressions of a theory,
then, we are free to quantify over only those domains of
objects whose general structural properties permit
interpretations of all the theory's predicates in their
preestablished logical relations, while making all sentences of
the theory come out true. Thus the objects we may choose to
see as satisfying the expressions of a theory are just those

80

domains that may serve as a model or true interpretation of
the theory in question.26

In other words, the definition of satisfaction by means of Tarski's
method does not hinge on generic ontological distinctions between
the correlated predicates over isomorphic domains, but only on the
broader structural features of a universe required for it to serve as

the model of a given theory.

3.4 Model-Theoretic Interpretation

So let us investigate the possibility of a "model-theoretic"
reworking of Tarski's definitions of 'satisfaction' and 'truth'. As
before, the theories to be discussed below are referred to as "theories
with standard formalization,"27 which may be briefly characterized
as theories that are formalized within the first-order predicate logic
with identity, but without predicate-variables. Let a theory T be,
then, a set of sentences in the first—order quantificational language,
closed under logical consequence; and what needs to be defined is
the locution "X is true under an interpretation 1" where X is any
sentence or formula of T. An interpretation or model I for a theory T
is an ordered pair <D, <b> consisting of any nonempty set D as its
domain, and an assignment function <1) which is a map from the
extralogical predicates, say 'P', and individual constants, say 't', onto
D such that <I>(t) is an element of D and d>(P) is a subset of D. To be

 

26 G. Romanos, Quine and Analytic Philosophy, MIT press, 1983, p.163.

27 A. Tarski, A. Mostowski, and R. M. Robinson, Undecidable Theories,
Amsterdam, North-Holland, 1953, p.5.

81

precise, the definition of 'truth' for a theory T relative to an
interpretation I will be as follows

(i) For "denotation" of terms of T: a denotation function 8 of <1) is

a map from the terms onto elements of D, for I = (D, (D), such that

5(I)(t) =

8(t) if t is an individual constant or name;

t if t is a variable.

(ii) For "valuation" of formulas of T: a valuation function to of (D
maps from the formulas onto the set {true, false}, for I, such that
(1) If X is of the form (t1 = t2), then
(I)(ti = t2) is true iff 5(I)(t1) = 5(1)(t2);
(2) If X is a predicate or open sentence '(Pt1 . . . tn)', then
(I)(Pt1 . . . tn) is true iff <8(I)(t1), . . ., 8(I)(tn)> e w(I)(P)

or <8(I)(t1). . . ., 5(1)(tn)> satisfies 'Pt1 . . . tn';
(3) If X is of the form -Y, then

(I)(—Y) is true iff m(I)(Y) is false

9

(4) If X is of the form (Y & Z), then

(I)(Y & Z) is true iff both m(I)(Y) and m(l)(Z) are true;
(5) If X is of the form (Vx)Y, then

(I)[(Vx)Y] is true iff (6(6', I)(Y) is true for every 8' such
that 8'(t) = 8(t) for every t except (possibly) x23

Georg Kreisel, in his "Models, Translations and Interpretations,"

expresses the usefulness of the concept of a model not only for a

given theory, but for the study of intertheoretic relationships as well.

According to him, "The notion of a model allows one to study

23 See ibid., pp.7-8; H. Hendry, Deductive Logic (draft: August 1995), East
Lansing, Michigan, Paper Image, pp.99-108.

82

relations between (axiomatizable and nonaxiomatizable) systems
which contain the same logical apparatus (classical quantification

theory)."29 For instance,

The familiar consistency proofs of various geometries, the
algebra of complex numbers, or—to take a modern case—of
general set theory, are obtained by means of models. This
notion, which Tarski calls "interpretation", may be defined for
systems of the first-order predicate calculus as follows: A
system (81) has a model in (82) if the nonlogical constants of
(81), i.e. its predicate symbols and function symbols, can be
replaced by expressions of (S2) in such a way that the axioms
of (81) go into theorems of (82).30

Here Kreisel is referring to the notion that Tarski himself terms a

theory's being interpretable in another":

Let now T1 and T2 be any two theories. First assume that T1
and T2 have no nonlogical constants in common. In this case we
say that T2 is interpretable in T1 if we can extend T1, by
including in the set of valid sentences some possible definitions
of the nonlogical constants of T2, in such a way that the
resulting extension of T1 turns out to be an extension of T2 as
well. Speaking precisely, T2 is interpretable in T1 if and only if
there is a theory T and a set 2 satisfying the following
conditions: (on) T is a common extension of T1 and T2, and every
constant of T is a constant of T1 or T2; ([3) 2 is a recursive set of
sentences which are valid in T and which are possible
definitions in T1 of nonlogical constants of T2; (7) each
nonlogical constant of T2 occurs in just one sentence of Z; (5)
every valid sentence of T is derivable (in T) from a set of
sentences each of which is valid in T1 or belongs to 2.31

 

29 G. Kreisel, "Models, Translations and Interpretations," in Thoralf Skolem et
al., Mathematical Interpretations of Formal Systems, Amsterdam, North-
Holland, 1955, p.30.

30 lbid., p.28.
31 A. Tarski, et. al., Undecidable Theories, pp.20-21.

83

Of course, Tarski's notion of 'interpretability' given here is a
syntactic notion, defined in terms of axioms, deductive 'validity' and
'derivability'. However, it would be no obstacle for our "semantic"
(i.e. ontological) purposes if we rephrase or redefine it, replacing the
syntactic notions with the corresponding semantic notions of 'logical
truth' and 'logical consequence', respectively. For it has been proved,
c.f. Gddel's Completeness theorem (1930), that the two method of
defining derivability and logical validity are entirely equivalent
when applied to elementary theories with "standard formalization".
That is, a sentence is valid if and only if it is logically true; a sentence
is derivable from some set of sentences if and only if it is a logical
consequence of the set. Therefore, Kreisel's notion of "82 has a model
in 81" is the same as Tarski's notion "T2 is interpretable in T1" but
with the possible definitions of Tarski which would serve as
translation-rules, belonging to a common extension T = (T1 u T2),
rather than as genuine formulas of either T1 or T2.

Tarski's concept of a possible definition of a given nonlogical
constant in a theory T to which this constant does not belong may be
defined as follows. Assume first that the only nonlogical symbols in
the theories concerned are predicate constants; we can eliminate
individual constants or functional symbols in favor of these
predicates in familiar ways, such as Russell's. A possible definition of
an n-ary predicate 'P' in a theory T to which P does not belong, then,

will be a universally quantified biconditional such as

(Vxl) . . . (Vxn)[P(x1, . . ., xn) <—-> ‘l’(x1, . . ., xn)]

84

where ‘P(x1, . . ., xn) is a formula of T with n-free variables.32 Clearly,
this whole formula belongs itself to neither T nor the original theory
where the predicate P is part of its vocabulary. Imagine that in
arithmetic the symbol '5' has not as yet been employed, but that one
wants to introduce it now into consideration. To achieve this,
granting that '>' belongs to the symbols already available, we may

write down a possible definition for 's', such that

(S) (Vx)(Vy)[x S y H -(x > y)]

By virtue of this definition, we can transform any simple or
compound sentence containing the symbol '5' into an expression
which no longer contains the symbol. For instance, by translating the
sentence 'if a s b and b s c, then a S c' in accordance with (S), we may
obtain another "equivalent" sentence 'if -(a > b) and —(b > c), then -(a
> c)‘, in which the symbol 5 does not occur at all.

This technique of a universalized biconditional for definition is
nothing new. As a matter of fact, we already saw one such in Frege's
definition of number according to which the number k is identifiable
as the class of all k-membered classes. But what is clearer now than
before in Frege‘s case is that, thanks to Tarski's object- and meta-
language hierarchy, the biconditional '(s)' is itself neither a sentence
in the original theory, say T2 of '5', nor a sentence in the new theory
T1 of '{>}'; rather it is a formula which belongs to every common
extension T of both T1 and T2. (Note that a theory T1 is an extension

of a theory T2 if every theorem of T2 is also a theorem of T1; a theory

 

32 lbid., p.20. See also his "The formulations of definitions and its rules,"
Introduction to Logic, Oxford, 1946, pp.33-36.

85

is uniquely determined by the set of all its valid sentences, so that
two theories are regarded as identical if their sets of valid sentences
coincide.) And the biconditional '(S)', as a translation-rule, makes
possible the inter-theoretic reduction or transformation between T1
and T2. Compare this with Frege's approach in his analysis of
number, whose definitions of number are themselves given in the
object language. For Frege believed in the universality of a logic
which would constitute what Tarski calls a 'semantically closed'
language, rather than a hierarchy of languages. It is this feature that
eventually leads to the "paradox" that Russell derived from Frege's
famous axiom V of his Grundgesetze.

In other words, Kreisel takes Tarski's notion of "possible
definitions" and transfers it to the metalanguage, in the sense that
such definitions can serve as translation-rules between interpreted
theories. Thus, we should be able to say that an interpretation or
model I for a theory T, where T is the minimal extension of T1 u T2 1.)
2, will be an interpretation of T2 in T] if and only if I is an ordered
pair <(D, 2:), (1» where (1) 2‘. is the set of possible definitions of those
nonlogical constants of T2 which do not occur in T1, satisfying
conditions (co-(8) above; (2) D is the universe of discourse of T which
is identified with the domain of 7‘] only; (3) (I) is a recursive function
on the set {2, D} such that, in brief,

(i) There is a translation function, ‘C of <1), that takes as
arguments the predicates of T2 (say, 'Qx1 .. . xn') and takes as values
some open sentences of T1 in accordance with )3 (say, 'Z(Q)x1 . . . xn').

(ii) There is a value-assignment function, u of (I), which maps

from the formulas of T onto elements and subsets of D, such that

86

(D01 = t2) is true iff (0)01) = (U)(t2);
(I)(Pt1 . . . tn) is true iff <(u)(t1), . . ., (u)(tn)> e (u)(P);
(I)(-A) is true iff (u)(A) is false;
(I)(A & B) is true iff both (u)(A) and (u)(B) are true;
(l)[(Vx)A] is true iff (u')(A) is true for every 11'
such that (u')(x) = (u)(x).

In essence, in such a model of T there are "consistent" submodels for
each of T1 and T2, which have as their universe of discourse a
subdomain of a model of T1, thereby generating the ontological
reduction in question. Let us say, therefore, that a theory is
interpretable in another just in case we can construct a metatheory
(or extension) for the two theories, whose model-theoretic
interpretation is to be specified in the manner defined above. Such
an interpretation for the metatheory, then, I believe, can explicate
adequately the so-called typical/standard conception of 'ontic
reduction' presented in the paradigm of Frege's logicism. For it
clearly intends to capture by a precise formal analysis the traditional
derivational paradigm of reduction, based on the explicit definability
of predicates or concepts involved in the reduction. In other words, it
provides in the "standard formalization" a general framework or
method for the two reductive conditions required for an adequate
ontic reduction—namely, the definability of concepts and the
derivability of truths.

For instance, Frege-Russell logicism requires that mathematics

M follow from logic or set theory L. To say that M follows from L

87

amounts to saying that each formula in M is also in L, or that under a
suitable translation M is a subset or subtheory of L. However, since
mathematics and logic have different vocabularies, each
mathematical theorem does not follow from logic in the sense of
already being a logical truth. Recall that the thesis of logicism, as
stated in the previous chapter, is a two fold thesis: mathematical
concepts can be defined in terms of logical concepts, and, given those
logical definitions, mathematical truths follow from logical truths.
Thus, it is not the case that M follows from L immediately; instead, M
must follow from L, taken together with the set of definitions which
serves as a set of intertheoretic translation-rules linking the
(primitive) terms of M and L. It can be said, therefore, that M follows
from L just in case there is such a set of definitions 2, such that M is
a subtheory of L u 2. (A theory T2 is a subtheory of a theory T1 if
every theorem of T2 is also a theorem in T1.) To accomplish that, we
may transfer, as it were, the definitions of the terms of M into the
metalanguage, so as to construct a translation-function, say 1: of d),
that takes as arguments the predicates of M and takes as values
some open sentences of L in accordance with Z. The function 1: will
make it possible, by virtue of the explicit biconditionals of 2, to
preserve truth values when (I) transforms the statements of M into
statements of L by help of such biconditionals.

It appears to me that this Tarskian notion of 'interpretability'
(albeit in the form of our "semanticized" modification) would fit into
scientific reductions as well. In general, reduction in the empirical
sciences has most often been considered a sort of intertheoretic

explanation such that when one theory reduces to another it is

88

explained by that reducing theory. For example, if statistical
mechanics reduces classical thermodynamics, the behavior of large-
scale masses (e.g. gases) is explained by the "statistical" account of
the behavior of small—scale masses (collection of molecules). In this
sense, scientific explanation is on a par with the logicistic reductive
attempt for the integration of logic and mathematics. Indeed, the
similarity between them is striking if we consider what Ernest Nagel

calls 'two necessary conditions' for reduction in science:

when the laws of the secondary science [the reduced theory] do
contain some term 'A' that is absent from the theoretical
assumptions of the primary science [the reducing theory], there
are two necessary formal conditions for the reduction of the
former to the latter: (1) Assumptions of some kind must be
introduced which postulate suitable relations between
whatever is signified by 'A' and traits represented by
theoretical terms already present in the primary science. . . the
'condition of connectability'. (2) With the help of these
additional assumptions, all the laws of the secondary science,
including those containing the term 'A', must be logically
derivable from the theoretical premises and their associated
coordinating definitions in the primary discipline . . . the
'condition of derivability'.33

And the derivability condition implies, Nagel further argues, the

condition of connectability, in the sense that "Connectability would
indeed assure derivability if . . . for every term 'A' in the secondary
science but not in the primary one there is a theoretical term 'B' in
the primary science such that A and B are linked by a biconditional:

A if and only if B."34 For instance, all statements involving the term

 

33 E. Nagel, The Structure of Science, Harcourt, Brace & World, 1961, pp.353-
354.

34 lbid., p.355n.

89

'temperature' in thermodynamics may be replaced salva veritate by
the corresponding sentences of statistical mechanics, in case that the
'temperature' of a gas is the same as the 'mean kinetic energy' of the
molecules which by hypothesis constitute the gas.

In any event, granting that we have a successful explication for
the standard/typical conception of 'ontic reduction' (explicandum) in
terms of the Tarskian formal analysis of 'interpretability' (explicans),
we may now ask what are some important consequences of the
explication in question, especially regarding the issues of ontic
reduction and commitment. In particular, what are the ontological
characteristics of Z—the set of "possible definitions"? How do we
single out one such set, if more than one is available or constructible,
for the purported ontic reduction? Must all reductions that are said
to be intuitively or presystematically acceptable as legitimate
reductions be subject to the strict requirement of "biconditionality"
in 2, that is, the requirement of "explicit" definitions? These are
important questions which deserve to be answered in detail, for the
answers to them will be directly connected to Quine's philosophical
standpoint regarding 'ontic commitment' and 'ontic reduction' (which
we have been seeking), and will therefore be a substantial concern of
the next two chapters, respectively. Before doing them justice in
more detail later, however, we may briefly characterize the ontic
significance of the questions.

In Tarski's terms, 2 is a set of possible definitions with exactly
one formula, of universalized biconditional form, for each predicate
occurring in the reduced but not the reducing theory. Since the

biconditionals of 2 provide a recursive link between the definiens

90

and the definiendum, the primitive predicates of the reduced theory
can be eliminated in favor of the vocabulary of the reducing theory
in the final analysis of the reduction. It appears then that, in
accordance with Quine's criterion of ontological commitment together
with 'ontic reduction' understood thus far, the ontic commitment or
ontology of the reduced theory can be dispensed with in favor of that
of the reducing theory. For instance, given a set-theoretic model of
arithmetic, the language of arithmetic is indeed in principle
eliminable; we can replace it with the language of sets, so that we
need not be troubled about the commitments of numbers sui generis
in addition to the ontology of sets. Moreover, as has been seen in
Tarski's analysis of 'satisfaction' previously, the nonlogical
expressions of a theory can be freely reinterpreted without affecting
the logical consequences of a truth definition, whereas the
interpretations of the logical constants to which the recursive clauses
of the definition apply must remain fixed. In other words, there can
be a number of different but equally truth-preserving satisfaction
definitions—namely, the multiplicity of E's—for interpreting or
reinterpreting a given theory. For instance, for the definition of
number, there are not only Frege's classes but Zermelo's and von
Neumann's classes to be identified with one and the same number.
This leads Quine to adopt and exploit the notion of 'ontological
relativity', in that ontology is immanent to a given theory and the
interpretation of it—the position of 'intemal realism'. This view of
Quine can be contrasted with Frege's platonism, the position of
'external realism', which commits Frege to the ultimate reality being

theory—transcendent and so discoverable, but not to be invented or

91

postulated. (This much will constitute the next chapter immediately
following.)

Regarding the adequacy condition of definability, associated
with Tarski's "possible definitions", Quine wants it to be relaxed to
the effect that the concepts or predicates of the reduced theory need
not be defined in terms only of the "already-significant" vocabulary
of the reducing theory. Quine believes that the reduction from the
mental to the physical, for instance, is an adequate ontic reduction,
but one which cannot fulfill Tarski's strict definability requirement.

For, as Quine remarks,

The reduction of the mental to the physical . . . invites the
gentler phrasing: the mental is explained in physical terms . . .
the assimilation seems reasonable as applied to pain and other
sensations and emotions. Where it perhaps seems less
compelling is in application to thinking . . . It is at this point
that we must perhaps acquiesce in the psychophysical dualism
of predicates, though clinging to our effortless monism of
substance. It is what Davidson has called anomalous monism.
Each occurrence of a mental state is still, we insist, an
occurrence of a physical state of a body, but the groupings of
these occurrences under mentalistic predicates are largely
untranslatable into physiological terms. There is token identity,
to give it a jargon, but type diversity . . . The point of
anomalous monism is just that our mentalistic predicate
imposes on bodily states and events a grouping that cannot be
defined in the special vocabulary of physiology.”

Hence Quine needs to provide an alternative criterion or conception

of 'ontic reduction', especially in connection with the role of 2. (This
and some associated issues are to be addressed and discussed in the

final chapter, Chapter 5.)

 

35 W. V. Quine, From Stimulus to Science, Harvard U., 1995, pp.86-88.

4. Ontological Relativity

In this chapter, we will see the multiplicity of set-theoretic
reduction schemes for arithmetic or the multiple interpretability of
arithmetic in various set theories. The multiplicity in question will be
presented through the point of view of Quine's famous thesis of
'indeterminacy of translation' and eventually will lead to his claim of
'ontological relativity'. Ontological relativity, thus, is a consequence of
adopting the Tarskian notion of 'interpretability' as representing
formally the "standard/typical" notion of 'ontic reduction'. And the
concept of ontic reduction properly understood, then, must imply the
notion of theory-immanent ontology or ontic commitment, against
the Frege-Russell conception of 'ontic commitment' which allows
none but one ontology as the ultimate reality. The argument against
the Frege-Russell view of 'ontic commitment' will be found in

Benacerraf's examination of whether "numbers are in fact classes".

4.1 The Predicament of Frege's System

It was a purpose of Frege's logicism to show that the ontology
or category of numbers can be dispensed with in favor of the
category of classes. The success of his logicism, therefore, depends in
part on the success of his theory of classes. Frege, after having laid
the philosophical foundations of logicism in his Grundlagen, spent the
next nineteen years carrying out this long and difficult programme.

It resulted in his two volumes of Grundgesetze der Arithmatik,

92

93

published separately in 1893 and 1903. In spite of its great value for
the logicist reduction of arithmetic, however, the system of Frege's
Grundgesetze proved to be inconsistent, a death blow for a "formal"
system. Shortly before Frege published the second volume of the
Grundgesetze , he received a letter from Russell announcing Russell's
paradox and showing that it could be derived in Frege's system of

classes:

For a year and half I have been acquainted with your

Grundgesetze der Arithmatik . . . I find myself in complete
agreement with you in all essentials . . . There is just one point
where I have encountered a difficulty . . . because of the

following contradiction. Let w be the predicate: to be a
predicate that cannot be predicated of itself. Can w be
predicated of itself? From each answer its opposite follows.
Therefore we must conclude that w is not a predicate.1

Classically, Frege's logicist system is based on the assumption
that every concept Fx has a class "x(Fx) as its extension, which

constitutes the principle of class-abstraction '(Vx)(x e "y(Fy) <-) Fx)‘,

or, in a modern set theory, the axiom of abstraction:
(3y)(Vx)(x e y (-> Fx),

determining class-membership by the context 'F'. The Russell
paradox is most easily seen from this axiom. We have only to
substitute x e x (i.e. x is not a member of itself) for Fx to get
'(3y)(Vx)(x e y (-) x e x)‘; and putting w in place of y we have

'(Vx)(x e w <-> x e x)’; and by an instantiation, we get

 

1 J. van Heijenoort (ed.), "Letter to Frege," From Frege to Godel, Harvard U.,
1967, pp.124-125.

94

(WEWH we w),

which is a contradiction. Frege immediately felt a disaster for his
logicist foundation of arithmetic, and hurriedly responded in an
appendix to that second volume. His response was to mend his fifth
axiom, i.e. "x(Fx) = "x(Gx) H (Vx)(Fx H Gx), to the effect that contrary
to the axiom, for some concepts F and G, we shall allow "x(Fx) and
"x(Gx) to be one and the same class even though the equivalence

(Vx)(Fx H Gx) is false of one particular object, that is, the class itself.

In other words, Frege was proposing
(VX)[X ¢“y(Fy) -> (xe "y(Fy) H FX)].

but which, according to Quine and others, unfortunately leads only to
another inconsistency.2

One might think that these results are mere technical
difficulties in Frege's system. But it is substantially more than some
formal predicaments in the presence of contradictions. It eventually
gives rise to a serious ontic problem in Frege's notion of a class as the

extension of a concept. As Quine explains,

the prefix "‘x' has been understood, to begin with, as 'the class
of all and only the objects x such that'. If we . . . taking "‘x(Fx) =
"x(Gx)' as true and '(x)(Fx .=. Gx)’ as false, then surely we have
departed from that original reading of "‘x', and left no clue as to
what the classes "x(Fx) and "x(Gx) are supposed to be which are

 

2 For the detailed discussion of it, see W. V. Quine, "On Frege's Way Out," in
Selected Logic Papers (enlarged ed.), Harvard U., 1995—especially, pp.150-153.

95

talked of in the allegedly true equation '(x)(Fx a Gx)’ . . . Similar
difficulties arise over the word 'extension' in Frege's account.3

That is to say that Frege's system, failing its original intention of his
logicism, will no longer be able to one-to-one correlate or map
(Fregean) concepts into (logical) objects. Perhaps, that may have been
expected since no system of notation ever has "expressive power" to
such an extent that each existing class is specifiable—that is,
specifiable by virtue of the abstraction principle. In fact, there is an
implicit presumption to the contrary. For it is a corollary of Cantor's
theorem that there can be no systematic way of assigning a different
expression (say, positive integers) to every subclass of such
expressions (reals for instance), whereas we can quite adequately
assign a different expression to every open sentence or predicate of
any given language.

Frege's notion of a class as the extension of a concept is of
particular interest to us, since it not only leads to the Russellian
paradoxes but is also opposed to the Cantorian or iterative notion of
sets which dominates the mathematical scene today. Frege assures us
that "I do, in fact, maintain that the concept is logically prior to its
extension; and I regard as futile the attempt to take the extension of
a concept as a class, and make it rest, not on the concept, but on
individual things."4 A principal basis for such assurance of Frege
seems to be that the Cantorian notion of set cannot explain the

existence of the empty class, save postulating it which is not

 

3 lbid., p.154.

4 P. T. Geach & M. Black (tr. & ed.), Translations From the Philosophical
Writings of Gottlob Frege, 1960, p.106.

96

explaining it. For, says Frege, "if it is elements that form a system
[class], then the system is removed at the same time as the
elements."5 Thus, to ensure a close tie between concepts and classes,
Frege introduces classes through a second level function (such as the
abstraction principle) that applies to concepts and yields objects as
its values.

Fregean objects are, as we've seen before, possible references
of singular terms, and Frege's class-abstracts are treated as such
(complex) singular terms in his system. And singular terms are
subject to identity conditions. For Frege, identity is a relation whose
field is the whole category of objects. It would seem impossible to
restrict the range of significance of identity, for Frege, because
"Identity is a relation given to us in such a specific form that it is
inconceivable that various kinds of it should occur."6 In other words,
Frege would not permit the category of objects to be divided into
subcategories so that each concept would take its arguments only
from a specified subcategory, as a theory of types would. Instead of
restricting the range of the quantifiers, Frege took an alternative
course in which the restricted range, say F, of quantification would
take the form of a conditional that for every object x, if x is an F,
then x is an G. Given this understanding of quantification, of course, it
is necessary to explain a concept for every comprehensible object as
its argument. But in doing so, Frege was deprived not only of the
theory of types but also of the iterative approach to set theory in

general, in which most of the well-known methods for dealing with

 

5 lbid., p.150.
6 lbid., p.235.

97

the paradoxes have since been developed, such as Zermelo's
limitation of class-size, von Neumann's distinction between sets and
classes, or Russell's type hierarchy and Quine's method of
stratification.7 As a consequence, not only do we lack a satisfactory
repair for Frege's system, but the prominent methods for avoiding
the paradoxes either fail to preserve a significant part of set theory
when applied to Frege's system or they are not compatible with

Frege's notion of sets or classes.

4.2 Multiplicity of Reduction or Model

The various systems of set theory which emerged after the
discovery of Russell's paradox differ considerably in their contents,
especially in the existence assumptions. For all the standard ways to
avoid the paradoxes, except for the theory of types, do so by
recognizing exceptions to the abstraction axiom. Allowing such
exceptions, they have made ad hoc existence premises expressed
within the theories that used them. Thus, says Quine, "A major
concern in set theory is to decide, then, what open sentences to view
as determining classes; or, if I may venture the realistic idiom, what
classes there are."3 For instance, in his set theory, Zermelo (1908)
replaced the abstraction axiom by postulating ad hoc families of
instances of that axiom—such as the existence of the null class, the
class of all subclasses of any given class, the unit class of any given

thing, and his famous Aussonderung axiom specifying that

 

7 See A. Fraenkel et al., Foundations of Set Theory (revised ed.), North-
Holland, 1973—especially, chapters 11 and 111; W. V. Quine, Set Theory and Its
Logic, Harvard U., 1963—especially, Part Three: Axiom Systems.

8 W. V. Quine, Set Theory and Its Logic, 1963, p.3.

98

(3y)(Vx)[x e y H (x e at & Fx)]

which makes possible the limitation of class-size, that is, all sets in
the system being elements of, hence smaller than, the class (1.9 The
limitation doctrine, according to Russell's words which predate
Zermelo's, is one that "there will be (so to speak) a certain limit of
size which no class can reach; and any supposed class which reaches
or surpasses this limit is an improper class, i.e. is a non-entity."10
Another modified form of the abstraction axiom is one in Quine's
"New Foundations". This axiom states that (3y)(Vx)(x e y H Fx)
where 'Fx' is a stratified formula in which y does not occur free.11 A
formula is stratified if it is possible to put numerals for the variables,
without appealing to types, in such a way that '6' comes to occur only
in contexts of the form 'n e n+1'. Note incidentally that the

abstraction axiom in any modified form, then, is a schema, not a
theorem for a given system, because of its schematic letter 'F' in the
axiom or axiom-schema. It belongs, therefore, not to the system itself
but to a metalanguage that can generate the instances of (T)-like
sentences yielding Tarskian "possible definitions" for the context 'F'

(e.g. an arithmetic statement) in terms of the vocabulary 'e' of a set-

theoretic system.

 

9 E. Zermelo (1908) "Investigations in the foundations of set theory," in J. van
Heijenoort (ed.), From Frege to Godel, pp.199-215.

10 B. Russell (1906) "On some difficulties in the theory of transfinite
numbers and order types," in D. Lackey (ed.), Essays in Analysis, George
Braziller, 1973, p.152.

11 W. V. Quine, "New Foundations for Mathematical Logic," From 0 Logical
Point of View (revised ed.), Harvard U., 1980, p.92.

99

In many cases, therefore, different set-theoretic systems use
different languages, and there is not always a natural translation of
the statements of one system to the statements of another system.
This would prompt the notion of 'indeterminacy of translation', an
eminent philosophical position of Quine. Indeterminacy of translation
results from what he calls a 'radical translation' wherein a field
linguist, unaided by an interpreter, is out to penetrate and translate
a language hitherto unknown in a remote foreign place. In such a
situation, the linguist must himself pioneer a translation manual that
would have its utility as an aid to communication with the native
community. Success in communication will be judged by smoothness
of conversation, by frequent predictability of verbal and non-verbal
reactions, and by coherence and plausibility of native testimony.
After a while, according to Quine, the linguist or a group of linguists
will end up with the outcome that "manuals for translating one
language into another can be set up in divergent ways, all compatible
with the totality of speech dispositions, yet incompatible with one
another."12 This is, in a succinct form, Quine's thesis of indeterminacy
of translation or of sentence meaning.

For some philosophers, however, translation offers a reason for
postulating the determinate meanings for sentences, for they believe
that translation consists in finding a sentence in one language which
has the same meaning as a given sentence in another language. This
is not, Quine argues, a correct account of translation, and he dismisses

it as "the myth of a museum in which the exhibits are meanings and

 

12 W. V. Quine, Word and Object, MIT press, 1960, p.27.

100

the words are labels. To switch languages is to change the labels."13
For, for Quine, it would make no sense to speak of the translation of a
single sentence of one language into a corresponding sentence of
another language apart from the holistic translation of all the
sentences one would make. Holism tells us that sentences do not
have their own range of conforming experiences, and the idea that
the meanings of individual sentences are mental or Platonic entities
must be abandoned. Instead, we must recognize that "our statements
about the external world face the tribunal of sense experience not
individually but only as a corporate body."14 We have to cease to
demand or expect of a sentence that it have its own separable
empirical meaning, and admit that "the unit of empirical significance
is the whole of science."15 This point may be illuminated more clearly
in the following example of multiple reduction schemes of arithmetic
into set theory.

Suppose that we want to translate an arithmetic sentence '2 is
an even prime' into our set-theoretic language. In discussing Frege's
definition of number and his reduction of number theory to set
theory, we saw that Frege's scheme met all the conditions that any
successful reduction of number theory should. In other words,
Frege's reduction scheme provided in fact a model for the Peano
axioms. We take zero and successor as primitives which are governed

by the axioms

 

13 W. V. Quine, "Ontological Relativity," Ontological Relativity and Other
Essays, New York, Columbia U. Press, 1969, p.27.

14 W. V. Quine, "Two Dogmas of Empiricism," From a Logical Point of View,
Harvard U., 1980 (rev. ed.), p.41.

15 lbid., p.42.

101

(A1) (VXXO is 300)
(A2) (VX)(VY)(S(X) = 30’) -> X = Y).

We can define, then, the natural numbers together with the less-than
and equicardinality relations, and can also develop the Peano axioms
and the theory of counting.16 Thus, all that remains to complete the
reduction is definitions of zero and successor that enable us to derive
the axioms (A1) and (A2). Frege gave us one definition as we saw
previously. Many other reduction schemes, however, meet these
reductive conditions. In Zermelo's scheme, with zero being defined as

the empty set, 0 = Q, one may identify each number with the unit set

of its predecessor, S(x) = {x}, such that
$6 N & (Vx)(xe N—9 {x}e N)

where N is the class of natural numbers. This yields Zermelo's axiom
of infinity, the most characteristically arithmetic part of set theory.
In von Neumann's scheme, each number may be identified with the

set of all smaller numbers, i.e. S(x) = x u {x}, such that
@e N & (Vx)(xe N—)(xu {x})e N).

An alternative reductive scheme has 0 = {O} and S(x) = {{O} u {x}},
and there are infinitely many others which can be obtained from,
say, Zermelo's progression by choosing any member in the

progression as the new zero and the members of the progression

 

16 See, for instance, W. V. Quine's Set Theory and Its Logic, pp.74-81.

102

following the new zero as the other new numbers, while adjusting
the successor relation so that zero will not be a successor.

Under these circumstances, to translate the arithmetic sentence
'2 is an even prime' into set theoretic language is after all not unlike
what a field linguist makes an effort to do in penetrating a hitherto
unknown jungle language. By diligent observation of the indigenous
speech behavior, he may be able to set up what Quine calls an
analytical hypothesis that "segments heard utterances into
conveniently short recurrent parts, and thus compiles a list of native
'words'."17 For our set theorist, this would amount to a hypothesis
that the natural number two—the reference of the numeral '2'—can
be identified either as the set {0, {O}} of von Neumann, or as {{Q}} of
Zermelo, or similarly as many other sets that would function as the
referent of the numeral '2' which recurs in various arithmetic
contexts. In other words, then, our arithmetic sentence '2 is even

prime' in question can be translated as either

{{9}} is an even prime
or
{Q, {OH is an even prime,

and so forth in accordance with different manuals or systems of such
analytical hypotheses.

Both schemes or manuals, von Neumann's and Zermelo's, satisfy
all the reasonable conditions that are placed on possible manuals of

translation from number theory into set theory. Both schemes are

 

~ 17 W. V. Quine, Word and Object , p.68. And Quine added that "The method of
analytical hypotheses is a way of catapulting oneself into the jungle language
by the momentum of the home language." (p.70)

103

relatively simple in their number-generating syntax, and are capable
of permitting translation of all sentences of number theory while
preserving truth. Yet they do not provide equivalent translations, as
witnessed above. That is to say that there are sentences for which
they provide translations that differ in truth value. Some sentences
even receive no truth value in number theory before translation into
set theory. Consider, for example, the sentence 'the number two has
exactly one member'. It is clear that the sentence has no truth value
in arithmetic. It is, however, translated into a false sentence by von
Neumann's scheme of translation, but comes out true on Zermelo's
scheme. Apart from some such set-theoretic scheme of translation,
therefore, it make no sense to ask what is the correct translation of
an isolated statement, e.g. '2 is an even prime', of number theory.
Translation must always proceed against the background of such a
translation scheme from the one language to the other. Without
reference to a scheme of translation—that is, a given set of set
theoretic axioms including the axiom of infinity—the notion of the
translation of an isolated sentence of number theory is
indeterminate. Nothing can decide between different translation
manuals, or different systems of analytical hypotheses, except the
purpose of the moment. Though a localized example involving formal
languages only, this is in essence Quine's thesis of 'indeterminacy of
translation' according to which there are no singly correct, but rather
multiple, yet equally correct, manuals of translation. They all are
equally legitimate and coherent translation manuals when
considered each separately, but become incompatible when

considered together intertheoretically or interlinguistically. Quine

104

claims that this kind of indeterminacy can be generalized to all
interesting cases of radical translation. And consequently the
postulation of (determinate) meanings for individual sentences is not
vindicated by the mere possibilities of translation from one language
into another.

At this point, there arises a serious objection to the multiplicity
of reductive schemes of arithmetic into set theory. In his "What
numbers could not be,"18 Benacerraf argues as follows. Frege
maintained that numbers are classes. But if they are classes, then
they must be particular classes. For, surely, it would make no sense
to say that the number 2 is a class, but that no particular class is
identical to it. Which class, then, is the number two? Frege said that
it was the class of all pairs. But why not identify it with the class or
set {Q, {OH to follow von Neumann, or the set {{9}} to follow Zermelo,
or indeed many other sets and classes depending on which set theory
we are referring to? It must be identical with at most one of these,
says Benacerraf, but which one? There would seem no mathematical
or other "cognitive" considerations that will, as we saw above, resolve
these questions. But if the number two, or any other number for that
matter, is not any particular class, then how can it be a class at all?
In other words, we have satisfactorily reduced one predicate to
others, certainly, if in terms of these others we have fashioned an
open sentence that is co-extensive with the predicate in question as
originally intended; i.e., that is satisfied by the same values of

variables. But this standard does not suit the Frege, von Neumann,

 

13 P. Benacerraf & H. Putnam, eds., Philosophy of Mathematics (2d ed.),
Cambridge U., 1983. PP.272-294.

105

and Zermelo reductions of number. For these reductions are all
adequate, yet not co-extensive with each other. Benacerraf has
argued from just this premise that the multiplicity of ontic
reductions of arithmetic to set theory precludes the possibility of
numbers really being sets, and that the uniqueness of a reduction is
a necessary condition for its having any philosophical significance. He
thus declared "that numbers could not be sets, 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)."19

The force and structure of Benacerraf‘s argument appears to be
so general that we may fear that no reductions, no interesting
ontological reductions at least, are ever possible. His argument can be
used to discredit many other reductions as well, mathematical or
otherwise, for the reason that any theory that has one model will
usually have infinitely many isomorphic ones. For example, real
numbers can be reduced to either Dedekind cuts of rationals or
Cauchy sequences or some variations on these methods. Therefore,
Benacerrafs argument would have us conclude that the reals are not
logical constructions from the rationals, and the same for other
constructions or interpretations such as an ordered pair being an
unordered class. The argument will be widely applicable against
many reductions of completely different sort, such as well known
scientific reductions from psychology to biology and from biology
eventually to physics. For it is to be expected that if one reduction of

the objects of kind A to the objects of kind B is available, then objects

 

19 Ibid.. pp.29o-291.

106

of other kinds will generally be forthcoming through isomorphic
interpretations. That is, if a theory T is multiply interpretable in
another T', then T cannot be reducible to T' in an important sense
that would have implications for ontology.

Since the interpretability or translatability relations holding
between theories, especially theories of abstract objects, are almost
invariably multiple, the argument can be easily generalized to deny
or discredit the claim that the relations on multiple interpretability
can ever have ontological significance. Even our seemingly clearest
cases of successful reduction—such as that of ordered pairs to
unordered sets—will then turn out not to have any ontological
significance at all. In fact, even if there is a rare case where a
reduction is apparently unique, it will be extremely difficult to prove
that no other mapping or set of possible definitions could do the
same work. We may interpret Benacerraf's position, then, as
demanding the addition of another condition to our account of
ontological reduction. We should add, if Benacerraf is correct, that
there must be a unique mapping or a unique set of possible
definitions which takes truths of one theory into truths of another.
This demand poses a challenge to our entire project of "adequate"
ontological reduction, for the probability for uniqueness of

interpretability of any theory seems vanishingly small.

4.3 Ontological Relativity
One way of responding to this challenge may be to dismiss the
seriousness of logicism, as Steiner did. He diagnosed the problem and

recommended his solution as follows:

107

Benacerraf and Frege both take logicism too seriously as a
philosophy of mathematics, and hence both overreact to its
downfall . . . When confronted by Russell's paradox, which
undermines logicism's epistemological pretensions, Frege sees
arithmetic itself as tottering. And Benacerraf sees, in the failure
of logicism to show that numbers are sets, the ontological doom
of arithmetic. The view of arithmetic that I proffer for
consideration, that of arithmetic as an independent science,
avoids both consequences.20

This, however, hardly answers Benacerraf's question. Steiner seems
content with a detour, avoiding the challenge, without any intention
to resolve or repair the original serviceable road that was damaged,
so to speak. If we take ontological reduction seriously, however,
which has been our position all along, the Benacerrafian challenge
deserves at least our earnest effort to "repair" or answer it.

Another response to Benacerraf would be to link it to the
problem of interpreting theories, as Russell did. One could argue that
we may single out one interpretation as being ontologically
significant and discard the others as philosophically uninteresting.
We can develop, Russell insisted, a unique analysis of number-
theoretic notions by considering not only arithmetic itself but
extramathematical or applied contexts as well, such as 'There are n
F's' or 'There are as many x's as y's', and so on. And he believed that

his own analysis is the proper explication of number, because

Our reasons for being specially interested in the axioms that
lead to ordinary arithmetic lie outside arithmetic, and have to
do with the application of number to empirical material. This
application itself forms no part of either logic or arithmetic; but

 

20 M. Steiner, Mathematical Knowledge, Cornell U. Press, 1975, p.92.

108

a theory which makes it a priori impossible cannot be right.
The logical definition of numbers makes their connection with
the actual world of countable objects intelligible; the formalist
theory does not.21

But this merely shows that Russell's response is ad hoc, because
different people may have different intuitions regarding which
feature of the natural number progression is the main characteristic
we are to reckon with. Russell's extramathematical contexts are
irrelevant or insignificant to decide such intuitions. For one might
argue for the intuitiveness of the Frege-Russellian version of number
which serves primarily to measure the size, or might argue
differently for the intuitiveness of von Neumann's version which
views the main function of number as counting. We would choose
either one at any different time to suit the job in hand.22

The proper response to Benacerrafs challenge would be, I
think, not to question the validity of his argument but rather to take
a closer look at the purpose of an ontological reduction and our
expectation from it. Regarding the purpose of the Wienerian
construction or definition of the notion of ordered pair, where the
ordered pair <x, y> is identified with the class {{x}, {y, OH, Quine says
that such a "construction is paradigmatic of what we are most
typically up to when in a philosophical spirit we offer an 'analysis' or
'explication' of some hitherto inadequately formulated 'idea' or

expression."23 The notion of ordered pair is a widely useful notion

 

21 B. Russell, Principles of Mathematics, 1903, p.vi; italics emphasized. See
also Introduction to Mathematical Philosophy, 1919, pp.8-9.

22 See, for instance, Quine's Word and Object , p.263; Ontological Relativity,
pp.44-45; The Ways of Paradox, pp.213-214.

23 Word and Object, p.258.

109

wanted for objects of any sort, in that an ordered pair will be treated
as a single object doing the work of two already recognized objects. A
typical use of the notion is in assimilating relations to classes, by
taking them as classes of ordered pairs. The uncle relation, for
instance, is the class of all ordered pairs <x, y> such that x is uncle of
y. We want <x, y> to be determined by x and y, in the way that it
differs from <y, x> unless y is x. Therefore, any construction subject

to the following single postulate
<x, y> = <z, w> if and only if x = z and y = w

will in effect be construed as an adequate definition of 'ordered pair'.
Accordingly, the notion of ordered pair has been explicated by
several different constructions or definitions of it, each such
definition satisfying the postulate. So Kuratowski showed, in a way
that is different from Wiener's version of ordered pairs, that the pair
<x, y> may also be identified with {{x}, {x, y}}. In fact, the notion of
ordered pair can be defined or explicated in an infinite variety of
ways, such as {2x, 3V} or {{x, O}, {x, O, y}}. all versions of the notion
satisfying the only required postulate.

A demand that there must be a unique correct analysis of the
"intended" notion of ordered pairs, then, may be said to be "wrong
headed". Such a demand has been often based on the mistaken
notion that philosophical analysis or explication must consist
somehow in the uncovering of hidden meanings and underlying
intensions. Indeed, some philosophers have argued that for a
philosophical explication to be informative at all we must beforehand

know the correct meanings of its terms, and hence already know that

110

the terms which it equates are synonymous. The argument, however,
merely reveals a misunderstanding about the informative nature of
'philosophical analysis', and could only lead to the so-called paradox
of analysis in that no philosophical explication would be
philosophically productive. According to Quine, such a lingering air of
paradox simply stems from a miscomprehension of the true function
of the analysis in question. Quine reminds us that when we are in a

position to generate philosophically a useful analysis or explication,

We do not claim synonymy. We do not claim to make clear and
explicit what the users of the unclear expression had
unconsciously in mind all along. We do not expose hidden
meanings, as the words 'analysis' and 'explication' would
suggest; we supply lacks. We fix on the particular functions of
the unclear expression that make it worth troubling about, and
then devise a substitute, clear and couched in terms to our
liking, that fills those functions. Beyond those conditions of
partial agreement, dictated by our interests and purposes, any
traits of the explicans come under the head of "don't cares".24

This concept of 'philosophical explication‘ does not claim a
uniquely correct way of traversing its terrain, to speak figuratively.
Nor, needless to say, can it ever claim finality or perfect
understanding for some particular exploration of its terrain.
Nevertheless, it captures what is really happening when a useful or
informative philosophical explication is being brought about. As a
condition or criterion for an adequate philosophical analysis, it is
neither necessary nor sufficient for us to know every detail of the
associated but irrelevant "don't cares" that come along with the

terms that attract our attention. It would certainly be a mistake to

 

24 Ibid.. pp.258-259.

111

identify such details as the essential constituents of the meanings of
the explicandum in question. In other words, given a confused but
philosophically serviceable concept, the only context or meaning that
is worthwhile to explicate would be the limited context that fits the
purported use or aim in hand. In the case of 'ordered pair'. for
instance, such limited contexts ought to be ones where the postulate
above can be exploited without difficulty. From the standpoint of this
enlightened view of 'philosophical analysis', then, artificiality or
multiplicity of the explications (Wiener's, Kuratowski's, and others)
for a concept in question ('ordered pair') will not be obstacles at all;
they will rather become the desirable features contributing to some
further philosophically useful goals.

Now, we can relate this to Benacerraf's challenge. It would
seem, indeed, that the question 'What is a number?‘ is on a par with
the corresponding question about ordered pairs. Recall that we were
previously forced to face the fact that there are many schemes for
reducing arithmetic to set theory, and that no single one such
scheme, to the exclusion of others, may be identified as the unique
correct reduction. But the uniqueness condition, according to
Benacerraf, is required because numbers are particular objects, and
so must be their corresponding set-theoretic representations.
Otherwise, the identity condition—the substitutivity of identicals—for
a given object will not be satisfied, and we accept no entity without
identity. In other words, if the number 2 is a class or set at all, it
must be some particular set. Why is this a difficulty? The problem is
simply that our reduction schemes are multiple and disagree. One

assigns {{9}} to the number two, while another assigns {9, {9}} to the

112

very same number; there are in fact infinitely many conflicting
assignments for each number. But the number two cannot, according
to the challenge, be both {{9}} and {9, {9}} on pain of inconsistency,
i.e. {{9}} = {9, {9}}, and we have in principle no way of deciding
which account fails in some way and which might be correct. So, our
purported schemes must fail to meet some proper criteria for the
adequacy of a reduction. Thus, Benacerraf concludes, numbers must
not be sets at all. Any attempt to reduce numbers to sets, therefore,
must be doomed owing to lack of relevant identification criteria
which, he insists, would constitute the only interesting sense of
'ontological reduction'.

This demand of an additional "uniqueness" condition to our
account of ontological reduction, however, seems to be an
unreasonably strong one if we consider what the purpose, and the
method, of reduction were in the first place. We have to realize at
once that the original purpose of our logicistic reduction was to show
that a number theory committing us to numbers can be restated in
set theory so that the category of numbers as distinct from that of
sets may be eliminated. For, after the reduction, we would not
mention numbers as some ultimate constituents in our complete
description of the world. In this sense, our set-theoretic explication
of arithmetic is not so much identification between numbers and sets
but elimination, so that "the corresponding objects of the new scheme
will tend to be looked upon as the old mysterious objects minus the
mystery."25 Ontological reduction actually does not and cannot

demonstrate the identity of objects from two different realms of

 

25 lbid., p.261.

113

existence. For, by an ontological reduction or intertheoretic
translation scheme, we do not prove that numbers are really sets,
but only show that we need not be ontologically committed to the
separation of those realms.

And the methodological question as to which translation
scheme of arithmetic into set theory is the correct one, thus, is
philosophically a wrong question based on the notion of identity
between numbers and sets, which in turn is based on the supposition
that there must be the uniquely right explication of 'number'.
Instead, we saw previously that each of Frege's, Zermelo's, and von
Neumann’s way of construing natural numbers is an adequate,
though not unique, explication of 'number'. Because each is a
structure-preserving model of the natural numbers, and that is a
sufficient adequacy condition for the reduction in question. All three
versions fulfill the axioms (A1) and (A2) which together are
sufficient to develop arithmetic: and those intertheoretic conflicts or
inconsistencies, such as {{9}} = {9, {9}}, arise only out of the "don't
cares". From this perspective, we are free to allow the explicans all
manner of novel connotations never associated with the
explicandum. For example, Frege's logicistic definition of number has
been called wrong because it says of numbers that they have classes
as members, which was not so before in arithmetic.

To say that the purported function of logicistic reduction is
eliminability of numbers in favor of classes, therefore, is to say that
there is no arithmetic fact of the matter regarding correct translation
into set theory. There remains only structures of arithmetic; indeed

"Arithmetic is, in this sense, all there is to number: there is no saying

114

absolutely what the numbers are; there is only arithmetic."26 In
consequence, arithmetic becomes that part of a particular set theory
which preserves intact all the necessary laws of arithmetic, such as
'p(m + n) = (pm + pn)‘, but which operates on a different "particular"
class N, that is, on Zermelo's N = {9, {9}, {{9}}, {{{9}}}, . . .} or on von
Neumann's N = {9, {9}, {9, {9}}, {9, {9}, {9, {9}}, . . .} and so on. There
appears to be no clear and definite way of answering questions as to
which among the several isomorphic universes that may serve as the
model of a given arithmetic we actually have in mind at any
particular time. Empirical considerations are no help, and our
attempts to clarify with words just what we mean seems only to

complicate matters further. Quine assesses the situation as follows:

We are finding no clear difference between specifying a
universe of discourse—the range of the variables of
quantification—and reducing that universe to some other . . . to
say more particularly what numbers themselves are is in no
evident way different from just dropping numbers and
assigning to arithmetic one or another new model, say in set
theory.27

Thus, although a theory is commonly understood to be a set of
"fully" interpreted sentences, it is impossible to specify, within a
theory, what model we intend for the theory to have. We question
meaningfully the reference of terms in a language only by recourse
to some "background" language that gives the query sense, if only

relative sense. In specifying a theory, Quine explains, "we do fully

 

25 W. V. Quine, "Ontological Relativity," Ontological Relativity and Other
Essays, New York, Columbia U. Press, 1969, p.45.

27 lbid., p.43.

115

interpret the theory, relative to our own words and relative to our
overall home theory which lies behind them. But this fixes the
objects of the described theory only relative to those of the home
theory; and these can, at will, be questioned in turn."28 In the home
language, with its vocabulary taken for granted, we fix reference
simply by saying that this is a rabbit and that is a number and that
is a desk—in just those words. It is meaningless to ask if 'rabbit',
'number' or 'desk' really refer in an absolute sense to a rabbit, a
number or a desk. It is meaningless to ask whether the numeral '2'
really refers to the number two rather than to ingeniously permuted
denotations such as Zermelo's {{9}} or von Neumann's {9, {9}}. In
general, thus, it makes no sense to say what the objects of a theory
are, beyond saying how to interpret or reinterpret that theory in
another—say, 'arithmetic' in 'set theory' or vice versa. A question
'What is an F?', according to Quine, can be answered only relative to
the "uncritical acceptance" of a further term G that an F is a G. Such is
the thesis of 'ontological relativity' that "reference is nonsense except
relative to a coordinate system."29 To inquire about reference in any
more absolute way would be like asking about absolute position or
velocity, rather than position or velocity relative to a given frame of
reference.

In providing a full semantic interpretation of some theory, in
terms of some selected background theory, we are indirectly
applying the laws and concepts of the object theory to the extent that

we are showing that they may be reduced, via our scheme for

 

28 lbid., p.51.
29 lbid., p.48.

116

translation, to those of the background theory whose applicability is
already presupposed. Quine explains the relativity of this
understanding by reference to the variously intertranslatable

theories of protosyntax, arithmetic, and set theory:

Expressions are known only by their laws, the laws of
concatenation theory, so that any constructs obeying those
laws—Godel numbers, for instance—are ipso facto eligible as
explications of expression. Numbers in turn are known only by
their laws, the laws of arithmetic, so that any constructs
obeying those laws—certain sets, for example—are eligible in
turn as explications of number. Sets in turn are known only by
their laws, the laws of set theory.30

When theories are reduced to or are interpretable in one another in
this fashion, there is simply no objective way of distinguishing the
ontology and doctrine of one from the ontology and doctrine of the
other. Just which objects and which truths we view as basic will
depend, so far as this makes any sense at all, upon which theory we
adopt as our background theory, or frame of reference, from which
all attempts at verbal explication must proceed. To say what the
objects of a theory are, therefore, turns out to be to say how
reference to those objects can be eliminated in favor of reference to
the objects of the background theory. In a sense, our attempts to deal
with the question of what there is are attempts to analyze or
explicate what there is not. We must think of a theory's ontological
commitments not as commitments per se, but as commitments
relative to the background language in which the interpretation of

the theory is given. For instance, when we reduce arithmetic to set

 

30 lbid., p.44.

117

theory or vice versa, we interpret arithmetic as being about sets or
set theory as being about numbers. The ontological commitment to
numbers or to sets, then, proves to be only prima facie in such
reductions; it ought to be a commitment relative to the background
theory of numbers or of sets, respectively.

Thus, for Quine what makes absolute ontological questions
meaningless is "not universality but circularity,"31 as opposed to
Carnap's "Allwdrter" of the linguistic frameworks that generates his
dichotomy of "internal" (empirical, theoretical) and "external"
(metaphysical, practical) statements of science. It is not a question of
universal predication that becomes meaningless when asked
absolutely of its ontic commitment. Rather, ontological talk for a
given theory is "meaningless insofar as the theory is considered in
and of itself."32 For, then, it is like asking what is the real or intended
model of a given theory when we are given no alternative models
save the already given interpretation as the only model for the
theory. The question of ontology for a theory makes sense only
relative to some translation of the theory into a background theory
in which some idioms of identity and quantification are available.
And Quine's translational indeterminacy tells us that there is always
a multiplicity of such background theories or translational manuals.
As we've seen above, Quine educates us that in order to determine
which terms in a sentence (if any) refer to objects, we must place the
sentence in the context of a theory or a system of interrelated

sentences. Talk of object, according to Quine, is meaningless apart

 

31 Ibid.. p.53.
32 lbid., p.64.

118

from such a system. Since the system is forever under-determined
by experience, a multiplicity of such systems is possible or
inevitable. Several of them might be consistent with all our
experience but inconsistent with one another. Therefore, experience
alone cannot provide a criterion for selecting among such systems.
Nor can anything else. So the indeterminacy or multiplicity of
translation is not contingent; it is a necessary result for any language
translation or theory-rebuilding in terms of the vocabulary other
than originally given. (Indeterminacy of translation is not the
intranslatibility of a sentence but the multiple translatibility of the

sentence.)

5. Proxy Function

In this chapter, as promised, we investigate Quine's alternative
conception of 'ontic reduction', based on his 'proxy function' criterion
that eventually provides a "structure-relative" view of ontology or
ontic commitment. In view of ontic relativity, however, we will first
consider what would constitute an adequate ontic decision for the
"ultimate reality" (if that makes sense at all). A comprehensive
criterion for such a decision would finally give us a clue to solve or
dissolve the dilemma in Quine's apparently "conflicting" philosophical
positions: namely, physicalism and pure set-theoretic ontology. One
criterion for such a decision would be 'ontic economy', and the neo-
Pythagoreanism based on the Skolem-deenheim theorem claims its
ontology to be the most economical one. Quine rejects neo-
Pythagoreanism on the ground that it is not really a viable ontology
for science, and will not meet his proxy-function requirement in its
reduction procedure. Quine's proxy-function criterion for 'ontic
reduction' not only provides a necessary and sufficient condition for
all preanalytically legitimate ontic reductions, but gives an insight
that what is essential in theory-reduction is not ontology but

Stl'llCtlll'C.

5.1 Ontic Decision
To break the ontological regress, we have to take the meta-

language or background language at face value, not asking about its

119

120

intended interpretation. We assume that we share enough of an
understanding of our intentions in using it to get by. However, since
every theory or language is subject to relativistic reinterpretation,
one might ask what do we take the phrase "at face value" to mean?
Quine sometimes seems to equate it with the identity-translation

based on the prior language practice:

what ontological relativity is relative to . . . is . . . a manual of
translation . . . And does the inscrutability or relativity extend
also somehow to the home language? In 'Ontological Relativity'
I said it did, for the home language can be translated into itself
by permutations that depart materially from the mere identity
transformation, as proxy functions bear out. But if we choose as
our manual of translation the identity transformation, thus
taking the home language at face value, the relativity is
resolved. Reference is then explicated in paradigms analogous
to Tarski's truth paradigm; thus 'rabbit' denotes rabbits,
whatever they are, and 'Boston' designates Boston.l

But sometimes Quine also intends it to be a matter of our
"deliberate convention", in a way that Carnap might have spoken. In
mathematics, for the present time at least, we commonly end our
attempts to clarify or explicate mathematical discourse by finally
acquiescing in set theory itself at face value. The grounds for doing
so are, perhaps, those of theoretical simplicity and ontological
economy. Yet in not requiring complete interpretation by means of
other theories, mathematical or otherwise, we do not thereby accept
set theory as merely an uninterpreted calculus or formal system. The
truth of set theory and the objects it purports to deal with must

somehow be understood and accepted on their own. How, then, are

 

1 W. V. Quine, "Three Indeterminacies," in R. Barrett & R. Gibson (eds.),
Perspectives on Quine, Basil Blackwell. 1990, p.6.

121

we to answer questions of truth and existence in set theory itself
once we realize that these questions cannot be profitably postponed
as questions relating to some background theory (say, category
theory) in terms of which we might seek to understand or interpret
our talk of sets? Oddly enough, Quine says, it is with respect to set

theory itself:

Set theory is pursued as interpreted mathematics, like
arithmetic and analysis; indeed, it is to set theory that those
further branches are reducible. In set theory we discourse
about certain immaterial entities, real or erroneously alleged,
viz., sets or classes. And it is in the effort to make up our minds
about genuine truth and falsity of sentences about these
objects that we find ourselves engaged in something very like
convention in an ordinary non-metaphorical sense of the word.
We find ourselves making deliberate choices and setting them
forth unaccompanied by any attempt at justification other than
in terms of elegance and convenience. These adoptions, called
postulates, and their logical consequences (via elementary
logic), are true until further notice.2

The point is that the ontology to which we subscribe for the
"true" account of the world depends on our deliberate but essentially
"free" choice of a background theory and thereby the ontology it is
committed to—i.e., either to stick to the old ontology or to switch to a
new ontology. The point can be seen in its relation to the dilemma
put at the beginning of this dissertation. For now, prima facie, with
the doctrine of ontological relativity at hand it appears to be an easy
task to explain why there is no real dilemma or conflict between

Quine's apparent commitments to physicalism and to a pure set-

 

2 "Carnap and Logical Truth," The Ways of Paradox and Other Essays (rev.
ed.), Harvard U., 1976, p.117.

122

theoretic ontology. For ontic relativity gives us room for choosing
either one of such ontologies as the ultimate reality to be identified
and described, provided the alternatives are not chosen at the same
time. However, although the relativity alleviates much of the tension
between choosing either of the ontologies in question, it still leaves
some questions unanswered, such as the following. To cut into the
knot of ontic regress or relativity, we must be able to ask or choose
what is to be reduced to what. It divides the traditional issue of
ontology into two separate questions: the question 'What does a
theory says there is?’ is a question of ontic commitment of a given
theory; while the question 'What really is there?‘ is a question about
how we are to adjudicate among rival ontologies, i.e., what categories
of objects we are to take as ultimate in our account of reality.

The latter question addresses the "primacy" of the direction of
ontological reduction, namely, what ought to be reduced to what,
contrasted to the question of what is reducible to what or what can
be reduced to what. And it may be called the question of our ontic
decision—the decision to take as the limning of ultimate reality a
background language at face value, out of which all other theories of
the world can eventually derive their "ultimate" meanings and
references. But when we are faced with mutually reducible or
interpretable theories, how can we decide which theory is to be
reduced to which? What would be appropriate constraints upon or
considerations relevant to such a decision to adopt one ontology
rather than another? Can there exist a single (background) language,
and thereby its ontology, adequate to all purposes of science? These

questions bring us to the heart of the dilemma and need to be

123

answered or clarified at least, prior to our exclusive choice between
physicalism and the abstract ontology of pure sets (since the ultimate

reality cannot be multiple).

5.2 The Pythagorean Challenge

Granted that ontic relativity is inevitable, there arises the
question 'How economical an ontology can we achieve and still have a
language adequate to scientific purposes?‘ A conspicuously
"economical" answer comes from a philosophical view, supposedly
derived from the ontological implications of the L'owenheim-Skolem
theorem, to the effect that we can adopt an all-purpose Pythagorean
ontology consisting exclusively of natural numbers. This is actually
on a par with the "brave new ontology" of pure set-theory (which
may be called 'all-purpose Cantorian ontology'), but which is more
ambitious in its economy. For, for both Pythagorean and Cantorian
ontologies, it is the goal of minimizing categories or primitive
predicates, not just the total number of objects, that makes
conspicuously for the simplicity of a theory and, hence, for its
ontological economy. Moreover, for the Pythagorean, in reducing set
theory to number theory instead of reducing number theory to set
theory, there seems to be an evident gain in economy since infinite
sets go. Anyway, the Lowenheim-Skolem theorem which allegedly
supports this (neo-) Pythagoreanism states that if a set of first-order
formulas has a model in a nonempty domain D, then it has a model in
a countable subdomain of D. But we know that any countable
ontology can be reduced to a numerical one, by an enumeration

function which we need anyway in order to demonstrate the

124

denumerability of the ontology. 80 according to the L-S theorem we
can model any first-order theory in the domain of natural numbers.
Therefore, on this "economical" view, we may be entitled to adopt the
Pythagorean ontology for any theory we hold, in the sense that any
theory can be reduced to or reinterpreted in a theory committing us
only to the ontology of natural numbers.

Michael Jubien objects to this Pythagorean move for the reason
that the L-S theorem is not directly applied to the question of
ontology, and accordingly "one who infers [Pythagorean ontology]
must be prepared to endorse the view that we are at liberty to
choose any model of a theory as the ontology of the theory. For
without this premise—or something very like it—[Pythagorean
ontology] is simply a non sequitur."3 However, according to him, even
if the adoption of the Pythagorean ontology proves to be consistent
with a given theory, we are not entitled to choose such an ontology.
For, to do so, one has to deny the independent considerations that
dictate the choice of ontology for the theory in the first place. For
example, if one's theory deals with some sensory evidence in a
physics or biology lab, then for him the notion that all models are on
an equal footing is visibly absurd. Since, for such a lab-scientist,
there must be some epistemological constraints placed upon his
theory; otherwise the evidence for his "physical" or "biological"
theory does not make sense to him. Thus, Jubien argues that in
assuming the all-purpose Pythagorean ontology ostensively provided

by the proof of the L-S theorem, one "overlooks the possibility of

 

3 M. Jubien, "Two Kinds of Reduction," The Journal of Philosophy, vol. 66
(1969). p.536.

125

there being compelling reasons for us to eliminate certain models
from ontological consideration."4 This is reminiscent of Russell's claim
that, in discussing Frege's reduction of arithmetic to set theory, more
is needed for a reduction than a truth-preserving model of
arithmetic. Russell had maintained that we must also construct a
method of translating "mixed contexts" in which arithmetical
expressions occur with nonmathematical expressions.

Dale Gottlieb likewise rejects the "blanket Pythagoreanism"
inferred from the L-S theorem, on the ground that "we must be
careful not to confuse the legitimacy of a reduction with its over-all
desirability."5 He demonstrated his point by the familiar case of set-
theoretic reductions of number theory, for example, of Peano
arithmetic based upon two primitives, 'x is zero' and 'y is the
successor of x'. As is well known, for instance, Zermelo interpreted
the Peano primitives as 'x is the empty set' and 'y = {x}'. However, on
another possible construction Peano primitives are to be interpreted
as, respectively, 'x is the empty set' and '[()’ = x & (3x)(x is a
unicorn)) v y = {x}]'. One might also propose infinite variations of the
latter, which involve nonarithmetic facts but which all are
extensionally equivalent to the Zermelo construction. Nevertheless,
unlike Zermelo's construction, we do not consider the Unicorn-
construction as an adequate model for Peano arithmetic. The reason
is that, for example, the Peano sentence '(Vx)-Sxx' (i.e. 'No number is

its own successor') can be interpreted in the Unicorn-construction as

 

4 lbid., p.540.
5 D. Gottlieb, "Ontological Reduction," The Journal of Philosophy, vol. 73
(1976). p.75.

126
(Vx)-([x = x & (3x)(x is unicorn)] v {x} = x),

which logically implies,

-(3x)(x is a unicorn)

But this sentence has no theoretical relation whatsoever, arithmetical
or otherwise, to the given Peano sentence 'No number is its own
successor'. Since arithmetical truths do not logically imply, or
ontologically commit us to, the existence or nonexistence of unicorns
(or for that matter any contingent nonarithmetic facts), the Unicorn-
construction or the likes are not to be counted as desirable reductive
models for elementary arithmetic.

Thus, for Gottlieb, the desirable direction for the mutually
reducible theories must emerge from our consideration of the
epistemic structure of the theories involved. An adequate ontic
reduction for him, then, will have to do more than just preserve
extensionality; it requires rather "a guarantee that the cognitive
usefulness of the theory to be reduced not be destroyed by the
reduction."6 Otherwise, according to him, we may expect cognitive or
epistemic anomalies as a result. In fact, the Pythagorean reduction of,
say, marine biology can be expected to sanction some pretty weird
translations (consisting entirely of numerals), which surely generate
epistemic or semantic inscrutabilities. Mark Steiner voices the same
opinion when he says that "explanation exists in mathematics just as
in the sciences. Two proofs of a theorem may differ in explanatory

value. And a mathematical explanation articulated in one theory

 

6 lbid., p.70.

127

might be destroyed when the theory is modeled in another."7
Therefore, for Jubien, Gottlieb and Steiner, ontic economy based on
the models that merely provide "consistent" reinterpretations is
illusory or meaningless, and does not contribute new beliefs and new
knowledge of the sort which can usually be gained in the continuing
practice of the old theory. In other words, for them, the justification
one has for his beliefs stated in the Pythagorean all-purpose
language will have to persistently rely upon evidence in the old
language or theory, so that we are left as dependent as before upon
the original ontology.

In his criticism of Gottlieb, Frederick Kroon downplays the
importance of epistemic features in ontological reduction. He asks us
to consider the reduction of mental states to the physico-neural
states of our body.8 Since, in this particular case, if anything is an
"essential" epistemic feature to be preserved in Gottlieb's sense, it
has to be by any account mental-state talk. If we are to have a
reductive scheme from the mental to the neural, then, in accordance
with the Gottlieb condition or "guarantee", it will have to preserve
the epistemic fact that a person, Joan, knows in an especially
intimate way that she is in pain. But the reducing 'neural' talk does
not and cannot warrant Joan to believe that she is also in the
corresponding physico-neural state. So in the reduction of the mental
to the physical we do not, and in fact cannot, preserve the "essential"

epistemic structure of the old theory in the new theory. One might

 

7 M. Steiner, "Quine and Mathematical Reduction," in R. Shahan & C. Swoyer,
Essays on the Philosophy of W. V. Quine, U. of Oklahoma, 1978, p.134.

8 F. Kroon, "Against Ontological Reduction," in Erkenntnis, vol. 36 (1992),
pp.53-81: see especially pp.62-66.

128

argue that the physico-neural reduction of the mental shouldn't be
construed as an adequate ontic reduction simply because it does not
preserve the essential epistemic features of the mental. But that
would be question-begging and provide no useful objection to the
neuro—physical reduction of the mental, especially given that the
reduction in question is the standard conception in our current
scientific studies of the mental.

Daniel Bonevac also argues against Gottlieb's idea that the
epistemic consideration of a theory must be an essential criterion for
an adequate ontic reduction. He maintains that "What we take as
essential to epistemic structure does not determine which reductions
we sanction as properly ontological; it is more nearly correct to say
that which reductions we sanction as ontological determines what we
take as [epistemically] essential. At the very least, there is a
symbiotic relationship between the two."9 In other words, for both
Bonevac and Kroon, talk of epistemology can be detached from talk
of ontic reduction, in the sense that in order to have a successful
intertheoretic reduction it is not necessary to preserve in the
reducing theory the epistemic structure of the reduced theory. For
there are cases, such as the reduction of the mental to the physical
and the reduction of the phenomenal gas laws to the kinetic theory
of molecules, in which the aim of reduction is simply to drop or to
eliminate the old talk. In such cases, the reducing theory is
concerned with neither the old ontology nor the old epistemic

structure of the reduced theory. For the epistemic evidence,

 

9 D. Bonevac, Reduction in the Abstract Sciences, Hackett Publishing, 1982,
p.132.

129

empirical or otherwise, that supposedly supports the truths of the
old theory would not remain so after the reduction in question; it
would no longer contribute or warrant the "same" evidence for the

truth of the new, reducing, theory.

5.3 Quine's Criterion of a Proxy Function

The Pythagorean claim by way of the L-S theorem, together
with philosophers' responses seen above, illustrate the depth of the
problem we are facing. We seem to have lost our grasp on what
ontological reductions are supposed to accomplish, and how they are
to be justified. Not only are we in search of an adequate criterion for
ontological reduction, but we are unclear concerning cases. Obviously,
though, any criterion or standard of ontic reduction must be judged
to be inadequate if it either excludes some plausible reduction, or
allows an implausible reduction. Quine, in his Word and Object,

carefully observes two basically different sorts of ontic reduction:

When Frege explains numbers as classes of classes, or
eliminates them in favor of classes of classes, he paraphrases
the standard contexts of numerical expressions into
antecedently significant contexts of the corresponding
expressions for classes; thus 'has . . . members' gives way to 'e',
and arithmetical operators such as '+' give way to appropriately
definable class-theoretic operators. But when we explain
mental states as bodily states, or eliminate them in favor of
bodily states, in the easy fashion here envisaged, we do not
paraphrase the standard contexts of the mental terms into
independently explained contexts of physical terms. Thus the
'Jones is in' of 'Jones is in pain', the 'Jones is' of 'Jones is angry',
remain unchanged, but merely come to be thought of as taking
physicalistic rather than mentalistic complements. The radical
reduction that would resolve the mental states into the

130

independently recognized elements of physiological theory is a
separate and far more ambitious program.”

We may call these, respectively, identificatory reduction (e.g. Frege's
reduction) and eliminative reduction (e.g. reducing the mental to the
physical). The terminology might be misleading here, since both
kinds purport the elimination of "old" objects by virtue of the "new"
objects. However, it is meant by 'eliminative reduction' that the "old"
theory will be dropped off and will be entirely replaced by the "new"
theory (e.g. eliminative materialism in philosophy of mind replacing
folk-psychology); whereas by 'identificatory reduction' it is meant
that the old theory remains even after its ontology is switched to the
ontology of the new theory (e.g. number theory founded on a set-
theoretic interpretation).

Eliminative reduction is the kind of reduction that will be
applied to what Quine calls 'entia non grata', such as meanings,
attributes, mental-states, unactualized possibles (e.g. 'Pegasus') and
other intensional entities. Discourse that is supposedly about such
alleged entities, in this eliminative reduction scheme, is to be
replaced by discourse about "corresponding" (but not identifiable)
extensional objects without any attempt to preserve the original
idioms intact. It is no aim of such an eliminative reduction that it be
able to define (in the form of biconditionals) the old concepts or
predicates in the new theory; nor to derive or preserve the truths of
the old theory in the reducing theory. Eliminative reduction also

seems to be applicable to the cases of some ideal entities, such as

 

10 Word and Object, MIT press. 1960, pp.265-266. Italics are all emphasized.

131

'infinitesimals' in Newton's or Leibniz's differential calculus or
geometric points as unextended spatial objects. Weierstrass, in his
theory of limits, showed how to paraphrase or explicate the language
and theory of infinitesimals in using only "proper" numbers as values
of variables, without impairing the utility of the calculus. In this
respect, Quine says, "explication is elimination" and "if there was a
question of [old] objects . . . the corresponding objects of the new
scheme will tend to be looked upon as the old mysterious objects
minus the mystery."11 The main point of 'eliminative reduction',
then, is that the original talk about 'entia non grata' was obscure and
confused, so it often misled in what are otherwise important and
intelligible assertions about the world. Accordingly, we are urged to
forsake the old talk in general, and keep only the "intelligible" part of
old assertions in terms of the new idioms or conceptual scheme.
Identificatory reduction, on the other hand, applies to the
reductive cases that identify the old ontology with the new, e.g.
numbers in terms of sets, "impure units" of measurement (e.g. 'mile',
'minute' or 'degree Fahrenheit') in terms of pure numbers, or
chemistry in terms of physics. It purports solely to provide a pure
ontological economy or a unity of science, for there is no suggestion
that the reduced theories, say elementary arithmetic for instance, are
unclear or misleading. In fact, from a semantic or intuitive point of
view, elementary arithmetic is much clearer than set theory; the so-
called "impure" unit-measures are much more intuitive and better
understood than the counterpart reductive translation (e.g. 'degree

Fahrenheit' explicated in terms of ordered pairs of physical objects

 

11 Ibid..p.26l.

132

and "pure" numbers); and the need for chemistry or chemical idioms
is indisputable, even though its ideology or vocabulary is all
explicable in physics. Therefore, all that is purported in this
identificatory form of reduction is modeling, in that we find a
"consistent" reinterpretation for a given theory that saves on
ontology or, what is the same thing, depends on less ideology
(achieved through explicit definition) and thereby involves less ontic
commitment.

Trapped between the two polar analyses or criteria for an
adequate ontic reduction—i.e. between mere truth-preserving
modeling (or "consistent" reinterpretation) and intertheoretic
definability (and derivability therewith), but neither of which
properly captures the other cases of reduction—Quine locates his
criterion in the existence of a proxy function for all the "legitimate"

ontic reductions:

All that is needed in either case, clearly, is a rule whereby a
unique object of the supposedly new sort is assigned to each of
the old objects. I call such a rule a proxy function. Then, instead
of predicating a general term 'P' of an old object x, saying that x
is a P, we reinterpret x as a new object and say that it is the f
of a P, where 'f' expresses the proxy function. Instead of saying
that x is a dog, we say that x is the lifelong filament of space-
time taken up by a dog [i.e. the case of 'identificatory
reduction']. Or, really, we just adhere to the old term 'P', 'dog',
and reinterpret it as 'f of a P', 'place-time of a dog' [i.e. the case
of 'eliminative reduction'].l2

 

12 ”Things and Their Place in Theories," Theories and Things , Harvard U.,
1981, p.19. Angled brackets and their contents are mine.

133

More generally, in' the level of theories, Quine characterizes his

criterion or standard for an adequate ontic reduction as follows.

The standard of reduction of a theory 6 to a theory 0' can now
be put as follows. We specify a function, not necessarily in the
notation of 9 or 6', which admits as arguments all objects in the
universe of 9 and takes values in the universe of 6'. This is the
proxy function. Then to each n-place primitive predicates of 6,

for each n, we effectively associate an open sentence of 6' in n
free variables, in such a way that the predicate is fulfilled by
an n-tuple of arguments of the proxy function always and only
when the open sentence is fulfilled by the corresponding n-
tuple of values.13

To be more explicit, Quine's standard of ontic reduction may be
restated formally as follows. A proxy function, f, is any effective
mapping that maps each and every object of the domain D1 (of the
reduced theory T1) into exactly one object in the domain D2 (of the
reducing theory T2). Then, T1 is said to be reducible to T2 just in case
there is a proxy function f such that, for any n-place primitive

predicate d) of T1, there exists an open sentence or formula <I>* of T2

such that

<D(X1. X2. . . .. xn) iff ¢*(f(X1). f(X2). . . .. f(xn)).

From the standpoint of Tarskian semantics, it may be said that there
is a sequence 0 = <x1, x2, . . ., xn> satisfying <1) under an intended
interpretation 11 on the domain D1 if, and only if, there exists a

sequence 0* satisfying <D* (under 12 on D2). And 0* is the sequence

 

13 W. V. Quine, "Ontological Reduction and the World of Numbers," The Ways
of Paradox and Other Essays, Harvard U., 1976, p.218.

134

obtained by taking the images of the elements of a under the proxy

function f—namely, 0* = <f(x1), f(x2), . . ., f(xn)>. Symbolically,

<x1, X2, . . ., xn> e I1(<I>) iff <f(x1), f(X2), . . ., f(xn)> e 12(<I>*).

The proxy function, f, itself belongs to neither T1 nor T2 but to a
background- or meta—theory of both.

Quine claims that on this account of ontic reduction, all known
"identificatory" reductions remain adequate, such as Frege's
reduction of arithmetic to set theory, as well as those of von
Neumann and Zermelo; various definitions or reductions of real
numbers using Dedekind cuts; and Carnap's reductions of impure
unit-numbers to pure numbers. For instance, in Frege's reduction of
number theory to set theory, the proxy function f will be the
function that maps any "genuine" number x to the class of all x-
member classes. Also Quine's standard appears to be equally
serviceable in applications to "eliminative" reduction. For instance,
Quine says, "now it is easily seen that dualism [of mind and body]
with or without interaction is reducible to physical monism, unless
disembodied spirits are assumed. For . . . for every state of mind
there is an exactly concurrent and readily specifiable state of the
accompanying body."14 In such an eliminative reduction the neural
state of a body will be specifiable as the physical state accompanying
the mind which is in that mental state; but it is so specifiable without
defining (in the form of biconditionals) the mental predicate in terms

of the physical vocabulary. We simply settle for the neuro-

 

14 W. V. Quine, "Things and Their Place in Theories," in Theories and Things,
Harvard U., 1981, p.18.

135

physiological states as denoting the alleged interaction between mind
and body, bypassing the mental states in terms of which we specify
the neuro-physiological states. Quine further maintains that
"Perceptions are neural realities, and so are the individual instances
of beliefs and other propositional attitudes . . . Physicalistic
explanations of neural events and states goes blithely forward with
no intrusion of mental laws or intensional concepts. What are
irreducibly mental are ways of grouping them."15 Thus, a proxy
function, which maps from distinctive mental states and events onto
certain correlative physiological states and events, may let us declare
no unbridgeable ontological differences between them, although
there may be irreducible ideological differences in the two sets of
vocabularies.

Therefore, the requirement of a proxy-function for an adequate
ontic reduction is a somewhat "liberal" standard, to the extent that it
does not also require an explicit definition or meaning equivalence
between intertheoretic concepts. It nevertheless turns out to be
"rigorous" enough to prevent the L-S trivialization of the subject of
ontology altogether. The reason is as follows. The force of the L-S
theorem, in essence, is that the logical structure of a theory—the
structure reflected in truth functions and first-order quantification,
in abstraction from any special terms—is insufficient to distinguish
its objects from the positive integers. It says that, in regards to truth
preservation of a theory, all but a denumerable part of an ontology
can be dropped and not be missed. The proof of the L-S theorem

assured us that for any consistent sentence there is a true

 

15 W. V. Quine, Pursuit of Truth, Harvard U., 1992 (rev. ed.), pp.71-72.

136

interpretation in the universe of positive integers, but it gave us no
way of finding and phrasing it. For the proof used the completeness
theorem, the proof of which appealed, after all, to the
"nonconstructive" method of assigning numerical values for the given
sentences. In other words, the L-S theorem does not provide an
effective mapping from the primitive predicates of any consistent
theory T of first-order sentences to predicates whose extensions are
purely numerical and which will make the members of T true. For
the L-S theorem does not in general determine which natural
numbers are to go proxy for the respective objects of T. A mere
nonconstructive assurance of the existence of a numerical model for
T is not enough to derive the Pythagorean ontology for T. As Gottlieb
complained, "A dictionary in Plato's heaven will not help us wield
Occams' razor."16

When, in conformity with the proof of the L-S theorem, we
reinterpret the primitive predicates of a theory T so as to make them
predicates of natural numbers, we do not in general make them
"arithmetical" predicates. To say that a theory comes out true under
some interpretation in the universe of positive integers, or has a
numerical model, is not to say that we reinterpret the primitive
predicates of a theory T in terms of sum, product, equality, and logic.
That is, we cannot write out suitable interpretations of 'Fx', 'ny', etc.
as formulas in arithmetical notation. One might very well regard this
latter claim as showing that no general reduction of theories to

numbers is possible, since on the one hand one wants to preserve all

 

15 D. Gottlieb, "Ontological Reduction," The Journal of Philosophy, vol. 73
(1976). p.60.

137

truths of a theory, while on the other hand a reduction to numbers
must mean a reduction to arithmetic—i.e. the new predicates must be
definable in terms of the "established" arithmetical predicates. In
fact, the reason that the various set-theoretic reductions of numbers
are regarded as "successful" is that the arithmetical relations are
defined purely in terms of the relations of an "antecedently
accepted" theory of sets. For Quine, however, insofar as a proxy
function exists and is specifiable, the reduction of a theory T to
numbers may very well entail new predicates over the
numbers—predicates perhaps definable only in terms of the old
predicates via the proxy function. This is an important point in
understanding Quine's position regarding an adequate ontic
reduction. Quine introduced the notion of proxy-function in order to
narrow the concept of ontological reduction, and says that under the
proxy-function requirement "there ceases to be any evident way of
arguing, from the L-S theorem, that ontologies are generally
reducible to natural numbers."17 But under the same requirement,
claims Quine, the serious cases of reduction such as the reduction
from the mental to the physical, even if no intertheoretic definition is
available, qualify as adequate ontic reductions. The latter can be
accomplished in such a way that, for instance, f'1 [f(5)] = 5, where
f(5) is the Frege class of all five-member classes; or f‘1 [f(Smith's
anger at time t)] = Smith's anger at t, where 'f(Smith's anger at t)' is
specified as a certain physiological state of Smith's body at t. Thus,

for the LS reduction, what is wanted is a translation of T's sentential

 

17 "Ontological Reduction and the World of Numbers," The Ways of Paradox
and Other Essays, p.218.

138

members into a purely "numerical" (if not "arithmetical") theory, in
that the sentences which have the unwanted ontic commitment of T
are to be replaced by the sentences or formulas committing only to
numbers. This means we need to be told how to find a translation
that is to be accomplished via a set of "constructive" definitions.

There is, however, a "constructive" improvement on the L-S
theorem due to Hilbert and Bernays that tells us how to find and
actually write out, for any consistent sentence, a true interpretation
in the universe of positive integers—and in purely arithmetic
notation at that, i.e., in the vocabulary consisting of 'plus', 'times’,
'equals', and the quantifiers and truth functions. This strengthened
L-S theorem interprets an arbitrary axiomatized theory T, but only
an "axiomatized" theory, to be explicitly defined within number
theory. It can be proved that any consistent first-order sentence (b of
T has a numerical model that assigns recursively enumerable or
complement-recursively enumerable extensions to all primitive
predicates in (D. Since, for every recursively enumerable or
complement-recursively enumerable set, there is an arithmetically-
definable predicate whose extension that set is, we know that there
are purely numerical predicates or open-sentences making <1) true.
Furthermore, the proof provides an effective procedure for finding
such predicates.

The problem, nonetheless, is this. If we are modeling merely
the theorems of an axiomatized system, then certainly we can get
arithmetical reinterpretations of the system, by the strengthened L-S
theorem. But that is not what Pythagoreanism is about. By a

'Pythagorean challenge', we are concerned rather to accommodate all

139

the truths of a given theory T—all the sentences, regardless of
axiomatizability, that were true under the original interpretation of
the predicates of T. What is needed for that purpose is a
reinterpretation of T that carries all these truths into truths about
natural numbers; but there may be no such interpretation in
arithmetical terms even by the strengthened L-S. The most that can
be said under this circumstance is that the numerical re-
interpretations are expressible in the notation of arithmetic plus the
truth predicate for T. For we know from Gddel and Tarski that the
truth predicate of T is expressible only in terms (belonging to a
metatheory of T) that are stronger in essential ways than any
originally available in T itself. Thus, says Quine, "on the whole the
reduction of a Pythagorean ontology exacts a price in ideology."13 In
short, neither the L-S theorem nor its strengthened version does by
itself imply Pythagoreanism. Of this weakened account of reducing

every ontology to natural numbers, it must be said that

On this score again, then, the relativistic proposition seems
reasonable: that there is no absolute sense in speaking of the
ontology of a theory. It very creditably brands this
Pythagoreanism itself as meaningless. For there is no absolute
sense in saying that all the objects of a theory are numbers, or
that they are sets, or bodies, or something else; this makes no
sense unless relative to some background theory.19

On the face of it, a Pythagorean reduction of an indenumerable
ontology is out of the question. For simply neither the L-S theorem

nor its strengthened version can yield a proxy function that will map

 

l8 lbid., p.216.
19 W. V. Quine, "Ontological Relativity," in Ontological Relativity, p. 60.

140

an indenumerable ontology, say the real numbers, into a
denumerable one. The proxy function for such a reduction would
have to be one-to-one, in order to provide distinct images or proxies
of distinct real numbers. And a one-to-one mapping of an
indenumerable domain into a denumerable one is a contradiction. In
some cases, of course, a proxy function need not be isomorphic or
one-to-one. Consider, for example, a theory whose domain is
constituted by income tax payers and whose predicates are incapable
of distinguishing between persons whose incomes are equal. In this
theory, the equivalence relation 'x has the same income as y' yields
the same substitutivity property as does the relation 'x = y' in the
background theory in which more can be said of personal identity
than equality of incomes. In such a case, a homomorphism suffices to
play the role of proxy function by assigning each person to his
income. For the reductive process merges the images of only such
individuals as never had been distinguishable by the predicates of
the original theory, so that distinct persons give way to identical
incomes salva veritate. However, the proxy function must of
necessity be one-to-one if the theory to be reduced is a theory in
which any two elements are distinguishable in terms of the theory's
vocabulary. For instance, the arithmetic of real numbers with the
predicate 'less than', when it is reduced to Zermelo-Fraenkel set
theory, will have to yield a one-to-one isomorphism between real
numbers and their corresponding Z-F sets. Therefore the
requirement that there be a proxy function for an adequate ontic
reduction rules out the possibilityof basing Pythagoreanism on the

L-S theorem. For it blocks any attempt to reduce a theory, whose

141

domain is indenumerable, to another theory which has a
denumerable domain.

As a matter of fact, the proxy-function criterion generally bars
any reduction of cardinality, at least in theories with identity. For, if
f is a proxy function and the cardinality of T1 (e.g. set theory) is
greater than that of T2 (e.g. number theory), then there must be
objects x and y in the domain of T1 such that x at y but f(x) = f(y).
And if '=*' plays the role of identity in T2, then f(x) =* f(y) since f(x)
= f(y). So f(x) ="‘ f(y) while x a: y, violating a condition the proxy
function must satisfy.20 On the other hand, one ontology is always
reducible to another when we are given a proxy function f that is
one-to-one. For there will be a guarantee for proxies in such a way
that where <I> is any predicate of the old theory, its work can be done
in the new theory by a new predicate <1>* which we interpret as true
of just those proxies f(x) of the old objects x that (I) was true of.

However, this built-in guarantee of a "successful" reduction
between theories of equicardinal ontology may give rise to a serious
objection. It is argued that when we are given two equicardinal
theories we can define, in accordance with Quine's standard of ontic
reduction, an isomorphic interpretation of one theory in another; but
is there, the question goes, a guarantee that we can characterize the
new interpretation without making reference to the old? Quine's
standard allows a new predicate (W (for the new interpretation 1*)

insofar as we can find a one-to-one mapping f from old (D, such that

I*(<I>) = <f(x1), f(X2), . . ., f(xn)> iff <x1, X2, . . ., xn> e I(<1>).

 

20 L. Tharp, "Ontological Reduction," Journal of Philosophy, 68 (1971), p.151.

142

Since, however, in characterizing the new model or predicate <D* we
have made essential reference to the old—namely <x1, x2, . . ., xn> of
<D, it is difficult to see what ontological economy we have achieved.
This situation would be, as Richard Grandy argues, "a case of
ontological disguise, not reduction,"21 in that the old ontology
remains. Charles Chihara made essentially the same point when he
argued that since for Quine dispensability of ontology is a necessary
consequence of reduction,22 and since "the very definition of the
reducing theory presupposes the ontology of the reduced theory,"
Quine's standard of ontic reduction should be restricted "to only cases
in which one theory is reduced to another that is specified without
reference to the reduced theory."23 But Quine's standard fails to be so
restricted, for it allows new predicates or vocabulary in the reducing
theory that are definable only in terms of the 01d predicates by the
help of a proxy function.

It must be pointed out, however, that Quine himself anticipated

this objection, and appropriately prepared an answer by saying

we cannot declare our new ontological economies without
having recourse to the uneconomical old ontology. This sounds,
perhaps, like a predicament: as if no ontological economy is

 

21 R. Grandy, "Ontology and Reduction," in R. Shahan & C. Swoyer (eds.)
Essays on the Philosophy of W. V. Quine, U. of Oklahoma, 1978, p.76. See also
Tharp (1971), op. cit., pp.158-159.

22 Quine says, in his Ontological Relativity that "A usual occasion for
ontological talk is reduction, where it is shown how the universe of some
theory can by a reinterpretation be dispensed with in favor of some other
universe, perhaps a proper part of the first"(p.55). From this quotation, it
would seem that a reduction for Quine is to show how an ontology can be
dispensed with in favor of another.

23 Charles Chihara, Ontology and the Vicious-Circle Principle, Cornell U.,
1973—respectively, p.132 and p.133.

143

justifiable unless it is a false economy and the repudiated
objects really exist after all. But actually this is wrong; there is
no more cause for worry here than there is in reductio ad
absurdum, where we assume a falsehood that we are out to
disprove.24

To see what exactly Quine's 'reductio ad absurdum' argument

amounts to, let us consider Carnap's clarification of measurement, or
reduction of impure numbers, where he construes 'the temperature
of x is n0C' in the fashion 'the temperature-in-degrees-centigrade of
x is n' and so dispenses with the impure numbers MC in favor of the

pure numbers 11. Suppose we are initially confronted with a theory 0
whose objects includes place-times x and impure numbers, say 0t in
the general case, and whose primitive predicates include a two-place
predicate 'H' of temperature such that 'H(x, nOC)'-—the temperature of
x was nOC. And we are to reduce the theory to a new one 9' whose
objects includes place-times and pure numbers, and whose
predicates include a new two-place predicate 'Hc' of temperature in
degrees Centigrade such that 'Hc(x, n)'—the temperature of x in-
Celsius is n. The crucial "reductive" step consists of explaining 'H(x,
nOC)' as 'Hc(x, n)'. This Carnapian reduction or explication is
successful, according to Quine, because three conditions are met. One
is, of course, that 'Hc(x, n)' and 'H(x, nOC)' under their intended
interpretations are coextensive, i.e., agree in truth values for all
values of x and n. A second condition is that, in the original theory,
all mention of impure numbers was confined or confinable to the
specific form of the context 'H(x, nOC)'; otherwise the switch to 'Hc(x,

n)' would not eliminate such mention. A third condition, finally, is

 

24 "Ontological Relativity." Ontological Relativity, p.58.

144

that an impure number, 01, can always be referred to in terms of a
pure number and a unit—cg. nOC, n-inches, n-seconds, and so on.
This third condition is, of course, the requirement that we be
able to specify a proxy function, which assigns one of the new objects
in the reducing theory to each of the old objects in the reduced
theory. The proxy function in Carnap's example is the function 'how
many degrees centigrade'—the function f such that f(nOC) = n. Quine
emphasizes, however, that it is required neither that such a function
be expressible or specifiable in the original theory 6 to which the
predicate 'H' belonged, nor that it be available in the final theory 6'
to which 'Hc' belongs. For Quine, the reduction of an ontology U of 9
to another V of 6' is the process occurring not in the original theory 9
nor in the resulting theory 6', but "out in the metatheory where we
are explaining and justifying the discontinuance of 9 in favor of 9'."25
Only within such a background- or meta-theory with its inclusive
universe, containing both U and V together, can we make sense of
the proxy function which will map all the old objects of U into their
new proxies of V—showing how every impure number or that was
intended in the old 6 effectively determines a pure number n of the
new 0'. Only relative to such a background theory and proxy function
can an ontic reduction adequately proceed to the conclusion that the
universe of U is excessive and that U can be replaced by V (and in
many cases V itself is a proper part of U). 80 we seem to be quite
within our rights in the metatheory to assume all of U and V for the
sake of the argument, and therein our ontological reduction is sealed

by reductio ad absurdum. Thus, in the Carnap's case we had to talk

 

25 "Ontological Reduction and the World of Numbers, Ways of Paradox, p.218.

145

as if there were such things as x0C, though we applaud Carnap's
repudiation of them—namely, 'H' had applied to putative things 01,
impure numbers, which come to be banished from the universe by
way of the proxy function f such that f(nOC) = 11. Their banishment
was Carnap's very purpose, explaining 'H(x, 0t) away by 'Hc(x, 11).
Such is Quine's 'reductio ad absurdum' argument for ontological
economy or his standard for an adequate ontic reduction.

Another objection to Quine's reduction standard has been as
follows. Quine's "definition" of ontological reduction closely resembles
the Tarskian syntactic notion of one theory being 'relatively
interpretable' in another—the concept of which, according to Grandy,
"might pretheoretically be thought to be a candidate for the
definition."26 A theory T1 is relatively interpretable in T2 iff there is
a mapping of predicates Rn of T1 into open formulas An with n free
variables and a one-place open formula Dx of T2 such that for any

sentence <1>(R1, . . ., Rk) of T1,
T1|- d>(R1,. . ., Rk) iff T2 1- (1)1)(A1, . . ., Ak)

where d>D(A1, . . ., Ak) is the sentence of T2 obtained from <D(R1, . . .,
Rk) by relativising all of the quantifiers to D and replacing each Ri by
Al. In other words, all T1 sentences of the form '(Vx)Fx' and '(3x)Fx'
will be translated or interpreted as T2 sentences of the form '(Vx)(Dx
—) Fx)‘ and '(3x)(Dx & Fx)’, respectively. This notion taken as a
criterion of ontological reduction, however, would appear to have a

serious defect. The apparent defect, the critique states, is that by the

 

26 R. Grandy, "Ontology and Reduction," in R. Shahan & C. Swoyer (eds.), op.
cit., p.70.

146

strengthened L-S theorem any theory T is 'relatively interpretable'
in number theory (with the addition of an axiom asserting the
consistency of T). Thus on this definition any theory would be
ontologically reducible to an extension of number theory.

However, Quine would have a short but decisive reply that
Grandy's paraphrase of his standard of ontic reduction in terms of
'relative interpretability' must not be accepted as a correct
understanding or interpretation of his ontic standard. For it is clear
that unlike 'relative interpretability' that allows a Pythagorean ontic
trivialization in the end, Quine's standard as it stands is already
protected from such a trivialization by the requirement that the
proxy function be expressible in the reduction. And we saw
previously that the strengthened L-S theorem tells us that any
consistent interpreted theory admits of a true interpretation that is
expressed in the notation of elementary number theory, but it does
not tell us, for each object of the original theory, what number is to
be its proxy. Thus, contrary to Grandy's interpretation of Quine, the
Tarskian concept of 'relative interpretability' cannot be viewed as a
pretheoretic explicandum for Quine's corresponding explicans which
would be Quine's criterion or standard for ontic reduction.

There also has been the objection that Quine's standard is
neither a sufficient condition nor a necessary condition for a
reduction to be an adequate ontic reduction. In other words, critics
claim that there are some analyses that are "intuitively" clear cut
cases of adequate ontic reductions but that do not satisfy Quine’s
criterion. Chihara argues that, in Weierstrass' reduction that

eliminates infinitesimals from the ontology of the infinitesimal

147

calculus, "the proxy-function requirement is not satisfied: nothing in
the Weierstrassian theory goes proxy for the infinitesimals of the
early theory. The beauty of the modern theory of limits is that we do
not have to postulate infinitesimals or even entities that go proxy for
them."27 However, this is a mistaken view of the 'proxy function'. For
the point of Quine's proxy-function demand for the adequacy of an
ontic reduction is that the proxy function must be expressible or
specifiable only in the metatheory. And such a function need not be
an ontologically committal kind, as we saw previously in the reductio
ad absurdum argument, in that it can be given as a virtual class or an
Open sentence consisting of two variables, one of which takes as
values the value (i.e. proxies of the reducing theory) of the function
for each value (object intended by the reduced theory) of the other
variable as argument.

But critics insist that the Quinean reply based on the notion or
the role of 'metatheory' may not be an acceptable answer to their
original intention or question. For example, Grandy considers a
theory K whose vocabulary consists of {0, S, =, <}, and whose domain
consists of the natural numbers plus another sequence of objects
arranged in the order type of the integers. In this theory K, then, 0 is
assigned zero, 8 is the successor relation for both kind of objects, = is
identity, < is the relation which is obtained by taking the less than
relation over each type of objects but which puts all the natural
numbers before any of the unnatural ones. This theory, K, will end
up with the result that exactly the same sentences are true as are

true in the usual version of number theory without the extra objects.

 

27 Ontology and the Vicious-Circle Principle, 1973, p.133.

148

Thus the extra integers are both unnecessary and unwanted; but the
question goes 'can we eliminate them on Quine's criterion'? In order
to eliminate such an excess baggage we would have to map it onto
the natural numbers. However, when we map some unnatural
numbers K onto a natural number n in the form of 'f(1<) = n', we must
also preserve the ordering under the predicate '<'. Since the ordering
'n +1 < x' holds in the original model K, it must also be the case that
f(n +1) < f(x) in the reducing theory, say K*. This is impossible,
however. For, from our primary arrangement of f(n) = n, for all n, to
preserve the successor relation, we can derive a consequence that
f(n) = n = f(x) where we should instead have 'f(n) < f(n +1) < f(x)’ in
K“. Thus, if K is reducible at all to ordinary number theory (and it in
fact is), then this 'reduction' will not be a genuine proxy-function
reduction. For, according to Grandy, "what we want to do in this case
is not a reduction of the excess entities but destruction of them."28
Quine admitted that his criterion or standard of ontic reduction
does call for a codicil to the effect that "besides such ontological
reduction as is provided by the proxy function, there is that which
consists simply in dropping objects whose absence will not falsify
any truths expressible in the notation."29 However, Quine continues
to emphasize the role of metatheory in ontological reduction, holding
that "in the background language we must, of course, be able to say
what class of objects is drOpped, just as in other cases we had to be

able to specify the proxy function."30 Quine would favor free use of

 

28 R. Grandy, "On What There Need Not Be," The Journal of Philosophy, vol. 66
(1969), p.810.

29 "Ontological Relativity," Ontological Relativity, p.68n. See also Quine,
"Replies to Eleven Essays," in Philosophical Topics, vol. 12 (1981), pp.233-235.

30 lbid., p.68n.

149

the resources of the metatheory, and typically they will include both
those of the theory undergoing reduction and those of the theory to
which it is being reduced. As a consequence, even such rare and
somewhat unlikely cases of "ontological inflation" (e.g. Grandy's
example above) can properly be managed by Quine's proxy-function
standard, though indirectly in a two-step procedure. That is, we first
”deflate" the theory, or what Grandy calls 'ontic destruction', by
dropping extra objects whose absence will not alter the structure of
the theory, sentential or predicate. This is possible, simply because in
the background language the objects eliminated under the codicil
must have constituted a specifiable class, at least a virtual one;
otherwise there existed no assumption of an 'adequate' reduction in
the first place. Next, then, we can put the resulting theory to the
application of Quine's proxy-function requirement. In other words, it
is required for an adequate ontic reduction that there not only exists
a proxy function, but such a function must be specifiable or definable
in the vocabulary given in the meta- or background-language.
Existence is one thing and specifiability another. For Quine, it is not
merely the existence of a proxy-function but its specifiability that is
what really counts for the adequacy of ontic reduction. For otherwise
the "ghost" of Pythagoreanism based on the L-S theorem will be still
with us, since the theorem guarantees the existence, though not the
specification, of a model in the natural numbers for any consistent

first-order theory to be reduced.

150

5.4 Ontology Naturalized

At this point, it is expedient to look into what really prompts
Quine's 'proxy-function' standard of ontic reduction and also into
some consequences of it, with which we might gain a helpful insight
to solve or dissolve the dilemma addressed at the beginning of this
dissertation. Regarding some criterion or standard for the adequacy
of ontic reductions, it is desired that such a criterion must be
sufficiently liberal to include all legitimate or intuitively valid cases
of 'eliminative' reduction (as well as 'identificatory' ones) and yet
rigorous enough to avoid the Pythagoreanism based on the L-S
theorem. And it was observed previously that both Tarskian
'interpretability' and mere 'modeling' of truth-preservation of one
theory into another would not adequately represent our preanalytic
notion of 'ontic reduction'. For they would not capture squarely the
correct extension of such preanalytically accepted cases of ontological
reduction—especially in the 'eliminative reduction' cases, for
instance, the reduction of the mental to the physical. Quine locates
his "alternative" criterion for all legitimate ontic reductions in the
existence of a 'proxy function' and its specifiability or definability in
the background language where the reduction in question is being
undertaken. And Quine emphasizes the role of the proxy-function for
ontic reductions: that it enables us, on the one hand, to take into
account as adequate all legitimate reductions of an 'eliminative' sort
which previously could not be so accounted because of failing to
satisfy the "explicit definability" condition; and on the other hand it

enables us to avoid Pythagoreanism, which is allegedly inferred from

151

the L-S theorem, that would otherwise "trivialize" any serious
ontological talk.

In fact, the requirement of a 'proxy function' would seem to
establish a necessary and sufficient condition for the adequacy of
ontic reduction. For apparently the existence of a proxy function
between two theories guarantees that there will be an "effective"
ontic reduction between them, while each of the preanalytically-
accepted ontological reductions that are known so far involves, or at
least is capable of producing, some proxy function. One might argue,
however, that, although Quine's proxy-function requirement for an
adequate ontic reduction succeeds from a technical point of
view—especially, in blocking Pythagoreanism—its independent
philosophical motivation is obscure or at least not well
documented.31 For, it may be argued, the very technical role the
proxy-function plays in blocking the Pythagorean reduction—i.e. the
demand of 1-1 correspondence between the domain of the reduced
theory and that of the reducing theory—is all that prompts Quine's
'proxy function' standard in the first place. But this is a mistaken
view, I believe. For Quine's motivation for adopting the 'proxy-
function' for his ontic standard or criterion is not based on its
technical success in this matter. The 'proxy function' criterion for
ontic reduction should rather be viewed in a broader background of
Quine's phiIOSOphy, particularly, his ontology. The requirement of the
proxy function for an adequate reduction, I would say, is a particular

form of his "higher" ontological thesis—namely, No entity without

 

31 See L. Tharp, "Ontological Reduction," Journal of Philosophy, 1971, p.156;
D. Bonevac, Reduction in the Abstract Sciences, 1982, p.144; C. Chihara,
Ontology and the Vicious-Circle Principle, 1973, p.126.

152

identity. The 'proxy function' requirement is specifically designed to
implement this higher thesis in the context of ontological reduction
and that only; while there may be other similar exemplifications of
the thesis in different philosophical contexts, such as the
requirement of 'substitutivity of identity' for testing referential
clarity or extensionality of a given sentential context.

The thesis 'no entity without identity' has often been adopted
in the form of the principle of 'substitutivity of identity', stating that
co-designative singular terms can be interchanged salva veritate in
any true statement.32 It is designed to test whether the statement in
question is "referentially" opaque or transparent. Statements
involving intensional terms would fail the 'substitutivity' test
because of their lack of identity conditions, and hence are
referentially opaque. Failure of the 'substitutivity' test in intensional
contexts, thus, will incline us to doubt the existence of intensional
entities. For, without an identity condition, we don't have a clear idea
of the individuals, whether intensional or extensional, under

discussion or of the universe of discourse—of what it is that the

variables range over. Quine notes that

It was only after somehow deciding what heathen locutions to
construe as identity and the like that our linguist could begin to
say which heathen words serve as terms and what objects they
refer to . . . And it is to the demand of identity still, specifically
the substitutivity of identity, that the adult speaker of our
language remains answerable as long as he may be said to be
using terms to refer.33

g

32 See, for instance, "Reference and Modality," From a Logical Point of View,
pp.139-159.
33 "Speaking of Objects," Ontological Relativity and Other Essays, p.18.

153

We will have no clue, in other words, as to how to recognize an object
except the circumstances under which the object may be said to be
the same or different. 80 there is a direct connection between Quine's
slogan 'to be is to be the value of a bound variable' and his thesis of
'no entity without identity', which in this particular context (i.e. topic
of 'referential opacity') will be formulated as the principle of
'substitutivity of identity'.

Likewise, Quine's proxy-function requirement can be viewed as
another form of his general thesis 'no entity without identity', in that
it is associated with the issue of adequacy for an ontological
reduction. The connection may be explained in the following way. If
we suppose that we have a clear conception of the ontology of the
original theory—say a theory that refers to 'numbers', and if we have
an explicit or "effective" structure-preserving mapping from that
domain to the domain of the reducing theory—say a theory that
refers to Zermelo sets or von Neumann sets—then we will have a
clear conception of identity conditions in the reduced theory.
However, if there is no "effective" proxy function in the reduction,
then we won't have a clear notion of identity, hence we won't have a
clear conception of our ontological commitment. In consequence, we
have no way of knowing or specifying which object is reduced to
which, hence we will not have an 'effective' or successful reduction at
all. For example, suppose we take it that we know what sets are—i.e.,
we know in particular what the identity conditions for sets are as
determined by the 'principle of extensionality'. By the waenheim-
Skolem theorem we know that there exists a model of set theory in

the domain of the natural numbers. However, there does not exist a

154

definable proxy function from the presystematic sets to their
analogues in the set of numbers. We know that there cannot be such
a definable mapping because it is provable (in the presystematic
theory) that there are non-denumerably many sets. This is why the
L-S theorem, failing the 'proxy function' test, doesn't constitute any
genuine ontological reduction, and hence cannot support
Pythagoreanism after all. In other words, though we can pare down
the size of an uncountable domain by way of the L-S theorem, we
cannot characterize by the theorem the resulting universe in a sense
that would satisfy Quine—that is, we can not specify which objects
are dropped and which are retained in the reduction.

In his over-all conception of 'ontic reduction' together with
'ontic commitment', then, Quine is thinking that, first, a theory is
given as interpreted in that its predicates have intuitive extensions
assigned to them and its intended universe is characterized in terms
of the resources of the metalanguage. In other words, a theory's
universe is given prefixed, in that its specification is integral to the
"metatheoretic" specification of the theory. In this sense, a theory is
ontologically committed to what its metatheory or metalanguage says
there is. In view of ontic relativity, however, the given theory's
ontology or ontological commitment can always by a reinterpretation
be dispensed with in favor of another, provided there exists a proxy
function between them which is specifiable in their common
background language. As a consequence, when we are discussing
ontological commitment of a theory from the point of view of ontic
reduction, there is no difference between isomorphic models of the

theory; or as Quine expresses it "we are finding no clear difference

155

between specifying a universe of discourse—the range of the
variables of quantification—and reducing that universe to some
other."34 If we are given such alternative models, there is nothing to
choose between them on the grounds of the truth values of the
sentences which would indicate that one rather than the other
represents the ontological commitments of the given theory. For it is
not necessary that the various domains of objects which make a
theory true should coincide, but it suffices that they should be
isomorphic in the sense that "different classes of objects quantified
over in defining the satisfaction relation . . . will produce no
discernible difference with respect to the total class of sentences
each determines as true."35

Thus, the truth of a theory remains unchanged no matter how
ontologies are switched and predicates reinterpreted, so long as the
proxy-function requirement is met in such reinterpretations of the
original theory. In this regard, the ontological commitments of a
theory must be taken to involve the structure (namely, sentential
and predicate structure) rather than objects themselves, such that
each model must contain not simply objects but objects that stand in
appropriate relations to each other, with respect to 'truth' and
'satisfaction'. The essential feature of ontological reduction of one
theory into another, therefore, must accordingly be the preservation
of the structure of the original theory in the reducing or resulting
theory. This point has been put decisively by Quine, who notes that

in a proxy-function reduction,

 

34 "Ontological Relativity," Ontological Relativity and Other Essays, p.43.
35 G. Romanos, Quine and Analytic Philosophy, MIT, 1983, p.162.

156

The original objects have been supplanted and the general
terms reinterpreted. There has been a revision of ontology on
the one hand and of ideology, so to say, on the other; they go
together. Yet verbal behavior proceeds undisturbed, warranted
by the same observations as before and elicited by the same
observations. Nothing really has changed . . . Structure is what
matters to a theory, and not the choice of its objects . . . The
objects, or values of variables, serve merely as indices along
the way, and we may permute or supplant them as we please
as long as the sentence to- sentence structure is preserved. The
scientific system, ontology and all, is a conceptual bridge of our
own making, linking sensory stimulation to sensory
stimulation.36

Structuralism has interesting consequences for the issue of
ontology. In mathematics, for instance, structuralism is usually
contrasted with traditional platonism. The platonist holds that the
subject matter of arithmetic, for example, is a collection of natural
numbers that have some sort of ontological independence, in a sense
that they are the same kind of objects as marbles but are just more
of them and they are abstract and eternal. The structuralist, on the
other hand, holds that the subject matter of arithmetic is a single
abstract structure, the pattern common to any infinite collection of
objects that has a successor relation with a unique initial object and
satisfies the "induction" axiom. A structure is the abstract form of a
system or theory, focusing on the interrelationships among the
objects, and ignoring any features of them that do not affect how
they relate to other objects in the system; hence, focusing on two or

more place predicate and ignoring one-place predicates. Mathematics

 

36 "Things and Their Place in Theories," Theories and Things, Harvard U.,
1981, pp.19—20.

157

is the study of structures as such, independent of whether they are
exemplified in the physical realm, or any realm for that matter.“
The arithmetic of Peano's axioms, for example, directly characterizes
the infinite natural number structure, without describing its subject
matter. For a structuralist, then, a number is a place in the number
structure. From that perspective it is nonsense to ask questions like
'Is JULIUS CAESAR = 2?’ and 'Does l e 2?’ For "it would be like trying
to imagine a shortstop independent of an infield,"38 where we can
identify a shortstop with all who have the roles of shortstop on each
team on game day. Such questions may be said to involve a category
mistake, for these are questions internal to the particular number
structure such as Zermelo's or von Neumann's. There is a difference
between a place in a structure and an object, like that between an
office and office-holder. To be meaningful, a statement of identity
must refer to the places in a single structure, so that there is no "fact
of the matter" as to whether 2 is in fact {{9}} or {9, {9}} or any other
set for that matter. From the structuralist's point of view, therefore,
there are no objects simpliciter and each object is part of a structure
and cannot be seriously considered apart from it.

However, Quine adopts a more 'global structuralism', asserting
that ”my own line is a yet more sweeping structuralism, applying to
concrete and abstract objects indiscriminately."39 For he sees all

objects as theoretical, more or less, in the sense that "The very notion

 

37 This view point, associated with so called 'Ramsey sentences', was
developed by Frank Ramsey in his The Foundations of Mathematics, Routledge,
1931, pp.216-236. See also Quine's From Stimulus to Science, op. cit., p.73 ff.

33 S. Shapiro, "Structure and Ontology," in Philosophical Topics, vol. 17 (Fall
1989L p149.

39 W. V. Quine, "Structure and Nature," The Journal of Philosophy, vol. 89
(1992). p.6.

158

of an object at all, concrete or abstract, is a human contribution, a
feature of our inherited apparatus for organizing the amorphous
welter of neural input."40 All objects, for Quine, are theoretical posits
or reifications, and their technical purpose seems to consist in a
tightening of implications between observations and high theory, by
reinforcing loose truth-functional relations.“ The reference and
ontology, in this structuralist scheme, recede thus to the status of
mere auxiliaries. What particular objects there may be is indifferent
to the support they lend to the observation sentences, to the success
of the theory in its predictions. What is significant in an ontology is
just its contribution of neutral nodes or reference points in the
structure that we have progressively elaborated as our theory of
nature. Two ontologies, if "explicitly" correlated one to one, are
empirically on a par, for the entire evidential support they provide
to the theory will remain unchanged.

With 'global structuralism', together with a proxy-function
reduction, we seem to be free to vary the ontic commitments of our
theory of the world or of ultimate reality. For there seems to be no
fact of the matter that would impose on such a decision one way or

another. In asking whether the question of what there is is really

 

40 lbid., p.6.

41 In his "Two Dogmas of Empiricism" Quine says: "As an empiricist I
continue to think of the conceptual scheme of science as a tool, ultimately, for
predicting future experience in the light of past experience. Physical objects
are conceptually imported into the situation as convenient intermediaries—
not by definition in terms of experience, but simply as irreducible posits
comparable, epistemologically, to the gods of Homer. . .[They] differ only in
degree and not in kind. Both sorts of entities enter our conception only as
cultural posits. The myth of physical objects is epistemologically superior to
most in that it has proved more efficacious than other myths as a device for
working a manageable structure into the flux of experience." (From a Logical
Point of View, p.44.)

159

deprived of its factuality, however, Quine answers that "the question
what there is is a substantive scientific question within our system of
the world, whereas the point about proxy functions is a point about
evidence, a point not of ontology but of the epistemology of
ontology.”2 Indeed, it would be a mistake to suppose that we can
recognize all the alternative ontologies as the true descriptions of our
world in their several ways. In his "Identity, Ostension, and

Hypostasis," Quine argues:

to talk about the world we must already impose upon the
world some conceptual scheme peculiar to our own special
language . . . we can improve our conceptual scheme . . . but we
cannot detach ourselves from it and compare it objectively
with an unconceptualized reality.43

Traditionally, language has often been seen as an essentially neutral
medium for the expression of beliefs, with room enough for all
possible diversity. But for Quine things are not so straightforward,
and the line between language and theory is a blurry one at best. For
in acquiring a language a child cannot detach it from a substantial
knowledge claim (or theory) of the world that we adult speakers of
the language take for granted. A child begins reasoning within the
inherited world theory as a going concern. While growing up, he/she
tentatively believes all of it but believes also that some unidentified
portions are wrong. To'understand, clarify, and improve the theory,
the child can only be "the busy sailor adrift on Neurath's boat,"

working from within. 80 language shades into theory; and the one big

 

42 "Ontology and ideology Revisited," The Journal of Philosophy, vol. 80
(1983). p.500.

43 From a Logical Point of View, op. cit., p.79.

160

language—the language we commonly speak—hues into the one
sweeping theory Quine sees as our immense 'background theory'. In
learning a scientific language, thus, we learn a theory of the world; in
particular, in learning the most comprehensive science or language
(i.e. physics) we are acquiring at the same time the most
comprehensive 'theory of the world' or ontology.

To suppose that we could stand aloof outside of all those
envisaged worlds warranted by the proxy-function reduction, thus,
would be a confusion of truth (ontology) with its evidential support
(epistemology). Truth is immanent and what counts as facts of the
matter is internal to 'our theory of the world'. For Quine, our theory
of the world ought to be what he terms 'naturalism'—namely, "the
recognition that it is within science itself, and not in some prior
philosophy, that reality is to be identified and described . . . it is

incoherent otherwise."44 Quine continues:

My global structuralism should not, therefore, be seen as a
structuralist ontology. To see it thus would be to rise above
naturalism and revert to the sin of transcendental metaphysics.
My tentative ontology continues to consist of quarks and their
compounds, also classes of such things, classes of such classes,
and so on, pending evidence to the contrary. My global
structuralism is a naturalistic thesis about the mundane human
activity, within our world of quarks, of devising theories of
quarks and the like in the light of physical impacts on our
physical surfaces.”5

This 'physicalism' pertains to Quine's philosophy and is to

explain why there is no real dilemma or conflict in Quine's ontology.

 

44 "Things and Their Place in Theories," Theories and Things (1981), p.21.
45 "Structure and Nature," The Journal of Philosophy, vol. 89 (1992), p.9.

161

which was addressed at the beginning of this thesis. A faithful
"naturalistic" empiricist, Quine seems to have decided to take the
physical as an ultimate category and thereby to stick to
physicalism.46 Despite its eminent virtue of ontic economy, pure set-
theoretic ontology is not an alternative to his physicalism. In fact,
from the standpoint of Quine's distinction between 'ontology' and its
epistemology, there arises no such dilemma in the first place. For it
may be said that these two ontologies, physicalism and the ontology
of pure sets, belong to different levels of ontic discourse. The contrast
would hold, as Paul Gochet points out, "between two approaches to
ontology, not between two ontologies."47 That is, physicalism is the
ontology Quine arrives at from within the scientific system of the
world; whereas the ontology of pure sets is reached by looking at
ontology from an epistemic point of view, by taking into account the
theory of evidence.

However, a question remains. Quine often tends to equate his
physicalism with his naturalism, expressing them in apparently
interchangeable ways. Naturalism, according to some of his
formulations, is a philosophical thesis: "it is within science itself . . .
that reality is to be identified and described"; or "abandonment of
the goal of a first philosophy prior to natural science"; or "to see

philosophy as natural science trained upon itself and permitted free

 

46 Physicalism, indeed, has persisted in Quine's philosophy for over four
decades. We can find almost the same passage as the above, already in his "The
Scope and Language of Science" (1954): "Our tentative ontology for science,
our tentative range of values for the variables of quantification, comes
therefore to this: physical objects, classes of them, classes in turn of the
elements of this combined domain, and so on up." (The Ways of Paradox and
Other Essays, p.244)

47 P. Gochet, Ascent to Truth: A Critical Examination of Quine's Philosophy,
Philosophia Verlag, p.89.

162

use of scientific findings.”8 However, this understanding of
'naturalism' is the original motivation that led Quine to the "brave
new ontology" of a pure set theory. It was essentially Quine's such
"free use of scientific findings," especially, of results in microphysics
that led him to encourage us to abandon the traditional notion of

'physical object' in favor of the field theory of physics:

Some findings known as the Bose-Einstein and Fermi-Dirac
statistics suggest how we might be led actually to repudiate
even the more traditional elementary particles as values of
variables, rather than retaining them and just acquiescing
provisionally in their mysterious ways. Those results seem to
show that there is no difference even in principle between
saying of two elementary particles of a given kind that they
are in the respective places a and b and that they are
oppositely placed, in b and a. It would seem then not merely
that elementary particles are unlike bodies; it would seem that
there are no such denizens of space-time at all, and that we
should speak of places a and b merely as being in certain
states, indeed the same state, rather than as being occupied by
two things.49

On this basis, physicalism turns out to be reduced to the ontology of
set theory, in that material contents disappear and physical objects
evaporate into mere space-time regions. And a space-time region, as
we saw previously, is to be identified with a quadruple of real
numbers (which is dense representing 'fields' rather than 'discrete'

particles) which in turn will be identified with a set in known ways.

 

48 Respectively, "Things and Their Place in Theories" (in Theories and
Things, p.21); "Five Milestones of Empiricism" (ibid., p.67); "Russell's
Ontological Development" (ibid., p.85).

49 Pursuit of Truth, op. cit., p.35. See also "Wither Physical Objects," in R.
Cohen, et. al. (eds.), Essays in Memory of Imre Lakatos, Dordrecht, Reidel, 1976,
pp.497-504: especially, pp.498-499.

163

So we seem to be back with the original dilemma, though in a much

clearer setting.

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