#### Craig’s Theorem and the Empirical

Underdetermination Thesis
Reassessed

**Christian List**

University of Oxford

The present paper proposes to revive the twenty-year old debate on the
question of whether Craig’s theorem poses a challenge to the empirical
underdetermination thesis. It will be demonstrated that Quine’s account of
this issue in his paper "Empirically Equivalent Systems of the World"
(1975) is mathematically flawed and that Quine makes too strong a
concession to the Craigian challenge. It will further be pointed out that
Craig’s theorem *would* threaten the empirical underdetermination
thesis *only if* the set of all relevant observation conditionals
*could* be shown to be recursively enumerable — a condition which
Quine seems to overlook —, and it will be argued that, at least within the
framework of Quine’s philosophy, it is doubtful whether this condition is
satisfiable.

**
****1. Introduction **

Theory can … vary though all possible observations be fixed. Physical
theories can be at odds with each other and yet compatible with all
possible data even in the broadest sense. In a word, they can be logically
incompatible and empirically equivalent (Quine, 1970).

Such is Quine’s empirical underdetermination thesis. Although the
question of whether this thesis is plausible is still far from settled —
even two decades after the philosophical debate on this subject was most
heated —, philosophers seem to have reached a (limited) consensus on one
particular aspect of this question: William Craig’s theorem (1953, 1956)
concerning the replacement of auxiliary expressions is seen as a
challenge, if only a theoretical one, to the empirical underdetermination
thesis. Jane English (1973) forcefully argues for this view, and Quine
himself concedes that, although "[Craig’s] result does not belie
under-determination", "it does challenge the interest of
under-determination" (Quine, 1975, p. 313).

In the present paper, I will argue that a reassessment of this view is
overdue, and that, at least within the framework of Quine’s philosophy,
Quine’s thesis can be defended against Quine’s own concession. In
particular, I will show that Quine’s concession is grounded in a
mathematically flawed use of Craig’s theorem. I will then argue that,
within the framework of Quine’s philosophy, it is highly doubtful whether
the conditions under which Craig’s result *would* pose a challenge to
the empirical underdetermination thesis are satisfiable.

**
**2. The Empirical Underdetermination Thesis

Using the traditional syntactic approach to scientific theories, we
define a *theory* to be a deductively closed set of sentences of a
formal language. Given a theory T, a *theory formulation* of T is a
(usually finite) subset S of T, often interpreted as the set of basic
axioms, such that the deductive closure of S is the whole of T. How are
theory and observations related? Typically, observation sentences of the
form "The liquid in this vessel is blue!" are not *directly* entailed
by a theory (i.e. contained in the set T), because, first, they lack the
generality characteristic of a theory and, second, their truth-value —
unlike the truth-value of a typical theoretical sentence — is dependent
upon the occasion of utterance. Quine portrays the link between a theory
and such sentences as a two-step relation. As a first step, observation
sentences are pegged to specific spatio-temporal co-ordinates so as to
make their truth-value independent of the occasion of utterance. But as
theories typically make little reference to particulars, pegged
observation sentences are still insufficiently general to be directly
entailed by a theory. Rather, theories imply particulars *via* other
particulars, that is to say, *via *boundary conditions. But, instead
of saying that the *conjunction* of a theory and a set of boundary
conditions implies certain pegged observation sentences, we may simply say
— and this is the second step of the two-step relation — that the theory
implies appropriate *observation conditionals*, where an
*observation conditional* is a conditional sentence whose antecedent
is a conjunction of pegged observation sentences and whose consequent is a
pegged observation sentence (Quine, 1975).

In this terminology, two theories are defined to be empirically
equivalent if they entail the same body of observation conditionals. Now
the empirical underdetermination thesis requires that there exist rival
theories which are empirically equivalent, but logically incompatible.
However, if the empirical underdetermination thesis is to be an
interesting and nontrivial claim, it must actually require more than that.
It must, firstly, rule out the possibility that two purportedly
irreconcilable rival theories simply turn out to be divergent extensions
of a single theory, which itself is not subject to empirical
underdetermination. And it must, secondly, rule out the possibility that
two such rival theories turn out to be notational variants of each other,
where one is, for example, the result of interchanging the terms
‘electron’ and ‘neutron’ throughout the other. To accommodate the first
point, the underdetermination thesis must require not just that there are
*some* pairs of suitable rival theories, but that, for *any
*theory, a suitable rival exists. To accommodate the second point, we
introduce a general method of constructing new notational variants of a
theory by defining a *reconstrual of predicates* to be a function
whose domain is the set of predicates of the relevant language and whose
converse domain is the set of open sentences of the language such that
each n-place predicate is mapped to a sentence with n free variables. Now
the empirical underdetermination thesis can be stated thus: given
*any* scientific theory, there exists a rival theory which is
empirically equivalent to the given theory and which cannot be rendered
logically compatible with it by means of a reconstrual of
predicates^{(2)} (Quine, 1975).

At first sight, we may find this claim puzzling. Surely, we may ask, if
our ‘irreconcilable’ rival theories are empirically equivalent, they must
have a considerable number of sentences in common, in particular all the
observation conditionals which each of the theories implies. So, given a
set of rival theories, can we not simply take their intersection as a new
theory^{(3)} which is both empirically equivalent to each of the
given theories and immune to empirical underdetermination? Indeed, if the
set of desired observation conditionals to be entailed by a theory is
finite, we can easily define the theory to be the deductive closure of the
conjunction of these observation conditionals; such a theory is obviously
unique and unaffected by empirical underdetermination. Similarly, if the
set of observation conditionals is infinite, but exhibits so much
structure that its deductive closure can be expressed as the deductive
closure of a finite theory formulation, it is also possible to construct a
tightly fitting theory. In its most general form, the empirical
underdetermination thesis is therefore wrong.

However, given the complexity of the world, we may expect that, in many
cases, (the deductive closure of) the set of all relevant observation
conditionals is not axiomatizable in terms of a conceptually neat, let
alone finite, theory formulation. Rather, any conceptually manageable (in
particular, finite) theory formulation may well entail (the deductive
closure of) the desired observation conditionals *as well as some other
(non-observational) sentences*. And this is precisely why we can
envisage that the problem of empirical underdetermination may crop up.

These considerations are certainly plausible. But are they
incontrovertible? At this point we should turn our attention to Craig’s
theorem, since, as we have indicated in the introduction, this result is
often viewed as providing a method of constructing a conceptually
manageable (at least, theoretically), though not usually finite, theory
formulation which entails *exclusively* (the deductive closure of)
the desired set of observation conditionals.

**
**3. Craig’s Theorem

Some technical preliminaries are due. A set S is said to be
*recursively enumerable* if it can be written as a sequence S =
{s_{1}, s_{2}, s_{3}, …} which can be generated by
means of an effective mechanism (i.e. by means of an appropriate Turing
machine). A set S is said to be *recursive* (or *decidable*) if
there exists an effective mechanism which can determine in a finite number
of steps whether or not any given entity is a member of S. A set of
sentences of a formal language is said to be *recursively
axiomatizable* if it is the deductive closure of a recursive set of
axioms. It is important to note that there are sets which are countably
infinite (i.e. ‘enumerable’), but not recursively
enumerable^{(4)}, and that there are sets which are recursively
enumerable, but not recursive^{(5)}.

The basic insight underlying Craig’s theorem is the following: every
theory which can be expressed as the deductive closure of a recursively
enumerable set of axioms is recursively axiomatizable. Let us briefly go
through the proof of this proposition. Suppose that T is the deductive
closure of the recursively enumerable set of sentences S = {s_{1},
s_{2}, s_{3}, …}. We shall construct a recursive set of
axioms for T. Define S' to be the set {s_{1}, (s_{2 Ù } s_{2}), ((s_{3 Ù }s_{3}) Ù
s_{3}), …} such that, for each s_{i} in S, S'
contains a self-conjunction of s_{i} of length i. The question of
whether or not a given sentence f is a member of
S' is mechanically decidable in a finite number of steps. Given f , the unique readability of sentences of our formal
language implies that there exist a unique sentence y and a unique number n (possibly y = f and n = 1) such that
f is a self-conjunction of y of length n. We then consider the n^{th}
element of the sequence {s_{1}, s_{2}, s_{3}, …},
which is, by assumption, mechanically accessible in a finite number of
steps, and we compare y with s_{n}. If
these two sentences are identical, we conclude that f is a member of S', and if they are distinct, we
conclude that it isn’t. So S' is a recursive set. Furthermore, S and S'
are clearly logically equivalent, and hence they have the same deductive
closure, namely T. Thus S' is a recursive set of axioms for T as required.

Now Craig’s theorem concerning the replacement of auxiliary expressions
is actually a special case of the result we have just proved.

Let T be a theory expressed in a formal language L, where T has a
recursive theory formulation, and let P be the set of all predicate
symbols of the language L. Consider any recursive subset P* of P,
interpreted, for instance, as the subset of all ‘essential’ (as opposed to
‘auxiliary’) predicates of L. Let L* denote that part of the language L
which contains all the sentences that can be expressed in the restricted
vocabulary contained in P*. Craig’s theorem states that the restriction of
T to L* — i.e. the set of all those sentences of T in which only
predicates in P* (e.g. ‘essential’ ones) occur — is recursively
axiomatizable.

This follows immediately from the basic insight we have just proved; to
apply this insight, it is sufficient to show that the restriction of T to
L* is recursively enumerable; its recursive axiomatizability will then
follow. We first note that the set of *all* sentences of T is
recursively enumerable: using the standard method of Gödel-numbering, we
can effectively generate all well-formed strings of symbols of L, and,
since T has a recursive theory formulation, it is possible to determine in
a finite number of steps whether or not a given string of symbols of L
constitutes a *deduction* of a sentence of T from the theory
formulation of T. By selecting the last line of each such proof, our
mechanical enumeration of *all proofs* of sentences of T can easily
be converted into a mechanical enumeration of all sentences of T. But as
P* is a recursive subset of P, there exists a mechanical procedure for
deciding in a finite number of steps whether or not a given sentence of T
belongs to the restriction of T to L*: the procedure simply needs to check
whether all predicate symbols that occur in the given sentence are
contained in P*. Using this decision procedure, our effective procedure
for enumerating all sentences of T can be transformed into an effective
procedure for enumerating all sentences of the restriction of T to L*. The
recursive axiomatizability of the restriction of T to L* now follows from
its recursive enumerability, as indicated above.

Let us return to our original question. We have seen that the empirical
underdetermination thesis is parasitic upon the claim that, given a
sufficiently complex set of observation conditionals, (i) the deductive
closure of this set is not axiomatizable in terms of a conceptually neat
theory formulation, and (ii) any conceptually manageable theory
formulation will entail (the deductive closure of) the given set of
observation conditionals *as well as some other (non-observational)
sentences*. In what way could Craig’s theorem pose a challenge to this
claim?

**
**4. Quine’s Use of Craig’s Theorem

Let us quote Quine in detail:

Consider any [theory] formulation, and *any* [my italics] desired
class of consequences of it. For our purposes these consequences would be
observation conditionals, but for Craig they can be *any* [my
italics] sentences. Then Craig shows how to specify a second or Craig
class of sentences which are visibly equivalent, one by one, to the
sentences of the desired first class; and the remarkable thing about this
second class is that membership in it admits of a mechanical decision
procedure.

In the cases that matter, these classes are infinite. Even so, the
second or Craig class evidently makes the original finite [theory]
formulation dispensable, by affording a different way of recognizing
membership in the desired first class. Instead of showing that a sentence
belongs to it by deducing it from the finite [theory] formulation, we show
it by citing a visibly equivalent sentence that belongs, testably, to the
Craig class.

This result does not belie under-determination, since the Craig class
is not a finite [theory] formulation, but an infinite class of sentences.
But it does challenge the interest of under-determination, by suggesting
that the finite [theory] formulation is dispensable; and indeed the Craig
class, for all its infinitude, is an exact fit, being a class of visible
equivalents of the desired class. … Each sentence in the Craig class is
simply a repetitive self-conjunction, ‘ppp…p’, of a sentence of the
desired class. …

Why, when the desired class is undecidable, should this Craig class of
its repetitive self-conjunctions be decidable? The trick is as follows.
Each of the desired sentences (each of the desired observation
conditionals, in our case) is deducible from the original finite [theory]
formulation. Its proof can be coded numerically, Gödel fashion. Let the
number be n. Then the corresponding sentence in the Craig class is the
desired sentence repeated in self-conjunction n times. The resulting Craig
class is decidable. To decide whether a given sentence belongs to it,
count its internal repetitions; decode the proof, if any, that this number
encodes; and see whether it is a proof of the repeated part of the given
sentence. (Quine, 1975, pp. 324-325)

Obviously, the basic idea is to invoke Craig’s theorem to establish the
existence of a recursive (but possibly infinite) set of sentences which is
logically equivalent to the desired set of observation conditionals and
which can be used as a tightly fitting theory formulation for the
deductive closure of our set of observation conditionals.

However, Quine’s argument is logically flawed. Although Quine is of
course especially concerned with sets of observation conditionals, he
insists that, for "*any* desired class of consequences" (my italics)
of a theory formulation, we can specify a second class of sentences which
is decidable and whose elements are logically equivalent to the elements
of the given class. Let us examine this rather general claim first. In the
next section, we shall then turn to Quine’s more specific claim regarding
the application of Craig’s result to classes of observation conditionals.

The see whether the former claim is tenable, let us choose any finite
theory formulation which has an infinite set of consequences, say T, and
let us construct a particular subset of T, for which we will subsequently
try to specify the corresponding ‘Craig class’ as explained by Quine. As
before, we note that T is a recursively enumerable set. So T can be
expressed as a mechanically producible sequence {t_{1},
t_{2}, t_{3}, …}. Let M be any subset of the natural
numbers which is *not *recursively enumerable^{(6)}. Define S
:= {t_{n} : n Î M}. If S were
recursively enumerable, we could use our enumeration mechanisms for S and
for T to construct an enumeration mechanism for M. But there exists no
such enumeration mechanism for M; and, in consequence, S cannot be
recursively enumerable.

Using the method proposed by Quine, we shall define S' to be the ‘Craig
class’ corresponding to S. Quine’s claim is that S' is decidable. We may
already be puzzled here. Why? Given any sentence f of T and assuming that the Gödel number of its proof
is n, define y to be a self-conjunction of f of length n, and use Quine’s suggested decision
procedure to determine whether or not y is a
member of S'. As Quine demands, we "count its internal repetitions" — the
answer is n —, we "decode the proof, if any" — the result is a proof of
f —, and we "see whether it is a proof of the
repeated part of the given sentence" — the answer is ‘yes’! And this
answer follows irrespective of whether or not the original sentence f is contained in S and also irrespective of whether or
not the sentence y is contained in S'. Something
must have gone wrong.

And indeed, we shall now see that Quine’s claim that S' is decidable
gives rise to a contradiction. So let us begin with the assumption that
"membership in [S'] admits of a mechanical decision procedure". Given an
effective mechanism for generating the sequence of all well-formed
formulae of our formal language, we can go through this sequence of
well-formed formulae, one by one, mechanically testing whether or not each
of the enumerated formulae is contained in S'. In this manner, we can
mechanically enumerate all members of S'. But, of course, each element of
S' is simply a certain self-conjunction of a corresponding element of S.
So we can easily transform our mechanical enumeration procedure for S'
into a mechanical enumeration procedure for S. This implies that S is
recursively enumerable, a contradiction!

So what has gone wrong? Recalling our exposition of Craig’s theorem, we
can easily see that Quine’s claim is simply too strong. Instead of
starting off with "*any* desired class of consequences" (my italics)
of a given theory formulation, he should have started off with a
*recursively enumerable* set of consequences. The required "visibly
equivalent" ‘Craig class’ could then be constructed in the manner
explained in our proof of the insight preceding Craig’s theorem.

Bearing these observations in mind, we should now turn to Quine’s more
specific point, namely that Craig’s result, applied to the class of all
desired observation conditionals, may "challenge the interest of
under-determination".

**
**5. Does Craig’s Theorem "Challenge the Interest of
Underdetermination"?

Clearly — and as Jane English (1973) argues conclusively —, if the set
of all relevant observation conditionals could somehow be shown to be
recursively enumerable, we would immediately be in a position to infer
that the deductive closure of that set was recursively axiomatizable. In
this case, there would indeed exist a tightly fitting theory formulation
which could be regarded as ‘conceptually manageable’ so long as our notion
of ‘conceptual manageability’ were to admit not only finite theory
formulations, but also recursive ones. In particular, there would be no
room for empirical underdetermination. For the present purposes, let us
concede all this and focus upon the observation that the claim that
Craig’s theorem challenges the empirical underdetermination thesis hinges
crucially upon the recursive enumerability of the set of observation
conditionals. In the remaining part of this paper, I will argue that, at
least within the framework of Quine’s philosophy, the question of whether
the relevant set of observation conditionals is recursively enumerable is
likely to receive a negative answer.

Essentially, there are two strategies through which one might hope to
establish the recursive enumerability of this set. One strategy would be
to try to construct an explicit enumeration mechanism directly, and the
other strategy would be to invoke a suitable subdivision of our language
into ‘theoretical’ and ‘observational’ terms, in the manner of Craig’s
theorem. We shall discuss each strategy in turn.

Presumably, an effective mechanism for enumerating all observation
conditionals would have to be a combination of an effective mechanism for
enumerating *all* sentences of our theory, which, as we know, exists,
and a mechanical procedure for determining in a finite number of steps
whether or not each such sentence is an observation conditional. The task
of designing the latter procedure is tantamount to the task of designing a
mechanical procedure for determining in a finite number of steps whether
or not a given sentence is an observation sentence; for, the relation
between observation sentences and observation conditionals (via pegged
observation sentences) seems sufficiently systematic to be tractable by a
mechanical procedure. But can this new task be performed?

Quine defines observation sentences in terms of a behavioural
criterion: an observation sentence is an occasion sentence — i.e. a
sentence whose truth-value depends upon the occasion of utterance — on
which all competent speakers of the relevant language "give the same
verdict when given the same concurrent stimulation" (Quine, 1969, p.
87)^{ (7)}. This definition not only relies on contingent facts
about the behaviour of the relevant group of competent speakers, but it
is, in particular, ‘community-specific’ in the sense that the question of
what occasion sentences are regarded as observation sentences may depend
upon the community of witnesses that is taken to be relevant, and upon the
witnesses’ background conceptual frameworks. Given all this and the fact
that the cognitive and behavioural sciences are still in their infancy,
the sheer idea of designing a *mechanical* procedure for determining
in a finite number of steps whether or not a given sentence is an
observation sentence according to the stated behavioural criterion appears
to be highly implausible.

As a result, the second strategy, namely the search for a suitable
subdivision of our language into ‘theoretical’ and ‘observational’ terms,
may seem to be a more promising way of establishing the recursive
enumerability of the set of all relevant observation conditionals. If we
could capture the observational part of a theory by devising such a
linguistic subdivision, the premises of Craig’s theorem would be met, and
(the deductive closure of) the set of observation conditionals would be
axiomatizable in a tightly fitting way.

Jane English, for instance, recognises the difficulties involved in the
task of devising the required linguistic subdivision, but holds that "[i]f
… science is recursively axiomatized, the problem of saying which of the
system’s terms are observational is tractable" (1973, p. 454). Whether or
not there are philosophical views according to which the present strategy
would be regarded as promising, I will here point out that, from a Quinean
perspective, it won’t. First and foremost, a Quinean would strongly resist
the idea of drawing a principled distinction between ‘theoretical’ and
‘observational’ terms. As Quine is more than ready to argue, the smallest
individually significant units of language are statements or sentences
rather than terms (1953, section 5; and 1960). Observationality, for
Quine, is a property of sentences or statements, but not of individual
terms: the same term can occur in a broad range of fundamentally different
sentences, observational and non-observational ones, and no sense can be
made of claims about the alleged ‘observational’ or ‘theoretical’ nature
of a term *on its own*.

But even if, contrary to Quine’s position, a principled distinction
between ‘theoretical’ and ‘observational’ terms could be drawn, it would
remain highly unclear whether we would be in a better position to utilise
Craig’s theorem to challenge the empirical underdetermination thesis. Let
me explain. It is well known that a restriction of the vocabulary of our
language is not a particularly accurate or successful method of
pinpointing sentences with an observational content. The fact that all the
predicate symbols occurring in a given sentence are ‘observational’
predicate symbols does not guarantee that the sentence itself is an
‘observational’ sentence. As van Fraassen argues, "[t]he empirical import
of a theory cannot be isolated in this syntactical fashion, by drawing a
distinction among theorems in terms of vocabulary. If that could be done,
T/E [i.e. the restriction of a theory to that part of our language in
which there are only ‘observational’ predicate symbols] would say exactly
what T says about what is observable and what it is like, and nothing
more. But any unobservable entity will differ from the observable ones in
the way it systematically lacks observable characteristics. As long as we
do not abjure negation, therefore, we shall be able to state in the
observational vocabulary (however conceived) that there are unobservable
entities, and, to some extent, what they are like. The quantum theory,
Copenhagen version, implies that there are things which sometimes have a
position in space, and sometimes have not. This consequence I have just
stated without using a single theoretical term. Newton’s theory implies
that there is something (to wit, Absolute Space) which neither has a
position nor occupies a volume. Such consequences are by no stretch of the
imagination about … the observable world" (van Fraassen, 1980, pp. 54-55).

So even on an optimistic view on whether the empirical import of a
theory can be captured by means of a suitable restriction of its
vocabulary, we would have to acknowledge that the set of all relevant
observation conditionals was only a *proper* subset of the set of all
sentences of the theory expressible in the restricted vocabulary. In
consequence, even a suitably constructed Craig reduction would not
constitute a perfectly tightly fitting theory, for it would entail
*all* sentences of the theory expressible in the restricted
vocabulary *and not just the genuinely observational ones*; again,
there would be logical space for empirical underdetermination.

The following objection could be raised: is it not conceivable that,
although the set of all relevant observation conditionals is indeed a
*proper* subset of the set of all sentences of the theory expressible
in the restricted vocabulary, the former set logically determines the
latter, in the sense that, once truth-values are assigned to all
observation conditionals, this will immediately imply an assignment of
truth-values to *all* sentences expressible in the restricted
vocabulary (or, in other words, once we know what sentences are contained
in the former set, we have no degrees of freedom in choosing what
sentences are to be included in the latter one)?

However, van Fraassen’s above cited argument provides a straightforward
counterexample to this objection. As we know, the totality of genuine
observation conditionals of Newtonian mechanics is logically compatible
both with the statement that there exists such a thing as absolute space
and with the statement that there doesn’t exist such a thing. But as
demonstrated by van Fraassen, such statements can be expressed solely in
terms of the ‘observational’ vocabulary. Hence we see that the set of
genuine observation conditionals of Newton’s theory by no means logically
determines the set of all those sentences of the theory that are
expressible in the ‘observational’ vocabulary. The objection must
therefore be rejected.

We conclude that, from a Quinean viewpoint, it is doubtful whether the
conditions under which Craig’s theorem *would *pose a challenge to
the empirical underdetermination thesis can be satisfied. Neither the idea
of directly designing an effective mechanism for enumerating all
observation conditionals, nor the idea of devising a suitable subdivision
of the terms of the relevant language into ‘theoretical’ and
‘observational’ ones are particularly promising strategies for
establishing, as required, that the set of all observation conditionals
entailed by a theory is recursively enumerable.

**
**6. Conclusion

From the perspective of Quine’s philosophy, I have argued for a
revision of the view that Craig’s theorem poses a (theoretical) challenge
to the empirical underdetermination thesis. I have shown that Quine’s
concession to the Craigian argument can probably be traced back to the
fact that Quine invokes Craig’s theorem in a mathematically flawed way.
Indeed, once the requirement that the set of all observation conditionals
be recursively enumerable is recognised, Quine’s empirical
underdetermination thesis can be defended against the Craigian challenge.
This is not to say, however, that the thesis is correct. There may (or may
not) be independent reasons why the empirical underdetermination thesis is
untenable.

Christian List

University of Oxford

Nuffield College

Oxford
OX1 1NF, U.K.

christian.list@nuffield.oxford.ac.uk

**
**

**References **

Craig, William (1953) "On Axiomatizability within a System", *Journal
of Symbolic Logic*, XVIII.

Craig, William (1956) "Replacement of Auxiliary Expressions",
*Philosophical Review*, 65.

Davidson, Donald (1990) "Meaning, Truth and Evidence" in Barrett, R.,
& Gibson, R. (eds.), *Perspectives on Quine*, Oxford, Blackwell.

English, Jane (1973) "Underdetermination: Craig and Ramsey", *Journal
of Philosophy,* 70.

Quine, W.V. (1953) "Two Dogmas of Empiricism", in *From a Logical
Point of View*, Cambridge, Mass., Harvard Univ. Press.

Quine, W.V. (1960) *Word and Object*, Cambridge, Mass., MIT Press.

Quine, W.V. (1969) "Epistemology Naturalized", in *Ontological
Relativity and Other Essays*, New York, Columbia University Press.

Quine, W. V. (1970) "On the Reasons for Indeterminacy of Translation",
*Journal of Philosophy,* 67.

Quine, W. V. (1975) "Empirically Equivalent Systems of the World",
*Erkenntnis*, 9.

van Fraassen, Bas C. (1980) *The Scientific Image*, Oxford, Oxford
University Press.

#### Notes

- I wish to express my gratitude to Simon Saunders and to an anonymous
referee for comments and suggestions, to Daniel Harbour for many
fruitful discussions, and to the Scatcherd European Scholarship
Foundation of Oxford University as well as to the (British) Economic and
Social Research Council for financial support.
- To be precise, we shall say that two theories can be
*rendered
logically compatible by means of a reconstrual of predicates* if
there exists a reconstrual of predicates under which one of the two
theories is mapped onto a (not necessarily proper) subset of the other.
- Since each of the given theories is deductively closed, so is their
intersection.
- Consider, for instance, the set of all non-theorems of first-order
Peano-arithmetic.
- Consider, for instance, the set of all theorems of a first-order
system of the predicate calculus.
- In note (4), I have cited the set of all non-theorems of first-order
Peano-arithmetic as an example of a set that is countably infinite, but
not recursively enumerable. Call this set A. We can use A to generate
the required set M as follows. A is clearly a subset of the recursively
enumerable set of
*all* well-formed formulae of an appropriate
first-order language. Call the latter set B. Then B can be expressed as
the sequence {b_{1}, b_{2}, b_{3}, …}. Now
define M = {n Î N :
b_{n Î } A}. If M were
recursively enumerable, we could easily combine our enumeration
mechanism for M with that for B so as to construct an enumeration
mechanism for A. But this would imply the recursive enumerability of A,
a contradiction! Hence M is a subset of the natural numbers which is
*not* recursively enumerable.
- As Davidson (1990) argues, Quine vacillates between two different
criteria for defining observation sentences. The definition I state here
makes use of the
*proximal* criterion, Quine’s ‘official’ one,
according to Davidson. Under the *distal* criterion, the subclause
"when given the same concurrent stimulation" would have to be replaced
with the subclause "when presented with the same intersubjectively
shared situation". Whilst the two accounts have rather distinct
philosophical consequences in many respects, the important point to note
in the present context is that both definitions are essentially
behavioural.