Realism and Empirical Equivalence

The main purpose of this paper is to investigate various notions of empirical equivalence in relation to the two main arguments for realism in the philosophy of science, namely the no-miracles argument and the indispensability argument. According to realism, one should believe in the existence of the theoretical entities (such as numbers and electrons) postulated by empirically adequate theories. According to the no-miracles argument, one should do so because truth is the the best explanation of empirical adequacy. According to the indispensability argument, one should do so because the theoretical terms employed in the formulation of an empirically adequate theory are practically indispensable for the formulation of its empirical content. The no-miracles argument might be refuted if one can establish an underdetermination thesis to the effect that every theory has empirically equivalent rivals that are incompatible with each other. Insofar as truth cannot explain the empirical adequacy of two incompatible theories, there is an obvious conflict between the underdetermination thesis and the no-miracles argument. I show that, under certain assumptions, some but not all notions of empirical equivalence support the underdetermination thesis. The indispensability argument might be refuted if one can establish a dispensability thesis to the effect that, for any theory with a practical formulation (e.g. for any axiomatizable theory), there is an empirically equivalent theory with a practical formulation in purely empirical terms. I show that (using axiomatizability as the measure of practicality) some but not all notions of empirical equivalence support this thesis.


Introduction
Scientific theories help predict the observable behavior of ordinary things such as planets, bridges, and light-bulbs with great accuracy and precision. They typically do so by postulating non-observable things such as electrons, energy, or dark matter. Roughly, if a theory successfully predicts the observable behavior of ordinary things, we say that it is empirically adequate. But does empirical adequacy provide reason for thinking that the theory is true and that all non-observable things it postulates exist?
Realists in the philosophy of science argue that it does. According to one of the most influential arguments for realism, the no-miracles argument, 1 the best explanation for why a theory is empirically adequate is that it is true. According to a second influential argument, the indispensability argument, 2 the theoretical terms employed in the formulation of an empirically adequate theory are practically indispensable for the formulation of its empirical content.
Two theories that make the same empirical predictions are said to be empirically equivalent. At least some versions of the no-miracles argument can be refuted if one can establish an underdetermination thesis to the effect that (1) For any theory T , there are theories T 1 and T 2 empirically equivalent to T such that T 1 and T 2 are incompatible.
Insofar as truth cannot explain the empirical adequacy of two incompatible theories, there is an obvious conflict between this thesis and the no-miracles argument. The status of the thesis remains a matter of dispute, however. 3 Under certain assumptions, presented in Section 2, it has been shown that the classical notion of empirical equivalence doesn't support it, but rather the opposite: underdetermination becomes impossible [6, p. 373]. For an exposition, see Section 3. In Section 4, we show that a weaker notion of empirical equivalence still does. A third notion of empirical equivalence is considered in Section 5, but we have not been able to show whether it supports the thesis or not. The indispensability argument can be refuted if one can establish a dispensability thesis to the effect that (2) For any theory with a practical formulation (e.g. for any axiomatizable theory), there's an empirically equivalent theory with a practical formulation in purely empirical terms.
In Section 6, we show that (using axiomatizability as the measure of practicality), some but not all notions of empirical equivalence support this thesis.

Empirical and Non-Empirical Vocabulary
Following Carnap [4], I shall assume that the distinction between empirical and nonempirical consequences of a theory can be drawn by dividing the vocabulary in which the theory is formulated into an empirical and non-empirical part. Canonically, 'dog', 'table' and 'bigger than' are examples of empirical terms, whereas 'number', 'electron' and 'oppositely charged' are examples of non-empirical (i.e. theoretical) terms.
On the classical notion of empirical equivalence, two theories are empirically equivalent just in case they entail the same empirical sentences, where an empirical sentence is one containing only logical and empirical terms (e.g. 'All dogs bark'). As we shall see, other notions of empirical equivalence are possible. But we shall assume throughout that the vocabulary L T of any theory T can be split into two parts, an empirical part E T and a theoretical part L T − E T . For present purposes, I shall take this distinction as basic, and not attempt to explicate or define it. By theory, I shall mean a set of sentences, not necessarily deductively closed. Unless otherwise stated, all theories are assumed to be first-order. Admittedly, the distinction between empirical and non-empirical consequences is controversial, as is the distinction between what is directly observable and what is merely deducible from such observations together with some background theory. If one equates empirical consequences with what is directly observable, and requires direct observations to be infallible, it follows that the empirical consequences of a theory basically only pertain to the distribution of patches of color in ones visual field. On these assumptions, there is no epistemological difference between tables and electrons (they are both theoretical entities), and no distinction between empirical and non-empirical consequences that seems relevant to the debate about realism and anti-realism in the philosophy of science.
However, one need not require direct observations to be infallible. We may also grant that there is no metaphysical distinction to be made between tables and electrons, or even numbers. If they exist, they all exist in the same way, so to speak. But realism understood in the above manner is a thesis concerning human epistemology. For such a thesis, the distinction between what human beings considered as measuring instruments can and cannot (fallibly) detect will be relevant. As Ladyman [13, p. 190] puts it: ... realists do not see why our physical constitution, as a contingent feature of our evolution, has any philosophical significance whatsoever. One response to this is simply to restate the opposite intuition: what else but our (biologically determined) observational capacities would one consider relevant to our epistemology? Some realists do not want to recognize the distinction between empirical and nonempirical consequences in the first place. They seem to think that its mere recognition yields an easy win for anti-realism. As we shall see, that is not true. But there is an easy argument for the underdetermination thesis lurking in the background that might explain this sentiment. The argument, however, is deeply flawed. It runs as follows: take any theory T (not equivalent to its empirical consequences, presumably), and let T * consist of all the empirical consequences of T plus the assertion that T is false. Then, it is claimed, 4 T * will be empirically equivalent to T and yet jointly inconsistent with it. Hence, no matter what one takes the empirical consequences of a theory to be (as long as they don't encompass all the consequences of the theory), the underdetermination thesis is true. The argument is usually dismissed by realists because they do not think T * is a genuine alternative to T . But the real problem with the argument is that there is no guarantee that T * , constructed in this way, is empirically equivalent to T . To see why, let T E be a set of empirical consequences such that T E and T E → θ for some empirical sentence , e.g. 'Pigs can fly', and some theoretical sentence θ, e.g. 'Numbers exist'. Consider the theory T = T E ∪ { → θ }. Since T E has a model where is false and therefore → θ is true, it follows that T . Now, let T * consist of the empirical consequences of T plus the assertion that T is false. Since T * ∪ T E ∪ { → θ} does not have a model, and T * ∪ T E is equivalent to T * , it follows that T * ¬( → θ). Hence, T * , which means that T * is not empirically equivalent to T .
A more promising algorithm for generating empirically equivalent rivals is suggested by Quine [21, p. 319]: Take some theory formulation and select two of its terms, say 'electron' and 'molecule'. I am supposing that these do not figure essentially in any observation sentences; they are purely theoretical. Now let us transform our theory formulation merely by switching these two terms throughout. The new theory formulation will be logically incompatible with the old: it will affirm things about so-called electrons that the other denies.
Quine contends, however, that the example is not a genuine case of underdetermination. The quote continues: Yet their only difference, the man in the street would say, is terminological; the one theory formulation uses the technical terms 'molecule' and 'electron' to name what the other formulation calls 'electron' and 'molecule'. The two formulations express, he would say, the same theory.
The example suggest that joint inconsistency is not sufficient for incompatibility. But in order for the underdetermination thesis to have some bearing on the no-miracles argument, we shall assume it to be necessary. In other words, we shall assume that (3) Incompatible theories are jointly inconsistent.
In order to avoid the kind of triviality raised by Quine's example, there are several options. One option, advocated by Quine himself, is to assume that (4) Consistent incompatible theories are (in some sense) not inter-translatable.
The sense Quine had in mind was roughly that two theories T 1 and T 2 are intertranslatable just in case, for all predicates P 1 , ..., P n in the vocabulary of T 1 there are formulas ϕ 1 , ..., ϕ n in the vocabulary of T 2 such that replacing all occurrences of P 1 , ..., P n in T 1 by ϕ 1 , ..., ϕ n yields a theory equivalent to T 2 , and vice versa. But other notions of inter-translatability are possible. 5 Another option is to assume that inter-translatability preserves incompatibility: If T 1 and T 2 are incompatible, and T 2 and T 2 are inter-translatable, then T 1 and T 2 are incompatible as well.
This principle, which I take to be plausible enough to merit attention, allows for certain simplifications. Clearly, (3) and (5) already entail (4). Moreover, on any reasonable notion of empirical equivalence and inter-translatability, 6 they also entail that (6) If T 1 and T 2 are incompatible, then there is a theory T 2 empirically equivalent to T 2 such that T 1 and T 2 are incompatible and do not share any theoretical vocabulary.
That means we can assume, without loss of generality, that incompatible theories do not share any theoretical vocabulary. This assumption can be motivated by the idea, most famously advocated by Feyerabend [9], that theoretical terms have no meaning outside of the theoretical context in which they occur. Indeed, that this is part of what makes them theoretical rather than empirical in the first place. In this paper, I shall therefore assume (5). The underdetermination thesis under consideration will in effect be the following: For any theory T , there are theories T 1 and T 2 empirically equivalent to T such that T 1 and T 2 are jointly inconsistent and do not share any theoretical vocabulary.

Strong Empirical Equivalence
Arguably, the classical notion of empirical equivalence corresponds to what we shall call strong empirical equivalence. To define it, we first introduce the following notion: Empirical equivalence is then defined as follows: Definition 3.2 (Strong empirical equivalence) Two theories are strongly empirically equivalent just in case they have the same E-fragments. 5 Cf. Barrett and Halvorson [1]. 6 All it takes is that the result of replacing every theoretical predicate P in a theory T with a new theoretical predicate P * yields a theory T * such that T and T * are both empirically equivalent and inter-translatable. As far as empirical equivalence goes, all notions thereof considered in this paper satisfy this constraint.
Granted that the only vocabulary rival theories have in common is empirical, it can be shown that strongly empirically equivalent consistent theories are always jointly consistent, using a theorem by Robinson [2, p. 264]: Theorem 3.1 (Robinson's joint consistency theorem) If T is a complete theory in L 1 ∩ L 2 , and T 1 and T 2 are satisfiable extensions of T in L 1 and L 2 respectively, then T 1 ∪ T 1 is satisfiable. Corollary 3.1 If T 1 and T 2 are strongly empirically equivalent consistent theories sharing no theoretical vocabulary, then T 1 ∪ T 2 is consistent.
Proof Assume, towards contradiction, that T 1 and T 2 are (i) consistent theories that are (ii) equivalent over their common vocabulary L, and that (iii) T 1 ∪ T 2 is inconsistent. Since T 1 is consistent, it has a complete and consistent extension T * 1 . The L-sentences of T * 1 forms a complete and consistent L-theory T * such that T * ⊆ T * 1 . Assume, towards contradiction, that T * 2 = T 2 ∪ T * is consistent. By Robinson's joint consistency theorem, it follows that T * 2 ∪ T * 1 is consistent. That means T 1 ∪ T 2 is consistent, contradicting (iii). Hence, T 2 ∪ T * is inconsistent. That means there is an L-sentence ϕ such that T * ϕ and T 2 ¬ϕ. By (ii), it follows that T 1 ¬ϕ. Hence, T * ¬ϕ, contradicting the fact that T * is consistent.
Hence, on the strong notion of empirical equivalence, the underdetermination thesis is false. Demopoulos [6, p. 373] essentially makes the same observation using Craig's Interpolation Lemma. He observes that if one takes the content of a theory to be given by its Ramsey sentence, then one can assume without loss of generality that competing theories do not share any theoretical vocabulary, and hence that strongly empirically equivalent consistent theories are jointly consistent. Similarly, English [8, p. 460] observed that if the Ramsey sentences of two theories are jointly inconsistent, then they cannot be empirically equivalent.
Lastly, we show that Corollary 3.2 If T 1 and T 2 are strongly empirically equivalent consistent theories sharing no theoretical vocabulary, then they are strongly empirically equivalent to Proof Assume that T 1 and T 2 are strongly empirically equivalent consistent theories. Suppose there is a sentence ϕ of their common empirical vocabulary such that T 1 ϕ (and hence T 2 ϕ). That means T 1 ∪ {¬ϕ} and T 2 ∪ {¬ϕ} are consistent. Moreover, they are strongly empirically equivalent. For assume, towards contradiction, that there is sentence ψ of their common empirical vocabulary such that T 1 ∪ {¬ϕ} ψ but T 2 ∪ {¬ϕ} ψ (or vice versa). That would mean T 1 ¬ϕ → ψ but T 2 ¬ϕ → ψ (or vice versa), contradicting our assumption. By Corollary 3.1, it now follows that Together, these two corollaries suggest that, if two consistent theories are strongly empirically equivalent and we believe one of them, we might just as well believe both.

Weak Empirical Equivalence
The classical notion of empirical equivalence has been challenged, most famously by Van Fraassen [26, pp. 54-55]: The 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 [the E-fragment of T , in my terminology] 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 what there is in the observable world, nor about what any observable thing is like. However, as argued by Turney [25], this objection does not pertain to every syntactic notion of empirical equivalence. Assuming, with Van Fraassen, that the notion of observability can be expressed in the empirical (observational) vocabulary of each theory, a syntactic notion of empirical equivalence can be defined that is immune his objection. Essentially following Turney [25], such a notion will now be defined, called weak empirical equivalence. This notion will only apply to theories containing a monadic empirical predicate O, the intended interpretation of which is observable. This is no limitation in principle, however. In his quote, Van Fraassen assumes that, for any theory T of the relevant kind, there is an E T -formula ϕ(x) such that the sentence ∃x¬ϕ(x) states that there are unobservable entities. Now, suppose we have two theories of the relevant kind that are, in some reasonable sense, empirically equivalent. At the very least, they would have to contain the same empirical vocabulary. Arguably, that means the empirical expressions in each theory are intended to apply to the same things. So if an empirical formula ϕ(x) in one theory is intended to apply to all and only observable things, the same formula in the other theory is intended to do the same. That means we can introduce the predicate O in both theories, and regard it as an abbreviation of said formula. Alternatively, we may extend the empirical vocabulary of both theories with the predicate O, and extend the old theories with the defining axiom ∀x(Ox ↔ ϕ(x)). The result will be a conservative extension with respect to the old vocabulary, where the intended interpretation of O is observable. The important thing is that, either way, every formula containing only empirical expressions and the predicate O may be regarded as an empirical formula. 7 If two theories contain this predicate, then (in sense to be made precise shortly) they may entail the same things with respect to observable objects without being strongly empirically equivalent. As we shall see, one theory may entail that all objects are observable, while the other theory entails that some objects are not observable (Theorem 4.1). Intuitively, however, the theories seem to be empirically equivalent.
In order to ensure that O can be interpreted as observable, we shall require that the theories in question are O-theories, in the following sense: In order to give a precise syntactic definition of when two O-theories say the same thing about observable objects, we introduce the following notion: If ψ = Q[ϕ] for some formula ϕ, we say that ψ is an O-qualified formula.
For convenience, we also introduce the notion of the O-fragment of an O-theory, which is a subset of its E-fragment: Although it is perhaps intuitively clear already, in order to verify that O-qualified formulas only talk about observable objects, we introduce the following notion: The following result, which we shall also make heavy use of later on, verifies our claim: Proof Let M be a model of an O-theory T , and let M be an O-variant of M. Let X be the set of variables. Since T Oc for any constant c ∈ E T , it follows that, for any E T -term t, (8) For any assignment g : Moreover, since T ∀x 1 ...x n (P x 1 ...x n → Ox 1 ∧ ... ∧ Ox n ), it follows that, for any E T -predicate P other than identity, For any assignment g : by induction on the complexity of ϕ: (8) and (9), t Assuming with Van Fraassen that any theory of the relevant kind can be construed as an O-theory, the result establishes that weak empirical equivalence supports the underdetermination thesis.

Van Fraassen Equivalence
In order to define Van Fraassen's notion of empirical equivalence, we introduce the following notion:  Observe that this does not yet establish that the notion of Van Fraassen equivalence supports the underdetermination thesis. We include the result here merely to show that Van Fraassen equivalence does not entail strong empirical equivalence. It does, however, entail weak empirical equivalence. To see that, we introduce to following notion: Elementary equivalence (making the same first-order sentences true) is strictly weaker than isomorphism. Clearly, Van Fraassen equivalence entails elementary Van Fraassen equivalence. We go on to establish the identity between elementary Van Fraassen equivalence and weak empirical equivalence:

Lemma 5.1 Let M be the O-part of a model M of some O-theory T . Then, for any
Proof It is straightforward to show, by induction on the complexity of ϕ, that for any E T -formula ϕ and assignment g, M , g ϕ iff M , g Q[ϕ]. The following result shows that weak empirical equivalence does not entail Van Fraassen equivalence:

Theorem 5.3 Some O-theory is not Van Fraassen equivalent to its E-fragment (nor to its O-fragment).
Proof Let N, < be the natural number structure with the binary relation <, and let O, ≺ be a structure isomorphic to it under bijection h :

Corollary 5.1 Some strongly (and hence weakly) empirically equivalent O-theories are not Van Fraassen equivalent.
Proof Follows from Theorem 5.3 and the fact that every theory is strongly empirically equivalent to its E-fragment.  To sum up, weak empirical equivalence does not entail Van Fraassen equivalence. Moreover, both strong empirical equivalence and Van Fraassen equivalence entail weak empirical equivalence, but neither of the two entails the other. The question is whether Van Fraassen equivalence supports the underdetermination thesis, i.e. whether the following conjecture is true:

Conjecture 5.1 For any O-theory T , there are O-theories T 1 and T 2 Van Fraassen equivalent to T such that T 1 and T 2 do not share any theoretical vocabulary and
In view of Theorem 5.3, the answer is not obvious. For some O-theories, it is certainly possibly to find Van Fraassen equivalent rivals that are jointly inconsistent. Take, for instance, the O-fragment T of an arbitrary O-theory T . By Theorem 5.1, T will have Van Fraassen equivalent rivals T ∪{∀xOx} and T ∪{¬∀xOx}. However, by Theorem 5.3, these need not be Van Fraassen equivalent to T .

Indispensability
Quine and Putnam famously argued that we should believe that theoretical terms refer because these terms are indispensable for the practical formulation of empirically adequate theories. In response to the second part of their claim, the following results are sometimes mentioned: Theorem 6.1 (Craig) Every recursively enumerable set of sentences is axiomatizable (i.e. equivalent to some recursive set of sentences). Remark 6.1 Not every recursively enumerable set of sentences is finitely axiomatizable, however. To see why, let T be any finitely axiomatizable consistent theory having only infinite models (e.g. Robinson arithmetic), and let T = be the set of theorems of T containing no non-logical vocabulary. Then T = is recursively enumerable. Let be the set of sentences saying, for each natural number n, that there are at least n objects. We know that is complete with respect to the logical vocabulary, but not finitely axiomatizable. Since T = is consistent and ⊆ T = , it follows that T = is equivalent to , and hence not finitely axiomatizable.

Corollary 6.1 For any axiomatizable theory T , there is a strongly empirically equivalent axiomatizable theory T in the language of E T .
The relevance of this result was noted, for instance, by Hempel [10]. It shows that, insofar as axiomatizability is the relevant notion of practicality, the strong notion of empirical equivalence supports the dispensability thesis. On the other hand, if finite axiomatizability is what matters for practicality, it follows from Remark 6.1 that strong empirical equivalence does not support the thesis (as witnessed by T = considered as the E-fragment of T ). For various other reasons, Hempel resists the conclusion that theoretical terms are dispensable. List [15] also resists the dispensability thesis in spite of Corollary 6.1, but rather because he takes the strong notion of empirical equivalence to be irrelevant (citing Van Fraassen). More specifically, he argues that the relevant empirical consequences of an axiomatizable theory may not be recursively enumerable. In response to List, we make the following observation:

Lemma 6.2 The O-fragment of an axiomatizable theory is recursively enumerable.
Using Theorem 6.1 and Lemma 6.2, together with Theorem 4.1, it follows that The result shows that, assuming weak empirical equivalence to be the relevant notion of empirical equivalence, and axiomatizability to be the relevant notion of practicality, the dispensability thesis is true. Again, in view of Remark 6.1, assuming finite axiomatizability to be the relevant notion of practicality instead, it is not hard to construct a counterexample to dispensability also on the weak notion of empirical equivalence.
Lastly, we show that Van Fraassen equivalence together with axiomatizability does not support indispensability. To do that, we shall use the following result, as reported by Keisler [11, p. 177, Corollary 10.8]: Theorem 6.2 (Keisler-Shelah isomorphism theorem) Two models are elementary equivalent just in case, for some ultrafilter, the corresponding ultrapowers of the two models are isomorphic.
Using that result, we first prove the following lemma: It can be shown that D U is a partitioning of D I into equivalence classes [f ] U for each f ∈ D I . The models are well-defined because it can be shown that, for any f 1 , ..., f n ∈ D I and g 1 ∈ [f 1 ] U , ..., g n ∈ [f n ] U , {i ∈ I : P M (f 1 (i), ..., f n (i))} ∈ U iff {i ∈ I : P M (g 1 (i), ..., g n (i))} ∈ U . Let M 1 and M 2 be two L-models with the same domain D such that D ⊆ D . Assume that, for each predicate P ∈ L, P M 1 = P M 1 and P M 2 = P M 2 . The domain D U of both U M 1 and U M 2 is given by (15) where, for each f ∈ D I , the equivalence class [f ] U is given by (16) [f ] U = {g ∈ D I : {i ∈ I : f (i) = g(i)} ∈ U } U M 1 and U M 2 are defined so that, for each n-place predicate P and f 1 , ..., f n ∈ D I , (17) a.
Assuming the Axiom of Choice, 8 Hence, by (14), for any f 1 , ..., f n ∈ D I , Moreover, since P M 1 = P M 1 and P M 2 = P M 2 , we have for each f 1 , ..., f n ∈ D I and i ∈ I , .., f n (i)).
Observe that, by (13) and (16), for any f, g ∈ D I , By (22-a), we can define a function a : D I → D I such that, for each f ∈ D I , , since only one of them will contain a member of D I . Hence, we get two cases: Hence, Hence, Finally we show that, for any f 1 , ..., f n ∈ D I , . We get two cases: 1. There are g 1 , ..., g n ∈ D I such that by (17-a), {i ∈ I : P M 1 (g 1 (i), ..., g n (i))} ∈ U iff, by (21-a), {i ∈ I : P M 1 (g 1 (i), ..., g n (i))} ∈ U iff, by (20), {i ∈ I : P M 2 (a(g 1 )(i), ..., a(g n )(i))} ∈ U iff, by (21b), {i ∈ I : P M 2 (a(g 1 )(i), ..., a(g n )(i))} ∈ U iff, by (23), {i ∈ I : P M 2 (a (g 1 )(i), ..., a (g n )(i))} ∈ U iff, by (17-b), P U M 2 ([a (g 1 )] U , ..., [a (g n )] U ) iff, by (24), .., f n (i))} ∈ U and {i ∈ I : P M 2 (a (f 1 )(i), ..., a (f n )(i))} ∈ U by upwards closure of U . Hence, by (17) and (24) Proof Let L * = {O, ≺, <, 0, s, f }, and let T * consist of the following groups of axioms. The first group says that ≺ is a strict total order on observable objects: The second group says that < is a strict total order on unobservable objects: The third group says that 0 is the smallest unobservable object: The fourth group says that s is the successor function on the unobservable objects: The fifth group says that f is an order-preserving bijection from unobservable to observable objects: Since T * ⊆ T , we only need to show the direction from left to right. Assume, towards contradiction, that there is an L * -sentence ϕ such that T ϕ but T * ϕ. That means there is a finite subset ⊆ such that T * ∪ ∪ {¬ϕ} does not have a model but T * ∪ {¬ϕ} has. Let M be a model of T * ∪ {¬ϕ}. Let t be the longest term such that t < c ∈ . Let M be an expansion of M such that c M = s(t) M . Then M is a model of T * ∪ . By locality, M is also a model of ¬ϕ. Hence, M is a model of T * ∪ ∪ {¬ϕ}, contrary to our assumption.
It follows from (30) that the O-fragments of T and T * are identical. But N, < is a model of this fragment, and non-isomorphic to the O-part of any model of T . Hence, T is not Van Fraassen equivalent to its O-fragment. By Corollary 6.3, it follows that T is not Van Fraassen equivalent to any O-theory in the language of E T . Remark 6.2 In a way, this result strengthens that of Melia [17], who shows (in my terminology) that some axiomatizable theory T * is not Van Fraassen equivalent to its E-fragment T . Melia's aim is essentially to show that the trivial nominalizationstrategy suggested by Craig's theorem does not work if one adopts Van Fraassen's notion of empirical equivalence. My aim is to show that, for some theories, no nominalization-strategy works in that case. It is not obvious from his result that no other E T * -theory is Van Fraassen equivalent to his theory T * , but it can be shown using Corollary 6.3.
As mentioned in the beginning, all these result pertain to first-order theories only. I suspect that, for any theory T , one always can find a Van Fraassen equivalent higher-order theory in the empirical vocabulary of T . For instance, if the theory is finite, it is enough to consider its Ramsey sentences. If so, the choice between Van Fraassen equivalence and weak empirical equivalence (i.e. elementary Van Fraassen equivalence) boils down to whether or not one takes the empirical content of a theory to include the higher order empirical consequences of the theory. Lutz [16] has shown that, in the context of higher-order logic, Van Fraassen's notion of empirical equivalence can be defined syntactically. Thus, if one thinks that the higher order empirical consequences are important, then one might opt for Van Fraassen equivalence. If, on the other hand, one thinks that sentences of higher-order logic do not even express empirical propositions (since they quantify over non-observable entities such as relations), one might opt for weak empirical equivalence.

Conclusion
We take the underdetermination thesis to say that, for any theory T , there are theories T 1 and T 2 empirically equivalent to T such that T 1 and T 2 do not share any theoretical vocabulary and are jointly inconsistent. Is it true? On the classical notion of empirical equivalence (Definition 3.2), where two theories are empirically equivalent just in case they entail the same things with respect to their shared empirical vocabulary, the answer is no (Corollary 3.1). However, on a weaker understanding of empirical equivalence (Definition 4.4), where two theories are empirically equivalent just in case they entail the same things about observable objects, the answer is yes (Theorem 4.1). Using this notion of weak empirical equivalence, it can be argued further that theoretical terms are dispensable for science (Corollary 6.2). However, if one adopts Van Fraassen's slightly stronger notion of empirical equivalence (Definition 5.2), where two theories are empirically equivalent just in case they are satisfied by the same types of observable structures, dispensability cannot be established on the same grounds (Theorem 6.3). The question is whether this notion still supports underdetermination (Conjecture 5.1).