Ainsworth 2012

The Third Path to Structural RealismAuthor(s): Peter M. AinsworthSource: HOPOS: The Journal of the International Society for the History of Philosophy ofScience, Vol. 2, No. 2 (Fall 2012), pp. 307-320Published by: The University of Chicago Press on behalf of the International Society for theHistory of Philosophy of ScienceStable URL: http://www.jstor.org/stable/10.1086/664775 .

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THE THIRD PATH TO
STRUCTURAL REALISM
Peter M. Ainsworth

In addition to the “upward” and “downward” paths to structural realism, there is a third
path, scouted by GroverMaxwell. The purpose of this article is to critically examine thisthird path. Maxwell explicitly notes that his argument relies on a neo-Russellian prin-ciple of acquaintance. It also tacitly relies on the assumption that a predicate is “obser-vational” if it applies to an observable object. It is argued here that (i) this assumption isevidently false if one adopts an intuitive notion of an observational term but that (ii) theassumption seems plausible if one works with a notion of an observational term like thatMaxwell himself seems to have had inmind but that (iii) if one works with this notion ofan observational term then the principle of acquaintance leads to absurdity.
1. Introduction

Roughly speaking, epistemic structural realism is the view that (i) we do haveknowledge of the unobservable world (the realist thesis) but that (ii) our knowledgeof the unobservable world is limited to knowledge of its structural (mathematical)properties (the structuralist thesis). Psillos (2001) distinguishes two paths tostructural realism: Russell’s (1927) “upward path” and Worrall’s (1989)“downward path.”1 Russell’s upward path involves starting from a neoskepticalposition and “building up” to structural realism. Russell posits two foundational

Contact Peter M. Ainsworth at University of Bristol, Department of Philosophy, 43 Woodland
Road, Bristol BS8 1UU, United Kingdom ([email protected]).
I gratefully acknowledge the financial support of the Leverhulme Foundation. I am also very gratefulto Frank Arntzenius, Roman Frigg, Jeff Ketland, and John Worrall, who read and commented on anearly draft of this article, and to everyone in the Bristol Foundations of Structuralism project forstimulating discussions on related issues.

1. The subject of this article is the epistemic structural realism of Russell (1927), Maxwell (1968,1970a, 1970b), Worrall (1989), and others, not the ontic structural realism of Ladyman (1998) andothers. The term “structural realism” is used here to mean exclusively “epistemic structural realism.”

HOPOS: The Journal of the International Society for the History of Philosophy of Science, vol. 2 (Fall 2012).2152-5188/2012/0202-0002$10.00. © 2012 by the International Society for the History of Philosophy of Science.All rights reserved.

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epistemological principles (which have since been called the “Helmholtz-Weyl”

HO PO S l Third Path to Structural Realism

principle and the “mirroring relations” principle; see Psillos 2001; Votsis 2005)and argues that if our knowledge of the unobservable world is based on theseprinciples, then we can have knowledge of the structure of the unobservableworld (and no more). Worrall’s downward path involves starting from a con-ventional scientific realist position and “working down” to structural realism.He argues that structural realism is the strongest form of realism compatiblewith the history of science.

Both of these paths have been extensively discussed in the literature, andboth face serious objections. So it is of interest to learn that there is a third pathto structural realism, which was scouted by Maxwell (1968, 1970a, 1970b).The purpose of this article is to critically examine this third path. While theconclusion will be that the path is blocked, Maxwell’s argument is nonethelessinteresting and subtle, and it is strange that it has been overlooked, given therecent interest in structural realism.2

Like Russell’s upward path, this path is based on a foundational principle,although in this case it is a semantic (rather than epistemological) principle (infact a Russellian semantic principle: the principle of acquaintance). However,in other ways it has more in common withWorrall’s downward path: what theargument purports to establish is not that we can have knowledge of the struc-ture of the unobservable world grounded on the principle of acquaintance butrather that structural realism is the strongest form of realism that is compatiblewith the principle of acquaintance.

In section 2, structural realism and the Ramsey-sentence approach to thedoctrine are outlined. In section 3, the principle of acquaintance is discussed.In section 4, Maxwell’s argument for structural realism is presented and thencriticized: it is noted that the argument only goes through if we work with anobservable/unobservable distinction that has certain properties (properties thatseem peculiar, given a contemporary conception of the distinction, but arenonetheless reasonable, givenMaxwell’s conception of the distinction) and thatwhen one does this then the principle of acquaintance leads to absurdities.

2. Structural Realism and Ramsey Sentences

Although structural realism can be traced back to Russell, and possiblyfurther, the term “structural realism” was first used by Maxwell (1968,

2. Some people regard this interest as misguided, as they believe that structural realism is dead in thewater, killed off by the famous Newman objection. But others believe that this objection can be
answered and that structural realism is alive and well (see Ainsworth [2009] for a review of work onNewman’s objection).
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155).3 Maxwell also offered a novel argument for the position and was the

Ainsworth l F A L L 2012

first person to propose the modern “Ramsey-sentence” formulation of thedoctrine.4 The main task of this article is to critically examine Maxwell’sargument for structural realism, but before doing this it will be useful tooutline the Ramsey-sentence formulation of structural realism.

From a formal point of view, languages are built out of two types of terms:logical terms and nonlogical terms. In a language of second-order logic, thesetwo groups are as follows

1. Logical terms: (a) logical connectives (¬, &, etc.), (b) quantifiers(∀, ∃), (c) individual and predicate variables (x1, x2, … andX1, X2, …), and possibly (d ) the identity predicate (=)

2. Nonlogical terms: (a) a number of names (a1, a2, …) denotingobjects and (b) a number of predicates (P1, P2, …) denotingproperties and relations.

Let an observational term be a nonlogical term that refers to an observableobject, property, or relation and a theoretical term be a nonlogical term thatrefers to an unobservable object, property, or relation. The Ramsey sentenceof a theory is obtained from a sentence expressing the theory by first replacingthe theoretical terms (names and predicates) in the sentence with new vari-ables (using the same variable for each occurrence of the same term and dif-ferent variables for different terms). The resulting formula is then turned intoa sentence (the theory’s Ramsey sentence) by binding the variables with theappropriate existential quantifiers (placed at the start of the formula, so thatevery occurrence of the same new variable is in the scope of the same quan-tifier). Note that, in general (as long as the original sentence contains at leastone theoretical predicate), constructing the Ramsey sentence of a theory willrequire a language of second-order logic.

The Ramsey-sentence version of structural realism comes in two forms:

1. We are justified in believing that the Ramsey sentences of maturescientific theories are approximately true. We are not justified inbelieving that the un-Ramsey-fied theories are approximately true.

3. Worrall (1989) argues that the doctrine can be found in the writings of Poincaré (1903/2001) andDuhem (1906/1911).

4. In fact, Maxwell suggests that the argument is Russell’s. Arguably, something like Maxwell’sargument does appear in embryonic form in chaps. 3 and 5 of The Problems of Philosophy (Russell1912), but in The Analysis of Matter (Russell 1927)—the work in which Russell most clearlypropounds structural realism—Russell’s argument for the position seems to be quite different (theaforementioned “upward path”).

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2. The true logical form of a theory is its Ramsey sentence, and we are


justified in believing that (the Ramsey sentences of ) mature scientifictheories are approximately true.

The first of these is suggested by Worrall’s seminal article on structural real-ism (1989), in which he argues that while theories are often abandoned inscientific revolutions, their Ramsey sentences are inevitably (approximately)preserved.5 As we shall see, the second seems to be closer to what Maxwellhad in mind.6

Notice that, when it is elaborated this way, structural realism has the follow-ing corollary:

The knowledge provided by our mature scientific theories consists (in its ulti-mate form) of statements constructed using only logical and observational terms.

This corollary is a claim the instrumentalist would also endorse. Structuralrealism remains distinct from instrumentalism because the instrumentalistclaims that our theories provide knowledge only about the observable world,whereas the structural realist maintains that the Ramsey sentences of our the-ories provide knowledge about both the observable world and the structure ofthe unobservable world. As we shall see, even if Maxwell’s argument for struc-tural realism went through, it would in fact only establish this corollary.

3. The Principle of Acquaintance

As noted, Maxwell’s argument for structural realism is founded on Russell’s(1911/1917, 1912) principle of acquaintance. The principle of acquaintanceis based on the view that there are two types of nonlogical terms: primitiveterms and nonprimitive terms. There are correspondingly two routes of refer-ence: acquaintance and description. What is acquaintance? For Russell, “Wehave acquaintance with anything of which we are directly aware, without theintermediary of any process of inference or any knowledge of truths” (1912,25). In particular, “We have acquaintance in sensation with the data of theouter senses, and in introspection with the data of what may be called the innersenses—thoughts, feelings, desires, etc; we have acquaintance in memory withthings which have been data either of the outer senses or of the inner sense.

5. However, in informal discussionsWorrall has revealed that he now endorses something more likethe second of these.

6. Although Maxwell would not have expressed the doctrine quite this way. In particular, he makesno mention of mature scientific theories or approximate truth.

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Further, it is probable, though not certain, that we have acquaintance with the


Self, as that which is aware of things or has desires towards things” (28).Maxwell,perhapsmore straightforwardly, takes it that “observation and acquaintance [are]coextensive” (1970a, 182), although, as we shall see, he has—from a modernperspective—a rather unusual conception of what is observable.

The idea behind Russell’s “theory of descriptions” is that a speaker can use aprimitive term to refer, only if he is acquainted with the object, property, orrelation of reference. Nonprimitive terms should be taken as abbreviationsfor definite descriptions formed, ultimately, using only primitive (and logical)terms.7 A speaker can use a nonprimitive term to refer, only if he can use all theterms involved in the underlying definite description to refer: the term refers tothe unique object, property, or relation that satisfies the description (if there isone; if there is not, the term does not refer). This can be summarized as follows:

1. If x can use a primitive term to refer, then x is acquainted with thereferent.

2. If x can use a nonprimitive term to refer, then x can use all the termsinvolved in the underlying definite description to refer: the term refersto whatever satisfies the description.

However, although Russell’s theory of descriptions is primarily a theory ofreference, the principle of acquaintance, which is based on this theory, relates tounderstanding or, in Maxwell’s approach, meaningfulness. Russell formulatesthe principle of acquaintance as follows: “Every proposition which we can un-derstand must be composed wholly of constituents with which we are ac-quainted” (1912, 32). (This is not meant to imply that we cannot understandpropositions about things with which we are not acquainted but only that if wedo so, the proposition must contain a description, where the direct referents ofthe terms in the descriptions—the “constituents” of the proposition—arethings with which we are acquainted.) Similarly, Maxwell formulates the prin-ciple as follows: “All the descriptive (nonlogical) terms in any meaningful sen-tence refer to items with which we are acquainted” (1970a, 181).

7. A definite description (of an individual) behaves syntactically like a name and refers to anindividual but contains a formula. For example, if just one object has both the property P and theproperty Q, then we can form a definite description referring to this object from the formula Px & Qx,i.e., ix(Px & Qx), which is read, “the (object) x such that x is P and x is Q” (see Fitting and Mendelsohn[1998, chap. 12], for a formal introduction to definite descriptions of individuals). In second-order logic,one can form definite descriptions of properties and relations. For example, if there is just one property thattwo objects, a and b, share, then we can form a definite description referring to this property from theformula Xa & Xb, i.e., iX(Xa & Xb), which is read “the (property) X such that a is X and b is X.”

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We can state this principle more fully as follows:


1. If x understands a primitive term, then x is acquainted with the ref-erent of the term.

2. If x understands a nonprimitive term, then x understands all the termsinvolved in the underlying definite description.

3. If x understands a sentence, then x understands the nonlogical termsin the sentence.

Note that parts 1, 2, and 3 of the principle of acquaintance provide necessarybut not sufficient conditions for x to understand primitive terms, nonprimitiveterms, and sentences, respectively. The principle (as stated) is thus a partial, not acomplete, theory of understanding. (Likewise, Maxwell’s and Russell’s formula-tions of the principle state a necessary but not sufficient condition for a propo-sition to be understood [Russell] or for a sentence to be meaningful [Maxwell].)

However, this is all that is needed for Maxwell’s argument for structural re-alism. So it is no objection to the principle of acquaintance (at least, not in thiscontext) that it does not provide both necessary and sufficient conditions forunderstanding. Plausibly, if we wanted to elaborate the principle into a full the-ory of understanding, what we would need to add to 1 is the claim that x alsoknows that the relevant thing with which he is acquainted is the referent of theprimitive term in question, and what we would need to add to 2 is the claimthat x knows that the nonprimitive term in question is an abbreviation of therelevant definite description. However, as noted, these additions are superfluousin this context, and, moreover, neither Russell nor Maxwell suggests them.

It may be wondered on what basis we are to believe the principle of acquain-tance. In fact, both Russell and Maxwell seem to take it to be almost self-evident. Maxwell offers no justification for the principle whatsoever, whereasRussell argues that “the chief reason for supposing the principle true is that itseems scarcely possible to believe that we can make a judgement or entertain asupposition without knowing what it is that we are judging or supposingabout” (1911/1917, 209). It is not obvious how this is supposed to supportthe principle of acquaintance. If we read it as, “The chief reason for supposingthe principle true is that it seems scarcely possible to believe that we can make ajudgement or entertain a supposition without being acquainted with the referentsof the primitive terms that we use to make the judgement or supposition,” then itdoes support the principle of acquaintance. But with this reading, it seemsmorethan “scarcely possible” that the claim Russell rejects is true: it seems we canentertain the supposition that “Caesar was bald”without being acquainted withCaesar, and “Caesar” appears to be a primitive term in this supposition. Russell

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would say that “Caesar” is a nonprimitive name, which explains how we can en-


tertain suppositions about Caesar even though we are not acquainted with him,but as Russell’s only argument for the principle rests on an appeal to intuitionsabout what seems possible, it is worrying that these intuitions do not seem tosupport the principle.8

Moreover, the claim that a person can understand a primitive term only if heis acquainted with the referent is highly problematic when we consider prim-itive terms referring to properties and relations (assuming that there are suchterms). What can it even mean to be acquainted with a property or relation(rather than just instances of the property or relation)? Maxwell says nothingabout this matter. Russell was prepared to bite the Platonic bullet and thoughtthat we really do become “acquainted” with properties and relations (and notjust their instances). He claimed that “not only are we aware of particularyellows, but if we have seen a sufficient number of yellows and have sufficientintelligence, we are aware of the universal yellow” (1911/1917, 203) and againthat “in addition to our acquaintance with particular existing things, wealso have acquaintance with what we shall call, universals… such as whiteness,diversity, brotherhood, and so on” (1912, 28), but few people would be preparedto accept that these days (and if they were prepared to accept it, they ought toexplain how and in what sense we can be acquainted with a universal).

In fact, we can get around this problem by modifying the principle of ac-quaintance, making the condition for the understanding of a predicate not ac-quaintance with the universal to which it refers but only acquaintance with aninstance of the universal to which it refers. That is, we can make do with thefollowing version of the principle of acquaintance:

1. If x understands a primitive name, then x is acquainted with theobject to which the name refers.

2. If x understands a primitive predicate, then x is acquainted with atleast one of the objects (or n-tuples of objects) to which it applies.



8. It has been suggested to me that although we can entertain the supposition that “Caesar was bald”
without being acquainted with Caesar, we cannot judge it to be true or false. If the historical evidence isequivocal, then there may indeed be some difficulties involved in judging whether Caesar was bald.However, the claim that we cannot judge a statement to be true or false without being acquainted withthe subject of the statement does not in general appear to be true. I am not acquainted with Russell, but Ican (with a high degree of confidence) judge that the statement “Russell was bald” is false.
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This version of the principle of acquaintance does not imply that we must be


acquainted with universals in order to understand universal terms, and, as weshall see shortly, it is still strong enough for Maxwell’s argument for structuralrealism to go through. The claim here is only that a necessary (but not suf-ficient) condition for one to understand a universal term is that one must beacquainted with at least one object (or n-tuple of objects) to which it applies.On the face of it, this has some plausibility: it does seem reasonable to sup-pose that a person could not understand the word “yellow” (for example),unless he was acquainted with at least one yellow object (it will be arguedthat this is in fact false, but at least it seems plausible at first sight).9

4. Maxwell’s Argument for Structural Realism

As noted, for Maxwell, “observation and acquaintance [are] coextensive”(1970a, 182). The principle of acquaintance thus reduces as follows:

1. If x understands a primitive name, then x is has observed the object towhich the name refers.

2. If x understands a primitive predicate, then x has observed at least oneof the objects (or n-tuples of objects) to which it applies.



We can now reconstruct Maxwell’s argument as follows: consider any statementthat we know to be (approximately) true. Presumably, we must understand the

9. However, note that it is not plausible to say that this version of the principle of acquaintance couldbe modified to provide necessary and sufficient conditions for x to understand all primitive terms. What
could be added to 2 to provide a sufficient condition for x to understand a primitive predicate? Intuitively,what we require to make the condition sufficient is not just that the speaker is acquainted with a yellowthing but also that he can pick out the yellowness of the thing as the property to which “yellow” issupposed to refer. However, (part of ) the point of Wittgenstein’s private language argument (1953/1967, sec. 243) is that we beg the question if we suppose that the speaker can pick out yellowness inthis way. Suppose a speaker sees a yellow object and says to himself, “I shall call things like that ‘yellow.’”If this is all he says to himself—if he does not further specify the way in which things must be like theobject in question in order for them to be called “yellow”—then he will call everything “yellow” becauseeverything is similar to the object in question in some respect. However, if he can tell himself what therelevant respect is (with a statement of the form “I shall call things like that in respect X ‘yellow’”), then(since the relevant respect is yellowness) he can already refer to yellow (with the term X), in which casethis explanation of how someone can refer to yellow with the word “yellow” presupposes that he canalready refer to yellow with some other word, and it is in this sense question begging. However, althoughthese problems are serious for someone who wants to elaborate the principle of acquaintance into a fulltheory of understanding, they do not affect Maxwell’s argument for structural realism.
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statement. So, by 4, we must understand all the nonlogical terms in the state-


ment. There are only two types of nonlogical terms: primitive terms and non-primitive terms. By 1 we can understand a primitive name only if it refers toan object that we have observed, and by 2 we can understand a primitive pred-icate only if it applies to an object (or n-tuple of objects) that we have observed.If we assume that a name is an observational term if it refers to an observableobject and that a predicate is an observational term if it applies to at least oneobservable object (or n-tuple of observable objects), then it follows that anyprimitive term that we understand is an observational term. Hence, any prim-itive term in the statement must be observational. But by 3 we can understand anonprimitive term only if it is an abbreviation of a description formed usingultimately only logical and primitive terms (that we understand). Hence, inits unabbreviated form, the statement contains only logical and primitiveterms. Hence, in its unabbreviated form, the statement contains only logicaland observational terms. Hence, any statement we know to be (approximately)true is constructed using only logical and observational terms. This impliesthat:

The knowledge provided by our mature scientific theories consists (in its ulti-mate form) of statements constructed using only logical and observationalterms.

As noted in section 2, this is actually only a corollary of the Ramsey-sentenceformulation of structural realism, so the argument does not really establish(this form of ) structural realism. However, the argument does suggest (if itis sound) that something like (this form of) structural realism is probably thestrongest form of realism that could reasonably be upheld. (It is hard to seehow we could know anything more than the Ramsey sentences of our besttheories, if our knowledge is restricted to statements constructed using onlylogical and observational terms.)

It will be noted that the above argument relies on the assumption “that aname is an observational term if it refers to an observable object and that a pred-icate is an observational term if it applies to at least one observable object (orn-tuple of objects).” But while this assumption might at first sight seem plau-sible, it in fact classifies far too many predicates as observational. For example,consider the predicate “being a collection of superstrings.” If superstringtheory is broadly right, then this predicate applies to some—in fact all—observable objects and, on the above account, would thereby qualify as anobservational predicate. But this, of course, is just the sort of predicate thatthe structural realist wants to treat as theoretical and Ramsey-fy. So it seems

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this assumption must be rejected, and without this assumption the argument10


does not go through.However, Maxwell has (from a modern viewpoint) a rather unusual under-

standing of what is observable. He holds that “all items should be consideredtheoretical unless they occur in direct experience; since I reject any form of di-rect realism, this means that the observable is instantiated only in inner eventsof observers” (1970a, 181).Maxwell also argues that the properties of objects inthe internal (mental) world are not exemplified by objects in the external (phys-ical) world: “colors are exemplified only in our private experiences and that thereis no reason to believe that they are ever properties of the material objects of theexternal environment. What holds for colors must be true for all of the first orderproperties that we perceive directly” (1970b, 19). This notion of observability (ifit is accepted) seems to undercut the above objection and suggests the view thatnot only is a predicate an observational term if it applies to at least one observ-able (i.e., internal) object (or n-tuple of observable objects) but moreover thatobservational predicates apply exclusively to observable (i.e., internal) objects(or n-tuples of observable objects).11

Let us suppose then that we adopt this understanding of “observable” andthat the foregoing characterization of an “observational term” is acceptable inthis case. ThenMaxwell’s argument for structural realism does go through. Butunfortunately (from Maxwell’s point of view) when we work with this charac-terization of “observational term,” the principle of acquaintance leads to absur-dities. In the appendix it is shown that:

Given two sets of objects, A and B, and a language of first-order logic whosenonlogical terms refer exclusively to objects in A (or sets of objects in A, or sets

10. Of course, the structural realist could just stipulate that he is going to treat as “observational” all
those predicates that apply to at least one observable object. But—if we take this as a purely stipulativedefinition of “observational”—then we are then bound to ask why we should treat these “observational”predicates so differently from “nonobservational” predicates: i.e., why do we need to Ramsey-fy“nonobservational” predicates but not “observational” predicates? For example, why do we need toRamsey-fy the predicate “being a superstring” but not the predicate “being a collection ofsuperstrings.” The original motivation for Ramsey-fying theoretical predicates but not observationalpredicates is, I take it, that we are on firmer epistemic ground when it comes to ascribingobservational properties to objects than when ascribing theoretical properties to objects. But thismotivation will clearly not work if we count predicates like “being a collection of superstrings” asobservational.
11. However, theoretical relations will not be those that apply exclusively to unobservable (i.e.,external) objects (or n-tuples of unobservable objects). Maxwell assumes that there are causalrelations instantiated by pairs of the form ⟨external object, internal object⟩. These would have to beclassified as theoretical because a binary relation is clearly not observable if we can only ever observeone of its relata.

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of tuples of objects in A), it is impossible to create a definite description of an12


object in B, if B contains more than one object.

A straightforward corollary of this is that:

It is impossible to create a definite description of an external object using alanguage of first-order logic whose nonlogical terms refer exclusively to internalobjects (or properties exclusively thereof or relations exclusively among internalobjects), if there is more than one external object.

What this shows is that, if we work with Maxwell’s characterization of “ob-servational term” and we think that reference to “unobservable” (i.e., exter-nal) objects is fixed in the way Maxwell thinks, then such reference is neveruniquely fixed. What is the significance of this? After all, the Ramsey sentenceof a theory does not purport to refer to unobservable (external) objectsuniquely. So it might seem that Maxwell could happily accept the conclusionthat it is impossible to refer to a unique external object. But what could thenbe said about terms that purport to refer to unique external objects (e.g., “BillyBob Thornton,” “the radium atom that decayed at (x, y, z, t)”)? There seem tobe two options: either (i) conclude that such terms fail to refer at all, in whichcase talk that purports to be about unique external objects is in the same bracketas talk about Santa Claus, or (ii) conclude that while such terms do refer, there isno fact as to whether they refer to what we (pretheoretically) think they refer toor to some other (arbitrary) external object. So, for example, there is no fact as towhether “the radium atom that decayed at (x, y, z, t)” refers to what we (pre-theoretically) think it refers to or whether it refers to what we (pretheoretically)think “Billy Bob Thornton” refers to. But both of these options are absurd.So the theorem appears to provide a reductio of the combination of Maxwell’sconception of the observable/unobservable distinction and the principle of ac-quaintance. And, as noted, Maxwell’s argument for structural realism cruciallydepends on this combination.

5. Summary of Argument

Maxwell’s argument for structural realism relies not only on the principle ofacquaintance but also on the assumption “that a name is an observationalterm if it refers to an observable object and that a predicate is an observational

12. It is trivial to extend the theorem to second-order logic.

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term if it applies to at least one observable object (or n-tuple of objects).” This


assumption is false if we adopt an intuitive characterization of the obser-vational term/theoretical term distinction. Consequently, if we adopt thischaracterization of the observational term/theoretical term distinction,Maxwell’s argument does not go through. However, if, like Maxwell,we hold the view that only the inner events of observers are genuinely ob-servable and we adopt a characterization of “observational term” that isappropriate to this view, then the assumption seems plausible, and the ar-gument does go through. But, when we work with this characterization,the principle of acquaintance leads to absurdities and should thus be rejectedas false. Consequently, Maxwell’s argument should be rejected, even bysomeone who shares Maxwell’s view that only the inner events of observersare genuinely observable.

Appendix

Theorem: Given two sets of objects, A and B, and a language of first-order logicwhose nonlogical terms refer exclusively to objects in A (or sets of objects in A orsets of n-tuples of objects in A), it is impossible to create a definite description ofan object in B, if B contains more than one object.

To prove this theorem, let’s suppose that we have two sets of objects, A and B,and a language of first-order logic whose nonlogical terms refer exclusively toobjects in A (or sets of objects in A or sets of n-tuples of objects in A). That is, letus suppose that (1) L is a first-order language, (2) M is a model of L withdomain D, (3) A is the set of elements in D that are in the field of some relationinM, (4) B is the set of elements inD that are not in the field of some relation inM, and (5) B contains at least two elements.

Consider an arbitrary element of B: let’s call it b. Let b′ be an element of Bsuch that b 6¼ b′ (by the assumption that B contains at least two elements,there must be such an element). Consider the bijection f (on domain D), whichis such that

f ðbÞ ¼ b′

f ðb′Þ ¼ b

f ðdÞ ¼ d otherwise

Consider an arbitrary one-place predicate of the language, P. Let PM be theextension of the predicate in M.

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LEMMA 1. For every element in D, d ∈ PM, if and only if fðdÞ ∈ PM.


To prove lemma 1, we must consider three cases:

1. d ¼ b. We know (from 4) that d ∉ PM and fðdÞ ∉ PM (as d ¼ b andfðdÞ ¼ b′ and both b and b′ are elements of B), so in this case d ∈ PM,if and only if fðdÞ ∈ PM.

2. d ¼ b′. We know (from 4) that d ∉ PM and fðdÞ ∉ PM (as d ¼ b′ andfðdÞ ¼ b and both b′ and b are elements of B), so in this case d ∈ PM,if and only if fðdÞ ∈ PM.

3. d 6¼ b and d 6¼ b′. In this case, fðdÞ ¼ d, so, trivially, d ∈ PM, if andonly if fðdÞ ∈ PM.

Now consider an arbitrary n-place predicate of the language, R. Let RM bethe extension of the predicate in M.

LEMMA 2. For every n-tuple of elements in D, hd1 . . . dni ∈ RM, if and only ifhfðd1Þ . . . fðdnÞi ∈ RM.

To prove lemma 2, we must consider three cases:

1. dj ¼ b, for some dj in the tuple hd1 . . . dni. We know (from 4) thathd1 . . . dni ∉ RM and that hfðd1Þ . . . fðdnÞi ∉ RM (as dj ¼ b andfðdjÞ ¼ b′ and both b and b′ are elements of B), so in this casehd1 . . . dni ∈ RM, if and only if hfðd1Þ . . . fðdnÞi ∈ RM.

2. dj ¼ b′, for some dj in the tuple hd1 . . . dni. We know (from 4) thathd1 . . . dni ∉ RM and that hfðd1Þ . . . fðdnÞi ∉ RM (as dj ¼ b′ andfðdjÞ ¼ b and both b′ and b are elements of B), so in this casehd1 . . . dni ∈ RM, if and only if hfðd1Þ . . . fðdnÞi ∈ RM.

3. dj 6¼ b and dj 6¼ b′, for every dj in the tuple hd1 . . . dni. In this case,fðdjÞ ¼ dj, for every dj in the tuple hd1 . . . dni, so, trivially,hd1 . . . dni ∈ RM, if and only if hfðd1Þ . . . fðdnÞi ∈ RM.

By lemmas 1 and 2, f is an automorphism of M.A definite description of an object, ix(φ), contains a formula, φ, with one

free variable, x, that is satisfied if and only if the object is assigned to x. Supposethat there is a definite description of b. Then there is a formula φ, with one freevariable, x, that is satisfied if and only if b is assigned to x. But, as f is an auto-morphism of M, if φ is satisfied when b is assigned to x, then φ is also satisfiedwhen b′ (=f(b)) is assigned to x. So there is not a definite description of b. As b is

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an arbitrary element of B, this means that there is not a definite description of any13


element of B.

REFERENCES

Ainsworth, Peter. 2008. “Structural Realism: A Critical Appraisal.” PhD diss., LondonSchool of Economics.

———. 2009. “Newman’sObjection.”British Journal for the Philosophy of Science 60:135–71.Duhem, Pierre. 1906/1991. The Aim and Structure of Physical Theory. Trans. P. P. Wiener.

Repr. Princeton, NJ: Princeton University Press.Fitting, Melvin, and Richard L. Mendelsohn. 1998. First-Order Modal Logic. Dordrecht:

Kluwer.Ladyman, James. 1998. “What Is Structural Realism?” Studies in History and Philosophy of

Science 29:409–24.Maxwell, Grover. 1968. “Scientific Methodology and the Causal Theory of Perception.”

In Problems in the Philosophy of Science: Proceedings of the International Colloquium inthe Philosophy of Science, London, 1965, vol. 3, ed. Imre Lakatos and Alan Musgrave,148–60. Amsterdam: North-Holland.

———. 1970a. “Structural Realism and the Meaning of Theoretical Terms.” In Analysesof Theories and Methods of Physics and Psychology, ed. Stephen Winokur and MichaelRadner, 181–92. Minneapolis: University of Minnesota Press.

———. 1970b. “Theories, Perception, and Structural Realism.” In Nature and Functionof Scientific Theories, ed. Robert G. Colodny, 3–34. Pittsburgh: University of PittsburghPress.

Poincaré, Henri. 1903/2001. Science and Hypothesis. Repr. in The Value of Science: EssentialWritings of Henri Poincaré, ed. Stephen J. Gould. New York: Random House.

Psillos, Stathis. 2001. “Is Structural Realism Possible?” Philosophy of Science 68 (Proceed-ings): S13–S24.

Russell, Bertrand. 1911/1917. “Knowledge by Acquaintance and Knowledge by Descrip-tion.” Repr. in Mysticism and Logic. London: Routledge.

———. 1912. The Problems of Philosophy. Oxford: Oxford University Press.———. 1927. The Analysis of Matter. London: Kegan Paul.Votsis, Ioannis. 2005. “The Upward Path to Structural Realism.” Philosophy of Science

72:1361–72.Wittgenstein, Ludwig. 1953/1967. Philosophical Investigations. 3rd ed. Trans. G. Elizabeth

M. Anscombe. Oxford: Blackwell.Worrall, John. 1989. “Structural Realism: The Best of BothWorlds?”Dialectica 43:99–124.

13. The theorem can also be proved bymathematical induction (see Ainsworth 2008), but the proofis much longer. Thanks to Jeff Ketland for suggesting this version of the proof to me.

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