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Philosophia Mathematica
, Volume 26 (1) – Feb 1, 2018

28 pages

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- Oxford University Press
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- Philosophia Mathematica (III) Vol. 00 No. 00 © The Authors [2016]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
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- 0031-8019
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- 1744-6406
- D.O.I.
- 10.1093/philmat/nkw030
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ABSTRACT Recent work in the philosophy of mathematics has suggested that mathematical structuralism is not committed to a strong form of the Identity of Indiscernibles (II). José Bermúdez demurs, and argues that a strong form of II can be warranted on structuralist grounds by countenancing identity properties, or haecceities, as legitimately structural. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as structural but that this concession does not warrant Bermúdez’s version of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects. 1. Introduction It is well-known that some earlier versions of mathematical structuralism [Resnik, 1997; Shapiro, 1997] appeared to be committed to a strong form of the principle of the Identity of Indiscernibles (II) on which distinct mathematical objects must differ in at least some of their structural properties. It is also well-known that the principle in question is falsified by the existence of structures like the complex field $$\mathfrak{C}$$ that admit of non-trivial automorphisms, or symmetries, as these structures contain distinct positions that nonetheless appear to share all of their structural properties. Subsequently, structuralists have provided strong reasons to think that, so long as primitive facts about identity (and, hence, difference) are taken to be legitimately structural, nothing beyond a trivial form of II is demanded of the structuralist. José Bermúdez [2007], among others, disagrees, arguing that, even with primitive identity, the challenge to structuralism posed by non-trivial symmetries remains: absent any property to distinguish distinct positions in a structure, there is no structural explanation of their difference. In response, Bermúdez in effect proposes a notion of discernibility in a structure and a corresponding version of II that can provide the needed explanations. The key to his proposal — as I will interpret it — lies in allowing identity properties, or haecceities, like being identical to $$c$$ (for an arbitrary complex number $$c$$, say) to count as structural properties. Typically, structuralists dismiss such properties as obviously non-structural and, hence, that any proposal along the lines of Bermúdez’s can be dismissed out of hand on structuralist grounds. But I think the issue is not quite that cut and dried. I will first argue, contrary to the typical structuralist view, that, on a strongly model-theoretic rendering of structuralism at least, haecceities can be viewed as properly structural. This conclusion might appear to warrant Bermúdez’s version of II as a legitimately structural answer to the challenge of non-trivial symmetries. However, second, I will argue that the sound structural intuition underlying Bermúdez’s proposal does not, in fact, lead to his notion of discernibility but, rather, a weaker notion of discernibility and a corresponding version of II that is still clearly falsified by structures like $$\mathfrak{C}$$ with non-trivial symmetries. I close with some reflections on reference and quantification vis-à-vis non-trivial symmetries. 2. Preliminaries First, a bit of stage setting. Although mathematical structures are perhaps not identifiable with model-theoretic structures in general — mathematical structures, for one thing, are identical if isomorphic — judicious use of model theory can bring a great deal of clarity to a variety of philosophical issues of mathematical structuralism, notably those I will be addressing here. In this preliminary section I will define some important (generally well-known) model-theoretic notions and connect these to corresponding structuralist notions. 2.1. Languages and $$\boldsymbol{L}$$-systems For purposes here, by a formal language $$L$$ I mean simply a vocabulary, i.e., a set of primitive (nonlogical) predicates and individual constants. It will be understood that there is a general standard grammar specifying how the formulas of any given language $$L$$ are constructed recursively from its vocabulary and that, in particular, the formulas of $$L$$ include identities $$\tau=\tau'$$ for all terms $$\tau$$, $$\tau'$$ of $$L$$. (I will use some familiar operators like ‘$$+$$’ as $$n$$-place function symbols in some formulas below but these formulas should be understood strictly as shorthand for formulas in which those operators occur as $$n+1$$-place predicates.1) I will also assume that languages are first-order unless otherwise specified, although nothing in my argument hinges on this. By an $$L$$-system$$\mathcal{A}$$ I mean a pair $$\langle A,V\rangle$$, where $$A$$ is a nonempty set (the domain of $$\mathcal{A}$$) and $$V$$ an interpretation of $$L$$ mapping each $$n$$-place predicate $$\pi$$ to an $$n$$-place relation $$\pi^{\mathcal{A}}$$ over $$A$$ (i.e., a subset of $$A^{n}$$) and each individual constant $$\kappa$$ to an element $$\kappa^{\mathcal{A}}$$ of $$A$$. The relations $$\pi^{\mathcal{A}}$$ and individuals $$\kappa^{\mathcal{A}}$$ are known as the distinguished relations and individuals of $$\mathcal{A}$$. I will take the satisfaction of a formula $$\varphi$$ in an $$L$$-system $$\mathcal{A}=\langle A,V\rangle$$ relative to a variable assignment $$s$$ ($$\mathcal{A}\models\varphi[s]$$) to be defined in the usual way. For variables $$\nu,\mu$$ of $$L$$, $$a\in A$$, and a variable assignment $$s$$, $$s_{a}^{\nu}(\mu)$$ is $$a$$ if $$\mu=\nu$$ and $$s(\mu)$$ otherwise. Where $$\varphi$$ contains free occurrences of exactly the (pairwise distinct) variables $$\nu_{1}$$, $$\dots$$, $$\nu_{n}$$ (in order of first occurrence), let $$\mathcal{A}\models\varphi[a_{1},\dots,a_{n}]$$ mean that $$\mathcal{A}\models\varphi[s]$$ for any variable assignment $$s$$ such that $$s(\nu_{m})=a_{m}$$, for $$1\leq m\leq n$$. Then we say that $$R_{\varphi}^{\mathcal{A}}=\{\langle a_{1},\dots,a_{n}\rangle:$$$$\mathcal{A}\models\varphi[a_{1},\dots,a_{n}]\}$$ is the ($$n$$-place) systemic relation over $$\mathcal{A}$$ determined by $$\varphi$$; when $$n=1$$, we say that $$R_{\varphi}^{\mathcal{A}}$$ is the systemic property over $$\mathcal{A}$$ determined by $$\varphi$$ and, if $$a\in R_{\varphi}^{\mathcal{A}}$$, we say that $$R_{\varphi}^{\mathcal{A}}$$ is a systemic property of$$a$$ in $$\mathcal{A}$$. $$R$$ is a systemic relation over$$\mathcal{A}$$ if it is the systemic relation determined by some formula of $$L$$ in $$\mathcal{A}$$. For $$n$$-tuples $$\overset{\rightharpoonup}{a}=\langle a_{1},\dots,a_{n}\rangle\in A^{n}$$, let $$\mathit{tp}_{\mathcal{A}}(\overset{\rightharpoonup}{a})=\{\varphi:R_{\varphi}^{\mathcal{A}}(\overset{\rightharpoonup}{a})\}$$ be the type of $$\overset{\rightharpoonup}{a}$$ (relative to $$\mathcal{A}$$). As $$\mathit{tp}_{\mathcal{A}}(a)$$ comprises everything true of $$a$$ that is expressible in terms of the resources available in $$L$$, it is useful to think of it as the ‘representational role’ that $$a$$ plays in $$\mathcal{A}$$. Suppose $$L\subseteq L_{E}$$, where $$E$$ (= $$L_{E}\setminus L$$) is a (not necessarily countable) set of lexical items (any mixture of constants and predicates) not in $$L$$. An $$L_{E}$$-system$$\mathcal{A}_{E}=\langle A,V_{E}\rangle$$ is an $$L_{E}$$-expansion of an $$L$$-system $$\mathcal{A}=\langle A,V\rangle$$, and $$\mathcal{A}$$ an $$L$$-reduct of $$\mathcal{A}_{E}$$, if $$V\subseteq V_{E}$$, that is, if $$V_{E}$$ agrees with $$V$$ on the semantic values assigned to the lexical items in $$L$$. (Note: When $$E=\{\varepsilon\}$$ is a singleton, we will write ‘$$L_{\varepsilon}$$’ rather than ‘$$L_{\{\varepsilon\}}$$’; similarly when $$E$$ is a pair.) $$\mathcal{B}$$ is an expansion (simpliciter) of $$\mathcal{A}$$ if $$\mathcal{B}$$ is an $$L_{E}$$-expansion of $$\mathcal{A}$$, for some $$E$$. The notion of an isomorphism from one $$L$$-system $$\mathcal{A}=\langle A,V\rangle$$ to another (not necessarily distinct) $$L$$-system $$\mathcal{B}=\langle B,W\rangle$$ is understood as usual as a bijection $$f:A\longrightarrow B$$ that ‘preserves structure’. Specifically, where, for $$n$$-tuples $$\overset{\rightharpoonup}{a}=\langle a_{1},\dots,a_{n}\rangle$$, $$f[\overset{\rightharpoonup}{a}]=\langle f(a_{1}),\dots,f(a_{n})\rangle$$, $$f:A\longrightarrow B$$ is an isomorphism from$$\mathcal{A}$$to$$\mathcal{B}$$, written $$\mathcal{A}\stackrel{f}{\cong}\mathcal{B}$$, iff, for every $$n$$-place predicate $$\pi$$, $$\overset{\rightharpoonup}{a}\in\pi^{\mathcal{A}}$$ iff $$f[\overset{\rightharpoonup}{a}]\in\pi^{\mathcal{B}}$$ and, for constants $$\kappa$$, $$f(\kappa^{\mathcal{A}})=\kappa^{\mathcal{B}}$$. An automorphism, or symmetry, on $$\mathcal{A}$$ is an isomorphism from $$\mathcal{A}$$ to $$\mathcal{A}$$. Equivalently, a symmetry is a permutation $$f$$ of the domain $$A$$ of $$\mathcal{A}$$ under which the distinguished relations and objects of $$\mathcal{A}$$ are invariant, that is, (i) for distinguished $$n$$-place relations $$R$$ of $$\mathcal{A}$$, $$R=\left\{ f[\overset{\rightharpoonup}{a}]:\overset{\rightharpoonup}{a}\in R\right\} $$, and (ii) for distinguished individuals $$a$$ of $$\mathcal{A}$$, $$a=f(a)$$. A symmetry on $$\mathcal{A}$$ is nontrivial if it is not the identity function on $$A$$, and $$\mathcal{A}$$ is rigid if there are no nontrivial symmetries on it. Objects $$a,b\in A$$ are said to be symmetric in$$\mathcal{A}$$ if, for some symmetry $$f$$ on $$\mathcal{A}$$, $$f(a)=b$$, and fully symmetric if, in addition, $$f(b)=a$$. 2.2. Systems, Structures, and $$\boldsymbol{L}$$-Structures $$L$$-systems can serve as powerful mathematical representations of what Shapiro calls systems simpliciter. A system $$\boldsymbol{A}$$ is a structured collection $$A$$ of objects (of any sort), that is, a collection of objects that have certain distinguished, or (as I shall call them) featured, properties and bear certain featured relations to one another; for example, an extended family with its blood and marital relationships or the pieces in a standard chess set with their possible configurations and possible moves [Shapiro, 1997, p. 73].2 An $$L$$-system $$\mathcal{A}=\langle A,V\rangle$$ then can be said to represent a system $$\boldsymbol{A}$$ just in case $$A$$ consists of the elements of $$\boldsymbol{A}$$ and, for every featured $$n$$-place relation $$\boldsymbol{R}$$ of $$\boldsymbol{A}$$ there is a unique $$n$$-place predicate $$\pi_{\boldsymbol{R}}$$ of $$L$$ such that, for all $$a_{1},\dots,a_{n}\in A$$, $$\boldsymbol{R}a_{1}\dots a_{n}$$ if and only if $$\langle a_{1},\dots,a_{n}\rangle\in\pi_{\boldsymbol{R}}^{\mathcal{A}}$$. A structure, for Shapiro, is ‘the abstract form of a system highlighting the interrelationships among the objects [of the system], and ignoring any features of them that do not affect how they relate to other objects in the system’ (ibid, p. 74). Many distinct systems, therefore, can exhibit the same structure. Each structure $$\mathfrak{A}$$ consists of a collection of ‘places’, or ‘positions’, or ‘offices’, each of which is occupied by exactly one object in any system that exhibits the structure. For an ante rem structuralist like Shapiro, positions are abstract objects in their own right. Structures are therefore themselves systems of a special sort — they are the systems that, quastructures, exhibit themselves. For any structure $$\mathfrak{A}$$ then, there is a corresponding $$L$$-system $$\mathcal{A_{\mathfrak{A}}}=\langle A,V\rangle$$ that represents it qua system; such an $$L$$-system I will refer to as an $$L$$-structure for$$\mathfrak{A}$$ and I will accordingly refer to the systemic relations over an $$L$$-structure $$\mathcal{A_{\mathfrak{A}}}$$ as the structural relations over $$\mathcal{A_{\mathfrak{A}}}$$. I will assume that, if $$\mathcal{A}$$ is an $$L$$-structure for structure $$\mathfrak{A}$$, then any structural relation that can be defined within $$\mathcal{A}$$ is reflected in a featured relation in $$\mathfrak{A}$$ and vice versa. An $$L$$-system $$\mathcal{B}$$ that is isomorphic to an $$L$$-structure $$\mathcal{A}$$ for $$\mathfrak{A}$$ is said to exhibit$$\mathfrak{A}$$. Two brief notes. First, I realize that there is both unclarity and controversy surrounding the question of what a structural relation is. I will address this to some degree below but, for now, I note that my notion here should simply be viewed as a terminological stipulation, and is not meant to indicate a fixed judgment about the nature of structural relations per se. Second, I do not believe that anything I say here involves any deep commitment to the truth of ante rem structuralism. Rather, I think that everything critical to my argument is preserved (though might need to be rephrased at points) if talk of structures in general is thought of simply as a convenient way of talking about isomorphic systems and that talk of this or that specific structure can be thought of as talk about an arbitrary instance of a class of isomorphic systems. 3. Burgess’s Challenge: Non-Trivial Symmetries John Burgess [1999] raised an important and, by now, well-known challenge to the strong version of mathematical structuralism found in [Shapiro, 1997], illustrated in passages like the following (p. 100): Every office [of a structure] is characterized completely in terms of how its occupant [in a system exhibiting the structure] relates to the occupants of the other offices of the structure $$\ldots$$ . As Shapiro [2008] himself subsequently notes, since according to ante rem structuralism a structure is itself a system whose elements are the offices of the structure (and hence, qua system, ‘occupy’ themselves), this passage suggests that the identity of every mathematical object, every office of every structure, is entirely determined by the properties it has — jointly, the object’s ‘complete characterization’ — simply in virtue of its being a part of that structure. This, in turn, appears to commit the structuralist to a strong form of the Identity of Indiscernibles: that offices within a structure can differ only if there is some such property that they do not share. Assuming the close connection between structures and their corresponding $$L$$-structures noted in the preceding section, following Ladyman et al. [2012] we can spell out the idea model-theoretically. Consider the following notion: AD Objects $$a,b\in A$$ are absolutely discernible in an $$L$$-structure $$\mathcal{A}=\langle A,V\rangle$$ if and only if there is a structural property of $$\mathcal{A}$$ they do not share, i.e., if and only, for some $$L$$-formula $$\varphi$$, $$R_{\varphi}^{\mathcal{A}}(a)$$ but not $$R_{\varphi}^{\mathcal{A}}(b)$$. $$a$$ and $$b$$ are absolutely indiscerniblein$$\mathcal{A}$$ if they are not absolutely discernible in $$\mathcal{A}$$.3 Then the version of the Identity of Indiscernibles that seems implicit in Shapiro’s quote amounts to the following:4 AII Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-structure. Then for all elements $$a,b\in A$$, $$a=b$$ iff $$a$$ and $$b$$ are absolutely indiscernible in $$\mathcal{A}$$. Burgess’s challenge to this version of structuralism is simply that there seem to be obvious counterexamples to AII, viz., structures like the complex field that admit of non-trivial symmetries: We have two roots to the equation $$z^{2}+1=0$$, which are additive inverses of each other, so that if we call them $$i$$ and $$j$$ we have $$j$$ = $$-i$$ and $$i$$ = $$-j$$. But the two are not distinguished from each other by any algebraic properties, since there is a symmetry or automorphism of the field of complex numbers, which is to say an isomorphism with itself, which switches $$i$$ and $$j$$. On Shapiro’s view the two are distinct, although there seems to be nothing to distinguish them. [Burgess, 1999, p. 288] Expressed in terms of the definitions in the previous section, letting $$L_{F}$$ be the usual language $$\{+,\times,0,1\}$$ of field theory, the complex field $$\mathfrak{C}$$ can be represented by the $$L_{F}$$-structure $$\mathcal{C}=\langle C,V\rangle$$, where $$C$$ is the set of complex numbers, $$+^{V}$$ and $$\times^{V}$$ are the obvious operations and $$0^{V}$$ and $$1^{V}$$ are the identities for those operations. As Burgess notes, the function $$g:C\longrightarrow C$$ that maps each complex number $$a+bi$$ to its complex conjugate $$a+bj$$ is a symmetry on $$C$$. Since an automorphism on a structure preserves all the structural properties of, and relations among, the elements of the structure, there appears to be nothing structural to distinguish the roots $$i$$ and $$j$$ of $$z^{2}+1=0$$, no structural property $$R_{\varphi}^{\mathcal{C}}$$ that the one has that the other lacks. Put another way, the types $$\mathit{tp}_{\mathcal{C}}(i)$$ and $$\mathit{tp}_{\mathcal{C}}(j)$$ of $$i$$ and $$j$$ are identical. Since $$i$$ and $$j$$ are provably distinct — that is, more exactly, since it is provable that there are exactly two distinct complex numbers with the property $$R_{(x\times x)+1=0}^{\mathcal{C}}$$ — this observation appears to be a counterexample to AII and, hence, insofar as it is taken to entail AII, to structuralism proper. 4. A Haecceitist Proposal I now turn to the disarmingly simple answer to Burgess’s challenge proposed by Bermúdez [2007]. 4.1. Robinson Expansions As I understand Bermúdez, the existence of a non-trivial symmetry for a given structure simply reveals that the language of the structure is expressively weak. In the case at issue in Burgess’s challenge, there are facts about specific complex numbers that will enable us to distinguish $$i$$ from $$-i$$, but those facts ‘cannot be “stated” in terms of the relations, elements, and functions available within the structure’ (ibid, p. 112). Bermúdez’s idea, then, very roughly put, is simply to add sufficient richness to the vocabulary to be able to express these implicit facts. Let $$L^{+}$$ be a language. Say that an $$L^{+}$$-system $$\mathcal{A}^{+}=\langle A,V^{+}\rangle$$ is a name expansion of an $$L$$-system $$\mathcal{A}=\langle A,V\rangle$$ if it is an expansion of $$\mathcal{A}$$ and $$L^{+}$$ is simply the result of adding some new individual constants to $$L$$, and that $$\mathcal{A}^{+}$$ is a Robinson expansion of$$\mathcal{A}$$ if, in addition, every $$a\in A$$ is named by some new constant $$\kappa_{a}$$.5 Name expansions, then, give us the expressive power to ‘transform’ structures with distinct absolute indiscernibles into structures in which there are none. Specifically, in the case at hand, let $$L_{F}^{+}$$ be the result of adding $$2^{\aleph_{0}}$$ new constants to $$L_{F}$$ and let the $$L_{F}^{+}$$-structure $$\mathcal{C}^{+}=\langle C,V^{+}\rangle$$ be a Robinson expansion of $$\mathcal{C}$$. Then both $$i$$’s haecceity$$R_{\nu=\kappa_{i}}^{\mathcal{C}^{+}}$$ — the property (over $$\mathcal{C}^{+}$$) that is true of $$i$$ alone6 — as well as $$j$$’s anti-haecceity$$R_{\nu\neq\kappa_{j}}^{\mathcal{C}^{+}}$$ — the property true of everything but$$j$$ — will be true of $$i$$ but not $$j$$. Likewise, $$j$$’s haecceity $$R_{\nu=\kappa_{j}}^{\mathcal{C}^{+}}$$ and $$i$$’s anti-haecceity $$R_{\nu\neq\kappa_{i}}^{\mathcal{C}^{+}}$$ will be true of $$j$$ but not $$i$$. Hence, $$i$$ and $$j$$ fail to share all of their structural properties in $$\mathcal{C}^{+}$$ and so, by AD, they are absolutely discernible in $$\mathcal{C}^{+}$$; likewise, of course, for any distinct pair $$a,b\in C$$. AII, therefore, is satisfied in any Robinson expansion of $$\mathcal{C}$$. Bermúdez sums up as follows: The general point is that expanding a structure can often allow us to say things that we could not say in the unexpanded structure. And we do this without changing the fundamental nature of the mathematical structure in question. When we expand a structure all that we do is name some elements that were not previously named $$\ldots$$ . In a very important sense the structure itself remains unchanged. [2007, p. 115] Now, for reasons I will lay out in detail in § 5.2 below, I disagree strongly with Bermúdez’s claim that name expansions add no new structure. Moreover, I think his chief insight is found in the first sentence in the above quote, that expansions enable us to express information that is already implicit in a structure (in a sense to be defined more precisely below). Thus, it seems to me that Bermúdez underplays his hand here a bit and mutes his proposal by advising the structuralist to ‘concede the point’ that structures with symmetries have absolute indiscernibles and note only that the point has been ‘blunted’ by the fact that it is ‘language-relative’ (ibid.). Rather, a stronger expression of the insight is that a Robinson expansion $$\mathcal{A}'$$ of a structure $$\mathcal{A}$$ uncovers implicit structure in $$\mathcal{A}$$ that grounds a notion of discernibility for some object pairs that are nonetheless absolutely indiscernible in $$\mathcal{A}$$, specifically: RD Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-structure. Objects $$a,b\in A$$ are Robinson discernible over$$\mathcal{A}$$ if and only if $$a$$ and $$b$$ are absolutely discernible in some name expansion $$\mathcal{A}'$$ of $$\mathcal{A}$$. $$a$$ and $$b$$ are Robinson indiscernible over$$\mathcal{A}$$ if they are not Robinson discernible over $$\mathcal{A}$$. This in turn yields a corresponding version of the Identity of Indiscernibles: RII Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-structure. Then for all elements $$a,b\in A$$, $$a=b$$ iff $$a$$ and $$b$$ are Robinson indiscernible over $$\mathcal{A}$$. Clearly, so long as $$L$$ includes identity (as we have stipulated), distinct objects $$a$$ and $$b$$ in an $$L$$-structure $$\mathcal{A}$$, even if (like $$i$$ and $$j$$ in $$\mathcal{C}$$) absolutely indiscernible in $$\mathcal{A}$$, are Robinson discernible over $$\mathcal{A}$$.7 Hence, as the converse is obvious, Robinson indiscernibility, viewed extensionally, simply is the identity relation. Thus, RII is valid and the Identity of Indiscernibles appears to be preserved in a structurally acceptable way. 4.2. Weak Discernibility is Not Enough There are some rather obvious prima facie objections to (my take on) Bermúdez’s proposal that I will address shortly. But first, let me take up a critical element of Bermúdez’s view. As is well known, Ladyman [2005] argued that a ‘more discerning’ but still robustly structural notion of discernibility than absolute discernibility yields a version of II that meets Burgess’s challenge — on a certain understanding of the challenge: WD Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-structure. Objects $$a,b\in A$$ are weakly discernible in$$\mathcal{A}$$ if and only if there is a structural relation (over $$\mathcal{A}$$) that one bears to the other but not to itself, i.e., iff for some $$L$$-formula $$\varphi$$, $$R_{\varphi}^{\mathcal{A}}(x,y)$$ but not $$R_{\varphi}^{\mathcal{A}}(x,x)$$, for $$x=a$$ and $$y=b$$ or vice versa. $$a$$ and $$b$$ are utterly indiscerniblein$$\mathcal{A}$$ if they are not weakly discernible in $$\mathcal{A}$$.8 Rendering the Identity of Indiscernibles accordingly, we have: WII Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-structure. Then for all elements $$a,b\in A$$, $$a=b$$ if and only if $$a$$ and $$b$$ are utterly indiscernible in $$\mathcal{A}$$. $$i$$ and $$j$$ are in fact weakly discernible by a number of relations, e.g., notably, the additive inverse relation (ibid., p. 220), $$R_{x+y=0}^{\mathcal{C}}$$, as we have $$i+j=0\neq i+i$$. It follows that they are distinguishable in $$\mathcal{C}$$ on purely structural grounds. Indeed, as Ketland [2006, p. 308] quickly observed, simply to express the additive inverse relation we need to make use of the identity predicate. Hence, any two distinct objects in an $$L$$-structure $$\mathcal{A}$$, where $$L$$ contains identity, are weakly discernible simply by the difference relation $$R_{x\neq y}^{\mathcal{A}}$$; in particular, we have $$R_{x\neq y}^{\mathcal{C}}(i,j)$$ but, obviously not $$R_{x\neq y}^{\mathcal{C}}(i,i)$$. So long as we interpret Burgess’s challenge as the demand that there be some valid, fully general, structurally legitimate discernibility principle, the existence of distinct but absolutely indiscernible objects like $$i$$ and $$j$$ present no problem for the ante rem structuralist and the challenge is met. Ketland’s observation raises an important issue in the structuralism debate. Keränen [2001, pp. 312 ff.] in particular, and perhaps also Burgess [1999, p. 100] and Hellman [2001, p. 202], understand Shapiro’s [1997] characterization of structuralism (as illustrated by the quote in § 3 above) to require an account of identity for mathematical objects that would preclude, or at least restrict, the use of a primitive identity predicate in the characterization of structural properties and relations;9 under such a requirement, Burgess’s challenge (as understood above) remains. For, as Ketland [2006, pp. 309 ff.] first pointed out, many structures, like those illustrated in Figure 1 cannot even be defined without presupposing primitive facts of identity and difference — $$\mathfrak{A}_{0}$$, for example, is the bare cardinality structure containing exactly two distinct but otherwise absolutely indiscernible positions. However, Shapiro [2008, p. 287] himself, while freely acknowledging that his earlier characterizations of ante rem structuralism were misleading, denies that he ever had such a strong view as Keränen’s in mind in his 1997 book, citing as evidence his ‘whole-hearted acceptance [at the time] of the finite cardinal structures’. Ketland [2006, pp. 311–312], Ladyman [2007, pp. 33 ff.], and Leitgeb and Ladyman [2008] have likewise all provided cogent arguments that identity and difference are, and indeed must be, presupposed in mathematical practice.10 Fig. 1. View largeDownload slide Simple structures with absolute indiscernibles Fig. 1. View largeDownload slide Simple structures with absolute indiscernibles Nonetheless, important as it may be, the structural legitimacy of identity and difference is not the issue here; for our purposes, their legitimacy can simply be viewed as a background assumption that I (and Ketland et al.) share with Bermúdez. For it is obvious that Bermúdez himself has no objection to the use of a primitive identity predicate — indeed, his own account depends on it — and he is well aware of the validity of WII for languages with identity. Rather, the crux of his position is that weak discernibility, even with primitive identity, is still not enough. For, since all structural relations are invariant under symmetries, we have more generally that every structural relation on distinct but absolutely indiscernible objects like $$i$$ and $$j$$ is symmetric and, hence, that $$\textit{tp}_{\mathcal{C}}(i,j)=\textit{tp}_{\mathcal{C}}(j,i)$$. The problem, therefore, ‘is not distinguishing $$i$$ and $$j$$ within $$\mathcal{C}$$, but rather explaining in what that distinctness consists’ [Bermúdez, 2007, p. 113, emphasis added]. Bermúdez’s point then, I take it, is that Burgess’s challenge requires an account, not of mere distinguishability — the difference relation serves well enough for that — but of individuation; as Shapiro [2006, p. 134] expresses it (quoting, apparently, from an unpublished talk of Keränen’s):11 [T]he $$\ldots$$ task is not merely to distinguish any pair of distinct objects from each other but to individuate each object. As Keränen puts it, the job is to specify, for each object $$a$$, ‘the fact of the matter that makes $$a$$ the object it is, distinct from any other object’ by ‘providing a unique characterization thereof’. Call the challenge to produce a general principle of individuation that distinguishes even absolute indiscernibles strong Burgess. Of the options surveyed thus far, it seems that only the notions of absolute discernibility and Robinson discernibility are strong enough, when they hold, to guarantee individuation: if $$a$$ and $$b$$ are either absolutely or Robinson discernible in an $$L$$-structure $$\mathcal{A}$$, there is something true of $$a$$ — expressible in $$\mathcal{A}$$ in the former case and in a name expansion $$\mathcal{A}'$$ of $$\mathcal{A}$$ in the latter — that is false of $$b$$ and vice versa. However, as we have seen, the principle AII based on absolute discernibility in general falls victim to strong Burgess, while the principle RII based on Robinson discernibility does not. So Robinson discernibility might well be the only game in town that can meet this stronger challenge to ante rem structuralism. 5. Haecceities and Structural Discernibility In this section I will try to put my finger on what is right about Bermúdez’s proposal and where it falls short vis-à-vis strong Burgess. 5.1. Impure Haecceities Shapiro [2008, p. 288] voices a natural and widespread objection to the suggestion that haecceities have any sort of genuinely structural role to play: If each mathematical object has an haecceity $$\ldots$$ then the job of individuation is done trivially, but at least it is done. The existence of the haecceity of an object $$a$$ provides the fact that makes $$a$$ the object it is, distinct from any other. Only $$a$$ has that particular haecceity. The problem, of course, is that since it is virtually analytic that haecceities are not structural properties, the ante rem structuralist cannot invoke this trivial resolution of the individuation task. The critical claim here — that haecceities are not structural properties — needs some clarification. Say that a relation $$R^{\prime}$$ of $$\mathcal{A}$$ is qualitative just in case it is definable solely in terms of the predicates of $$L$$; that is, just in case there is a formula $$\varphi$$ of $$L$$ containing no individual constants such that $$R^{\prime}=R_{\varphi}^{\mathcal{A}}$$. In the non-modal context of classical mathematics, it is difficult (though, typically, unnecessary) to distinguish coextensional properties. And in the more formal context of standard, classical model theory, there is simply no choice but to identify co-extensional properties within an $$L$$-structure $$\mathcal{A}=\langle A,V\rangle$$. Hence, for any $$a\in A$$, any definable property $$R_{\varphi}^{\mathcal{A}}$$ of $$\mathcal{A}$$ that is true only of $$a$$ is identical with $$a$$’s haecceity, regardless of whether or not there is an individual constant $$\kappa$$ of $$L$$ such that $$\kappa^{\mathcal{A}}=a$$. So even if it is possible to express a haecceity by means of an identity $$\nu=\kappa$$, it does not follow that that haecceity is not structural, as it might be alternatively definable in qualitative terms. Noteworthy examples are found in our $$L_{F}$$-structure $$\mathcal{C}$$, where $$R_{\nu+\nu=\nu}^{\mathcal{C}}=R_{\nu=0}^{\mathcal{C}}$$ and $$R_{\nu\times\nu=\nu}^{\mathcal{C}}=R_{\nu=1}^{\mathcal{C}}$$. The view does however suggest that, insofar as names are introduced into the language of a structure, their use in the identification of structural relations should be eliminable. Hence, all genuinely structural relations are purely qualitative; if $$R_{\varphi}^{\mathcal{A}}$$ is a structural relation, there is a formula $$\varphi'$$ containing no individual constants such that $$R_{\varphi}^{\mathcal{A}}$$=$$R_{\varphi'}^{\mathcal{A}}$$. This seems to be the idea underlying Shapiro’s [1997] expression of the structuralist intuition in terms of distinguished relations (hence, also, functions) only: there is ‘no more to the [offices of a structure] “in themselves” than the relations they bear to one another’ [Shapiro, 1997, p. 73].12 A more reasonable way of understanding Shapiro’s quote above, then, is not as an objection to haecceities per se playing any individuating role but rather to nominal haecceities, that is, haecceities that can only be expressed by means of a formula $$\varphi$$ of $$L$$ that contains a name.13 Hence, to introduce names, as in Bermúdez’s proposal, solely for the purpose of individuating otherwise non-individuable objects — in effect, making them featured objects of the structure by fiat — is a cheat. However, an obvious variation on Burmúdez’s proposal is still available that gets around the letter, at least, of Shapiro’s objection, as haecceities for all the elements of a structure can of course be introduced just as well by means of predicates as by individual constants — a move famously suggested by Quine [1948],14 albeit to rather different ends. Specifically, for an $$L$$-structure $$\mathcal{A}=\langle A,V\rangle$$, define an $$L'$$-structure $$\mathcal{A}'=\langle A,V'\rangle$$ to be a monadic expansion of an $$L$$-structure $$\mathcal{A}=\langle A,V\rangle$$ just in case $$L'$$ is the result of adding some new monadic predicates to $$L$$ and that $$\mathcal{A}'$$ is a Quine expansion of $$\mathcal{A}$$ if, in addition, for every $$a\in A$$, there is a new predicate $$\pi_{a}\in L'$$, whose interpretation in $$\mathcal{A}'$$ is $$\{a\}$$. Then, of course, for every such predicate $$\pi_{a}$$, the ‘individual essence’ $$R_{\pi_{a}x}^{\mathcal{A}'}$$ is a haecceity that distinguishes $$a$$ qualitatively (in the sense defined above15) from every other element of $$\mathcal{A}'$$. And thus an obvious definition and, concomitantly, an obviously valid version of II: QD Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-structure. Objects $$a,b\in A$$ are Quine discernible over$$\mathcal{A}$$ if and only if $$a$$ and $$b$$ are absolutely discernible in some Quine expansion $$\mathcal{A}'$$ of $$\mathcal{A}$$. $$a$$ and $$b$$ are Quine indiscernible over $$\mathcal{A}$$ if they are not Quine discernible over $$\mathcal{A}$$. QII Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-structure. Then for all elements $$a,b\in A$$, $$a=b$$ iff $$a$$ and $$b$$ are Quine indiscernible over $$\mathcal{A}$$. Hence, $$i$$ and $$j$$ and, indeed, any two distinct elements of $$\mathcal{C}$$ are Quine discernible over $$\mathcal{C}$$, in accordance with QII. Intuitively, of course, the ‘qualitative’ haecceities of a Quine expansion are no more (or less) structural than the nominal haecceities of a Robinson expansion and, hence, are no more (or less) effective as the basis for a solution to strong Burgess. But, why, exactly, should such ‘impure’ haecceities — those that can only be picked out by means of a name or a Quinean predicate — be ruled out in the task of individuation? Something admittedly seems right about Shapiro’s assertion that ‘[impure] haecceities are not structural properties’ in cases like that of $$i$$ and $$j$$ in the complex field, but rejecting them in general a priori fails to satisfy. Indeed, it seems to me that there are grounds for considering all impure haecceities to be structural; at the same time, the intuition that they are not structural is, I believe, grounded in the fact that they are not up to the task of providing a genuinely structural solution to strong Burgess via RII or QII. Let me argue these points. 5.2. The Structural Legitimacy of Impure Haecceities With regard first to the structural nature of impure haecceities, consider the simple structures $$\mathfrak{A}_{1}$$, $$\mathfrak{A}_{2}$$, and $$\mathfrak{A}_{3}$$ depicted left to right, respectively, in Figure 2. $$\mathfrak{A}_{1}$$ involves a single binary relation $$\boldsymbol{R}$$ that the two ‘top’ positions bear to the single ‘bottom’ position — the structure exhibited, say, by a system $$\boldsymbol{A}_{1}$$ comprising a father with two children, where child of is the only featured relation. Intuitively, $$\mathfrak{A}_{1}$$ is a substructure of $$\mathfrak{A}_{2}$$, which, in addition, includes a property $$\boldsymbol{P}$$ that is shared by one of the two top positions and the bottom position — the structure exhibited by, say, a system $$\boldsymbol{A}_{2}$$ comprising the same individuals as in $$\boldsymbol{A}_{1}$$ but where the father and one of his children have blue eyes and the property being blue-eyed is a featured (1-place) relation. Of course, this is an intrinsic property of individuals in the system rather than a relation between them, but surely the featured properties of a system count as much as part of its structure as do its featured ($$>$$1-place) relations; they characterize the system no less than the relations do. Thus, by reflecting a featured property common to two of the elements of a system, $$\mathfrak{A}_{2}$$ clearly exhibits additional structure that is missing from $$\mathfrak{A}_{1}$$, a property that is not definable in terms of $$\boldsymbol{R}$$ (definable, that is, in a language containing only a predicate for $$\boldsymbol{R}$$). But now consider a system $$\boldsymbol{A}_{3}$$ comprising, say, a father and his two offspring, where being a boy and being a girl are the featured properties. This system exhibits a structure $$\mathfrak{A}_{3}$$ that differs from $$\mathfrak{A}_{1}$$ only in that it features two distinct properties $$\boldsymbol{H}$$ and $$\boldsymbol{H'}$$, each true of exactly one of the two ‘top’ positions. As with $$\boldsymbol{P}$$ in $$\mathfrak{A}_{2}$$, neither property is expressible in terms of $$\boldsymbol{R}$$ in the structure.16 More generally, it is difficult to see any compelling reason for denying that $$\boldsymbol{H}$$ and $$\boldsymbol{H'}$$ are fully structural in $$\mathfrak{A}_{3}$$ that would not simultaneously rule out $$\boldsymbol{P}$$ in $$\mathfrak{A}_{2}$$ as well. Fig. 2. View largeDownload slide Three structures Fig. 2. View largeDownload slide Three structures Now, in response one might argue that the fact that we have used a real-world example involving multiply exemplifiable properties — being a boy and being a girl — that just happen to be instantiated by one thing in the real-world system in question somehow vitiates the argument here; that the appeal to properties that are not themselves essentially uniquely exemplifiable does not really serve as adequate justification for the structural legitimacy of the modally rigid thisnesses of modern-day haecceitism like being Socrates or being the complex number $$i$$.17 However, in reply, we note once again that, in the extensional model-theoretic framework in which we are working, such modal metaphysical subtleties have no real purchase.18 The fact is that some of the featured properties within a given system of interest might have only a single instance and, hence, when that system’s structure is distilled into a model-theoretic form, any such structural property is, by definition, a haecceity, regardless of how the original system was intuitively conceptualized. Second, though, it seems to me that we can argue for the structural legitimacy of haecceities on purely logical grounds alone. So consider the structure $$\mathfrak{A}_{1}$$ of Figure 2 again. Let $$a_{1}$$ and $$a_{2}$$ be the two ‘top’ positions and $$a_{3}$$ the ‘bottom’ position. Intuitively, of course, $$a_{1}$$ and $$a_{2}$$ are (absolutely) indiscernible; more exactly, in a language containing only a 2-place predicate for the binary relation in the structure, $$a_{1}$$ and $$a_{2}$$ will share all of their structural properties. However, if we allow (as is the current consensus) that identity and, hence, distinctness are themselves structural, so that $$a_{1}$$’s being distinct from $$a_{2}$$ and $$a_{3}$$, $$a_{1}\neq a_{2}$$ and $$a_{1}\neq a_{3}$$, are themselves structural facts about $$a_{1}$$, then, intuitively, simply by logic alone it would seem to follow that $$a_{1}$$’s having the properties being distinct from $$a_{2}$$ — $$[\lambda x\,x\neq a_{2}]a_{1}$$ — and being distinct from$$a_{3}$$ — $$[\lambda x\,x\neq a_{3}]a_{1}$$ — are structural facts about $$a_{1}$$ and, hence, that being distinct from $$a_{2}$$ and being distinct from $$a_{3}$$ are structural properties of $$a_{1}$$.19 And jointly, of course, in the context of $$\mathfrak{A}_{1}$$, they are exactly (equivalent to) the haecceity being $$a_{1}$$, $$[\lambda x\:x=a_{1}]$$.20 However, notwithstanding the fact that these properties are not among the featured relations of $$\mathfrak{A}_{1}$$, their existence is a logical consequence of facts about the structure of $$\mathfrak{A}_{1}$$. We say therefore that they are implicit in $$\mathfrak{A}_{1}$$ (and, derivatively, in any $$L$$-structure for $$\mathfrak{A}_{1}$$). So the general charge that impure haecceities are not legitimately structural does not really seem to stick; not only can we not rule them out as non-structural a priori, but also there are grounds for taking them to be structural outright. But, as I noted, something also seems right about Shapiro’s skepticism about invoking (impure) haecceities to meet strong Burgess. In the following subsection I will attempt to identify what this is. I will in particular try to distinguish cases where such haecceities are genuinely individuating from those cases where they are not. On this basis I will define a corresponding notion of discernibility, whose corresponding indiscernibility principle, it turns out, is not valid — thus justifying Shapiro’s skepticism. 5.3. Overspecification and Structural Discernibility The general idea is this. As just illustrated, some relations can be thought of as implicit in a structure $$\mathfrak{A}$$ — while not among the featured relations of $$\mathfrak{A}$$, their existence follows logically from structural facts about $$\mathfrak{A}$$. Say more generally that a (property or) relation $$\boldsymbol{R}$$ is potentialin$$\mathfrak{A}$$ if $$\mathfrak{A}$$ is a substructure of a further structure in which $$\boldsymbol{R}$$ is explicitly featured; for example, the structure $$\mathfrak{A}_{2}$$ in Figure 2 shows that the property $$\boldsymbol{P}$$ is potential in $$\mathfrak{A}_{1}$$. Expressed model-theoretically, a property or relation $$\boldsymbol{R}$$ is potential in an $$L$$-structure $$\mathcal{A}$$ for $$\mathfrak{A}$$ if it is definable in some expansion of $$\mathcal{A}$$, i.e., somewhat more intuitively, if it is either definable directly or becomes so by the addition of further vocabulary to $$L$$ — not just names, which only give us the ability to express implicit relations, but predicates as well.21 (Say that $$\boldsymbol{R}$$ is merely potential in $$\mathcal{A}$$ if it is potential in $$\mathcal{A}$$ but not definable in $$\mathcal{A}$$, i.e., if one must introduce new vocabulary to define $$\boldsymbol{R}$$.) As I understand him, then, Bermúdez’s insight is that the idea of potential structure might provide us with a basis for defining a notion of discernibility sufficient for meeting strong Burgess. We have seen this general strategy play out in a simple form in the notion RD of Robinson discernibility and its corresponding indiscernibility principle RII. But I think the proposal fails. Specifically, I will argue that Robinson discernibility is not in fact the notion of discernibility warranted by Bermúdez’s insight about potential structure and, indeed, that it is not a legitimately structural notion of discernibility at all and hence that RII is not a legitimately structural rendering of the Identity of Indiscernibles. Rather, the insight in question leads us to a rather weaker notion of discernibility: in order for two objects to be discernible by any potential structure, it must be possible to identify a structural role (in a sense to be defined) that only one of them can play. And, as we shall see, unlike RII, the corresponding indiscernibility principle for this notion is invalid. Consider again the structures in Figure 2. Let $$L=\{\rho\}$$, where $$\rho$$ is a 2-place predicate and consider the $$L$$-system $$\mathcal{A}_{1}=\langle A_{1},V_{1}\rangle$$ for the structure $$\mathfrak{A}_{1}$$, where $$A_{1}=\{a_{0},a_{1},a_{2}\}$$ is (or represents22) the set of positions of $$\mathfrak{A}_{1}$$ and let $$\rho^{\mathcal{A}_{1}}=\{\langle a_{1},a_{0}\rangle,$$$$\langle a_{2},a_{0}\rangle\}$$. Likewise let $$\mathcal{A}_{2}=\langle A_{1},V_{2}\rangle$$ be an $$L_{\pi}$$-expansion of $$\mathcal{A}_{1}$$ exhibiting the structure $$\mathfrak{A}_{2}$$ and, hence, where $$\pi^{\mathcal{A}_{2}}=\{a_{0},a_{2}\}$$, say. Now, the fact that we can expand $$\mathcal{A}_{1}$$ to represent the property $$\boldsymbol{P}$$ illustrates the idea that $$\boldsymbol{P}$$ was ‘potential’ in $$\mathfrak{A}_{1}$$. Critically, however, while $$\mathfrak{A}_{2}$$ does indeed distinguish two of its positions as bearers of the featured property $$\boldsymbol{P}$$, $$\boldsymbol{P}$$ is captured model-theoretically in an expansion of $$\mathcal{A}_{1}$$ by selecting either of the pairs $$\{a_{0},a_{1}\},\ \{a_{0},a_{2}\}$$ to play the representationalrole of the bearers of property $$\boldsymbol{P}$$. Thus, qua expansion of $$\mathcal{A}_{1}$$, $$\mathcal{A}_{2}$$overspecifies$$\boldsymbol{P}$$, insofar as it represents $$\boldsymbol{P}$$ as $$R_{\pi\nu}^{\mathcal{A}_{2}}=\{a_{0},a_{2}\}$$ and therefore as a property of two specific positions $$a_{0}$$ and $$a_{2}$$ of $$\mathfrak{A}_{1}$$; the isomorphic $$L_{\pi}$$-expansion $$\mathcal{A}'_{2}=\langle A_{1},V'_{2}\rangle$$ of $$\mathcal{A}_{1}$$ in which $$\pi$$ is mapped to $$\{a_{0},a_{1}\}$$ instead of $$\{a_{0},a_{2}\}$$ would have served just as well. The pairs $$\{a_{0},a_{2}\}$$ and $$\{a_{0},a_{1}\}$$, therefore, could have ‘switched roles’, so to say, without any change of structure.23 Thus, the additional information that $$\mathcal{A}_{2}$$ carries specifically about $$a_{0}$$ and $$a_{2}$$ — that they, rather than $$a_{0}$$ and $$a_{1}$$, are the bearers of the property $$\boldsymbol{P}$$ — is not structural information but, rather, a mere representational artifact; it cannot be thought of as providing a structural basis for individuating the pair$$\{a_{0},a_{2}\}$$. The same, it seems, can — and typically (but not always, as we shall see) does — go for name expansions and monadic expansions, i.e., expansions that at most introduce new impure haecceities. Consider the $$L_{\kappa_{1},\kappa_{2}}$$-expansion $$\mathcal{A}_{3}=\langle A_{1},V_{3}\rangle$$ of $$\mathcal{A}_{1}$$ corresponding to the structure $$\mathfrak{A}_{3}$$, where $$\kappa_{1}^{\mathcal{A}_{3}}=a_{1}$$ and $$\kappa_{2}^{\mathcal{A}_{3}}=a_{2}$$. As with our $$L_{\pi}$$-expansion $$\mathcal{A}_{2}$$ of $$\mathcal{A}_{1}$$, an $$L_{\kappa_{1},\kappa_{2}}$$-expansion of $$\mathcal{A}_{1}$$ such as $$\mathcal{A}_{3}$$ must share $$\mathcal{A}_{1}$$’s domain. Likewise, qua$$L_{\kappa_{1},\kappa_{2}}$$-system, it must choose particular elements of $$A_{1}$$ to play the representational roles of the bearers of the featured haecceities $$\boldsymbol{H}$$ and $$\boldsymbol{H'}$$. $$\boldsymbol{H}$$ and $$\boldsymbol{H'}$$ are indeed haecceities and therefore single out particular positions of the structure$$\mathfrak{A}_{3}$$. But, on pain of overspecification, neither $$a_{1}$$ nor $$a_{2}$$ can justifiably be identified with either of the particular outer positions of $$\mathfrak{A}_{3}$$, since both $$a_{1}$$ and $$a_{2}$$ can ‘play the role’ of instantiating either haecceity. For suppose we use the formula ‘$$x=\kappa_{1}$$’ to express the haecceity $$\boldsymbol{H}$$ in our structure $$\mathfrak{A}_{3}$$, so that $$R_{x=\kappa_{1}}^{\mathcal{A}_{3}}$$ represents $$\boldsymbol{H}$$ in our $$L$$-system $$\mathcal{A}_{3}$$ and, hence, that $$a_{1}$$ represents the position of $$\mathcal{A}_{3}$$ exemplifying $$\boldsymbol{H}$$. Now let $$\mathcal{A}'_{3}=\langle A_{1},V'_{3}\rangle$$ be an $$L_{\kappa_{1},\kappa_{2}}$$-expansion of $$\mathcal{A}_{1}$$ such that $$\kappa_{1}^{\mathcal{A}_{3}'}=a_{2}$$ and $$\kappa_{2}^{\mathcal{A}_{3}'}=a_{1}$$. The function $$f:A\longrightarrow A$$ that maps $$a_{0}$$ to itself and $$a_{1}$$ and $$a_{2}$$ to each other is obviously an isomorphism from $$\mathcal{A}{}_{3}$$ to $$\mathcal{A}'_{3}$$ — in $$\mathcal{A}'_{3}$$, $$a_{1}$$ and $$a_{2}$$ switch representational roles — in $$\mathcal{A}_{3}$$, $$a_{1}$$ and $$a_{2}$$ represent the bearers of $$\boldsymbol{H}$$ and $$\boldsymbol{H'}$$, respectively, and in $$\mathcal{A}'_{3}$$, these roles are reversed. As above, therefore, the introduction of the names $$\kappa_{1}$$ and $$\kappa_{2}$$ notwithstanding, the additional information that $$\mathcal{A}_{3}$$, qua expansion of $$\mathcal{A}_{1}$$, carries about those specific positions $$a_{1}$$ and $$a_{2}$$ in $$\mathcal{A}_{1}$$ is not structural information that can legitimately be invoked to individuate them ‘over’ $$\mathcal{A}_{1}$$ in the way that RII would have it.24 More generally, then, to identify any individuating structural information that might be potential in a given $$L$$-structure $$\mathcal{A}=\langle A,V\rangle$$ — and, hence, to capture Bermúdez’s actual insight faithfully — it is necessary to filter out its overspecificity by looking, not simply at a single expansion $$\mathcal{A}_{E}$$ of $$\mathcal{A}$$, but at expansions that are isomorphic to it. For such expansions $$\mathcal{A}'_{E}$$ share the additional structure that $$\mathcal{A}_{E}$$ introduces but can differ from $$\mathcal{A}_{E}$$ in the members of $$A$$ they choose to play the new representational roles required by that additional structure. So whenever an element $$a$$ of $$\mathcal{A}$$ plays a certain representational role in $$\mathcal{A}{}_{E}$$ — which, recall, we can identify with its type $$\textit{tp}{}_{\mathcal{A}{}_{E'}}(b)$$ — but some other element $$b$$ of $$\mathcal{A}$$ can play that role in an isomorphic expansion $$\mathcal{A}'_{E}$$ — and, hence, whose type $$\textit{tp}{}_{\mathcal{A}{}_{E'}}(b)$$ in $$\mathcal{A}'_{E}$$ is exactly $$a$$’s type in $$\mathcal{A}_{E}$$ — then we know that $$a$$’s role in $$\mathcal{A}_{E}$$, qua expansion of $$\mathcal{A}$$, even if played by $$a$$ alone, does not distinguish $$a$$ from $$b$$structurally and, hence, fails to individuate$$a$$ structurally in $$\mathcal{A}_{E}$$. To express this a little more formally, note first that it is an elementary theorem of model theory (see, e.g., [Enderton, 2001, p. 96]) that if $$f:\mathcal{A}\longrightarrow\mathcal{B}$$ is an isomorphism from one $$L$$-system $$\mathcal{A}=\langle A,V\rangle$$ to another $$\mathcal{B}=\langle B,V'\rangle$$ and $$f(a)=b$$, then $$\textit{tp}_{\mathcal{A}}(a)=\textit{tp}{}_{\mathcal{B}}(b)$$, i.e., in $$\mathcal{B}$$, $$b$$ plays the role that $$a$$ plays in $$\mathcal{A}$$. Given this, we have the following notion of (in)discernibility: SD Let $${\mathcal{A}}=\langle A,V\rangle$$. Objects $$a,b\in A$$ are structurally discernible over$$\mathcal{A}$$ iff (i) there is a language $$L_{E}\supseteq L$$ and an $$L_{E}$$-expansion $$\mathcal{A}{}_{E}=\langle A,V\!{}_{E}\rangle$$ of $$\mathcal{A}$$ in which $$a$$ and $$b$$ are absolutely discernible, and (ii) there is no expansion $$\mathcal{A}'_{E}=\langle A,V'_{E}\rangle$$ of $$\mathcal{A}$$ such that, for some isomorphism $$f$$ from $$\mathcal{A}{}_{E}$$ to $$\mathcal{A}'_{E}$$, $$f(a)=b$$. $$a$$ and $$b$$ are structurallyindiscernibleover$$\mathcal{A}$$ if they are not structurally discernible over $$\mathcal{A}$$. That is, $$a$$ and $$b$$ are structurally indiscernible over $$\mathcal{A}$$ iff, for every expansion $$\mathcal{A}_{E}$$ of $$\mathcal{A}$$ in which $$a$$ and $$b$$ are absolutely discernible (hence, at the least, for every Robinson or Quine expansion), there is another $$\mathcal{A}'_{E}$$ that is isomorphic to $$\mathcal{A}_{E}$$ but in which $$b$$ plays the role that $$a$$ played in $$\mathcal{A}_{E}$$. It is straightforward to show that absolute discernibility in $$\mathcal{A}$$ entails structural discernibility over $$\mathcal{A}$$ (see Theorem 1 in the Appendix). However, the question might arise as to whether the converse is also true, in which case structural discernibility is a trivial notion. For if all structurally discernible pairs of objects over an $$L$$-system $$\mathcal{A}$$ are already absolutely discernible in$$\mathcal{A}$$, there is nothing to the idea of an $$L$$-system containing ‘potential’ structure that, via an appropriate expansion of $$\mathcal{A}$$, enables us to individuate entities that are not already individuated by means of structural information that is already explicit in $$\mathcal{A}$$. And, in fact, it is straightforward, though tedious, to show that, for finite structures, the two notions of discernibility coincide (see Theorem 2 in the Appendix). There are, however, non-trivial examples of infinite structures where they come apart. To illustrate, let $${\boldsymbol{Q}=\langle Q,\boldsymbol{+}_{\boldsymbol{Q}},\boldsymbol{<}_{\boldsymbol{Q}}\rangle}$$ and $$\boldsymbol{R}=\langle R,\boldsymbol{+}_{\boldsymbol{R}},\boldsymbol{<}_{\boldsymbol{R}}\rangle$$ be two (disjoint) systems that are instances of the rational and real group structures $$\mathfrak{Q}$$ and $$\mathfrak{R}$$, respectively, possessing in addition the corresponding dense total orderings on $$Q$$ and $$R$$. Consider now the system $$\boldsymbol{QR}$$ resulting from joining these two systems together. The featured relations $$\boldsymbol{+}_{\boldsymbol{QR}}$$ and $$\boldsymbol{<}_{\boldsymbol{QR}}$$ of $$\boldsymbol{QR}$$, therefore, are simply the union of the operations/relations on $$\boldsymbol{Q}$$ and $$\boldsymbol{R}$$, that is, $$a\boldsymbol{+}_{\boldsymbol{QR}}b=c$$ iff either $$a\boldsymbol{+}_{\boldsymbol{Q}}b=c$$ or $$a\boldsymbol{+}_{\boldsymbol{R}}b=c$$ and $$a\boldsymbol{<_{QR}}b$$ iff either $$a\boldsymbol{<_{Q}}b$$ or $$a\boldsymbol{<_{R}}b$$. Call any such system a QR-system.25 Such a system is pictured (sans binary operations) in Figure 3, where $$0_{Q}$$ and $$0_{R}$$ are the additive identities of the constituent groups. Let $$\mathfrak{S}$$ be the structure exhibited by a QR-system. Now, let $$L=\{+,<\}$$ and let $$\mathcal{S}=\langle Q\cup R,V\rangle$$ be the corresponding $$L$$-structure for $$\mathfrak{S}$$, so that $$Q$$ and $$R$$ contain the positions in $$\mathfrak{S}$$ of (or corresponding to26) the respective substructures $$\mathfrak{Q}$$ and $$\mathfrak{R}$$, and $$+^{V}=\boldsymbol{+}_{Q}\cup\boldsymbol{+}_{R}$$ and $$<^{V}=\boldsymbol{<}_{Q}\cup\boldsymbol{<}_{R}$$, where those are the corresponding operations and relations on those substructures. There are some obvious structural analogies between $$\mathcal{S}$$ and the complex field $$\mathcal{C}$$. Notably, in $$\mathcal{S}$$, we are capable of expressing the property $$R_{x+x=x}^{\mathcal{S}}$$ = $$\left\{ 0_{Q},0_{R}\right\} $$ of being a (partial) additive identity element of the structure,27 just as we are capable of picking out the property $$R_{x^{2}+1=0}^{\mathcal{C}}=\left\{ i,j\right\} $$ of being a square root of $$-1$$ in $$\mathcal{C}$$. But, like $$i$$ and $$j$$ in $$\mathcal{C}$$, the two identity elements $$0_{Q}$$ and $$0_{R}$$ themselves are absolutely indiscernible in $$\mathcal{S}$$; in $$\mathcal{S}$$, each has every systemic property of the other. Fig. 3. View largeDownload slide An $$L_{QR}$$-system Fig. 3. View largeDownload slide An $$L_{QR}$$-system However, there are structural properties that are potential in our $$L$$-system $$\mathcal{S}$$ that can (only) be revealed in an appropriate expansion and by virtue of which it becomes possible to individuate $$0_{Q}$$ and $$0_{R}$$ in a structurally meaningful way. For let $$\bullet_{Q}$$ and $$\bullet_{R}$$ be multiplication operations on $$Q$$ and $$R$$ such that $$\langle Q,\boldsymbol{+}_{Q},\bullet_{Q}\rangle$$ and $$\langle R,\boldsymbol{+}_{R},\bullet_{R}\rangle$$ are fields. Let $$L_{\times}=\{\times,+,<\}$$ and let $$\mathcal{S}_{\times}=\langle Q\cup R,V_{\times}\rangle$$ be an $$L_{\times}$$-expansion of $$\mathcal{S}$$ such that $$\times^{\mathcal{S}_{\times}}=\bullet_{Q}\cup\bullet_{R}$$. Then, where $$\psi$$ is the formula $$\exists z(z\times z=z\land z<y\land y\times y=z+z)$$,28$$\psi$$ is true in $$\mathcal{S}_{\times}$$ only of (positive) $$\sqrt{2}$$ — i.e., the $$r\in R$$, call it $$\sqrt{2}_{R}$$, occupying the $$\sqrt{2}$$ position of (the real subfield of) $$\mathcal{S}_{\times}$$ . Moreover, $$\psi$$ is true of no other $$a\in Q\cup R$$ in any $$L_{\times}$$-expansion $$\mathcal{S}_{\times}'=\langle Q\cup R,V_{\times}'\rangle$$ isomorphic to $$\mathcal{S}_{\times}$$.29 Hence, no other element of $$Q\cup R$$ can play the representational role that $$\sqrt{2}_{R}$$ plays in $$\mathcal{S}_{\times}$$. $$\sqrt{2}_{R}$$ is can thus be individuated in $$\mathcal{S}$$ ‘from above’, i.e., it is structurally discernible from every other $$a\in Q\cup R$$. $$0_{Q}$$ and $$0_{R}$$ are thereby also so discernible from one another as, of the two, only $$0_{R}<^{\mathcal{S}_{\times}}\sqrt{2}_{R}$$ and, hence, the formula $$x+x=x\land\exists y(x<y\land\psi)$$ is in $$\textit{tp}_{\mathcal{S}_{\times}}\!(0_{R})$$ but not in $$\textit{tp}_{\mathcal{S}_{\times}}\!(0_{Q})$$, i.e., the two entities play different representational roles in $$\mathcal{S}_{\times}$$. In this sort of case, then, a name expansion can do structurally significant work. Specifically, let $$\mathcal{S}'=\langle Q\cup R,V'\rangle$$ be an $$L_{\lambda_{1},\lambda_{2}}$$-expansion of $$\mathcal{S}$$, where $$\lambda_{1}^{\mathcal{S}'}=0_{Q}$$ and $$\lambda_{2}^{\mathcal{S}'}=0_{R}$$. $$\mathcal{S}'$$ adds names for our two additive identity elements and, hence, the ability to discern them via their haecceities $$R_{x=\lambda_{1}}^{\mathcal{S}'}$$ and $$R_{x=\lambda_{2}}^{\mathcal{S}'}$$. But, because $$0_{Q}$$ and $$0_{R}$$ play structural roles in the expansion $$\mathcal{S}_{\times}$$ that neither can play in any isomorphic expansion, the claim that haecceities in this case have the power to individuate the objects in question structurally is warranted. My analysis, therefore, is this: Both RD and its ‘qualitative’ counterpart QD rightly allow impure haecceities to count as structural insofar as singleton properties can be among the featured relations of a system — impure haecceities represent the contribution of these properties to the structure exhibited by those systems. However, because both notions fail to filter out overspecification, they do not meet strong Burgess — they endow some impure haecceities with an individuating power which, qua structural properties, they do not possess. When we filter out overspecification as we have in the notion of structural (in)discernibility (definition SD), the corresponding form of the identity of indiscernibles is invalid: SII Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-structure. Then for all elements $$a,b\in A$$, $$a=b$$ if and only if $$a$$ and $$b$$ are structurally indiscernible over$$\mathcal{A}$$. SII is obviously falsified by such simple examples as the finite $$L$$-structure $$\mathcal{A}_{1}$$ (specifically, by the positions $$a_{1}$$ and $$a_{2}$$) as well as, and for essentially the same reason as, far more complicated infinite structures like the $$L_{F}$$-structure $$\mathcal{C}$$. The complex numbers $$i$$ and $$j$$ are of course Robinson discernible over $$\mathcal{C}$$, as they are absolutely discernible in any Robinson expansion of $$\mathcal{C}$$. But they are not structurally discernible. For let $$g$$ be the symmetry on $$\mathcal{C}$$ that takes every complex number to its complex conjugate and, for any Robinson expansion $$\mathcal{C}^{+}=\langle C,V^{+}\rangle$$ of $$\mathcal{C}$$, let $$\mathcal{C}_{g}^{+}=\langle C,V_{g}^{+}\rangle$$ be the corresponding Robinson expansion.30$$g$$ is obviously an isomorphism from $$\mathcal{C}^{+}$$ to $$\mathcal{C}_{g}^{+}$$ in which $$i$$ and $$j$$ switch representational roles, i.e., in which $$\textit{tp}_{\mathcal{C}_{g}^{+}}(i)=\textit{tp}{}_{\mathcal{C}^{+}}(j)$$ and $$\textit{tp}_{\mathcal{C}_{g}^{+}}(j)=\textit{tp}{}_{\mathcal{C}^{+}}(i)$$; in particular, they even swap ‘haecceitist’ roles: $$R_{x=\kappa_{i}}^{\mathcal{C}_{g}^{+}}=\left\{ j\right\} $$ and $$R_{x=\kappa_{j}}^{\mathcal{C}_{g}^{+}}=\left\{ i\right\} $$. Thus, as the existence of an isomorphism between two $$L$$-systems is the paradigm of shared structure, I take this to show that Robinson expansions do not reveal any underlying structural information that is simply implicit within $$\mathcal{C}$$ but which individuates $$i$$ and $$j$$ when it is made explicit in an appropriate expansion. This should be unsurprising at this point. Both our simple $$L$$-structure $$\mathcal{A}_{1}$$ and the complex $$L_{F}$$-structure $$\mathfrak{\mathcal{C}}$$ admit of non-trivial symmetries under which $$a_{1}$$ and $$a_{2}$$ and $$i$$ and $$j$$, respectively, are symmetric (indeed, fully symmetric) and it is a straightforward theorem that symmetric pairs in an $$L$$-system are structurally indiscernible (see Theorem 4 in the Appendix). But isomorphisms generally are exactly what we take to reveal the structural similarities and differences between things in different systems — if, in particular, $$a$$ and $$b$$ are symmetric counterparts under an automorphism, then there is simply nothing structural, beyond mere difference, that distinguishes them and nothing whatever that individuates them. Expansions, as employed in SD in a manner that captures Bermúdez’s insight, far from exploiting haecceities to yield a robust notion of discernibility, in fact reveal rigorously why haecceities fall short (in SII) as the foundation for a general structural solution to strong Burgess. A haecceity may indeed be an important part of the structure exhibited by an $$L$$-system. However, it may not pick out a structural role that only one position of the structure can play — for that, the structural role played by a given position in a structure must meet the stronger criterion expressed in SD — over and above exemplifying a unique haecceity in a given expansion, there must not be anything else in the structure that can play the same role in an isomorphic expansion, as in the case of $$0_{Q}$$ and $$0_{R}$$ in the QR-structure. Hence, symmetric positions in an $$L$$-structure should turn out to be structurally indiscernible, the explicit existence of haecceities notwithstanding — any property $$R_{\varphi}^{\mathcal{A}_{E}}$$ of $$a$$, say, that absolutely discerns it from $${b}$$ in an expansion $$\mathcal{A}_{E}$$ of an $$L$$-structure will not in fact represent $$a$$’s haecceity per se but, rather, only a representational role that is equally well played by $$b$$ in an isomorphic expansion $$\mathcal{A}'_{E}$$ where $$R_{\varphi}^{\mathcal{A}_{E}'}$$ is true of $${b}$$ instead. The complex $$L_{F}$$-structure $$\mathcal{C}$$ is of course a special case of an $$L$$-structure with a nontrivial symmetry and our two roots $$i$$ and $$j$$ jointly are a special case of symmetric counterparts. The failure of Robinson expansions to provide a genuinely structural way to individuate $$i$$ and $$j$$ is therefore simply a consequence of a more general structural phenomenon concerning the (distinct) symmetric counterparts of a non-trivial automorphism. In a nutshell, haecceities are structural; to acknowledge them in a system is to identify additional structure. But they are not of themselves structurallyindividuating; for a haecceity genuinely to individuate an object or position in a structure, there must be underlying structure that can be ‘drawn out’ in an expansion in such a way that a position plays a structural role that nothing else can play in an isomorphic system. But symmetric counterparts cannot be so individuated, and thus the failure of Bermúdez’s program as a structurally legitimate answer to strong Burgess. As we have seen, Bermúdez claims that, absent a structural principle of individuation, there is no explanation of the distinctness of absolutely indiscernible objects like $$i$$ and $$j$$. But so long as one follows Ketland et al. and accepts the structural legitimacy of primitive facts of identity and difference (see § 4.2 above), it is hard to see why anyone would ever have thought that the capacity to supply such explanations should be an essential part of the structuralist project.31 For the lack of such explanations — the lack of any structurally individuating properties for structural indiscernibles — is exactly what is characteristic of, and indeed interesting about, non-trivial symmetries. From this perspective, then, Burgess’s challenge, understood simply as the demand for a structurally warranted principle that distinguishes any two distinct objects, is fully met by appeal to the weak discernibility principle WII and the difference relation. And strong Burgess is, ultimately, incoherent; in requiring not just discernibility but individuation, it demands that objects that differ structurally by nothing other than the fact that they are distinct be distinguishable structurally by something other than the fact that they are distinct. 5.4. Reference, Quantification, and Structural Indiscernibles I close with some reflections on reference and quantification vis-à-vis structural indiscernibles. Say that a property or relation is weakly potential in a structure $$\mathfrak{A}$$ if it is potential in $$\mathfrak{A}$$ but not implicit in it, that is, if (unlike a haecceity) its existence is not a (meta)logical consequence of structural facts about $$\mathfrak{A}$$. The die-hard defender of RII might argue that the ante rem structuralist has one last card to play, namely, that overspecification only arises with respect to properties and relations that are weakly potential in a structure. Consider, for example, the superstructure $$\mathfrak{C}'$$ of the complex field $$\mathfrak{C}$$ that adds a property $$\boldsymbol{E}$$ true of those positions $$c$$ such that, for some non-negative real $$b$$, $$c=bi$$.32 That is, in $$\mathfrak{C}'$$ — thought of in terms of its representation in the Euclidean plane — $$\boldsymbol{E}$$ is true of those positions that are on the ‘non-negative’ half of the imaginary axis. $$\boldsymbol{E}$$, it should be clear, is not implicit in $$\mathfrak{C}$$ and, hence, is only weakly potential in it.33 Let $$\varepsilon$$ be a 1-place predicate and consider the $$L_{F\cup\{\varepsilon\}}$$-expansion $$\mathcal{C}'=\langle C,V'\rangle$$ of $$\mathcal{C}=\langle C,V\rangle$$ that exhibits the structure $$\mathfrak{C}'$$, so that $$V'(\varepsilon)$$ is the relevant subset of $$C$$ that represents $$\boldsymbol{E}$$, i.e., $$\left\{ c\in C:\exists b\in\mathbb{R}(b\geq0\land c=bi)\right\} $$. (Recall that $$\mathcal{C}=\langle C,V\rangle$$ is an $$L_{F}$$-structure for $$\mathfrak{C}$$ and, hence, its domain $$C$$ consists of the positions of $$\mathfrak{C}$$, i.e., the elements of the complex field.) Now, note first that, on ante rem structuralist grounds, the set $$C_{\mathfrak{C}'}$$ of positions of the structure $$\mathfrak{C}'$$cannot be identical with the set of positions of $$\mathfrak{C}$$, i.e., with the set $$C$$.34 For consider the $$L_{F\cup\{\varepsilon\}}$$-expansion $$\mathcal{C}''=\langle C,V''\rangle$$ of $$\mathcal{C}$$ such that $$\varepsilon^{\mathcal{C}''}=\boldsymbol{\overline{E}}$$, where $$\boldsymbol{\overline{E}}$$ is the set of complex conjugates of members of $$\boldsymbol{E}$$. Obviously, as $$\mathcal{C}'$$ and $$\mathcal{C}''$$ are isomorphic (under the function mapping every element of $$C$$ to its complex conjugate), they both exhibit $$\mathfrak{C'}$$; there is no underlying fact of the matter that makes one more correct than the other. Hence, on pain of overspecification, we cannot identify the actual extension of $$\boldsymbol{E}$$ in $$\mathfrak{C}'$$ with either $$\varepsilon^{\mathcal{C}'}$$ or $$\varepsilon^{\mathcal{C}''}$$. But if the positions of $$\mathfrak{C}'$$ were the positions of $$\mathfrak{C}$$, the extension of $$\boldsymbol{E}$$ would have to be one or the other, $$\varepsilon^{\mathcal{C}'}$$ or $$\varepsilon^{\mathcal{C}''}$$, which, as just noted, is impossible. So the positions of $$\mathfrak{C}'$$, with its additional structure, must be distinct from the positions of the basic complex field $$\mathfrak{C}$$. The preceding observation — the ante rem structuralist continues — suggests that overspecification arises in an expansion $$\mathcal{A}_{E}$$ of an $$L$$-structure $$\mathcal{A}$$ for a structure $$\mathfrak{A}_{\mathcal{A}}$$ only if the positions of the structure $$\mathfrak{A}_{\mathcal{A}_{E}}$$ of the expansion cannot be identified with those of $$\mathfrak{A}_{\mathcal{A}}$$. But when $$\mathcal{A}_{E}$$ is an ‘impure’ (i.e., a name or monadic) expansion of $$\mathcal{A}$$, it is plausible that they can be. For there is something right about Bermúdez’s idea (ibid., p. 115) that, when we consider an impure expansion of an $$L$$-structure, ‘[i]n a very important sense the structure itself remains unchanged’. In fact, as I have argued, I do not think that that is strictly the case; the explicit recognition of a haecceity for a position in a structure yields a new structure. Nonetheless, as noted above, the existence of impure haecceities seems to follow simply as a matter of logic. Hence, there is reason to think that the structure $$\mathfrak{A}_{\mathcal{A}_{E}}$$ exhibited by an impure $$L_{E}$$-expansion $$\mathcal{A}_{E}$$ of an $$L$$-structure$$\mathcal{A}$$, while strictly different from the structure $$\mathfrak{A}_{\mathcal{A}}$$ due to the definability in $$\mathcal{A}_{E}$$ of properties that were absent from $$\mathfrak{A}_{\mathcal{A}}$$, nonetheless consists of exactly the same positions. For, in $$\mathfrak{A}_{\mathcal{A}_{E}}$$, we are only making explicit certain properties — haecceities — that were already implicit in $$\mathfrak{A}_{\mathcal{A}}$$, properties of certain positions of $$\mathfrak{A}_{\mathcal{A}}$$ whose existence (unlike weakly potential properties like $$\boldsymbol{E}$$ in the preceding paragraph) follows as a mere logical consequence of the existence of those positions. Thus, in particular, the positions of $$\mathfrak{A}_{3}$$ are, for the ante rem structuralist, exactly those of $$\mathfrak{A}_{1}$$. From this — the ante rem structuralist concludes — it seems to follow that, by choosing our initial impure $$L_{\kappa_{1},\kappa_{2}}$$-expansion $$\mathcal{A}_{3}$$, the referents of the names $$\kappa_{1}$$ and $$\kappa_{2}$$ are fixed in such a way as to delegitimize $$\mathcal{A}'_{3}$$, the $$L_{\kappa_{1},\kappa_{2}}$$-expansion in which the references $$a_{1}$$ and $$a_{2}$$ of $$\kappa_{1}$$ and $$\kappa_{2}$$, respectively, are switched. For once we fix the referents of $$\kappa_{1}$$ and $$\kappa_{2}$$ in $$\mathcal{A}{}_{3}$$, $$\mathcal{A}'_{3}$$ is rendered in an important sense incorrect, the fact that it is isomorphic to $$\mathcal{A}_{3}$$ notwithstanding. For, assuming that $$\kappa_{1}$$ denotes $$a_{1}$$ and $$\kappa_{2}$$ denotes $$a_{2}$$, we intend that ‘$$x=\kappa_{1}$$’ expresses $$a_{1}$$’s identity property $$[\lambda x\,x=a_{1}]$$ in $$\mathfrak{A}_{3}$$ and that ‘$$x=\kappa_{2}$$’ expresses $$a_{2}$$’s, $$[\lambda x\,x=a_{2}]$$. If so, then only the $$L_{\kappa_{1},\kappa_{2}}$$-expansion $$\mathcal{A}_{3}$$ correctly represents $$\mathfrak{A}_{3}$$; $$\mathcal{A}'_{3}$$ — in which $$\kappa_{1}$$ denotes $$a_{2}$$ and $$\kappa_{2}$$ denotes $$a_{1}$$, represents the extensions of the two intended identity properties incorrectly. A similar argument can be applied to Robinson expansions and the constants $$\kappa_{i}$$ and $$\kappa_{j}$$ denoting $$i$$ and $$j$$, respectively. This suggests that RD is in fact a structurally legitimate notion of discernibility after all and, hence, that RII is a structurally defensible indiscernibility principle. I think we can concede that the positions of $$\mathfrak{A}_{3}$$ are indeed those of $$\mathfrak{A}_{1}$$. But that alone is not enough to avoid overspecification. For, even with the concession noted, the reasoning above would be compelling only if the metalinguistic expressions ‘$$a_{1}$$’ and ‘$$a_{2}$$’ (and, consequently, ‘$$\mathcal{A}{}_{3}$$’ and ‘$$\mathcal{A}'_{3}$$’) were being used as names with fixed referents in the preceding paragraph. But it is a mistake to think that these expressions are functioning as anything akin to names at all. Shapiro [2008, p. 300] suggests that constants like ‘$$a_{_{1}}$$’ and ‘$$a_{2}$$’, or, in the context of the complex field, ‘$$i$$’, function like the arbitrary names introduced into a natural deduction proof by some versions of Existential Instantiation: The mathematical community first notes that there is only one algebraic closure of the real numbers, up to isomorphism. Members of the community decide to study or otherwise discuss this algebraic closure. They note that, in this structure, there is at least one square root of $$-1$$: $$(\exists x)(x^{2}=-1)$$. So they let $$i$$ be one such square root, and go on from there. We have that $$i^{2}=-1$$. It follows that $$-i$$ is the only other square root of $$-1$$. One might note, in line with existential elimination, that there is nothing to be said about $$i$$ that does not hold of every square root of $$-1$$. This is as it should be, since the two roots are indiscernible. I think this analysis is more or less correct as far as it goes but should be pushed a bit farther. For Shapiro (ibid., p. 297) allows that ‘$$i$$’ is, ‘at least prima facie’, a genuine proper name.35 It seems to me, however, that we use ‘$$i$$’, not as a name, which requires a fixed referent, but as a variable that is bound by an existential quantifier with, typically, very broad scope — most of an entire text on complex analysis, for example. That is, in any given discussion of the complex field it is at some point noted that there is a square root of $$-1$$and some other stuff — perhaps lots of other stuff — we want to say about it; formally, where $$\varphi$$ is that other stuff: $$(\exists i)(i^{2}=-1\land\varphi(i))$$. Pragmatically, there might be what appears to be an act of baptism — ‘Let $$i$$ be one such square root!’ — but, as there are provably more than one, nothing fixes ‘$$i$$’ semantically to either of them. Instead, the act is better thought of, not as a naming in any sense, but as a reminder of the kind of thing we are talking about and that what we are about to say holds true of any of them in virtue of being a thing of that kind. Returning, then, to the argument above for the thesis that one of the two $$L_{\kappa_{1},\kappa_{2}}$$-expansions $$\mathcal{A}_{3}=\langle A,V_{3}\rangle$$ and $$\mathcal{A}'_{3}=\langle A,V'_{3}\rangle$$ of the $$L$$-structure $$\mathcal{A}_{1}$$ for $$\mathfrak{A}_{1}$$ is in some way privileged: we note that the argument depends on treating ‘$$a_{1}$$’ and ‘$$a_{2}$$’ as names that we take to refer determinately to the two top positions of $$\mathfrak{A}_{1}$$ and, hence that, in our initial choice of the $$L_{\kappa_{1},\kappa_{2}}$$-expansion $$\mathcal{A}_{1}$$, $$\kappa_{1}$$ and $$\kappa_{2}$$ determinately refer specifically to those positions, respectively, and hence that ‘$$x=\kappa_{1}$$’ and ‘$$x=\kappa_{2}$$’ determinately express their identity properties. In fact, however, ‘$$a_{1}$$’ and ‘$$a_{2}$$’ are (metalinguistic) variables and there is no determining the referents of $$\kappa_{1}$$ and $$\kappa_{2}$$ at all. Rather, following the analysis above, the situation — more carefully expressed — is this: there are two ‘top’ positions $$a_{1},a_{2}$$ of the structure $$\mathfrak{A}_{1}$$ and, hence, also two $$L_{\kappa_{1},\kappa_{2}}$$-expansions $$\mathcal{A}_{3}=\langle A,V_{3}\rangle$$ and $$\mathcal{A}'_{3}=\langle A,V'_{3}\rangle$$ of the $$L$$-structure $$\mathcal{A}_{1}$$ for $$\mathfrak{A}_{1}$$, where $$V_{3}(\kappa_{1})=a_{1}$$ and $$V_{3}(\kappa_{2})=a_{2}$$, and where $$V'_{3}$$ reverses those assignments. Otherwise put, we note only that there are functions $$V_{3}$$ and $$V'_{3}$$ mapping the constants to the two top positions of $$\mathfrak{A}_{1}$$ and hence that there are two corresponding expansions $$\mathcal{A}_{3}$$ and $$\mathcal{A}'_{3}$$; but it is never the case that any of these — the top positions, the functions, the $$L$$-systems — is determinately named, only quantified over. So viewed, we have only that there are two $$L_{\kappa_{1},\kappa_{2}}$$-systems which, qua expansions of $$\mathcal{A}_{1}$$, are entirely on a par; neither bears any privileged relation to $$\mathcal{A}_{1}$$ that the other does not. We cannot refer determinately to either of them any more than than we can refer determinately to either of the two ‘top’ positions in the structure $$\mathfrak{A}_{1}$$. That we have two isomorphic $$L_{\kappa_{1},\kappa_{2}}$$-expansions of $$\mathcal{A}_{1}$$ (and, given that the positions of $$\mathfrak{A}_{3}$$ are those of $$\mathfrak{A}_{1}$$, two $$L_{\kappa_{1},\kappa_{2}}$$-structures for $$\mathfrak{A}_{3}$$) is not a structural fact but a simple combinatorial consequence of the fact that there are two ‘top’ positions and two individual constants in $$L_{\kappa_{1},\kappa_{2}}$$ and, hence, two ways of assigning the latter uniquely (one-to-one) to the former. †Special thanks to my colleagues Michael Hand and José Bermúdez for spirited discussion of the issues discussed in this paper. Thanks also to Jeffrey Ketland for a number of very helpful discussions and e-mail exchanges, to Guillermo Badia for several corrections and improvements, to Ed Zalta and Bernie Linsky for discussion and encouragement, to a perceptive anonymous referee, to the attendees of a talk on this material that I gave at the Munich Center for Mathematical Philosophy, and to Hannes Leitgeb and the Humboldt Foundation (through its support of the MCMP) for making my visit to MCMP possible. Finally, my sincere thanks are due to Ken Dykema of Texas A&M’s mathematics department for his patience in answering my questions about abstract algebra and the complex field, many of which I fear struck him as either shockingly elementary or confused. 1 Thus, for example, ‘$$x+y=z$$’ is to be understood as shorthand for ‘$$\forall w(+(x,y,w)\leftrightarrow z=w)$$’ (or equivalent). 2 ‘Possible’ configurations and moves here could be cashed out (perhaps as higher-order entities of some ilk) in terms of worlds in one manner or another. E.g., a possible configuration might be a property of worlds, viz., those worlds in which that configuration is reached in some chess match (in which the rules of chess are not violated). 3Ketland [2006, 2011] calls this monadic indiscernibility, in contradistinction to polyadic indiscernibility (which turns out to be one expression of the strongest form of indiscernibility short of identity). See [Caulton and Butterfield, 2012, p. 51] on this choice of definition for absolute indiscernibility in light of the fact that there are stronger notions of indiscernibility in the conceptual neighborhood (notably, the negation of weak discernibility, discussed below, which Ketland calls strong indiscernibility and Ladyman et al. [2012] call utter indiscernibility). 4 In his [2008, p. 86], Shapiro repudiates the hard structuralism of his [1997] and rejects AII. Our purpose here, however, is to investigate a defense of a particular version of structuralism; so we want a clear statement of it. 5 That is, if, for all $$a\in A$$, there is a new constant $$\kappa_{a}\in L^{+}$$ such that $$\kappa_{a}^{\mathcal{A}^{+}}=a$$. This is of course a common sort of construction in model theory, although the name ‘Robinson expansion’ appears to be Bermúdez’s own label. He also defines the notion somewhat differently (though equivalently). 6 More philosophically put, a haecceity is a property that (a) is essential to its bearer, and (b) could not have been exemplified by anything other than that bearer. Such properties were made especially prominent in the late twentieth century by Plantinga; see esp. his [1979]. Somewhat surprisingly, I have found in conversation with Bermúdez that he is not particularly friendly toward haecceities. Instead, with regard to his proposed solution to Burgess’s challenge, he prefers only to focus on the the anti-haecceities $$R_{\nu\neq\kappa_{j}}^{\mathcal{C}^{+}}$$ and $$R_{\nu\neq\kappa_{i}}^{\mathcal{C}^{+}}$$, presumably because, unlike haecceities, they are multiply-exemplifiable. I myself cannot see anything other than two sides of a single coin here. (The presence of identity in the language is of course also crucial — see [Ladyman et al., 2012, § 6.2] for more on this point.) 7 To verify this rather obvious point: suppose that $$a\neq b$$. Let $$L^{+}=L\cup\left\{ \kappa_{a},\kappa_{b}\right\} $$ and let the $$L^{+}$$-structure $$\mathcal{A}^{+}=\langle A,V^{+}\rangle$$ be a name expansion of $$\mathcal{A}$$ such that $$V^{+}(\kappa_{a})=a$$ and $$V^{+}(\kappa_{b})=b$$. Since $$a\neq b$$, $$V^{+}(\kappa_{a})\neq V^{+}(\kappa_{b})$$. Hence $$\mathcal{A}^{+}\models x=\kappa_{a}[a]$$ and $$\mathcal{A}^{+}\not\models x=\kappa_{a}[b]$$; so $$a$$, but not $$b$$, has the property $$R_{x=\kappa_{a}}^{A^{+}}$$. So $$a$$ and $$b$$ are absolutely discernible in the name expansion $$\mathcal{A}^{+}$$ of $$\mathcal{A}$$ and, hence, by RD, they are Robinson discernible in $$\mathcal{A}$$. 8 I follow Ladyman et al. [2012, p. 165] in the choice of ‘utterly indiscernible’ here. Ketland [2011, p. 174] uses ‘strongly indiscernible’ but ordinary usage would suggest that strong (= utter) indiscernibility is weaker than absolute indiscernibility when, in fact, it is stronger — as pointed out in the paragraph immediately following WII, $$i$$ and $$j$$ are absolutely indiscernible in $$\mathcal{C}$$ but not utterly so. 9 Thus the extensive studies by Ketland [2011], Ladyman et al. [2012], and Caulton and Butterfield [2012] all explore indiscernibility via both languages with identity and languages without it. 10Button’s [2006, pp. 218–220] is a dissenting voice here. He argues that there are no primitive identity facts and, as a consequence, that in fact there can be no genuine indiscernibles. To account for their apparent existence, he proposes instead (pp. 220–221) a ‘hybrid’ solution that distinguishes between basic structures like, on the one hand, the natural-number structure or the set-theoretic hierarchy that are rigid and, hence, contain no structural indiscernibles, and, on the other hand, constructed structures like $$\mathfrak{A}_{0}$$ and $$\mathfrak{C}$$, talk about which can be treated ‘eliminativistically’. (Parsons [2004, pp. 68–72] defends a similar idea, though allows that some basic structures might not be rigid.) See [Leitgeb and Ladyman, 2008, pp. 394–395] for a response to Button’s attacks on primitive identity. 11MacBride [2006, pp. 66–67] also appears to agree with Keränen and Bermúdez here. 12 Shapiro is speaking specifically of the natural numbers in this passage but, as I read it, only as an example of the more general claim. 13Keränen [2001, pp. 316-317] argues that genuinely structural properties are those that can be picked out by formulas that are free of names in his reconstruction of ante rem structuralism, although his justification is rather broader than simply the exclusion of haecceities. 14Caulton and Butterfield [2012, p. 35] discuss this move as well. 15 Most defenders of haecceities (including the originator of the idea, Scotus himself — see [Cross, 2009, esp. § 3]) do not think of them as qualitative; so my appropriation of the term in my definition of ‘purely qualitative’ above might seem tendentious. However, the definition is purely stipulative and plays no essential role in my broader argument. 16 Though of course either is definable in terms of the other; that is, given a language with predicates for $$\boldsymbol{P}$$ and $$\boldsymbol{H}$$, $$\boldsymbol{H'}$$ is definable, and likewise given predicates for $$\boldsymbol{P}$$ and $$\boldsymbol{H'}$$. But that is neither here nor there for our purposes. 17 Haecceitism is nicely characterized by Kaplan [1975, pp. 722–723] as ‘[t]he doctrine that holds that it does make sense to ask — without reference to common attributes and behavior — whether this is the same individual in another possible world $$\ldots$$ and that a common “thisness” may underlie extreme dissimilarity or distinct thisnesses may underlie great resemblance.’ 18Ladyman et al. [2012] implicitly acknowledge this point. They introduce the notion of a haecceity informally as ‘the property of being identical with a particular object’ (fn. 1) but later (p. 172) define a haecceity within an $$L$$-structure for a language formally simply to be a formula$$\psi_{a}(x)$$ of $$L$$ that is true of exactly one thing $$a$$ — in effect, the definition adopted here. 19 I am assuming here the intuitive validity of the principle of $$\lambda$$-conversion, $$\varphi_{a}^{x}\leftrightarrow[\lambda x\,\varphi]a$$. The $$\lambda$$-predicates in this paragraph are of course metalinguistic, not part of any particular object language. 20 This argument is similar to one given by Caulton and Butterfield [2012, pp. 36–37] to support the conclusion that commitment to transworld identity — that is, to the possibility of objects that exist in more than one possible world — commits one to the existence of haecceities, and vice versa. Unlike them, I focus on facts of distinctness rather than identity, as it seems to me that they are more structurally ‘salient’ — notably, such facts are in a sense constitutive of simple cardinality structures like $$\mathfrak{A}_{0}$$ of Figure 1. Of course, the existence of haecceities follows directly by the same sort of argument as the one given for anti-haecceities — from bare identity facts $$a=a$$ it follows that $$[\lambda x\,x=a]a$$. 21 Generalizing the argument in the preceding subsection, in a finite structure, all possible $$n$$-place relations over a structure are implicit, as any such relation can be abstracted from a disjunction of conjunctions of identities. Thus, for example, letting $$a$$ and $$b$$ be the positions in the simple 2-element structure $$\mathfrak{A}_{0}$$ in Figure 1, the binary relation $$\boldsymbol{R}=\left\{ \langle a,a\rangle,\langle a,b\rangle\right\} $$ is implicit in the structure, as its existence follows logically from the logical truth $$(a=a\land a=a)\lor(a=a\land a=b$$). For, by $$\lambda$$-conversion, the preceding proposition entails that exactly the pairs $$\langle a,a\rangle$$ and $$\langle a,b\rangle$$ stand in the relation $$[\lambda xy\,(x=a\land y=a)\lor(x=a\land y=b)]=\boldsymbol{R}$$. Relations on infinite structures are of course not in general implicit in this sense, but merely potential (assuming standard first-order logic in the metalanguage). 22 I remind the reader that, although I have been using the language of ante rem structuralism, the truth of ante rem structuralism is an orthogonal issue; it does not really matter to my argument whether or not we identify$$a_{0}$$, $$a_{1}$$, and $$a_{2}$$ with the actual positions of $$\mathfrak{A}_{1}$$ or simply take $$\mathcal{A}_{1}$$ to be an arbitrary $$L$$-system exhibiting the structure (however understood ontologically) in question. That said, a response to the arguments here on the assumption of ante rem structuralism is considered in § 5.4. 23 It seems to me that this observation constitutes an argument that no proper substructure $$\mathfrak{A}'$$ of a structure $$\mathfrak{A}$$ is in any literal sense a part of the superstructure $$\mathfrak{A}$$ and in particular that the positions of a substructure are not identical to the corresponding positions in the superstructure. Some structuralists have argued that there is no fact of the matter about this, notably, Resnik [1981, pp. 536–537] and an earlier temporal stage of Shapiro [1997, pp. 79–81], a view he later came explicitly to reject in [Shapiro, 2006, pp. 28 ff.]). 24Shapiro [2008, fn. 6] hints at this idea in his discussion of names of indiscernibles switching referents. 25QR-systems without $$+$$ are introduced in a proof that absolute indiscernibility does not imply symmetry within a fixed $$L$$-system in [Ladyman et al., 2012, p. 182]. I arrived at the idea of using QR-systems to show that absolute indiscernibility does not imply structural indiscernibility long before discovering [Ketland, 2011], [Ladyman et al., 2012], and [Caulton and Butterfield, 2012] but only became reasonably clear on the surrounding philosophical and technical issues after studying these important papers. 26 It seems clear that the ante rem structuralist must deny that a proper substructure is a proper part of the corresponding superstructure and indeed this denial is part of the ante rem argument in § 5.4, but the issue is a live one only on the assumption of ante rem structuralism. 27 Strictly speaking, $$+$$ has to be considered a 3-place predicate rather than 2-place function symbol, lest $$q+^{V_{E}}r$$ be undefined for $$q\in Q,\ r\in R$$ (and recall that our model-theoretic framework as defined in § 2.2 does not include function symbols anyway). The functional identity $$x+y=z$$ above should therefore be thought of as a more familiar rewriting of $$+\!(x,y,z)$$ and, hence, $$x+x=x$$ a rewriting of $$+\!(x,x,x)$$. 28 As noted with regard to $$+$$ in the preceding footnote, $$\times$$ has to be a considered 3-place predicate rather than a 2-place function symbol. 29 This follows directly from the fact that any $$L_{\times}$$-expansion $$\mathcal{S}_{\times}'=\langle Q\cup R,V_{\times}'\rangle$$ of $$\mathcal{S}$$ that is isomorphic to $$\mathcal{S}_{\times}$$ must in fact be identical to$$\mathcal{S}_{\times}$$. See Theorem 3 in the Appendix for a proof. 30 That is, the Robinson expansion $$\mathcal{C}_{g}^{+}$$ such that, for every constant $$\kappa$$ of the expanded language $$L_{F}^{+}$$ of $$\mathcal{C}^{+}$$, $$\kappa^{\mathcal{C}{}_{g}^{+}}=g(\kappa^{\mathcal{C}{}^{+}})$$. Since $$g$$ is a symmetry and $$L_{F}^{+}$$ only contains new constants, for $$n$$-place predicates $$\pi$$ of $$L_{F}^{+}$$, $$\pi^{\mathcal{C}_{g}^{+}}=\pi^{\mathcal{C}^{+}}$$. 31 Thus Shapiro [2008, p. 287]: ‘Frankly, I am not sure what is being demanded. The fact that it is a theorem of complex analysis that $$- 1$$ has two distinct square roots seems to be enough to distinguish them, or at least enough to convince us that there are two, and not just one. What else is required?’ And in [Shapiro, 2012, p. 381] he insists outright that ‘there is no individuation $$\ldots$$ requirement at all’ in ante rem structuralism. 32 As pointed out in fn. 21, all $$n$$-place relations are implicit in finite structures; so an infinite structure like $$\mathfrak{C}'$$ is required to provide an example of a weakly potential property. 33 In particular, the property of being a real number is not definable in the complex field and, hence, there is no way to express a complex number’s standard form $$a+bi$$, for reals $$a$$ and $$b$$. 34 Another way to put this is that, while the $$L_{F\cup\{\varepsilon\}}$$-system $$\mathcal{C}'$$exhibits the structure $$\mathfrak{C}'$$, it is not an $$L_{F\varepsilon}$$-structure for $$\mathfrak{C}'$$ since its domain is not the set of positions of $$\mathfrak{C}'$$. See the definitions in § 2.2. 35 Or, following Brandom [1996, pp. 313–314], perhaps more weakly, ‘merely distinguishing terms’, although I have no clear fix on how such terms are supposed to differ from proper names. Nodelman and Zalta [2014, pp. 31ff] also suggest a solution similar to Shapiro’s, where ‘$$i$$’ is an arbitrary name introduced in mathematical reasoning by Existential Introduction but with radically different ontological underpinnings, as on their view the two roots of $$-1$$ are not elements of the complex field structure $$\mathfrak{C}$$. 36 The existence of such an $$f$$ given the absolute indiscernibility of $$a$$ and $$b$$ follows by Theorem 2 of [Caulton and Butterfield, 2012, p. 58] if we restrict ourselves to finite languages $$L$$. My thanks to Guillermo Badia for pointing out that this restriction is unnecessary if we use Hodges’s Theorem. 37$$f\!\upharpoonright\!X$$ is the restriction of the function $$f$$ to the subset $$X$$ of its domain. 38 At the least, there will always be expansions that introduce names for $$a$$ and $$b$$, of course; so the existence of $$\mathcal{A}_{E}$$ is unproblematic. Appendix Theorem 1. Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-system. Objects $$a,b\in A$$ are absolutely discernible in $$\mathcal{A}$$ only if they are structurally discernible over $$\mathcal{A}$$. Proof. Suppose $$a$$ and $$b$$ are absolutely discernible in $$\mathcal{A}$$, so that there is some formula $$\varphi$$ of $$L$$ such that $$R_{\varphi}^{\mathcal{A}}(a)$$ but not $$R_{\varphi}^{\mathcal{A}}(b)$$ and, hence, $$\mathit{tp}_{\mathcal{A}}(a)\neq\mathit{tp}_{\mathcal{A}}(b)$$. Let $$E=\varnothing$$, so that $$\mathcal{A}=\mathcal{A}_{E}$$; this satisfies clause (i) of the definition of structural discernibility. Regarding (ii), as $$\mathcal{A=A}_{E}$$, the only expansion $$\mathcal{A}'_{E}$$ of $$\mathcal{A}$$ that can be isomorphic to $$\mathcal{A}_{E}=\mathcal{A}$$ is $$\mathcal{A}$$ itself. So let $$f$$ be any isomorphism from $$\mathcal{A}$$ to $$\mathcal{A}$$ (i.e., any symmetry on $$\mathcal{A}$$). Then, since $$R_{\varphi}^{\mathcal{A}}(a)$$, by a simple result of Ladyman et al. [2012, p. 180, Theorem 9.2] we must have $$R_{\varphi}^{\mathcal{A}}(fa)$$. But, by assumption, not $$R_{\varphi}^{\mathcal{A}}(b)$$, so it cannot be that $$f(a)=b$$. □ Theorem 2. For finite $$L$$-systems $$\mathcal{A}$$, $$a$$ and $$b$$ are absolutely discernible in $$\mathcal{A}$$ if and only if they are structurally discernible over $$\mathcal{A}$$. Proof. That absolute discernibility in $$\mathcal{A}$$ entails structural discernibility over $$\mathcal{A}$$ generally was shown in Theorem 1. For the converse of the theorem, suppose $$a$$ and $$b$$ are absolutely indiscernible in $$\mathcal{A}$$. Then $$\mathit{tp}_{\mathcal{A}}(a)=\mathit{tp}_{\mathcal{A}}(b)$$ and, hence, $$a$$ and $$b$$ have the same complete type over $$\varnothing$$ in the sense of [Hodges, 1993, p. 277]. Since finite structures are all $$\lambda$$-big for any infinite $$\lambda$$ (ibid., pp. 480, 483), by Hodges’s Theorem 6.3.2(b) (p. 279), there is a symmetry $$f$$ on $$\mathcal{A}$$ such that $$f(a)=b$$.36 By Theorem 4 below, it follows that $$a$$ and $$b$$ are structurally indiscernible over $$\mathcal{A}$$. □ Theorem 3. For any $$L_{\times}$$-expansion $$\mathcal{S}_{\times}'=\langle Q\cup R,V_{\times}'\rangle$$ of $$\mathcal{S}=\langle Q, V_{\times}\rangle$$, if $$\mathcal{S}_{\times}'$$ is isomorphic to $$\mathcal{S}_{\times}$$, it is identical to $$\mathcal{S}_{\times}$$. Proof. Let $$f$$ be an isomorphism from $$\mathcal{S}_{\times}$$ to $$\mathcal{S}_{\times}'$$. Note first that, since $$\mathcal{S}_{\times}$$ and $$\mathcal{S}_{\times}'$$ are both expansions of $$\mathcal{S}$$, $$+^{\mathcal{S}{}_{\times}}=+^{\mathcal{S}_{\times}'}=+^{\mathcal{S}}=\boldsymbol{+_{Q}}\cup\boldsymbol{+_{R}}$$; hence, since $$f$$ is an isomorphism, $$f(0_{Q})$$ must be one of the additive identity elements $$0_{Q}$$ or $$0_{R}$$. But it cannot be the latter. For suppose otherwise, i.e., that $$f(0_{Q})=0_{R}$$. Since $$R$$ is uncountable and $$f$$ is one-to-one, there must be elements of $$R$$ that are not the value of $$f$$ for any $$q\in Q$$. Let $$r$$ be such an element and let $$r'\in R$$ be such that $$f(r')=r$$. Since $$r\in R$$, either $$\mathcal{S}'_{\times}\models z<y[r,0_{R}]$$ or $$\mathcal{S}'_{\times}\models z<y[0_{R},r]$$. Since $$f$$ is an isomorphism, it follows that either $$\mathcal{S}{}_{\times}\models z<y[f^{-1}(r),f^{-1}(0_{R})]$$ or $$\mathcal{S}{}_{\times}\models z<y[f^{-1}(0_{R}),f^{-1}(r)]$$, i.e., either $$\mathcal{S}{}_{\times}\models z<y[r',0_{Q}]$$ or $$\mathcal{S}{}_{\times}\models z<y[0_{Q},r']$$. But both are impossible. For since $$<^{\mathcal{S}{}_{\times}}=<^{\mathcal{S}}=\boldsymbol{<_{Q}}\cup\boldsymbol{<_{R}}$$, we can have $$\mathcal{S}_{\times}\models z<y$$[$$a,b$$] only if $$a,b\in Q$$ or $$a,b\in R$$. Hence, it must be that $$f(0_{Q})=0_{Q}$$. It follows that $$f(0_{R})=0_{R}$$ as well (since $$f$$ is one-to-one and $$f(0_{R})$$ must be an additive identity element). Given these facts about $$f$$, similar reasoning about $$<^{\mathcal{S}_{\times}}$$ shows that, for all $$q\in Q$$, $$r\in R$$, $$f(q)\in Q$$ and $$f(r)\in R$$. It follows that $$f\!\upharpoonright\!Q$$ and $$f\!\upharpoonright\!R$$ must be automorphisms.37 However, there are no nontrivial automorphisms on the rational and real fields. Hence, $$\mathcal{S}'_{\times}=\mathcal{S}_{\times}$$. □ Theorem 4. Let $$\mathcal{A}=\langle A,V\rangle$$ be an $$L$$-system. $$a$$ and $$b$$ are symmetric in $$\mathcal{A}$$ if and only if they are structurally indiscernible over $$\mathcal{A}$$. Proof. Suppose $$a$$ and $$b$$ are symmetric in $$\mathcal{A}$$ and let $$\mathcal{A}_{E}=\langle A,V_{E}\rangle$$ be an $$L_{E}$$-expansion of $$\mathcal{A}$$ in which they are absolutely discernible. As in Theorem 2, to show that $$a$$ and $$b$$ are structurally indiscernible over $$\mathcal{A}$$, we need to show that there is an $$L_{E}$$-expansion $$\mathcal{A}'_{E}$$ of $$\mathcal{A}$$ that is isomorphic to $$\mathcal{A}{}_{E}$$ but in which $$b$$ plays the role that $$a$$ plays in $$\mathcal{A}_{E}$$, i.e., that there is an isomorphism $$f$$ from $$\mathcal{A}_{E}$$ to $$\mathcal{A}'_{E}$$ such that $$f(a)=b$$. But this is straightforward. By assumption, $$a$$ and $$b$$ are symmetric in $$\mathcal{A}$$; so let $$f:A\longrightarrow A$$ be a symmetry on $$\mathcal{A}$$ such that $$f(a)=b$$ and let $$\mathcal{A}'_{E}=\langle A,V'_{E}\rangle$$ be the $$L_{E}$$-expansion of $$\mathcal{A}$$ such that $$\pi^{\mathcal{A}'_{E}}=f[\pi^{\mathcal{A}_{E}}]$$, for predicates $$\pi\in E$$ and $$\kappa^{\mathcal{A}'_{E}}=f(\kappa^{\mathcal{A}_{E}})$$ for constants $$\kappa\in E$$. By definition, then, $$f$$ meets the condition for being an isomorphism from $$\mathcal{A}{}_{E}$$ to $$\mathcal{A}'_{E}$$ on the new vocabulary in $$E$$. Since $$f$$ is a symmetry on $$\mathcal{A}$$, by definition $$\pi^{\mathcal{A}}=f[\pi^{\mathcal{A}}]$$ and $$\kappa^{\mathcal{A}}=f(\kappa^{\mathcal{A}})$$ for predicates $$\pi\in L$$ and constants $$\kappa\in L$$. But, as both $$\mathcal{A}_{E}$$ and $$\mathcal{A}'_{E}$$ are by definition expansions of $$\mathcal{A}$$, $$V_{E}$$ and $$V'_{E}$$ do not alter the values of $$V$$ on $$L$$, i.e., $$V_{E}\upharpoonright L=V'_{E}\upharpoonright L=V$$. Hence, for such $$\pi,\kappa\in L$$, $$\pi^{\mathcal{A}'_{E}}=f[\pi^{\mathcal{A}_{E}}]$$ and $$\kappa^{\mathcal{A}'_{E}}=f(\kappa^{\mathcal{A}_{E}})$$. So $$f$$ is an isomorphism from $$\mathcal{A}_{E}$$ to $$\mathcal{A}'_{E}$$ and, by choice of $$f$$, $$f(a)=b$$. For the converse, suppose $$a$$ and $$b$$ are structurally indiscernible over $$\mathcal{A}$$. 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Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

Philosophia Mathematica – Oxford University Press

**Published: ** Feb 1, 2018

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