On Kinds of Indiscernibility in Logic and Metaphysics

Using the Hilbert-Bernays account as a spring-board, we first define four ways in which two objects can be discerned from one another, using the non-logical vocabulary of the language concerned. (These definitions are based on definitions made by Quine and Saunders.) Because of our use of the Hilbert-Bernays account, these definitions are in terms of the syntax of the language. But we also relate our definitions to the idea of permutations on the domain of quantification, and their being symmetries. These relations turn out to be subtle---some natural conjectures about them are false. We will see in particular that the idea of symmetry meshes with a species of indiscernibility that we will call `absolute indiscernibility'. We then report all the logical implications between our four kinds of discernibility. We use these four kinds as a resource for stating four metaphysical theses about identity. Three of these theses articulate two traditional philosophical themes: viz. the principle of the identity of indiscernibles (which will come in two versions), and haecceitism. The fourth is recent. Its most notable feature is that it makes diversity (i.e. non-identity) weaker than what we will call individuality (being an individual): two objects can be distinct but not individuals. For this reason, it has been advocated both for quantum particles and for spacetime points. Finally, we locate this fourth metaphysical thesis in a broader position, which we call structuralism. We conclude with a discussion of the semantics suitable for a structuralist, with particular reference to physical theories as well as elementary model theory.

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1 Introduction

1.1 Prospectus

This is one of several papers about identity. Their overall topic is quantum theory: where there has been discussion over many decades about the treatment of indistinguishable (also known as: identical) particles.333Castellani [1998] is a valuable collection of classic and contemporary articles. French and Krause [2006] is a thorough recent monograph. This discussion has been invigorated in the last few years, principally by Saunders’ revival of the Hilbert-Bernays account of identity (also briefly discussed by Quine) and his application of it to quantum theory.

In short, Saunders saw that there is an error in the consensus of the previous philosophical literature. That literature had shown that for any assembly of indistinguishable quantum particles (fermions or bosons), and any state of the assembly (appropriately (anti)-symmetrized), and any two particles in the assembly: the reduced density matrices (reduced states) of the particles (and so all probabilities for single-particle measurements) were equal; and so were appropriate corresponding two-particle conditional probabilities. This result strongly suggests that quantum theory endemically violates the principle of the identity of indiscernibles: for any two particles in the assembly are surely indiscernible.

444This consensus seems to have been first stated by Margenau ([1944], pp. 201-3) and it is endorsed and elaborated by e.g. French and Redhead ([1988], p. 241), Butterfield ([1993], p. 464), and French ([2006], §4). Massimi ([2001], pp. 326-7) questions these authors’ emphasis on monadic properties, but agrees that quantum mechanics violates the identity of indiscernibles when the quantum state is taken to codify purely relational properties.

Saunders’ basic insight ([2006], p. 7) is that the Hilbert-Bernays account provides ways that two objects can be distinct, which are not captured by these orthodox quantum probabilities: and yet which are instantiated by quantum theory. Thus on the Hilbert-Bernays account, two objects can be distinct, even while sharing all their monadic properties and their relations to all other objects. For they can be distinguished by either:
(1): a relation between them holding one way but not the other (which is called ‘being relatively discernible’);
(2): a relation between them holding in both directions but neither object having to itself (‘being weakly discernible’).
Saunders then argues that the second case, (2), is instantiated by fermions: the prototype example being the relation = ‘… has opposite value for spin (in any spatial direction) to ’ for two spin- fermions in the singlet state.

Saunders’ proposals have led to several developments, including exploring the parallel between quantum particles and spacetime points. But these will be the focus of other papers. Here we will have enough to do, answering some (broadly taxonomic) questions that arise from the Hilbert-Bernays account, and relating our answers to some associated (even venerable!) metaphysical theses—without regard to quantum theory.

More specifically, our aim here will be to define four different ways in which objects can be discerned from one another (two corresponding to (1) and (2) above), and to relate these definitions: (i) to the idea of symmetry; (ii) to one another, and (iii) to some metaphysical themes. These four ways of being discerned will be (our formulations of) definitions made by Quine and Saunders in terms of the Hilbert-Bernays account.

The Hilbert-Bernays account is a reductive account, reducing identity to a conjunction of statements of indiscernibility. So this account is controversial: many philosophers hold that identity is irreducible to any sort of qualitative facts, but nevertheless wholly unproblematic because completely understood.555For example, Lewis ([1986], pp. 192-193). More generally: in the philosophy of logic, the Fregean tradition that identity is indefinable, but understood, remains strong—including as a response to the Hilbert-Bernays account: cf. e.g. Ketland ([2006], p. 305 and Sections 5, 7). This latter view is certainly defensible, perhaps orthodox, once one sets aside issues about diachronic identity—as we will in this paper. But this is not a problem for us: for we are not committed to the Hilbert-Bernays account (and nor is Saunders’ basic insight about weak discernibility in quantum theory). On the contrary, we will eventually (in Section 5) unveil a metaphysical thesis which unequivocally denies the Hilbert-Bernays account, and which elsewhere we will apply to quantum theory. But these controversies do not undercut the rationale for our investigations into the kinds of discernibility; essentially because anyone, whatever their philosophy of identity or their attitude to the identity of indiscernibles, will accept that discernibility is a sufficient condition for diversity (the ‘non-identity of discernibles’).

So to sum up, our aim is to use the Hilbert-Bernays account as a spring-board so as to give a precise “logical geography” of discernibility. This logical geography will be in terms of the syntax of a formal first-order language. But we will also relate our definitions to the idea of permutations on the domain of quantification, and to the idea of these permutations being symmetries. These relations seem not to have been much studied in the recent philosophical literature about the Hilbert-Bernays account; and we will see that they turn out to be subtle—some natural conjectures are false.666We thank N. da Costa and J. Ketland for alerting us to their results in this area, which we (culpably!) had missed. Further references below.

The plan of the paper is as follows. After stating some stipulations about jargon (Section 1.2), we will state the Hilbert-Bernays account in terms of a first-order language (Section 2.1) and relate it to permutations on a domain of quantification (Section 2.2). With these preparations in hand, we will be ready to define four ways in which objects can be discerned from each other, using the predicates of the language (Section 3). These definitions will include precise versions of (1) and (2) above. Then in Section 4, we give several results relating permutations and symmetries to a kind of indiscernibility which we (following Quine and Saunders) will call ‘absolute’. We admit that one could instead define kinds of indiscernibility in terms of permutations and symmetries, rather than the instantiation of formulas. So in the second Appendix (Section 8) we explore this approach, emphasising the consequences for Section 4’s results. (The first Appendix, Section 7, surveys the logical implications between our four kinds of indiscernibility.) Finally, in Section 5

, we will use our four kinds of discernibility to state four metaphysical theses. (There will also be a fifth “umbrella” thesis called ‘structuralism’.) Three of these theses articulate two traditional philosophical themes: viz. the principle of the identity of indiscernibles (which will come in two versions), and haecceitism. The fourth is recent. Its most notable feature is that it makes diversity (i.e. non-identity) weaker than what we will call individuality (being an individual): two objects can be distinct but not individuals. For this reason, it has been advocated both for quantum particles and for spacetime points. We will classify and compare these theses in two ways: in terms of two philosophical questions (Section

5.1), and in terms of the conflicting ways in which they count possible structures (Section 5.2). We conclude (Section 5.3) by relating these theses to the position we call ‘structuralism’: which prompts a non-standard (though familiar) interpretation of, or even a revision of, traditional formalisms in both semantics and physical theories.

1.2 Stipulations about jargon

We make some stipulations about philosophical terms. We think that all of them are natural and innocuous, though the last one, about ‘individual’, is a bit idiosyncratic.

‘Object’, ‘identity’, ‘discernibility’:— We will use ‘object’ for the broad idea, in the tradition of Frege and Quine, of a potential referent of a singular term, or value of a variable. The negation of the identity relation on objects we will call (indifferently): ‘non-identity’, ‘distinctness’, ‘diversity’; (and hence use cognate words like ‘distinct’, ‘diverse’). When we have in mind that a formula applies to one of two objects but not the other, we will say that they are ‘discerned’, or that the formula ‘discerns’ them. We will also use ‘discernment’ and ‘discernibility’: these are synonyms; (though the former usefully avoids connotations of modality, the latter often sounds better). Their negation we will call ‘indiscernibility’.

‘Individual’:— Following a recent tradition started by Muller & Saunders [2008], we will also use ‘individual’ for a narrower notion than ‘object’, viz. an object that is discerned from others by a strong, and traditional, form of difference—which we will call ‘absolute discernibility’. Anticipating the following section, this usage may be illustrated by the fact a haecceitist (in our terms) would demand that all objects be individuals, in virtue of each possessing its own unique property.

We can see that it would be worth distinguishing cases according to whether the discernment is by an arbitrary language or an “ideal” one. That is: since the Hilbert-Bernays account will be cast in a formal first-order language, and such languages can differ as to their non-logical vocabulary (primitive predicates), our discussion will sometimes be relative to a choice of such vocabulary. So for example an object that fails to be an individual by the lights of an impoverished language may yet be an individual in an ideal language adequate for expressing all facts (especially all facts about identity and diversity). Therefore one might use the term ‘-individual’ for any object which is absolutely discerned from all others using the linguistic resources of the language . The un-prefixed term ‘individual’ may then be reserved for the case of ideal language. However, we will stick to the simple term ‘individual’, since it will always be obvious which language is under consideration. In Section 5 we will use ‘individual’ in the strictly correct sense just proposed, since there we envisage a language which is taken to be adequate for expressing all facts.

‘Haecceitism’:— Though the details will not be needed till Section 5, we should say what we will mean by ‘haecceitism’ (a venerable doctrine going back to Kaplan [1975] if not Duns Scotus!). The core meaning is advocacy of haecceities, i.e. non-qualitative thisness properties: almost always associated with the claim that every object has its own haecceity. But this core meaning is itself ambiguous, and authors differ about the implications and connotations of ‘haecceity’—about how “thick” a notion is advocated. Some discussion is therefore in order.

1.2.1 Haecceitism

At first sight, there are (at least) three salient ways to construe haecceitism.777We owe much of the discussion here to our conversations with Fraser MacBride. For each we give a description and one or more proponents. All three will refer to possible worlds, and will use the language of Lewis’s (especially [1986]) metaphysics (though it will not require a commitment his form of modal realism, or to any form of modal realism for that matter). The first is the weakest and the second and third are equivalent; we favour the second and third (and prefer the formulation in the third).

  1. Haecceitistic differences. Following Lewis ([1986], p. 221), we may take haecceitism to be about the way possible worlds represent the modal properties of objects. It is the denial of the following supervenience thesis: a world’s representation of de re possibilities (that is, possibilities pertaining to particular objects) supervenes on the qualitative mosaic, i.e. the pattern of instantiation of qualitative properties and relations, within that world.888This prompts the question, ‘What is a qualitative property or relation?’ That is not a question we will here attempt to answer. Thus a haecceitist allows that two worlds, exactly alike in their qualitative features, may still disagree as to which object partakes in which property or relation.999Note that this is a stronger position than one that just allows duplicate worlds, that is, several worlds exactly alike in their qualitative features. The existence of duplicate worlds does not entail Haecceitism in Lewis’s sense, since, for each individual, every world in a class of mutual duplicates may represent the same dossier of de re possibilities. Lewis ([1986], p. 224) himself refrains from committing either to duplicate worlds or a “principle of identity of indiscernibles” applied to worlds; though the question seems moot for anyone who does not take possible worlds to be concretely existing entities.

    This version of haecceitism makes no further claims as to what exactly is left out of the purely qualitative representations, and so it is the weakest of the three haecceitisms. But since two qualitatively identical worlds may disagree about what they represent de re, they must differ in some non-qualitative ways: ways which somehow represent (actual) objects in a way that does not rely on how things are qualitatively (whether accidentally or essentially), with those objects. There are two natural candidates for these representatives: the objects themselves (or some abstract surrogate for them), divorced from their qualitative clothing; or some non-qualitative properties which suitably track the objects across worlds. These two candidates prompt the second and third kinds of haecceitism (which, we argue below, are in fact equivalent). We see no sensible alternative to these two candidates for the missing representatives—though perhaps the difference could be taken as a primitive relation between worlds. But haecceitism in our first sense does not entail either of the haecceitisms below, though they each entail haecceitism in our first sense of the acceptance of haecceitistic differences.

  2. Combinatorial independence. Lewis’s definition was inspired by a definition by Kaplan ([1975], pp. 722-3); but Kaplan’s is stronger. It is phrased explicitly in terms of trans-world identification.101010Kaplan: [haecceitism is] “the 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, that individuals can be extended in logical space (i.e., through possible worlds) in much the way we commonly regard them as being extended in physical space and time, and that a common “thisness” may underlie extreme dissimilarity or distinct thisnesses may underlie great resemblance.” But we believe there is a version of haecceitism, clearer than Kaplan’s, defined in terms of combinatorial possibility; a version which is still stronger than Lewis’s, but does not commit one to claims about non-spatiotemporal overlap between worlds or trans-world mereological sums.

    According to this version of haecceitism, objects partake independently—that is, independently of each other and of the qualitative properties and relations—in the exhaustive recombinations which generate the full space of possible worlds. For example: with a domain of objects there are: -many possible property extensions (each of them distinct); -many distinct binary relation extensions (each of them distinct); so the number of distinct worlds111111Or distinct equivalence classes of world-duplicates! containing -objects, monadic properties and binary relations (and no other relations), is .

    Combinatorial independence would appear to favour the doctrine that objects “endure” identically through possible worlds, since it seems sensible to identify property extensions with sets of objects, and the same objects are added to or subtracted from the extensions in the generation of new worlds. But trans-world “perdurance”—the doctrine that trans-world “identity” (in fact a misnomer, according to the doctrine) is a relation holding between parts of the same trans-world continuant—can be accommodated without difficulty. (It is the commitment to there being a unique, objective trans-world continuation relation, and therefore something for a rigid designator to get a grip on, which distinguishes the perdurantist from those, like Lewis, who favour modal talk of world-bound objects: cf. Lewis ([1986], pp. 218-20).)

  3. Haecceitistic properties. Perhaps the most obvious form of haecceitism—and perhaps prima facie the least attractive—is the acceptance of haecceitistic properties. According to this view, for every object there is a property uniquely associated with it, which that object and no other necessarily possesses, and which is not necessarily co-extensive with any (perhaps complex) qualitative property. An imprecise (and even quasi-religious) reading of these properties is as “inner essences” or “souls” (hence the view’s unpopularity). Perhaps a more precise, and less controversial, reading is that each thing possesses some property which “makes” it that thing and no other: a property “in virtue of which” it is that thing (cf. Adams [1981], p. 13). Whether or not that in fact more precise, this reading implies a notion of ontological primacy—that the property comes in some sense “before” the thing—about which we remain silent. But we disavow this implication. But we intend our third version of haecceitism to be no more controversial than combinatorial independence; in fact we take this version of haecceitism to be equivalent to combinatorial independence.

    When we say that according to this view, there is a haecceitistic property corresponding to each object, by ‘object’ is not meant ‘world-bound object’, which would entail a profligate multiplication of properties. Rather, these haecceitistic properties are envisaged as an alternative means to securing trans-world continuation (understood according to either endurantism or perdurantism). The motivation is as follows.

    Throughout this paper, we are quidditists, meaning that we take for granted the trans-world identity of properties and relations. (We remain agnostic as to whether this trans-world identity is genuine identity, i.e. endurance of qualities; or whether there is instead a similarity relation applying to qualities between worlds (either second-order, applying directly to the qualities themselves; or else first-order, applying to the qualities indirectly, via the objects that instantiate them), i.e. perdurance.) So for the sake of mere uniformity, we may wish to accommodate haecceitism into our quidditistic framework by letting a trans-world monadic property do duty for each trans-world thing. Haecceitism and anti-haecceitism alike can then be discussed neutrally in terms of world-bound objects and trans-world properties and relations, the difference between them reconstrued in terms of whether or not there are primitive monadic properties which allow one to simulate rigid designation.121212Anti-quidditists (like Black [2001]) can still use our framework, by populating the domain of objects with properties and relations, now treated as the value of first-order variables, and using the new predicate or predicates ‘has’, so that ‘’ becomes ‘ has ’ (cf. Lewis [1970]). Anti-quidditism may then amount to what we later (Section 5) call ‘anti-haecceitism’, but applied to these hypostatized qualities. However, this purported anti-quidditism must be a “quidditist” about the ‘has’ relation(s), a situation that clearly cannot be remedied by hypostatizing again, on pain of initiating a Bradley-like regress. There is no space to pursue the issue here. But we endorse the view of Lewis [2002] that ‘has’ is a ‘non-relational tie’, and speculate that, if it is treated like a relation in the logic, then it is better off treated as one whose identity across worlds is not in question—that is: treated “quidditistically”.

    We can characterise this position syntactically as one that demands that, in a language adequate for expressing all facts, the primitive vocabulary contains a 1-place predicate ‘’ for each envisaged trans-world object .131313To simplify we assume that a haecceitist is a haecceitist about every object. This leaves some (uninteresting) logical space between haecceitism and what we later characterise as anti-haecceitism (see the end of Section 5.1, below). That a monadic property can do duty for a trans-world object—or, to rephrase in terms of the object-language, that a monadic predicate can do duty for a rigid designator—without any loss (or gain!) in expressive adequacy, is well known (cf. Quine [1960], §38). Given a haecceitistic predicate ‘’, one can introduce the corresponding rigid designator by definition: ; and conversely, given a rigid designator ‘’ and the identity predicate ‘=’, one can likewise introduce the corresponding haecceitistic predicate: . We therefore urge the view that the difference between the two stronger versions of haecceitism is merely notational, meaning that the addition of the haecceitistic predicate ‘’ to one’s primitive vocabulary ontologically commits one to no more, and no less, than the addition of the name ‘’, together with all the machinery of rigid designation.141414By saying that we intend the acceptance of haecceitistic properties to be equivalent to combinatorial independence, we do not intend to rule out as counting as a (strong) version of the acceptance of haecceitistic properties the view described above, viz. that the haecceities are in some sense ontologically “prior” to the objects, and serve to “ground their identity” (whatever that may mean). That view would entail combinatorial independence between objects and other objects, and between objects and the qualitative properties and relations (though not, of course, between objects and their haecceities), and would equally well be served by rigid designators as by the addition of haecceitistic predicates to the primitive vocabulary. The point is that our second and third versions of haecceitism match completely in logical strength, so they are equally consistent with more metaphysically ambitious views which seek to “ground” trans-world identity in non-qualitative properties.

From now on, we will in this paper take haecceitism in a sense stronger than the first, Lewisian version. The second and third, stronger versions are equivalent, but we favour the notational trappings of the third version, i.e. the acceptance of haecceitistic properties. Thus when we later ban names from the object-language (in Section 2.1), this ought to be seen not as a substantial restriction against the haecceitist, but merely a convenient narrowing of notational options for the sake of a more unified presentation. We simply require the haecceitist to express her position though the adoption of haecceitistic predicates, though all are at liberty to read each instance of the haecceitist’s ‘’ as ‘’.

A concern may remain: how can ontological commitment to a property be equivalent to ontological commitment to a (trans-world) object? There is no mystery, once we lay down some principles for what ontological commitment to properties involves. We take it that ontological commitment to a collection of properties, relations and objects entails a commitment to all the logical constructions thereof, because instantiation of the logical constructions can be defined away without residue in terms of instantiation and the existence of the original collection. (For example.: for any object, the complex predicate ‘’ is satisfied by that object just in case both ‘’ and ‘’ are satisfied by it.) So ontological commitment to logical constructions is no further commitment at all. Now, ontological commitment to certain objects is revealed clearly enough: one need only peer into the domain of quantification to see if they are there. Ontological commitment to complex properties and relations is equally straightforward, being a matter of commitment to their components. But what about ontological commitment to the properties and relations taken as primitive, i.e. not as logical constructions—what does that involve? Well, since ontological commitment to objects is clear enough, let us use that: let us say that ontological commitment to the primitive properties and relations is a commitment to their being instantiated by some object or objects.

With these principles laid down, we can now prove that commitment to the trans-world continuant and commitment to the haecceitistic property, being , entail each other. Left to right: We can take two routes. First route: Commitment to any object at all (i.e. a non-empty domain) entails commitment to the identity relation, which is instantiated by everything. So commitment to the trans-world continuant entails commitment to the identity relation, since . But the property being is just a logical construction out of the identity relation and the trans-world continuant , so commitment to the trans-world continuant entails commitment to the property being . Second route: Commitment to the trans-world continuant entails commitment to the property being being instantiated. But that is just to say that commitment to the trans-world continuant entails commitment to the property being . Right to left: Commitment to the property being entails either: (i) a commitment to its being instantiated (if taken as primitive); or (ii) a commitment to the entities of which it is a logical construction (if taken as a logical construction). If (i), then we are committed to something’s being , that is, the existence of . If (ii), and ‘being ’ is understood properly as containing a rigid designator, not as an abbreviated definite description à la Russell, then we are committed to its logical components, i.e. the identity relation, and itself. QED.

To sum up: the acceptance of haecceitistic differences (our first version of haecceitism, above) need not commit one to either combinatorially independent trans-world objects (our second version), or to non-qualitative properties that could do duty for them (our third version), though the two latter doctrines are perhaps the most natural way of securing haecceitistic differences, and may themselves be considered as notational variants of each other. We stipulate that we mean by ‘haecceitism’ one of the stronger versions, and for reasons purely to do with uniformity of presentation, we stipulate that the advocacy of this stronger version of haecceitism be expressed through the acceptance of haecceitistic properties. This version of haecceitism will be further developed, along with our three other metaphysical theses, in Section 5.

2 A logical perspective on identity

2.1 The Hilbert-Bernays account

What we will call the Hilbert-Bernays account of the identity of objects, treated as (the values of variables) in a first-order language, goes as follows; (cf. Hilbert and Bernays ([1934], §5: who in fact do not endorse it!), Quine ([1970], pp. 61-64) and Saunders ([2003a], p. 5)). The idea is that there being only finitely many primitive predicates enables us to capture the idea of identity in a single axiom. In fact, the axiom is a biconditional in which identity is equivalent to a long conjunction of statements that predicates are co-instantiated. The conjunction exhausts, in a natural sense, the predicates constructible in the language; and it caters for quantification in predicates’ argument-places other than the two occupied by and .

In detail, we proceed as follows. Suppose that is the th 1-place predicate, is the th 2-place predicate, and is the th 3-place predicate; (we will not need to specify the ranges of ). Suppose that the language has no names, or function symbols, so that predicates are the only kind of non-logical vocabulary. Then the biconditional will take the following form:

(For primitive predicates, we usually omit the brackets and commas often used to indicate argument-places.) Note that for each 2-place predicate, there are two biconditionals to include on the right-hand side; and similarly for a 3-place predicate. The general rule is: biconditionals for an -place predicate.151515We do not need to include explicitly on the right-hand side clauses with repeated instances of or , such as , since these clauses are implied by the conjunction of other relevant biconditionals. For example: From we have, in particular, that . And from we have, in particular, that . It follows that . A similar chain of arguments applies for an arbitrary -place predicate.

This definition of the Hilbert-Bernays account prompts three comments.

  1. Envisaging a rich enough language:— The main comment is the obvious one: since the right hand side (in square brackets) of (HB) defines an equivalence relation—which from now on we will call ‘indiscernibility’ (or for emphasis: ‘indiscernibility by the primitive vocabulary’)—discussion is bound to turn on the issue whether this relation is truly identity of the objects in the domain. Someone who advocates (HB) is envisaging a vocabulary rich enough (or: a domain of objects that is varied enough, by the lights of the vocabulary chosen) to discern any two distinct objects, and thereby force the equivalence relation given by the right hand side of (HB) to be identity. This comment can be developed in six ways.

    • Indiscernibility has the formal properties of identity expressible in first-order logic, i.e. being an equivalence relation and substitutivity. (Cf. eq. (2.1) in (i) of comment 3 below; and for details, Ketland ([2009], Lemma 12)).

    • Let us call an interpretation of the language, comprising a domain and various subsets of etc. a ‘structure’; (since in the literature on identity, ‘interpretation’ also often means ‘philosophical interpretation’). Then: if in a given structure, the identity relation is first-order definable, then it is defined by indiscernibility, i.e. the right hand side of (HB) (Ketland ([2009], Theorem 16)).

    • On the other hand: assuming ‘=’ is to be interpreted as the identity relation (cf. (i) in 3 below), there will in general be structures in which the leftward implication of (HB) fails; i.e. structures with at least two objects indiscernible from each other. We say ‘in general’ since from the view-point of pure logic, ‘=’ might itself be one of the 2-place predicates : in which case, the leftward implication trivially holds.

    • The last sentence leads to the wider question whether the interpretation given to some of the predicates etc. somehow presupposes identity, so that the Hilbert-Bernays account’s reduction of identity is, philosophically speaking, a charade. This question will sometimes crop up below (e.g. for the first time, in footnote 17); but we will not need to address it systematically. Here we can just note as an example of the question, the theory of pure sets. It has only one primitive predicate, ‘’, and the axiom (HB) for it logically entails the axiom of extensionality. But one might well say that this only gives a genuine reduction of identity if one can understand the intended interpretation of ‘’ without a prior understanding of ‘’.

    • There is the obvious wider question, why we restrict our discussion to first-order languages. Here our main reply is twofold: (a) first-order languages are favoured by the incompleteness of higher-order logics, and we are anyway sympathetic to the view that first-order logic suffices for the formalisation of physical theories (cf. Boolos & Jeffrey [1974], p. 197; Lewis [1970], p. 429); and (b) the Hilbert-Bernays account is thus restricted—and though, as we emphasized, we do not endorse it, it forms a good spring-board for discussing kinds of indiscernibility. There are also some basic points about identity in second-order logic, which we should register at the outset. Many discussions (especially textbooks: e.g. van Dalen [1994], p. 151; Boolos & Jeffrey [1974], p. 280) take the principle of the identity of indiscernibles to be expressed by the second-order formula : which is a theorem of (any deductive system for) second-order logic with a sufficiently liberal comprehension scheme. But even if one is content with second-order logic, this result does not diminish the interest of the Hilbert-Bernays account (or of classifications of kinds of discernibility based on it). For the second-order formula is a theorem simply because the values of the predicate variable include singleton sets of elements of the domain (cf. Ketland [2006], p. 313). And allowing such singleton sets as properties of course leads back both to haecceitism, discussed in Section 1.2, and to (iv)’s question of whether understanding the primitive predicates requires a prior understanding of identity. In any case, we will discuss the principle of the identity of indiscernibles from a philosophical perspective in Section 5.

    • Finally, there is the question what the proponent of the Hilbert-Bernays account is to do when faced with a language with infinitely many primitive predicates. The right hand side of (HB) is constrained to be finitely long, so in the case of infinitely many primitives it is prevented from capturing all the ways two objects can differ. Here we see two roads open to the Hilbert-Bernays advocate: (a) through parametrization, infinitely many -adic primitives, say may be subsumed under a single, new -adic primitive, say , where the extra argument place is intended to vary over an index set for the previous primitives; and (b) one may resort to infinitary logic, which allows for infinitely long formulas—suggesting, as regards (HB), in philosophical terms, a supervenience of identity rather than a reduction of it. (We will consider infinitary logic just once below, towards the end of Section 4.3.)

    To sum up these six remarks: we see the Hilbert-Bernays account as intending a reduction of identity facts to qualitative facts—as proposing that there are no indiscernible pairs of objects. This theme will recur in what follows. Indeed, the next two comments relate to the choice of language.

  2. Banning names:— From now on, it will be clearest to require the language to have no individual constants, nor function symbols, so that the non-logical vocabulary contains only predicates. But this will not affect our arguments: they would carry over intact if constants and function symbols were allowed.

    As we see matters, it is only the haecceitist who is likely to object to this apparent limitation in expressive power. But here, Section 1.2’s discussion of haecceitism comes into its own. For even with no names, the haecceitist has to hand her thisness predicates etc., with which to refer to objects by definite descriptions. Thus we propose, following Quine ([1960], §§37-39), that we, and in particular the haecceitist, replace proper names by 1-place predicates; (each with an accompanying uniqueness axiom; and with the predicates then shoe-horned into the syntactic form of singular terms, by invoking Russell’s theory of descriptions).161616Then any sentence containing the name , where the 1-place formula contains no occurrence of , may be replaced by the materially equivalent sentence , which contains no occurrence of the name . Note also that Saunders ([2006], p. 53) limits his inquiry to languages without names; but no Quinean trick is invoked. Robinson ([2000], p. 163) calls such languages ‘suitable’. And as emphasised in Section 1.2, etc. are to be thinly construed: the predicate commits one to nothing beyond what the predicate commits one to. Thus the presence of these predicates in the non-logical vocabulary means that the haecceitist should have no qualms about endorsing the Hilbert-Bernays account—albeit in letter, rather than in spirit.171717For the spirit of the Hilbert-Bernays account is a reduction of identity facts to putatively qualitative facts, about the co-instantiation of properties. Indeed: according to the approach in which ‘=’ is not a logical constant (cf. comment 3(ii)), acceptance of (HB) entails that the language with the equality symbol ‘=’ is a definitional extension of the language without it. However, representing each haecceity by a one-place primitive predicate, accompanied by a uniqueness axiom, assumes the concept of identity through the use of ‘=’ in the axiom: making this reduction of identity, philosophically speaking, a charade. By contrast, if only genuinely qualitative properties are expressed by the non-logical vocabulary, the philosophical reduction of identity facts to qualitative facts can succeed—provided, of course, that the language is rich enough or the domain varied enough.

    A point of terminology: Though we ban constants from the formal language, we will still use and as names in the meta-language (i.e. the language in which we write!) for the one, or two!, objects, with whose identity or diversity we are concerned. We will also always use ‘=’ in the meta-language to mean identity!

  3. ‘=’ as a logical constant?:— It is common in the philosophy of logic to distinguish two approaches by which formal languages and logics treat identity:

    • ‘=’ is a logical constant in the sense that it is required, by the definition of the semantics, to be interpreted, in any domain of quantification, as the identity relation. Then for any formula (open sentence) with one free variable , the formula

      (2.1)

      is a logical truth (i.e. is true in every structure). Thus on this approach, the rightward implication in (HB) is a logical truth.

      But the leftward implication in (HB) is not. For as discussed in comment 1, the language may not be discerning enough. More precisely: there are structures in which (no matter how rich the language!) the leftward implication fails.181818Such is Wiggins’ ([2001], pp. 184-185) criticism of the Hilbert-Bernays account as formulated by Quine ([1960], [1970]).

    • ‘=’ is treated like any other 2-place predicate, so that its properties flow entirely from the theory with which we are concerned, in particular its axioms if it is an axiomatized theory. Of course, we expect our theory to impose on ‘=’ such properties as being an equivalence relation (cf. Ketland [2009], Lemmas 8-10). We can of course go further towards capturing the intuitive idea of identity by imposing every instance of the schema, eq. 2.1. So on this approach, the Hilbert-Bernays account can be regarded as a proposed finitary alternative to imposing eq. 2.1: a proposal whose plausibility depends on the language being rich enough. Of course, independently of the language being rich: imposing (HB) in a language with finitely many primitives implies as a theorem the truth of eq. 2.1.191919Hilbert and Bernays ([1934], p. 186) show that ‘=’ as defined by (HB) is (up to co-extension!) the only (non-logical) two-place predicate to imply reflexivity and every instance of the schema, eq. 2.1. The argument is reproduced in Quine ([1970], pp. 62-63).

To sum up this Subsection: the conjunction on the right hand side of (HB) makes vivid how, on the Hilbert-Bernays account, objects can be distinct for different reasons, according to which conjunct fails to hold. In Section 3, we will introduce a taxonomy of these kinds. In fact, this taxonomy will distinguish different ways in which a single conjunct can fail to hold. The tenor of that discussion will be mostly syntactic. So we will complement it by first discussing identity, and the Hilbert-Bernays account, in terms of permutations on the domain of quantification.

2.2 Permutations on domains

We now discuss how permutations on a domain of objects can be used to express qualitative similarities and differences between the objects. More precisely: we will define what it is for a permutation to be a symmetry of an interpretation of the language, and relate this notion to the Hilbert-Bernays account.

2.2.1 Definition of a symmetry

Let be a domain of quantification, in which the predicates etc. get interpreted. So writing ‘ext’ for ‘extension’, ext() is, for each , a subset of ; and ext() is, for each , a subset of ; and ext() is, for each , a subset of etc. For the resulting interpretation of the language, i.e.  together with these assigned extensions, we write . We shall also call such an interpretation, a ‘structure’.

Now let be a permutation of .202020We note en passant that since a permutation is a bijection, the definition of permutation involves the use of the ‘=’ symbol; so that which functions are considered to be permutations is subject to one’s treatment of identity. But no worries: as we noted in comment 2 of Section 2.1, this definition is cast in the meta-language! We now define a symmetry as a permutation that “preserves all properties and relations”. So we say that is a symmetry (aka automorphism) of iff all the extensions of all the predicates are invariant under . That is, using as meta-linguistic variables: is a symmetry iff:

(2.2)

and similarly for predicates with four or more argument-places.212121So a haecceitist (cf. comment 2 in Section 2.1) will take only the identity map as a symmetry—unless they stipulate that haecceitistic properties are exempt from the definition of symmetry.

2.2.2 Relation to the Hilbert-Bernays account

Let us now compare this definition with the Hilbert-Bernays account. For the moment, we make just two comments, (1) and (2) below. They dispose of natural conjectures, about symmetries leaving invariant the indiscernibility equivalence classes. That is, the conjectures are false: the first conjecture fails because of the somewhat subtle notions of weak and relative discernibility, which will be central later; and the second is technically a special case of the first.

222222Our two comments agree with Ketland’s results and examples ([2006], Theorem (iii) and example in footnote 17; or [2009], Theorem 35(a) and example). But we will not spell out the differences in jargon or examples, except to report that Ketland calls a structure ‘Quinian’ iff the leftward implication of (HB) holds in it, i.e. if the identity relation is first-order definable, and so (cf. comment 1(ii) of Section 2.1) defined by indiscernibility, i.e. by the right hand side of (HB). In Section 4, we will discuss how the conjectures can be mended: roughly speaking, we need to replace indiscernibility by a weaker and “less subtle” notion, called ‘absolute indiscernibility’.

(1): All equivalence classes invariant?:— Suppose that we adopt the relativization to an arbitrary language: so we do not require the language to be rich enough to force indiscernibility to be identity. Then one might conjecture that, for each choice of language, a permutation is a symmetry iff it leaves invariant (also known as: fixes) each indiscernibility equivalence class. That is: is a symmetry of the structure iff each member of each equivalence class is sent by to a member of that same equivalence class.

In fact, this conjecture is false. The condition, leaving invariant the indiscernibility equivalence classes, is stronger than being a symmetry. We first prove the true implication, and then give a counterexample to the converse.

So suppose leaves invariant each indiscernibility class; and let be in the extension ext() of some -place predicate . Since leaves invariant the indiscernibility class of , it follows that is also in ext(). (For if not, (HB)’s corresponding conjunct, i.e. the conjunct for the first argument-place of the predicate , would discern and ). From this, it follows similarly that since fixes the indiscernibility class of , is also in ext(). And so on: after steps, we conclude that is in ext(). Therefore is a symmetry.

Philosophical remark: one way of thinking of the Hilbert-Bernays account trivializes this theorem. That is: according to comment 1 of Section 2.1, the proponent of this account envisages that the indiscernibility classes are singletons. So only the identity map leaves them all invariant, and trivially, it is a symmetry. (Compare the discussion of haecceitism in footnotes 17 and 21.)

Counterexample to the converse: Consider the following structure, whose domain comprises four objects, which we label to . The primitive non-logical vocabulary consists of just the 2-place relation symbol . is interpreted as having the extension

Now let ‘=’ be defined by the Hilbert-Bernays axiom (HB). From this the reader can check that ‘=’ has the extension

i.e.  and are indiscernible. So ‘=’ is not interpreted as identity (cf. the relativization of ‘=’ to the language, discussed in 3(ii) of §2.1). The relation ‘=’ carves the domain into three equivalence classes: and .

Now consider the permutation , whose only effect is to interchange the objects and ; i.e. . This permutation is a symmetry, since it preserves the extension of according to the requirement (2.2); yet it does not leave invariant the equivalence classes and (Fig. 1).232323 Two remarks. (1): Agreed, this counterexample could be simplified. A structure with just and , with ext has discernible, but the swap is a symmetry. But our example will also be used later. (2) Accordingly, in our counterexample, would work equally well: i.e. it also is a symmetry that does not leave invariant and . Looking ahead: Theorem 1 in Section 4.1 will imply that and are each subsets of absolute indiscernibility classes; in fact, each is an absolute indiscernibility class.

Figure 1: A counterexample to the claim that symmetries leave invariant the indiscernibility classes. In this structure, drawn on the left, the permutation which swaps and is a symmetry; but and form their own separate equivalence classes under ‘=’, as shown on the right.

(2): Only the trivial symmetry?:— Suppose now that in , the predicate ‘=’ is interpreted as identity; and that (HB) holds in . So we are supposing that the objects are various enough, the language rich enough, that indiscernibility in is identity. Or in other words: the indiscernibility classes are singletons. On these suppositions, one might conjecture that that the only symmetry is the trivial one, i.e. the identity map . (Such structures are often called ‘rigid’ (Hodges [1997], p. 94).)

In fact, this is false. Comment (1) has just shown that there are symmetries that do not leave invariant the indiscernibility classes. Our present suppositions have now collapsed these indiscernibility classes into singletons. So there will be symmetries which do not leave invariant the singletons, i.e. are not the identity map on . To be explicit, consider the structure, and its indiscernibility classes, drawn in Fig. 2.

Figure 2: A counterexample to the claim that if the indiscernibility classes are singletons, the only symmetry is the identity map. In this structure, the indiscernibles of Figure 1 have been identified. and are still distinct (they are discernible) but are swapped by the symmetry .

3 Four kinds of discernment

We turn to defining the different ways in which two distinct objects can be discerned in a structure. These kinds of discernment will be developed (indirectly) in terms of which conjuncts on the RHS of the Hilbert-Bernays axiom (HB) are false in the structure.242424Other authors, notably Muller and Saunders [2008], consider the discernment of two objects by a theory (say ), so that the RHS of (HB), applied to the two objects in question, is a theorem of ; whereas our concern is the discernment of two objects in a structure. Our focus on structures is necessary: given our ban on names, the sentence expressing the satisfaction of the RHS of (HB) by the two objects cannot even be written!

3.1 Three preliminary comments

(1) Precursors: Broadly speaking, our four kinds of discernment follow the discussions by Quine ([1960], [1970], [1976]) and Saunders ([2003a], [2003b], [2006]). Quine ([1960], p. 230) endorses the Hilbert-Bernays account of identity and then distinguishes what he calls absolute and relative discernibility. His absolute discernibility will correspond to (the disjunction of) our first two kinds—and we will follow him by calling this disjunction ‘absolute’. Besides, his relative discernibility will correspond to (the disjunction of) our third and fourth kinds. But we will follow Saunders ([2003a], p. 5; [2003b], pp. 19-20; [2006], p. 5) by reserving ‘relative’ for the third kind, and using ‘weak’ for the fourth. (So for us ‘non-absolute’ will mean ‘relative or weak’.)252525Quine ([1976], p. 113) defines what he calls grades of discriminability, which is a spectrum of strength. Saunders ([2006], pp. 19-20) agrees that there is such a spectrum of strength, although in his [2003a] (p. 5) he makes the three categories ‘absolute’, ‘relative’ and ‘weak’ mutually exclusive. We say ‘kinds’ not ‘categories’ or ‘grades’ to avoid the connotation of mutual exclusion or a spectrum of strength.

(2) Suggestive labels: We will label these kinds with words like ‘intrinsic’ which are vivid, but also connote metaphysical doctrines and controversies (e.g. Lewis 1986, pp. 59-63). We disavow the connotations: the official meaning is as defined, and so is relative to the interpretation of the non-logical vocabulary.

(3): Two pairs yield four kinds: Our four kinds of discernment arise from two pairs. We begin by distinguishing between a formula with one free variable (labelled 1) and a formula with two free variables (labelled 2).262626We thank Leon Horsten for the observation that the notion of discernibility may be parameterized to other objects (so that, e.g., we might say that is discernible from relative to ), which would involve formulas with more than two variables. The idea seems to us workable, but we will not pursue it here. Each of these cases is then broken down into two subcases (labelled a and b) yielding four cases in all: labelled 1(a) to 2(b). We will also give the four cases mnemonic labels: e.g. 1(a) will also be called (Int) for ‘intrinsic’.

The intuitive idea that distinguishes sub-cases will be different for 1 and 2. For 1, the idea is to distinguish whether discernibility depends on a relation to another object; while for 2, the idea is to distinguish whether discernibility depends on an asymmetric relation. Both these ideas are semantic, and even a bit vague. But the definitions of the sub-cases will be syntactic, and precise—and we will therefore remark that they do not completely match the intuitive idea. But we will argue in Section 8 in favour of adopting our syntactic definitions.

As announced in comment 2 of Section 2.1, and will be names in the meta-language (in which we are writing!) for the one or two objects with which we are concerned.

3.2 The four kinds defined

1(a) (Int)

1-place formulas with no bound variables, which apply to only one of the two objects and . This of course covers the case of primitive 1-place predicates, e.g. : so that for example, ext() but ext(), or vice versa. But we also intend this case to cover 1-place formulas arising by slotting into a polyadic formula more than one occurrence of a single free variable, while the polyadic formula nevertheless does not contain any bound variables.

The intent here is to exclude formulas which quantify over objects other than the two we are concerned with. So this case will include formulas such as: and ; ( primitive 2-place and 3-place predicates, respectively; not abbreviations of more complex open sentences). But it will exclude formulas such as , which contain bound variables.272727Recall our ban on individual constants ((2) in Section 2.1). If we had instead allowed them, this sub-case 1(a) would be defined so as to also exclude all formulas containing any constant, including and . The exclusion of formulas such as , which refer to a third object, is obviously desirable, given the intuitive idea of discernment by intrinsic properties. However, the exclusion of formulas involving only the constants and-or may be more puzzling. Our rationale is that, for any formula of the type , , etc. which is responsible for discerning two objects, there will be an alternative formula (either or ) which we would instead credit for the discernment, and which falls under one of the three other kinds.

We will say that two objects that do not share some monadic formula in this sense are discerned intrinsically, since their distinctness does not rely on any relation either object holds to any other. An everyday example, taking ‘is spherical’ as a primitive 1-place predicate, is given by a ball and a die. Another example, with the primitive 2-place predicate ‘loves’ (so that is the 1-place predicate ‘loves his- or herself’) is Narcissus ext, but (alas) Echo ext.

We shall say that any pair of objects discerned other than by 1(a)—i.e. discerned by 1(b), or by 2(a), or by 2(b) below—are extrinsically discerned.

1(b) (Ext)

1-place formulas with bound variables, which apply to only one of the two objects and . That is: this kind contains polyadic formulas that do contain bound variables. So it contains formulas such as . And and are discerned by such formulas if: for example, ext() but ext(). We will say that they have been discerned externally. An example, taking = ‘is a man’, = ‘admires’ is: Cleopatra ext() but Caesar ext(). (Recall that we reserve the term extrinsic to cover all three kinds 1(b), 2(a), 2(b): so external discernment is more specific than extrinsic.)

The intent is that in this kind of discernment, diversity follows from the relations the two objects and have to other objects. However, as we said in (3) of Section 3.1, the precise syntactical definition cannot be expected to match exactly the intuitive idea. And indeed, there are examples of external discernment where the relevant value of the bound variable in the discerning formula is in fact or , even though this is invisible from the syntactic perspective. (In the example just given, the universal quantifier in quantifies over a domain that includes Caesar himself.)282828As to our ban on individual constants: if we had instead allowed them, this sub-case 1(b) would be defined so as to also exclude formulas containing and , but to allow other constants etc., so as to capture the idea of discernment by relations to other objects. But as in the case of bound variables, it could turn out that the “third” object picked out is in fact or . Cf. footnote 27.

We will say that two objects discerned by a formula either of kind (Int) or of kind (Ext) are absolutely discerned. Note that a pair of objects could be both intrinsically and externally discerned. But since (Ext) is intuitively a “weaker” form of discernment, we shall sometimes say that a pair of objects that are externally, but not intrinsically, discerned, are merely externally discerned.

Interlude: Individuality and absolute discernment.  We will also say that an object that is absolutely discerned from all other objects is an individual or has individuality. Note that if an object is an individual, some or all of the other objects might themselves fail to be individuals (cf. figure 3).

Figure 3: An object’s being an individual requires its being absolutely discerned from all others, but not their being absolutely discerned from anything else. Here, is absolutely discerned from both and , e.g. by the formula , while and are themselves non-individuals.

Being an individual is tantamount to being the bearer of a uniquely instantiated definite description: where ‘tantamount to’ indicates a qualification. The idea is: given an individual, we take seriatim the formulas that absolutely discern it from the other objects in the structure, and conjoin them and so construct a definite description that is instantiated only by the given individual. The qualification is that in an infinite domain, there could be infinitely many ways that a given individual was absolutely discerned from all the various others: think of how a finite vocabulary supports arbitrarily long formulas, and so denumerably many of them. Thus in an infinite domain the above “seriatim” procedure might yield an infinite conjunction—preventing a finitely long uniquely instantiated definite description.292929 A terminological note: Saunders ([2003b], p. 10) says that an object that is the bearer of a uniquely instantiated definite description is ‘referentially determinate’, and Quine ([1976], p. 113) calls such an object ‘specifiable’. So, modulo our qualification about infinite domains, these terms correspond to our (and Muller & Saunders’ [2008]) use of ‘individual’. Cf. also comment 1(vi) in Section 2.1.

We will examine absolute discernibility in Section 4. For the moment, we return to our four kinds of discernibility: i.e. to presenting the last two kinds. End of Interlude.

2(a) (Rel)

Formulas with two free variables, which are satisfied by the two objects and in one order, but not the other. For example, for the formulas and , we have: ext(), but ext(); and ext(), but ext(). Here the diversity of and is an extrinsic matter (both intuitively, and according to our definition of ‘extrinsic’, which is discernment by any means other than (Int)), since it follows from their relation to each other. But it is not a matter of a relation to any third object. Following Quine ([1960], p. 230), we will say that objects so discerned are relatively discerned.

And as above, we will say that objects that are relatively discerned but neither intrinsically nor externally discerned, are merely relatively discerned. Merely relatively discerned objects are never individuals in our sense (viz. absolutely discerned from all other objects).303030Note that this kind of discernment does not require the discerning formula, for example , to be asymmetric for all its instances; i.e., we do not require . In this we agree with Saunders (e.g. [2003a], p. 5) and Quine ([1960], p. 230). Our rationale is that intuitively, this kind of discernment does not require anything about the global pattern of instantiation of the relation concerned. The same remark applies to (Weak) below, where now we differ from Saunders (e.g. [2003a], p. 5), who demands that the discerning relation be irreflexive. Nevertheless, we adopt Saunders’/Quine’s word ‘weak’.

2(b) (Weak)

Formulas with two free variables, which are satisfied by the two objects and taken in either order, but not by either object taken twice. For example, for the formulas and , we have: ext(), but ext(); and ext(), but ext(). (We say ‘but not by either object taken twice’ to prevent and being intrinsically discerned.) Again, the diversity of and is extrinsic, but does not depend on a third object; rather diversity follows from their pattern of instantiation of the relation . We call objects so discerned weakly discerned. And we will say that objects that are weakly discerned but neither intrinsically nor externally nor relatively discerned (i.e. fall outside (Int), (Ext), (Rel) above), are merely weakly discerned.

Objects which are discerned merely weakly are not individuals, in our sense (since they are not absolutely discerned). Max Black’s famous example of two spheres a mile apart ([1952], p. 156) is an example of two such objects. For the two spheres bear the relation ‘is a mile away from’, one to another; but not each to itself. The irony is that Black, apparently unaware of weak discernibility, proposes his duplicate spheres as a putative example of two objects that are qualitatively indiscernible (and therefore as a counterexample to the principle of the identity of indiscernibles).313131That is, assuming that space is not closed. In a closed universe, an object may be a non-zero distance from itself, so the relation ‘is one mile away from’ is not irreflexive, and cannot be used to discern. French’s ([2006], §4) and Hawley’s ([2009], p. 109) charge of circularity against Saunders [2003a] enters here: it seems that the irreflexivity of ‘is one mile away from’ relies on the prior guarantee that the two spheres are indeed distinct; but their distinctness is supposed, in turn, to be grounded by that very relation being irreflexive. The openness or closedness of space would decide the matter, of course, but that too seems to stand or fall with the irreflexivity or otherwise of distance relations—between spatial or spacetime points, if not material objects. Cf. also comment 1(iv) in Section 2.1.

We will also say that two objects that are not discerned by any of our four kinds (i.e. by no 1-place or 2-place formula whatsoever) are indiscernible. For emphasis, we will sometimes call such a pair utterly indiscernible. In particular, we will say ‘utter indiscernibility’ when contrasting this case with the failure of only one (or two or three) of our four kinds of discernibility. Of course, utter indiscernibles are only accepted by someone who denies the Hilbert-Bernays account.

This quartet of definitions prompts the question: What are the logical implications between these kinds of discernibility? We pursue this in the second Appendix (Section 7).

4 Absolute indiscernibility: some results

In Section 2.2.2, we saw that a permutation leaving invariant the indiscernibility classes must be a symmetry; then we gave a counterexample to the converse statement, and to a related conjecture that if indiscernibility is identity, there is only the trivial symmetry. But now that we have defined absolute discernibility (viz. as the disjunction, (Int) or (Ext)), we can ask about the corresponding claims that use instead the absolute concept. That is the task of this Section. (But its results are hardly needed for the discussions and results in later Sections.)

We will prove that with the absolute concept, Section 2.2.2’s converse statement is “resurrected”, i.e. a symmetry leaves invariant the absolute indiscernibility classes (Section 4.1). Then we will give some illustrations, including a counterexample to the converse of this statement (Section 4.2). Then we will show that for a finite domain of quantification, absolute indiscernibility of two objects is equivalent to the existence of a symmetry mapping one object to the other (Section 4.3).323232Absolute discernibility and individuality are closely related to definability in a formal language; (for example, an object that is definable is an individual in the sense of Section 3.2’s Interlude). The interplay between definability (and related notions) and invariance under symmetries is given a sophisticated treatment by da Costa & Rodrigues [2007], who consider higher-than-first-order structures. Some of their results have close affinities with our two theorems; in particular their theorems 7.3-7.7. As in footnote 6, we thank N. da Costa.

But first, beware of an ambiguity of English. For relations of indiscernibility, we have a choice of two usages. Should we use ‘absolute indiscernibility’ for just ‘not absolutely discernible’ (which will therefore include pairs of objects that are discernible, albeit by other means than absolutely)? Or should we use ‘absolute indiscernibility’ for some kind (species) of indiscernibility—as, indeed, the English adjective ‘absolute’ connotes? (And if so, which kind should we mean?)333333This sort of ambiguity is of course not specific to discernment: it is common enough: should we read ‘recalcitrant immobility’ as ‘not-(recalcitrant mobility)’ or as ‘recalcitrant not-mobility’?

We stipulate that we mean the former. Then: since absolute discernibility is a kind of (implies) discernibility, we have, by contraposition: indiscernibility implies absolute indiscernibility (in our usage!). Since both indiscernibility and absolute indiscernibility are equivalence relations, this implies that the absolute indiscernibility classes are unions of the indiscernibility classes; cf. Figure 4. With this definition, Section 4.1’s theorem will be: a symmetry leaves invariant the absolute indiscernibility classes.

Figure 4: Preview to Section 4.1’s theorem. A generic symmetry acting on a domain must preserve the absolute indiscernibility classes (thick broken lines), but may break the indiscernibility classes (thin broken lines). Note also that the object at centre-bottom, alone in its absolute indiscernibility class and therefore an individual, must be sent to itself under .

Besides, for later use, we make the corresponding stipulation about the phrases ‘intrinsic indiscernibility’, ‘external indiscernibility’ etc. That is: by ‘intrinsic indiscernibility’ and ‘intrinsically indiscernible’, we will mean ‘not-(intrinsic discernibility)’ and ‘not-(intrinsically discernible)’, respectively; and so on for other phrases.

We also admit that instead of Section 3’s syntactic approach to defining kinds of discernibility in terms of formulas, one could frame definitions in terms of the semantic ideas of permutations and symmetries. In the second Appendix (Section 8), we explore the consequences for alternatives, especially as regards this Section’s results.

4.1 Invariance of absolute indiscernibility classes

Recall that in Section 2.2.2, we saw that leaving invariant (fixing) the indiscernibility classes was sufficient, but not necessary, for being a symmetry. That is: Section 2.2.2’s counterexample showed that being a symmetry is not sufficient for leaving invariant the indiscernibility classes. This situation prompts the question, what being a symmetry is sufficient for. More precisely: is there a natural way to weaken Section 2.2.2’s sufficient condition for being a symmetry—viz. indiscernibility invariance—into being instead a necessary condition? In other words: one might conjecture that leaving invariant some supersets of the indiscernibility classes yields a necessary condition of being a symmetry.

In fact, our concept of absolute indiscernibility is the natural weakening. (N.B. Our ban on names is essential to its being a weakening: allowing names in a discerning formula makes (the natural redefinition of) absolute discernment equivalent to weak discernment.) That is: being a symmetry implies leaving invariant the absolute indiscernibility classes. Cf. Figure 4. We will first prove this, and then give a counterexample to the converse statement: it will be similar to the counterexample used in Section 2.2.2 against that Section’s converse statement.

Theorem 1: For any structure (i.e. interpretation of a first-order language): if a permutation is a symmetry, then it leaves invariant the absolute indiscernibility classes.

Proof: We prove the contrapositive: we assume that there is some element of the domain which is absolutely discernible from its image under the permutation , and we prove that is not a symmetry. (Remember that ‘’ and ‘’ are names in the metalanguage only; we stick to our ascetic object-language demands set down in comment 2 of Section 2.1.343434We are extremely grateful to Leon Horsten for making us aware of the problems with a previous version of this proof, in which we reintroduced names into the object language.) So our assumption is that for some object in the domain, with , there is some formula with one free variable for which ext() while ext(), or vice versa ( ext() while ext()):

The proof proceeds by induction on the logical complexity of the absolutely discerning formula . From the assumption that ext() iff ext(), we will show that, whatever the main connective or quantifier used in the last stage of the stage-by-stage construction of , there is some logically simpler open formula, perhaps with more than one (, say) free variable, , and some objects (not necessarily including , and not necessarily in number, since maybe for some ) such that we have: ext() iff ext(), where is the image of under the permutation . That is, we continue to break down to logically simpler formulas until we obtain some atomic formula whose differential satisfaction by some sequence of objects and the sequence of their images under the permutation directly contradicts ’s being a symmetry.

Our proof begins by setting (so to start with, the adicity of our formula equals 1 and our objects comprise only ). We then reiterate the procedure until we reach an atomic formula. Thus:—

Step one. We have that ext() iff ext(). (Remember that to start with, we set and . And the ‘iff’ means only material equivalence.)

Step two. Proceed by cases:

  • If , then we have
    ext() iff ext(); that is,
    ext() iff ext();
    so is our new, simpler .

  • If is a conjunction, then we can write
    ,
    where and , and and are injective maps. First of all, we recognise that
    ext() iff  ext() and ext().
    Then, given step one, namely
    ext() iff ext(),
    this is equivalent to
    ext() and ext()  iff
    ext() or ext(.
    That is:
    ext() iff ext(,  or
    ext() iff ext(.
    So either the formula or is our new formula ; with adicity , respectively , replacing the adicity ; and the objects , respectively replacing the objects

    . (This is an inclusive ‘or’: if either formula suffices, imagine that only one is chosen to continue the inductive procedure. Heuristic remarks. (1): It is only in this clause that the process can reduce the number of variables occurring in

    , and hence the number of objects under consideration. (2): The next three cases can be dropped in the usual way, if we suppose the language to use just as primitive connectives.)

  • If , then continue with .

  • If , then continue with .

  • If , then continue with

  • If , then we have, using ‘’ to mark the -component argument-place,
    ext() iff ext().
    So we have
    ext() and ext(), or
    ext() and ext().
    That is:
    ext() and ext(), or
    ext() and ext().

    • The first disjunct entails that there is some object in —call it —for which
      ext() and ext(), i.e.
      ext() and ext().
      (The second conjunct holds for , since it holds for all objects in .)

    • The second disjunct entails that there is some object in —call it —for which
      ext() and ext(), i.e.
      ext() and ext().
      (The first conjunct holds for , since it holds for all objects in .)

    But we can give the name ; so that we can recombine the disjuncts and conclude that, for some object in , ext() iff ext().
    So the formula is our new , is our new adicity, and are our new objects. (Heuristic remark: It is only in this clause that the process can increase, by one, the adicity of , and hence the number of objects under consideration.)

  • If , then continue with . (Heuristic remark: This case can be dropped in the usual way, if we suppose the language to use just as the primitive quantifier.)

  • If is an atomic formula, then:

    • either for some primitive 1-place predicate , in which case:

      ext()  iff  ext();

    • or for some primitive 2-place predicate , in which case:

      ext() iff  ext()

      (we emphasize that this case includes the 2-place predicate being ‘=’, i.e. equality);

    • and so on for any 3- or higher-place predicates.

    Each case directly contradicts the original assumption that is a symmetry (cf. (2.2)).

End of proof

Corollary 1: An individual is sent to itself by any symmetry.
Proof: Section 3.2 defined an individual as an object that is absolutely discerned from every other object. So its absolute indiscernibility class is its singleton set. QED.
This implies, as a special case, the “resurrection” of Section 2.2.2’s second conjecture. That is, we have

Corollary 2: If all objects are individuals, the only symmetry is the identity map.

4.2 Illustrations and a counterexample

We will illustrate Theorem 1 and Corollary 2, with examples based on those in Section 2.2.2. Roughly speaking, these examples will show how Section 2.2.2’s counterexamples to its two conjectures are “defeated” once we consider absolute indiscernibility instead of utter indiscernibility. Then we will give a counterexample to the converse of Theorem 1.

Theorem 1 illustrated:— In Section 2.2.2’s counterexample (1), and are absolutely indiscernible. Thus Figure 5 illustrates the theorem.

Figure 5: Illustration of Theorem 1, viz. that a symmetry leaves invariant the equivalence classes for the relation ‘is absolutely indiscernible from’. Here, the symmetry from Fig. 1 leaves invariant the absolute indiscernibility classes, shown on the right.

Corollary 2 illustrated:— We similarly illustrate Corollary 2 by modifying Section 2.2.2’s second counterexample, i.e. counterexample (2) (Figure 2) to Section 2.2.2’s second conjecture. The rough idea is to identify absolute indiscernibles; rather than just utter indiscernibles (as is required by (HB)). But beware: identifying absolute indiscernibles that are not utter indiscernibles will lead to a contradiction. Figure 2 (and also Figure 5) is a case in point: it makes true and , so that if you identify and , you are committed to the contradiction between and . But by increasing slightly the extension of , turning absolute indiscernibles into utter indiscernibles, we can give an illustration of Corollary 2, based on Figure 2, which avoids contradiction. Namely, we require that and ; this makes and utterly indiscernible, not merely absolutely indiscernible. Then we identify and , yielding Figure 6.

Figure 6: Illustration of Corollary 2. When ‘is not absolutely discernible from’ is taken as identity, the only symmetry for each structure is the identity map.

Against the Theorem’s converse:— We turn to showing that Theorem 1’s converse does not hold: there are structures for which there are permutations which preserve the absolute indiscernibility classes, yet which are not symmetries. Consider the structure in Figure 7.353535We thank Tim Button for convincing us of this, and for giving this counterexample. In this structure the relation has the extension ext. So as in Fig 1 (i.e. the counterexample in (1) of Section 2.2.2), are weakly discernible, and are indiscernible. But and are absolutely indiscernible. (Proof using Theorem 1: the permutation is a symmetry, so and must each be (subsets of) absolute indiscernibility classes.) Then the familiar permutation , which just swaps and , clearly preserves the absolute indiscernibility classes. Yet is not a symmetry, since e.g. ext, but ext.

Figure 7: A counterexample to the claim that if the absolute indiscernibility classes are left invariant by the permutation , then is a symmetry.

Figure 7 illustrates the general reason why the class of symmetries is a subset of the class of permutations that leave invariant the absolute indiscernibility classes. Namely: for a permutation to be a symmetry, it is not enough that it map each object to one absolutely indiscernible from it; it must, so to speak, drag all the related objects along with it. For example, in Figure 7, it is not enough to swap and ; the objects “connected” to them, namely and respectively, must be swapped too. (In more complex structures, we would then have to investigate the objects “connected” to these secondary objects, and so on).

To sum up: this counterexample, together with Theorem 1 and the results of Section 2.2.2, place symmetries on a spectrum of logical strength, between two varieties of permutations defined using our notions of utter indiscernibility and absolute indiscernibility. That is: for a given structure, we have:

leaves invariant the indiscernibility classes

is a symmetry

leaves invariant the absolute indiscernibility classes

4.3 Finite domains: absolute indiscernibility and the existence of symmetries

For structures with a finite domain of objects, there is a partial converse to Section 4.1’s Theorem 1: viz. that if and are absolutely indiscernible, then there is a symmetry that sends to . To prove this, we will temporarily expand the language to contain a name for each object. We will also use the Carnapian idea of a state-description of a structure (Carnap [1950], p. 71). This is the conjunction of all the true atomic sentences, together with the negations of all the false ones. But for us, the state-description should also include the conjunction of all the true statements of non-identity between the objects in the domain. This will ensure that a map that we will need to define in terms of the state-description is a bijection (and thereby a symmetry).

We see no philosophical or dialectical weakness in the proof’s adverting to these non-identity statements. But we agree that the reason it is legitimate to include them is different, according to whether you adopt the Hilbert-Bernays account of identity or not. Thus the opponent to the Hilbert-Bernays account will include in the state-description all the non-identity sentences holding between any two objects in the domain; for the state-description is to be a complete description of the structure, so these non-identity facts should be included. For the proponent of the Hilbert-Bernays account, on the other hand, facts about identity and non-identity are entailed by the qualitative facts, in accordance with (HB). So a description of a structure (in particular, a Carnapian state-description) can be complete, i.e. express all the facts, without explicitly including all the true non-identity sentences. But it is also harmless to include them as conjuncts in the state-description.363636Harmless, that is, provided the HB-advocate is clear-headed. Recall from comment 1 of Section 2.1, that the proponent of the Hilbert-Bernays account assumes that the language is rich enough, or that the domain is varied enough, for each object to be discerned in some way (maybe: relatively or weakly) from every other. Thus one could also argue that this assumption involves no loss of generality: for if it does not hold for a structure, then the indiscernible objects are to be identified; or else—on pain of contradiction for the HB-advocate—the primitive vocabulary is to be expanded so as to discern them, and the proof is then run again with a structure of discerned objects.

We will state the Theorem as a logical equivalence, although one implication (the leftward one) is just a restatement of Section 4.1’s Theorem 1, and so does not need the assumption of a finite domain.

Theorem 2: In any finite structure, for any two objects and :
and are absolutely indiscernible there is some symmetry such that .

Proof: Leftward: This direction is an instance of Section 4.1’s Theorem 1, that symmetries leave invariant the absolute indiscernibility classes.
Rightward: Consider an arbitrary finite structure with distinct objects, in its domain , and any two absolutely indiscernible objects in that domain, (so we set ). We temporarily expand the language to include a name for each object .373737Some readers—especially those worrying about our sudden use of names in the object-language—may like to convince themselves that one achieves the same results if, within each absolute indiscernibility class, names are permuted among their denotations. Therefore the proof is invariant under permutations of non-individuals. Then:

  1. Construct the state-description of the structure. Thus for example, if the language has just one primitive 1-place predicate and one primitive 2-place predicate , we define:

  2. Define the -place formula from by replacing all instances of each name with an instance of a free variable . Writing for the substitution of by etc., we define:

    Note that is .

  3. From , we define one-place formulas, the th being the “finest description” of the th object in , i.e. , by existentially quantifying over all but the th variable, which is itself replaced by another variable, say :

  4. The structure clearly makes true , since it is entailed by . But by our assumption, is absolutely indiscernible from . This means that and satisfy the same one-place formulas. So the structure also makes true . So also entails .

  5. We apply existential instantiation to every existential quantifier in . Formally, this introduces new names, say . But we know (from the last conjunct in ) that there are at most objects in ; so each of the must name one of the . And the non-identity conjuncts in also entail that any pair of names among the denote distinct objects, all of which are distinct from . Thus we infer that entails the sentence

    where the bijective map is defined by the fact that for each the new name refers to .

  6. We now construct a map on using the sentences and as follows:

    That is: maps to , and to , and to , and so on for all the objects in .

  7. Then is a bijection, since all the are distinct. And is the range of . So is a permutation. But by construction is also a symmetry. This is because: (i) it induces a map from the state-description to another state-description ; (ii) the latter sentence is entailed by the former and, since both are state-descriptions, both sentences have maximal logical strength; so (iii) the sentences are materially equivalent; and (iv) this material equivalence, together with their maximal logical strength, entails that the extensions of all primitive predicates are preserved under . Thus is the symmetry sought.

End of proof

We now sketch two examples showing the need for finiteness in the statement of Theorem 2. The first uses a countable domain and is sufficient on its own to prove the need for finiteness in Theorem 2; the second, which involves an uncountably infinite domain, is unnecessary, but illustrative.

The countable example is familiar (e.g. Boolos & Jeffrey [1974], p. 191). We use the fact that first-order arithmetic, in the language with primitive symbols ,383838Remember (comment 2 in Section 2.1) that we ban names from primitive vocabularies, so we take not to contain , the name assigned to 0 (the number zero) in the standard model of arithmetic, as a primitive. The standard results which we use here are not affected, for we can introduce by description, in terms of the other primitive vocabulary, i.e. . is not -categorical, by finding two elementarily equivalent but non-isomorphic structures, which we then use to create a single structure with an absolutely indiscernible pair not related by any symmetry. Take the standard model of arithmetic and a non-standard model , where consists of an initial segment, whose first element is an object and which is isomorphic to , followed by countably infinitely many -chains, densely ordered without a greatest or least -chain. (So has the same order type as .)

We then create a new structure ; (let us assume that the domains and are disjoint, so that ). Now, the structures and are elementarily equivalent, so any one-place formula in which is true of in is also true of in , and vice versa. So then any one-place formula in which is true of in will also be true of in , and vice versa.393939That is of course not to say that the same propositions will be true of in as in (and similarly for and ). In particular, the description proposed in fn. 38 as a definition of , namely , is false of in , because the uniqueness claim fails; but of course the description is also false of in . In that case, and are absolutely indiscernible in . But and are not isomorphic; therefore has no symmetries which map to or vice versa. So we have two objects which are absolutely indiscernible but which are not related by any symmetry.404040If we expand to the infinitary language (which allows countably infinite conjunction and disjunction, but only finitary quantification), and become absolutely discernible, since we now have the linguistic resources to distinguish between the order types of and . Specifically, if we define the relation , and recursively define the function , then and are absolutely discerned by the formula , since but . Intuitively, the formula says that every element greater than lies only finitely far away from .

Now we provide an example of an uncountably infinite structure in which two objects and are absolutely indiscernible, but no symmetry maps one to the other. In this example, there is just one 2-place relation . We require that bears to each of denumerably many distinct ; and each of these bears only to itself.414141So the s are weakly discerned from each other by ; and so will be distinct even for a proponent of the Hilbert-Bernays account. We also require that bears to each of continuum many distinct ; and each of these bears only to itself. Thus we have:

with

So and are each like the centre of a wheel, with relations to the