On a Certain Antinomy: Properties, Concepts and Items in Space
[ from Philosophical Perspectives: Vol 10; Metaphysics, 1996, J. Tomberlin, ed., Blackwell Publishers, (Cambridge, MA & Oxford; 1996), pp. 357-83]

In this essay, I want, in a preliminary and exploratory way, to address myself to a certain antinomy. I say “in a preliminary and exploratory way” because I do not propose, in this essay, to resolve the antinomy, but only to examine the alternatives with which it confronts us. The antinomy sets us, I think, some hard choices, none of which ab initio sits entirely comfortably. Mapping the terrain in this way is a necessary first step toward getting comfortable with one or another of those choices. By the time I have finished, it will perhaps at least have become clearer what further steps need to be taken in order to feel sufficiently comfortable with one of them that we are prepared to call it the antinomy’s resolution.

the antinomy

The antinomy I have in mind forms the point of departure of an admirable and challenging book, Subjektivität in Raum und Zeit, by Anton Friedrich Koch.[1] Rather than adhering strictly to Koch’s order of exposition, however, I want first to set out the antinomy in a different order and in slightly different terms. (Later I shall compare my version with his.) I shall consequently take as the first moment of my exposition a principle that Koch himself treats as purely logical[2], Leibniz’s principle of the identity of indiscernibles. Exactly what this principle says and how it is properly to be expressed are, in fact, questions that the antinomy itself will throw into sharp relief, but for the moment I shall provisionally formulate it as:

(LL) Necessarily, an entity, a, is identical to an entity, b, if and only if every property of entity a is a property of entity b, and conversely.

Let us call entities that differ in at least one property discernible. Inter alia, then, (LL) implies that

(DP) Distinct (i.e., non-identical) entities are necessarily discernible.

Thus understood, Leibniz’s principle captures the intuitive idea that entities cannot simply differ (full stop), but must necessarily differ in some respect. The “species” of identity at issue in the principle is typically termed “numerical identity”, and similarly one finds “numerical distinctness”, but fewer questions are prejudged by speaking, at least at the beginning, simply of identity — identity tout court or identity sans phrase. The notion of a property correlatively corresponds, in the first instance, to the notion of a way in which entities can resemble or differ from one another. In what follows, I will presuppose and make use of a traditional distinction between intrinsic and extrinsic properties. An intrinsic property of an item is, very roughly, a property whose instantiation by that item does not depend on the existence of any other items; it is one of the item’s “inner determinations”. A property is extrinsic just in case it is not intrinsic. In the general spirit of Koch[3], let us say that entity a and entity b are intrinsically identical just in case every intrinsic property of a is a property of b, and conversely. If, in addition, every extrinsic property of a is also a property of b, and conversely, we will say that a and b are attributively identical. Leibniz’s principle (LL) can consequently be understood as expressing the coincidence of identity (tout court) and attributive identity:

(LL*) Necessarily, an entity, a, is identical to an entity, b, if and only if entity a is attributively identical to entity b.

The second moment of our antinomy introduces the notion of a concept. While Leibniz’s principle thus understood belongs prima facie to ontology, i.e., to a story about entities that make this or that judgment true, the notion of a concept is initially at home in the theory of judgment per se. As Kant stressed, the most fundamental form of judgment, whether expressed in thought or speech, involves the subsumption of an individual item under a general concept. A concept is thus, in the first instance, a one that contrasts with the many constituted by its (actual and possible) instances, i.e., items that do or could fall under that concept. Every de facto application of a concept thus presupposes the notion of such a plurality of instances. A concept is essentially a principle of unity for its instances and so, a fortiori, cannot also be the ground of their plurality or multiplicity. Our use of general concepts (in judgments), in other words, does not give rise to the notion of the (actual or possible) plurality of such instances but rather presupposes a distinct principle of “pre-conceptual” multiplicity. (Cf. SRZ, 16-17)

Let us say that an entity, a, and an entity, b, are distinguishable just in case there is some general concept under which one entity falls and the other does not, some general concept which applies to one entity and not to the other. To say that our use of general concepts presupposes a principle of pre-conceptual multiplicity is then to say that it is possible for distinct entities to fall under all and only the same general concepts. That is,

(C) Possibly, there are entities, a and b, such that entity a is not identical to entity b, and (nevertheless), for every general concept C, entity a falls under C if and only if entity b falls under C.

To put it briefly,

(DC) Distinct entities are not necessarily distinguishable.

The third and final moment of our antinomy should now be obvious. It consists in the claim (surely initially plausible) that properties and concepts are correlative to one another in the sense that

(CP) Necessarily, for every property, P, there is a general concept, C, such that P is a property of an entity, a, if and only if a falls under the concept C, and conversely (i.e., for every general concept C, there is a property, P, such that an entity, a, falls under the concept C if and only if P is a property of a).

The principle (CP) thus has the effect of collapsing discernibility and distinguishability. That is,

(DD) Necessarily, entities are discernible if and only if they are distinguishable.

In the context of (CP), in other words, commitment to a principle of pre-conceptual multiplicity is a commitment to precisely the possibility that (LL*) rules out, namely, the possibility of distinct entities that are attributively identical. Correlatively, in the context of (CP), (LL) implies that there is something incoherent about the very notion of a “principle of pre-conceptual multiplicity”. It contradicts what Koch calls (SRZ, 19) “the supervenient character of identity”. It is thus easy enough to see that (LL), (C), and (CP) — or, equivalently, (DP), (DC), and (DD) — form an inconsistent triad.

What makes this inconsistent triad an antinomy is the fact that the “principle of pre-conceptual multiplicity” to which it adverts is not only, so to speak, a “something, I know not what”, theoretically demanded by our use of general concepts in judgment, but also, as Koch insists, actually at hand. “[S]pace and time are principles — or, taken together, the principle — of pre-conceptual numerical multiplicity.” (SRZ, 17) Indeed, to grasp what is prima facie involved in (C), it is only necessary to suppose that the world could be spatially completely symmetric (e.g., in the sense of Strawson’s “chessboard universe”[4]) or temporally infinitely repetitive (e.g., in the sense of Nietzsche’s “eternal recurrence”). Acknowledging such possibilities, Koch argues, implies that “even if all [attributively] identical entities are also de facto numerically identical, the concept of numerical identity with respect to space and time must be distinguished from the concept of [attributive] identity.” (SRZ, 23) It may not be possible for two entities to simply to differ (full stop), but what (C) evidently implies is that, in a suitably symmetric and repetitive world, it is possible for two entities to differ only in spatio-temporal location.

the original version

Koch’s own exposition of the antinomy differs from the one I have just given primarily in that it largely takes for granted the relationship between properties and concepts explicitly thematized in (CP) above and so, correlatively, makes no distinction between what I have called “discernibility” (in terms of properties) and “distinguishability” (in terms of concepts). In place of (CP), one finds two supplementary premises (SRZ, 20):

(1) Discernible (unterscheidbare) entities have distinct properties.

(2) Both intrinsic and relational properties are completely general, i.e., one attributes properties to a thing (Sache) by specifying the general concepts under which the thing falls.

Koch then proceeds to introduce notions of “qualitative identity” and “descriptive identity” directly in terms of non-relational and relational general concepts:

An entity a is qualitatively identical to an entity b just in case a is indistinguishable (ununterscheidbar) from b with respect to the non-relational general concepts under which a and b fall. On the other hand, an entity a is descriptively identical to an entity b just in case a is also indistinguishable from b with respect to all the relational general concepts under which a and b fall. (SRZ, 20)

The stage is then set for his original version of the antinomy:

The identitas indiscernibilium [i.e., Leibniz’s principle (LL)], understood as a purely logical principle, and the assumptions (1) and (2), regarded as conceptual (non-empirical) truths, together imply that identity as such, numerical identity, and descriptive identity conceptually coincide. In other words, they imply that there cannot be entities that are both numerically distinct and descriptively identical. (Only entities that are both numerically distinct and qualitatively identical are still allowed.) A principle of pre-conceptual or pre-descriptive multiplicity, on the other hand, is by definition a principle which separates the concepts of numerical and descriptive identity. It guarantees that there can be entities that are both numerically distinct and descriptively identical.
The contradiction thus obtains between the identitas indiscernibilium and the assumptions (1) and (2) on the one side and the fact of our concept-using per se on the other. And not only that. If that were all, one might be tempted to challenge the claim that the use of concepts per se assumes and requires such a pre-conceptual multiplicity. But in space and time we have really found such a principle of pre-conceptual multiplicity, so that the contradiction equally obtains between the actual existence of space and time on the one side and the identitas indiscernibilium and the assumptions (1) and (2) on the other. (SRZ, 21)

It is reasonably clear, I think, that Koch’s contradiction is, so to speak, only notationally different from the antinomy in the form in which I have presented it, but it is also clear, I hope, that his formulation blurs at least one crucial conceptual nexus which mine sets sharply in the foreground. The difficulty is that, whereas Koch’s premise (1) is framed in terms of properties, his notions of qualitative and descriptive identity are framed in terms of (non-relational and relational general) concepts. Premise (2) then, inter alia, presupposes something like the crucial connecting hypothesis explicitly formulated in my (CP), yet it remains unclear precisely what relationship or degree of correlativity of concepts and properties is being assumed. But then, by discursively introducing Leibniz’s Law only as a “purely logical principle”, i.e., in terms of its symbolic representation in the second-order predicate calculus, Koch is able to avoid confronting the absolutely central question of what, metaphysically or ontologically speaking, Leibniz’s Law actually says. For, as I shall later have occasion to stress, logical symbolism is not self-interpreting, and, within the context of our antinomy, two equally salient interpretations immediately suggest themselves. The symbolic formulation of Leibniz’s Law might correspond to a metaphysical principle regarding concepts, or it might correspond to an ontological principle regarding properties. In formulating the antinomy, I have elected to come down on the side of properties, but the chosen formulation is not what is important. What is important is to make it explicit that there is a choice here, i.e., to explictly thematize the question of the relationship between properties and general concepts. One virtue of developing the antinomy in my terms, I want to suggest, is that it brings into the foreground the implicit tension between an essentially Kantian theory of judgments and an essentially Platonistic understanding of what makes such judgments true. In particular, framing the antinomy as I do renders it especially acute for the sort of “linguistic nominalist” who proposes to interpret Platonistic talk of properties as a specialized (“material mode”) form of metalinguistic discourse and, correlatively, to understand a person’s mastery of a concept in terms of her having a suitably-functioning predicative item in her representational repertoire, inclinations which I am myself on record as sharing.

a test case

To focus our discussion, it will be useful to have before us a candidate realization of the possibility asserted in (C), i.e., a putative case of distinct but indistinguishable entities, potentially answering to the notion of a “pre-conceptual multiplicity”. The locus classicus for an ostensible description of two entities which differ only in spatio-temporal location is the minimalist state of affairs envisioned by Max Black. [5] Call it “Black’s World” (“World B” for short).

Isn’t it logically possible that the universe should have contained nothing but two exactly similar spheres? We might suppose that each was made of chemically pure iron, had a diameter of one mile, that they had the same temperature, color, and so on, and that nothing else existed. Then every quality and relational characteristic of the one would also be a property of the other. (II, 83)

Is Black’s World a possible world? Apart from claiming that what he describes is “logically possible”, Black himself never explicitly pursues the question, although “A”, one of the two alphabetic characters in his imaginary dialogue, does raise a question regarding the meaningfulness of Black’s description on verificationist grounds that are nowadays likely to strike us, at least initially, as quaintly old-fashioned. Given the claim of “logical possibility”, it is doubtless tempting to turn to logic for guidance, but, pacé Koch (and others), this is arguably one of the junctures at which formal logic itself is in need of guidance that prima facie only metaphysics can provide.

Consider Black’s claim that, in World B, every quality and relational characteristic of one sphere would also be a property of the other. If we think of this remark as imposing a constraint on the stipulation of Black’s World, his description, framed in the symbolic idiom of formal logic, will then presumably need to imply that

(B1) ($x)($y)[(x¹y) & ("F)(Fx º Fy)]

Now it’s perfectly clear that any world that satisfies (B1), if any does, cannot also satisfy

(L1) ("x)("y)[("F)(Fx º Fy) ® (x=y)]

(B1) and (L1) are contradictories. But (L1), of course, is also one symbolic formulation of the interesting half of Leibniz’s Law, the Identity of Indiscernibles,

(LL1) Necessarily, if every property of an entity, a, is a property of an entity, b, and conversely, then entity a is identical to entity b.

Our question thus becomes: Does every possible world validate (L1)?

Now it is certainly reasonable to take it as an ironclad constraint on possible worlds that they universally validate the uninteresting half of Leibniz’s Law, the Indiscernibility of Identicals:

(L0) ("x)("y)[(x=y) ® ("F)(Fx º Fy)]

(LL0) Necessarily, if an entity, a, is identical to an entity, b, then every property of entity a is a property of entity b, and conversely.

But whether there’s an interesting and relevant sense of ‘possible’ for which it is reasonable to require that every possible world must validate (L1) is surely what’s at issue. In particular, it will be reasonable to impose such a requirement on “possible worlds” just in case (L1) is “expressively complete” in the sense that the properties falling within the scope of the second-order quantifier ‘("F)’ exhaust the potential individuators of x and y.

But suppose that the scope of the second-level quantifier is limited to what we might call “general properties”, that is, to properties corresponding one-for-one with general concepts. If we accept (CP), then the collection of all such general properties will be coextensive with the collection of all properties sans phrase. That’s just what (CP) says. And so, if we accept (CP), the second-order quantifier in (L1) will not, so to speak, “leave anything out”, and it will be reasonable to regard it as giving formal expression to an additional constraint that any “possible world”must satisfy. In that case, assuming that (B1) would correctly describe Black’s World, we should have to conclude that Black’s scenario does not succeed in consistently describing a possible world.

One option for coming to terms with our antinomy, however, is to reject (CP), or, more specifically, to reject the interesting half of (CP), for, like (LL), (CP) is also equivalent to the conjunction of two separable theses:

(CP0) Necessarily, for every general concept C, there is a property, P, such that an entity, a, falls under the concept C if and only if P is a property of a,

and

(CP1) Necessarily, for every property, P, there is a general concept, C, such that P is a property of an entity, a, if and only if a falls under the concept C.

(CP0) arguably resembles (LL0) in point of triviality. (CP0) begins, so to speak, by directing our attention to the collection of general concepts. Now, whatever else may be true of general concepts, it seems undeniable that they play what is, on the face of it, a predicative role in judgment. If we continue to allow ourselves the characteristic idioms and metaphors of traditional Platonistic ontology, then, we seem committed to the conclusion that the fundamental job of a general concept is to represent a predicative entity, i.e., a property. In short, corresponding to each general concept, C, there will be a property, represented by that concept, which an item will exemplify just in case it falls under that concept. (CP0) thus appears to be inescapable but, correlatively, uninteresting.

(CP1), however, does not share this status of trivial inescapability. (CP1), so to speak, begins with the collection of (all) properties and advances the substantial thesis that these properties are all general properties, i.e., that to each there corresponds a general concept which, a fortiori, represents it. And while it may be trivially true that there is a (represented) property corresponding to every (representing) general concept, it is at least not obviously true that there must be a (representing) general concept corresponding to every property. If we reject this assumption and, at the same time, interpret the scope of the second-order quantifier ‘("F)’ appearing in (L1) as encompassing only general properties, i.e., those which do correspond to representing general concepts, we will be in the position of acknowledging the possibility of “possible worlds” which do not validate (L1).

the first option: dispensing with properties

Our discussion to this point has yielded sufficient resources to allow us to formulate the first of a series of potential resolutions of our antinomy. This resolutive strategy follows the general lines of our most recent reflections regarding (LL) and (CP), but, rather than formulating Leibniz’s principle of the identity of indiscernibles as (LL) and then interpreting (LL) in terms of quantification restricted to general properties, it interprets Leibniz’s Law directly, ab initio, as an unrestricted principle regarding general concepts:

(LLC) Necessarily, an entity, a, is identical to an entity, b, if and only if, for every general concept C, entity a falls under C if and only if entity b falls under C.

In traditional terms, this strategy treats the second-order quantifier in (L0) and (L1) as “ranging over” such concepts.

If we thus entirely eschew (Platonistic) talk of properties, the connecting principle (CP) drops out as superfluous, for Leibniz’s principle, understood as (LLC), and the principle of pre-conceptual multiplicity

(C) Possibly, there are entities, a and b, such that entity a is not identical to entity b, and (nevertheless), for every general concept C, entity a falls under C if and only if entity b falls under C.

now contradict each other immediately and directly, as do (C)’s rephrasing as

(DC) Distinct entities are not necessarily distinguishable.

and the corresponding rephrasing of (LLC) as

(DCC) Distinct entities are necessarily distinguishable.

On this strategy, then, we regard the example of Black’s World precisely as establishing the truth of (C) and, a fortiori, as a demonstration of the falsehood of (LLC). And since

(LLC0) Necessarily, if an entity, a, is identical to an entity, b, then, for every general concept C, entity a falls under C if and only if entity b falls under C.

is unobjectionable, we are led specifically to reject

(LLC1) Necessarily, if, for every general concept C, an entity, a, falls under C if and only if an entity, b, falls under C, then entity a is identical to entity b.

Our first option for coming to terms with the antinomy, in short, consists in adopting the view that Leibniz’s Law is a false principle about general concepts.

The obvious objection to this strategy is that it “resolves” our original antinomy not by solving or dissolving it, but only by evading it. Whatever we may ultimately want to say about general concepts, once we accept Black’s scenario as describing a possibility, we surely cannot avoid agreeing that there must be some respect in which the two spheres in Black’s World differ. It seems utterly natural simply to equate this claim with the claim that there must be at least one property which one sphere has that the other lacks — and once we have the notion of a property back on the table, the inevitable question regarding the cogency of a corresponding version of Leibniz’s principle for properties (in contrast to general concepts) immediately threatens to reinstate the antinomy in its original form. If we hope to learn anything useful from our antinomy, then, the most fruitful course appears to be to insist on holding true to its original formulation as the inconsistent triad (LL), (C), and (CP) — or, equivalently, (DP), (DC), and (DD). That brings us to the other, more interesting, option suggested by our earlier explorations of Black’s scenario.

the second option: singular properties

On this option, we retain the original interpretation of Leibniz’s Law as an unrestricted principle about properties, i.e.,

(LL) Necessarily, an entity, a, is identical to an entity, b, if and only if every property of entity a is a property of entity b, and conversely,

and treat it as formulating an ontological constraint on “possible worlds”. We require, in short, that a possible world validate (L1) as well as (L0). We agree also, however, that Black’s scenario successfully describes one such possible world, and, indeed, a possible world which answers to the specifications laid down in (C). We agree, that is, that the two spheres constituting (the whole of) World B are non-identical objects each of which falls under all and only the same general concepts as the other. Since (LL), (C), and (CP) form an inconsistent triad, our antinomy will be resolved, on this option, just in case we reject (CP). Since, as I earlier suggested, there appears to be nothing objectionable about

(CP0) Necessarily, for every general concept C, there is a property, P, such that an entity, a, falls under the concept C if and only if P is a property of a,

this strategy encourages us specifically to reject

(CP1) Necessarily, for every property, P, there is a general concept, C, such that P is a property of an entity, a, if and only if a falls under the concept C.

Our second strategy for coming to terms with the antinomy, in short, consists in adopting the view that there are more properties than there are general concepts.

On this construction, Wittgenstein’s remark, in the Tractatus, that

5.526 We can describe the world completely by means of fully generalized propositions, i.e., without first correlating any name with a particular object.
Then, in order to arrive at the customary mode of expression, we simply need to add, after an expression like, ‘There is one and only one x such that …’, the words, ‘and this x is a’.

apparently comes out false. For the “fully generalized proposition”

(B1A) ($x)($y)[(x¹y) & ("F)(Fx º Fy) & ("z)(z=x v z=y)]

surely bids fair to offer, at least in one sense of the phrase, a “complete description” of Black’s World, but it does not issue in a pair of uniquely individuating descriptions which might serve to fix the referents of two proper names, ‘a’, and ‘b’. Let us, for instance, assume for expository convenience that we have to do in Black’s World with only three (monadic) general properties, C, D, and G, and one irreflexive and symmetric relational property, R (e.g., “… is at a distance of thirty miles from …”). Now in the Tractarian Wittgenstein’s “customary mode of expression” a “complete description” of the world has the form of an enumeration of all the Sachverhalten (atomic states of affairs) by means of (atomic) propositions which picture them. Such a “complete description” of Black’s World thus resembles:

(B0) Ca, Cb, Da, Db, Ga, Gb, Rab, Rba

But while there is indeed an inferential route from (B0) to something like (B1)[6], there is no converse route from (B1) to any propositions of the form

($x)[Fx & ("y)(Fy º (y=x)]

to which we might add ‘& (x=a)’ and thereby be on the road to successfully recovering something like (B0).

This second strategy apparently asks us to acknowledge properties that cannot be represented by general concepts, i.e., by predicates. To put it differently, it evidently asks us to acknowledge singular properties. But what is a singular property? As far as I can see, there are exactly two candidates, tropes and hæcceities. Our second option consequently prima facie divides into two sub-options. I shall consider each in turn.

singular properties, the first candidate: tropes

Tropes, as they have come to be called, are difficult to locate on traditional categorial maps coordinatized in terms of “particulars” and “universals”. Roughly, they share with particulars spatio-temporal localizability, and in this respect appear to be “concreta”, but like general properties they have a form of dependent existence, and so in this respect appear to be “abstracta”. They have, indeed, been called both “concrete universals” and “abstract particulars”, although neither appellation sits quite comfortably. In the context of the present discussion, it is perhaps most useful to think of them, at least initially, as individual instances of general properties, a label that reflects our only workable linguistic device for referring to individual tropes. Since we lack any primitive representational apparatus which stands to tropes in a relation analogous either to that between proper names and particulars or to that between predicates and general properties, the best we can do is to combine an abstract noun phrase with reference to a determinate particular — e.g., “the shape of the dome of St. Peter’s cathedral”, “the blueness of Paul Newman’s eyes” — and even this idiom is easily misinterpreted.

Specifically, when thinking in terms of tropes, it is important not to regard such equalities as

(te1) the shape of the dome of St. Peter’s is the same as the shape of the dome of St. Paul’s

or (te2) the blueness of Paul Newman’s eyes is the same as the blueness of Frank Sinatra’s eyes

as derived from more basic contexts of property ascription, thus

(pa1) There is a shape, S, such that the dome of St. Peter’s is S and the dome of St. Paul’s is S

or (pa2) There is a shade of blue, B, such that Paul Newman’s eyes are B and Frank Sinatra’s eyes are B,

and so as having the sense of property identities,

(pi1) the shape of the dome of St. Peter’s = the shape of the dome of St. Paul’s

and (pi2) the blueness of Paul Newman’s eyes = the blueness of Frank Sinatra’s eyes.

A true trope theorist rather regards trope sameness (equality, exact resemblance) as a fundamental, unanalyzable, “basic” or “internal” relationship between tropes.[7] “Qualitative sameness” or resemblance of particulars, in turn, is analyzed in terms of trope sameness rather than property sharing. On this view, in fact, a concrete particular (individual substance) is nothing but a nexus of tropes — that is, a bundle or collection of “compresent” tropes, or a structure of tropes, or a group of tropes together with a bare particular (substratum) functioning as their bearer and principle of their (substantial) unity. A full-fledged trope theorist, in other words, proposes to do without general properties entirely; in this respect, he is a “trope nominalist”. An individual trope, correlatively, is, so to speak, “bound” to the concrete particular or individual substance of which it is a constituent, a circumstance corresponding to its non-repeatability, its existential dependency, and its spatio-temporal localizability.

All this makes tropes an odd choice to press into service as the singular properties we need on the second strategy to complement the properties represented by our general (predicative) concepts. For a trope nominalist, in fact, our current strategy for resolving the original antinomy takes on quite a different form.

To see this, consider how a trope nominalist would describe World B. Since he is committed to treating Black’s two qualitatively indistinguishable spheres as nexes of distinct tropes, i.e., as (in essence) consisting of non-overlapping collections of tropes[8], he will need to parse, for instance, Black’s claim regarding general properties, that “every quality and relational characteristic of the one would also be a property of the other”, as the claim that every trope constituent of either sphere is the same as some trope constituent of the other. Similarly, a consistent trope theorist will also interpret talk of general concepts as adverting to tropes, although at one further remove. Roughly, a predicative concept picks out a family of equal (exactly resembling) tropes. An individual falls under such a general concept just in case one of the tropes constituting that individual belongs to the trope family picked out by that concept. It follows that two entities will fall under a single general concept just in case some trope constituent of one is the same as some trope constituent of the other.

In this case, in short, both (C) and (LL) will be reinterpreted directly in terms of tropes, and our original linking principle (CP) drops out of the picture as irrelevant. Specifically, in accordance with our current strategy’s commitment to treat Black’s World as illustrating a principle of pre-conceptual multiplicity, (C) now becomes

(CT) Possibly, there are entities, a and b, such that entity a is not identical to entity b, and (nevertheless), for every family of equal tropes F such that some trope constituent of entity a belongs to F, there is a trope constituent of entity b that belongs to F, and conversely,

or symbolically — leaving the modality tacit and letting ‘x’ and ‘y’ range over particulars (individual substances), ‘t_i’ over tropes, ‘F’ over families of equal (exactly resembling) tropes, and representing the constitutive trope-particular relation by ‘p’ —

(CT*) ($x)($y)[(x¹y) & ("F)[($t_i )(t_iÎF & t_i p x) « ($t_j)(t_jÎF & t_jp y)].

Given the definitional principle

(FT) ("t_i )("t_j)[($F)(t_iÎF & t_jÎF) « t_i» t_j ],

where ‘»’ represents trope equality (sameness, exactly resemblance), it is easy enough to demonstrate that (CT) entails

(CTE) Possibly, there are entities, a and b, such that entity a is not identical to entity b, and (nevertheless) every trope constituent of entity a is the same as some trope constituent of entity b, and conversely,

or, symbolically,

(CTE*) ($x)($y){(x¹y) & [("t_i )(t_i p x ® ($t_j)(t_jp y & t_i» t_j) .&.
t_i p y ® ($t_j)(t_jp x & t_i» t_j))]}.

On the other hand, given our current strategy’s commitment to interpreting Leibniz’s Law as a true purely logical principle, (LL) can be interpreted only as the claim that entities are identical just in case they have identical trope-constitutions, i.e.,

(LLT) Necessarily, an entity, a, is identical to an entity, b, if and only if every trope constituent of entity a is identical to some trope constituent of entity b, and conversely.

or, symbolically — again leaving the modality tacit —

(LLT*) ("x)("y)[(x=y) « [("t_i)(t_i p x « t_ip y].

Recasting (LLT*) in a form that more closely echos the verbal formulation (LLT), i.e., as

(LLT*I) ("x)("y)[(x=y) « [("t_i )(t_i p x ® ($t_j)(t_jp y & t_i= t_j) .&.
t_i p y ® ($t_j)(t_jp x & t_i= t_j))],

highlights the fact that, thus interpreted, (CT) and (LLT) will appear to be inconsistent just in case trope equality (sameness, exact resemblance) is confused with trope identity.

This version of our second strategy, then, indeed offers a resolution of our original antinomy. I want to suggest, however, that it does so only at the price of relocating that antinomy’s problematic within the theory of tropes. We began by thinking of Leibniz’s principle of the identity of indiscernibles as capturing the intuitive idea that particulars (individual substances) cannot simply differ (full stop), but must necessarily differ in some respect, and by thinking of properties as the respects in which particulars resemble and differ. Our puzzlement arose from the observation, apparently borne out by Black’s thought-experiment, that non-identical particulars could nevertheless be “exactly the same”, i.e., conceptually indistinguishable. The trope theorist proposes that concrete particulars consist of tropes and proceeds to ground the distinctness (discernibility) of such particulars on the non-identity of their trope-constituents and the (conceptual) indistinguishability of such particulars on the equality (exact resemblance) of their trope-constituents. In short, when confronted with the question of how it is possible for there to be non-identical particulars which are nevertheless “exactly the same”, the trope theorist answers that such particulars consist of tropes which are both non-identical and nevertheless “exactly the same”.[9]

The problematic of our antinomy thus reemerges within trope theory when we begin to wonder about the “compresence” relation in virtue of which collections of tropes constitute or compose concrete individual substances (particulars). Since substantial alteration can consist only in the replacement of one trope by another, acknowledging the possibility of such alterations amounts to recognizing that the compresence relation is contingent. Unlike trope equality (sameness, exact resemblance), in other words, trope compresence cannot consistently be treated as another primitive “internal” relation, one founded in the nature or essence of its relata. As Simons writes,

It would seem that what we need to link two accidentally compresent tropes is their common relation of dependence to the larger bundle of which both are elements. But this cannot serve as the definition of compresence, since it presupposes what the relation of compresence is supposed to itself accomplish, namely the welding of a collection of tropes into a whole. (PPC, 560)

And the apartness of tropes is no less problematic than their togetherness. For what prevents there being several equal (exactly resembling) tropes constituent in a single concrete particular “bundle”? We seem to need a further principle to guarantee that coexisting distinct but indistinguishable (non-identical but equal) tropes must at least be in different places — and so we are inevitably brought around again to the themes of space and time which came to the fore in the original formulation of our antinomy. Before addressing them, however, we need to take a brief look at our second candidate.

singular properties, the second candidate: hæcceities

I won’t spend much time discussing hæcceities, since they strike me as even more problematic than tropes. To say that each of Black’s spheres differs from the other by virtue of its unique “thisness” is both to say too much and to say too little. Hæcceitism says too much in that what it says suffers from the same prima facie fatal shortcomings as trope theory in general. A resolution that appeals to hæcceities similarly replicates the problematic core of our original antinomy at a finer-grained level. Like trope nominalism, that is, hæcceitism proposes to explain the possibility of distinct but indistinguishable concrete particulars by positing further items — “individual essences” — “constituent” in such particulars, which are themselves distinct without being distinguishable. Hæcceitism also says too little, however, in that it offers no constructive account either of how its “individual essences” are related to the further general properties of the particulars whose essences they are or of how to integrate such non-repeatable singular “thisnesses” constituent in concrete particulars with the repeatable abstract spatio-temporal properties and relations obtaining among those particulars. If the “thisnesses” constituent in each of Black’s two spheres are not ab initio regarded as located in different places, it is difficult to see how the hæcceitist proposal can be brought to bear on issues regarding the spatial disposition of the spheres themselves.

the third option: ontologizing subjectivity

In order to proceed, it will prove useful at this point to ask just what it is that creates the original presumption that what Black has described is a possible world. One obvious answer is that we can evidently readily conceive of or visualize or imagine a world answering to his description. But is that so? Can we in fact imagine Black’s World? There can be no question, I think, that we both can and do imagine something in connection with Black’s description. What is much less clear, however, is whether what we imagine is what Black purports to describe, a world containing “nothing but two exactly similar spheres … made of chemically pure iron”, each one mile in diameter and each having the same temperature, color, etc. as the other. Let me approach this question somewhat indirectly.

Recall our Tractarian description of Black’s World (given the simplifying assumption that it instantiates only three monadic general properties, C, D, and G, and one irreflexive and symmetric relational property, R):

(B0) Ca, Cb, Da, Db, Ga, Gb, Rab, Rba,

I earlier suggested that (B0) represents one sort of “complete description” of World B. We can reestablish contact with Koch’s original text by filling out (B0) into a slightly different candidate “objective and complete description of [Black’s] spatio-temporal universe” (SRZ, 26), namely a Carnapian state-description.

On our simplifying assumptions, the two names ‘a’ and ‘b’ suffice to denote the two particulars in Black’s World, and the monadic predicates ‘C’, ‘D’, and ‘G’, together with the dyadic predicate ‘R’, suffice to express all the general concepts under which those two particulars are subsumed. To arrive a Carnapian state-description of World B, then, we need only add to our Tractarian description the negations of the remaining two atomic sentences formulable in the given vocabulary:

(BC) Ca, Cb, Da, Db, Ga, Gb, Rab, Rba, ~Raa, ~Rbb

Given the assumptions we have made, [this] state-description would be not only a complete, but also a completely general and objective world-description. Someone in command of it would be in a position of non-participatory omniscience with respect to the spatio-temporal world.
(SRZ, 27)

But when we imagine World B, we do not (indeed, we cannot) do so from a position of non-participatory omniscience. We are situated, perspectival beings, not omniscient beings. When we imagine Black’s World, in other words, we unavoidably do so from a point of view. Consequently, in that world as we imagine it, the two iron spheres, which ex hypothesi share all their monadic properties and a single general dyadic property R (e.g., … is at a distance of thirty miles from …), will necessarily instantiate at least one further dyadic indexical spatial property. In World B as we imagine it, one sphere will be to the left of the other, or in front of the other, or above the other, or closer than the other, or …. In short, the spheres will stand in some relation belonging to what Koch (following Gareth Evans) calls an “egocentric” spatio-temporal system. (Cf. SRZ 38-9)

On Koch’s view, however, far from being an adventitious artifact of our imagining, such indexicality is an a priori constraint on the very possibility of Black’s World, and in fact constitutes the solution to the antinomy that we have been seeking.

[We thereby] recognize that something can be true of a spatio-temporal entity or a particular — whether or not we choose to call it a property — which is not ascribable to it by saying under which general concepts it falls. Besides descriptive truths, we acknowledge a second set of truths, which, in addition to a perhaps indispensible descriptive element [Moment], must also possess a further element. … Alongside the element of descriptiveness [Deskriptivität], indexicality must be acknowledged as a second element of what can be true of a particular.
This, then, in abstracto, is the resolution of the contradiction discovered earlier. (SRZ, 22-3)

It is important to see that Koch interprets what might at first appear to be a merely epistemological constraint as entailing a genuinely ontological conclusion. Our reflections may indeed begin epistemologically, by asking “how a cognizing subject can separate two descriptively [i.e., attributively] identical particulars in thought, univocally intend the one or the other and [so] make identifying reference to it in thought and speech”. (SRZ, 24) The straightforward, Strawsonian, answer to this question is that any and all such identifying reference is ultimately subject-centered, resting in the last analysis on the demonstrative identification of items sensibly present to the cognizing subject. On Koch’s view, however, the subject-centeredness of identification that here comes to the fore is not merely contingent [nicht zufällig]. The subject-centeredness of identification is a consequence of an ultimate subject-centeredness of identity.

As we now see, two descriptively identical particulars a and b are distinguishable from and relative to a cognizing subject, itself necessarily a particular and so an entity in space and time, and indeed relative to this subject insofar as it is capable of indexical thinking. Determinate indexical truths hold of a which do not hold of b and conversely. In this way, the identity of indiscernibles can be reconciled with the coming apart of numerical and descriptive [attributive] identity for particulars. If everything non-indexically true of a is also true of b [and conversely], a and b are descriptively identical. When furthermore everything indexically true of a is also true of b [and conversely], a and b are numerically identical. (SRZ, 25)

“A philosophical theory of subjectivity,” claims Koch, “reaches not only into epistemology but also into ontology”. (SRZ, 26) We can see this, he suggests, by examining more carefully the ideal of “non-participatory omniscience” represented by the notion of a Carnapian state-description. That ideal is in principle unreachable, not simply for the reason that, prima facie, any state-description of the actual world could be framed only in an unlearnable language, one containing infinitely many individual constants and/or predicate constants.[10]

The ideal is fundamentally unreachable for reasons which weigh much more heavily. … In reality, it concerns a thought which, because it is contradictory, cannot [even] be entertained.
This in any event follows from our argumentation, which has yielded the result that indexical truths must hold of particulars not just for the sake of their cognizability but already for the sake of their numerical identity.[11] The neglect of indexical truths definitive of the ideal of non-participatory omniscience leads back into the contradiction between the identity of indiscernibles and the [principle of] pre-conceptual multiplicity. (SRZ, 27)

The upshot of this line of thought is that a commitment to subjectivity is conceptually inseparable from the very idea of particulars located in time and space.

Subjectivity belongs essentially to the system of particulars, i.e., the spatio-temporal universe. There would be neither space nor time nor particulars if there were not, somewhere and somewhen in space and time, particulars within which subjectivity was embodied. (SRZ, 28)

Koch, in short, sees our antinomy as a compelling argument for what Kant called the transcendental ideality of space and time, a conclusion already hinted at by Black’s imaginary alphabetic character “B”:

A. If you are right, nothing unobserved would be observable. For the presence of an observer would always change it, and the observation would always be an observation of something else.
B. I don’t say that every observation changes what is observed. My point is that there isn’t any being to the right or being to the left in the two-sphere universe until an observer is introduced, that is to say until a real change is made. (II, 91)

The thesis of the transcendental ideality of space and time is the thesis that such “egocentric” spatio-temporal systems are ontologically basic or fundamental. This is not to say that we cannot operate with purely formal representations of space and time “as mathematicians do”. But it is to claim that we can do so only by “one-sidedly” abstracting from something ontologically essential to them.

What can favor … the appearance that space and time possess a primitive capacity to produce exceptions to the identitas indiscernibilium is the predominance of a purely theoretical, mathematical outlook. A thoroughgoing strategy of isolation or “divide and conquer” … is characteristic for theoretical science. One can also call this the strategy of abstracting from determinate essential characteristics (Wesensmerkmale) of a thing in order to control it theoretically. In the case of space and time, what falls victim to this abstractive outlook — treating space geometrically and then time by analogy to space — is precisely their essential connection with subjectivity. They continue to appear as principles of separation for numerical and descriptive [attributive] identity, but we can no longer see the [indexical] grounds which make this possible …. Consequently, space and time present themselves as conditions of the possibility of exceptions to the law of the identity of indiscernibles. (SRZ, 31-2)

These last remarks highlight the fact that the thesis of the transcendental ideality of space and time is prima facie incompatible with scientific realism, the conviction that, as Sellars put it,

… in the dimension of describing and explaining the world, science is the measure of all things, of what is that it is, and of what is not that it is not.[12]

To frame the point less metaphorically, scientific realism holds that the methods of inquiry characteristic of theoretical natural science suffice in principle to yield a world story that is explanatorily closed in the sense that it includes an account of what there is, i.e., of what is real, which suffices for the explanation of all truths about how things appear, in short, that the (ideal) outcome of scientific inquiry is a representation of reality that is adequate to “save” all the “phenomena”. But this order of explanation is necessarily also an ontological order. We explain the (merely) apparent in terms of the real; things seem as they do because things are as they are. The ideal explanatory closure of science’s world-story is thus also its ultimate ontological authority. What there is, in the last analysis, is what, in the last analysis, science says there is.

What is especially troublesome about the hypothesis that space and time are “transcendentally ideal” is that it appears to invert this order of explanation. For, from the standpoint of natural science, it is surely (in part) the relative positions of objects and observer in physical space and time that explains the perspectival (indexical) relationships among items experienced as obtaining within that observer’s “egocentric” spatio-temporal system. My “perspectival” situatedness as an experiencing subject is explanatorily parasitic — and, hence, ontologically parasitic — on my objective situatedness as an organism, an object among objects. As I put it elsewhere:

I discover myself as res in the world — and, ultimately, this explains the possibility of [subjectivity]. For what, normatively (semantically) viewed, are my representings, ontologically viewed turn out to be complex inner states of an empirical organism … — matter-of-factual states which are “inner” only in the literal and unproblematic sense of occurring inside the skin. “Inner sense” thus emerges as nothing but a manifestation of the de facto causality of such inner states — the … power of one such state, normatively a representing, to evoke another, normatively a representing of that representing.[13]

Thus, while Koch’s strategy of ontologizing subjectivity does, in one sense, resolve our antinomy, like the proposals we investigated earlier, it does so only at a significant philosophical price. For the moment, at least, it is a price that I am personally am reluctant to pay.

one more try: transcendental ambiguity

But is there any reasonable alternative? Or are we condemned to choose between an ontology which incorporates the essential structure of our antinomy as an internal feature of its basic entities and one which instead takes us the first steps along a road that threatens to lead inexorably to absolute idealism? Well, to paraphrase Sherlock Holmes, once we’ve eliminated the merely unlikely, it’s time to begin thinking seriously about the really improbable. It is at this point that Johanna Seibt offers[14] the intriguing suggestion that the source of our problems lies in “transcendental ambiguity”, specifically, in “the false presupposition that individuality and countable particularity: sameness and oneness, are one and the same”. (NCI, 1) She thus proposes to argue

that [such] duplication scenarios [as Black’s thought-experiment] falsify [Leibniz’s Principle] as a principle of oneness but not as a principle of sameness. (NCI, 19)

Seibt’s hypothesis is that our dialectical troubles result from an ambiguity in the notion of identity. The term ‘identity’ is sometimes equivalent to ‘oneness’ (vs ‘plurality’), and sometimes to ‘sameness’ (vs ‘distinctness’), a relation which can obtain between two entities. In this context, then, it is most perspicuous simply to eschew the term ‘identity’ entirely, and that is the course I will adopt. Subject to these terminological conventions, then, Seibt’s proposal is that our original formulation of Leibniz’s Law,

(LL) Necessarily, an entity, a, is identical to an entity, b, if and only if every property of entity a is a property of entity b, and conversely,

is ambiguous between

(LLO) Necessarily, a and b are one entity, if and only if every property of a is a property of b, and conversely,

which is false, and

(LLS) Necessarily, a and b are the same entity, if and only if every property of a is a property of b, and conversely,

which is true.

Since our antinomy’s second leading principle, the requirement of “pre-conceptual multiplicity”, also mobilizes the notion of identity, viz

(C) Possibly, there are entities, a and b, such that entity a is not identical to entity b, and (nevertheless), for every general concept C, entity a falls under C if and only if entity b falls under C,

it will presumably also be “transcendentally ambiguous” — between

(CO) Possibly, there are a and b, such that a and b are not one entity, and (nevertheless), for every general concept C, a falls under C if and only if b falls under C,

and

(CS) Possibly, there are a and b, such that a and b are not the same entity, and (nevertheless), for every general concept C, a falls under C if and only if b falls under C.

Seibt’s strategy thus locates the source of the antinomy along an axis of distinctions orthogonal to any hypothesis regarding the relationship of properties and general concepts, and she consequently remains free simply to accept the strict correlativity posited by (CP). In the context of that posit, (CO) will be tantamount to the negation of (LLO) and so, on her view, true, while (CS) will be equivalent to the negation of (LLS) and thus false. One the face of it, then, Seibt’s proposal successfully provides conceptual resources that are nominally adequate to underwrite a resolution to our antinomy.

[Duplication] scenarios [like Black’s World] show that the same set of properties may be instantiated by more than one thing; they do not show that they may be instantiated by distinct things. That is, duplication scenarios show that [the Principle of the Identity of Indiscernibles] fails as a principle of oneness. But they do not show that [it] fails as a principle of individuality or sameness …. (NCI, 24)

The challenge at this point is to transform this suggestion into something more than a merely terminological proposal. As Seibt herself points out, this is not too easy to do.

The claims:
[1] the names ‘A’ and ‘B’ refer to two entities but to the same entity
[2] the names ‘A’ and ‘B’ refer to one entity but to distinct entities
strike one as non-sensical to a degree which borders on indecency. Any competent hearer of utterances of sentences [1] and [2] would seriously doubt the speaker’s linguistic competence, sincerity, or sanity. (NCI, 2)

One reason for this impression, she contends, is that such “transcendentals” as particularity and discreteness — general features under which items belonging to diverse ontological categories (objects, properties, events, states, etc.) can be subsumed — have traditionally been a blind spot of logical analysis. Commenting on Quine’s influential discussion of the notion of identity in Methods of Logic, she points out that,

First, the notion of identity [is introduced as] the notion of sameness simpliciter …. Second, the relation of sameness is said to be functionally exhausted in the indication of coreference of names; … all entities that stand in the relation of sameness are thus said to be particulars. Third, the relation of sameness is read as the relation of numerical oneness; this … amounts to a commitment that all entities which stand in the relation of sameness are countable. Altogether, entities standing in the relation of sameness, that is, individuals in the transcendental sense, are said to be countable particulars. But there are individuals which are neither countable nor particular entities. By incorporating the equation of individuality and countability into logical grammar, the latter is importantly restricted in its range of applicability. (NCI, 11)

Standard logical tools, Seibt concludes, veil transcendental ambiguity. Any explanation of the difference between oneness and sameness, in short, is going to have to be informal and discursive.

Whatever the ultimate details, oneness is prima facie likely to prove the easier notion, since we have available both the operation of counting and the question “How many …?” as potential elucidatory tools. Sameness, in contrast, looks more difficult. Seibt offers little in the way of an informal discursive elucidation. Her potentially most helpful account appeals to the idea of an equivalence class:

The notion of an equivalence class oscillates between two concepts: the class as the same (not ‘one’!) and the class as many. Of course, in common cases the members of an equivalence class can be distinguished by means of their remnant properties. But relative to its comprehension property the members of an equivalence class cannot be further individuated, and precisely therefore they can be treated as the same; every member of the class is the same individual as the arbitrarily chosen so-called ‘representative’ of that equivalence class. From this point of view the duplication scenarios accomplish nothng more than an illustration of the thesis that any object may be considered as the representative of an equivalence class with a maximally complex comprehension property, i.e., as a plurality of items which are all the same individual. (NCI, 21-2)

To frame the distinction in yet another idiom, sameness stands to oneness as “type identity” stands to “token identity”. Seibt’s proposal, in other words, enjoins us to think of World B’s two spheres, so to speak, as two (entity) tokens of one (entity) type. On this reading, our antinomy results from confusing a true principle regarding types (roughly, LLS or CO) with a false principle regarding tokens (roughly, LLO or CS).

But having come this far with Seibt, it becomes appropriate to ask whether accepting her proposal really leaves us any better off. The basic intuition underlying the principle of the identity of indiscernibles, we should recall, was precisely that entities cannot differ solo numero, and it was to this basic intuition that I appealed when criticizing our very first resolutive proposal, that Leibniz’s Law is a false principle regarding general concepts:

Whatever we may ultimately want to say about general concepts, once we accept Black’s scenario as describing a possibility, we surely cannot avoid agreeing that there must be some respect in which the two spheres in Black’s World differ. It seems utterly natural simply to equate this claim with the claim that there must be at least one property which one sphere has that the other lacks.

The philosophical price of accepting Seibt’s proposal, it should now be clear, is that we must abandon this fundamental intuition root and branch. We must rather be prepared to concede that there is (and need be) no respect in which the two spheres in Black’s World differ — for we must be prepared to concede that they do not differ at all. On Seibt’s account, items differ just in case they are not the same, but Black’s two spheres, although not one, nevertheless are the same. They are, as it were, “two and the same”.

In particular, we cannot take refuge in the suggestion that the two spheres in Black’s World differ by virtue of being in different places. What holds for the spheres holds equally for the places they occupy. The two spheres are not, of course, in one place. But it does not follow from this that they are not in the same place. Places, on Seibt’s account, are individuated Leibnizianly, “by membership in an equivalence class of regions having certain spatial coordinates”, where the coordinate system in question is, so to speak, “implicitly defined” by the relations obtaining among the items “in” space. “[Thus] one can consistently maintain that two spheres occur in the same place as much as one can say that 50 million Americans read the same newspaper article.” (NCI, 22) Like sameness of objects, in short, sameness of places is “type-identity”. Places, too, can be “two and the same”.

When the chips are down, then, Seibt’s proposal shares with trope nominalism and hæcceitism what we might call a certain resignation. In her own way, that is, Seibt too replicates the problematic core of our original antinomy. In each case we are counseled simply to accept the possibility of two indiscernible, (absolutely) indistinguishable, things. Seibt’s proposal differs from those earlier suggestions principally in that it prohibits us from drawing the further conclusion that two such indiscernible things are distinct (i.e., not the same), and thereby denies a presupposition of the question “How, then, do they differ?” In doing so, however, it also cuts us off from the onto-genetic question “In virtue of what are the two spheres two?” In Black’s World, plurality is left without an ontological ground — but only in Black’s World and its thoroughgoingly symmetrical like. For although plurality may not imply distinctness, distinctness surely must imply plurality. The onto-genetic question is not, then, one that Seibt rejects generally and in principle. “[In] common cases the members of an equivalence class can be distinguished by means of their remnant properties”. (NCI, 21) She denies the question application only where, on her account, its characteristic presupposition fails — but that is also precisely where it is likely to be most acutely felt.[15]

final thoughts

At the beginning of this essay, I promised no resolution but only a series of difficult choices. I submit that I have at least made good on that promise. If I am right, the antinomy we have been exploring confronts us with (at least) three relatively outré ontological alternatives: tropes (or hæcceities), irreducible subjectivity, and transcendental ambiguity.[16] For the reasons I have briefly given, I cannot yet rest comfortably with any of these options — but neither, alas, do I have any further alternative to propose. That is an unsatisfying way to leave a philosophical question, I grant, but our revels now are ended, and, for the moment at least, like Prospero, “my ending is despair”.

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