The Identity of Indiscernibles: Some Tractarian Reflections

[FROM: Acta Analytica, 21, 1998, pp. 11-29.]

Max Black’s well-known thought-experiment:

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 the locus classicus for an ostensible counterexample to Leibniz’s principle of the Identity of Indiscernibles:

(LP) 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.

Leibniz’s principle captures the intuitive ideas that distinct entities must somehow differ and that entities cannot merely differ (full stop), but must necessarily differ in some respect. If what are ostensibly distinct items differ in no respect, then, we may legitimately conclude that the items are two "in name only", i.e., that we have to do with one and only one item. The "species" of identity at issue in this principle is typically termed "numerical identity", and similarly one finds "numerical distinctness", but fewer questions are prejudged by speaking simply of identity, tout court or sans phrase. Call the minimalist state of affairs envisioned in this thought-experiment "Black’s World" or "World B" for short.

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 this is arguably one of the junctures at which formal logic itself is in need of guidance that perhaps only metaphysics can provide. Intuitively unproblematic informal descriptions of events, circumstances, or states of affairs can rest on structures of unarticulated presuppositions that in fact embody deep-seated and well-concealed incoherencies. Taken at face value, however, Black’s description certainly seems at least to be formally consistent, and this judgment is evidently confirmed by the prima facie ease with which we can construct a complete and consistent formal specification of World B.

What might such a specification look like? For the sake of expository convenience, let us assume that we have to do in Black’s World with only three (monadic) general properties, C, D, and G, and one dyadic relational property, R (e.g., "… is at a distance of nine sphere-diameters from …"). One potentially interesting form which, on the face of it, such a "complete formal specification" of World B can take is what we might call its "basic Tractarian world-story", i.e., an enumeration of all its elementary states of affairs — its Sachverhalten, as Wittgenstein calls them in the Tractatus — by means of the elementary sentences (Elementarsätze) which represent (picture) them. "2.04: Die Gesamtheit der bestehenden Sachverhalte ist die Welt." If we call the two spheres in Black’s thought experiment ‘a’ and ‘b’, then, given our expository assumptions, it is plausible to suppose that, in first approximation, a complete Tractarian description of World B looks like this:

                                       (BT)         Ca, Cb, Da, Db, Ga, Gb, aRb, bRa

Does (BT) describe a prima facie counterexample to Leibniz’s Principle (LP), presumably given logistically by

(L1)     (" x)(" y)[(" F)(Fx º Fy) ® (x=y)].

Well, it does if it entails

(BW) ($ x)($ y)[(x¹ y) & (" F)(Fx º Fy)]

which contradicts (L1), and, in the context of the expository assumptions we’ve made, it certainly seems as if (BT) ought to entail (BW). For, on the face of it, what we’ve assumed amounts to

(E1)          a¹ b .&. (" x)(x=a v x=b) and

(E2)    (" x)[($ F)Fx .º . Cx v Dx v Gx v ($ y)(xRy v yRx)],

and, taken together, (BT), (E1), and (E2) surely do entail (BW).

It is worth noticing, however, that, from the perspective of the Tractatus, (E1) represents an attempt to say what, in a properly perspicuous notation (richtige Begriffsschrift), can only be shown. In such a notation, what (E1) tries to say is shown by the fact that two and only two different lower-case signs from the beginning of the alphabet, ‘a’ and ‘b’, appear in the complete formal description, (BT), of World B. Indeed, in a properly perspicuous Tractarian notation, the sign ‘=’ does not appear at all — identity and distinctness of represented objects being shown by sameness and difference of representing signs — and so, like (E1), neither (L1) nor (BW) can strictly speaking be formulated. Thus whether a world is Leibnizian or Black (as I shall put it) will apparently itself be something shown. One can only, so to speak, "read it off" a complete formal Tractarian world-description, on the model of (BT). For the moment, however, I shall simply bracket and set aside such Tractarian strictures, and avail myself of the full notational might of what Sellars called PMese. It will turn out that perspicuous notation is in fact what this essay is about, but that certainly isn’t obvious, and so it will take a while for us to get around to the topic.

What I want to do meanwhile, however, is to consider an objection to the claim that (BT) — either in the context of, or straightforwardly conjoined with, (E1) and (E2) — represents, and correlatively, that World B therefore constitutes, a counterexample to Leibniz’s Principle (LP). The objection — call it ‘Objection O’ — runs roughly like this:

(BT) may indeed be, in one sense, a complete formal description of World B, but the fact that it represents an entirely symmetrical distribution of that world’s basic relations and simple intrinsic properties does not imply that the world is Black rather than Leibnizian. For the two items in World B also have derived, extrinsic or relational properties, and these do not occur symmetrically. Sphere a, for instance, has the property being at a distance of nine sphere diameters from sphere b, a property that sphere b plainly lacks. Formally, if we define a predicate, P, according to the schema
(DfP) Px =df xRb,
then it will be true that Pa, but it is not true that Pb. Given (E1), however, the truth of ‘Pa & ~Pb implies the falsity of (BW). But if (BW) is false, then (L1) must be true, and World B turns out to be Leibnizian after all.

According to Objection O, then, we were too quick to accept (E2). C, D, and G may be all the primitive (intrinsic) monadic properties of a and b, but they do not exhaust the monadic properties of the two spheres, primitive and derived (extrinsic or relational). (E2) is consequently false — but taken together with (E1) alone, (BT) does not entail (BW).

One way to begin thinking about this objection is to inquire into the reasons for supposing that ‘Pa’ is true and ‘Pb’ false. The answer to the first of these questions is easy. ‘Pa’ abbreviates ‘aRb’, which is an immediate consequence of (BT). The second question, however, is trickier. ‘Pb’ abbreviates ‘bRb’, and, whether taken alone or conjoined with (E1), (BT) simply does not entail the falsity of ‘bRb’. What evidently does (in some sense of the word) imply the falsity of ‘bRb’, however, is the presumed completeness of (BT)’s enumeration of World B’s elementary states of affairs. Speaking Tractarianly, in other words, the fact that neither ‘aRa’ nor bRb’ occurs in (BT) shows that neither is true in World B.

Faced with the task of arguing for the Leibnizianity of Black’s World, however, our objector may well propose a different paradigm for a "complete formal specification" of World B, namely a Carnapian state-description. Unlike a basic Tractarian world-story, which enumerates all and only those elementary states of affairs which obtain, a Carnapian state-description of a (possible) world undertakes to say how that world stands with respect to the obtaining or not obtaining of all elementary states of affairs. To arrive at such a description of World B, then, we would need to add to our Tractarian description the negations of the remaining two elementary sentences formulable in our expository vocabulary:

(BC) Ca, Cb, Da, Db, Ga, Gb, aRb, bRa, ~aRa, ~bRb

~Pb’ (abbreviating ‘~bRb’) then follows from (BC) as directly as does ‘Pa’.

Surely, however, one can also say Tractarianly that neither aRa nor bRb obtains in World B. Although neither is an elementary sentence, of course, both ‘~aRa’ and ~bRb’ are well-formed sentences of the extensional, truth-functional language that Wittgenstein sketches in the Tractatus. Is there any reason at all, then, to suppose that World B’s situation with respect to the possible states of affairs aRa and bRb might be better (more perspicuously) represented Tractarianly, by the absence of ‘aRa’ and bRb’ in (BT), than Carnapianly, by the occurrence of both ‘~aRa’ and ~bRb’ in (BC)? Curiously enough, there is.

We can bring the issues here into sharper focus by noting that there is one sense in which we may well be inclined to say that aRa and bRb are not possible states of affairs. Given the intended interpretation of ‘R’, i.e., "… is at a distance of nine sphere diameters from …", it is not implausible to hold no item could stand in the relation R to itself. One might reasonably maintain, in other words, that R is (necessarily) an irreflexive relation, and the absence of ‘aRa’ and bRb’ in a complete enumeration of World B’s Sachverhalten can correlatively be interpreted as reflecting this fact. Analogously, the occurrence of both ‘aRb’ and bRa’ in (BT) can be seen as mirroring R’s necessary symmetry.

From the Tractarian perspective, such formal properties of relations among objects are among the very paradigms of what can only be shown. I have already remarked that, in the perspicuous notation outlined in the Tractatus, the identity or distinctness of represented objects would itself be represented (shown) only by the sameness and difference of the signs used to represent those objects. What is here apposite, however, is one of Wittgenstein’s further requirements for such a notation, viz, that the formal properties of the relations (represented as) obtaining among objects be themselves represented (shown) only by formal properties of relations (actually) obtaining among the signs representing those objects. (Cf. 4.122) We could then no more say that a relation is irreflexive and symmetric than we can now say in our customary notation that it is dyadic.

Now it is obvious that the representational scheme adopted for (BT) is not thoroughgoingly perspicuous in this strict Tractarian sense. Nevertheless, there is arguably a way in which it comes closer than the alternative Carnapian scheme deployed in (BC). Viewed through Tractarian eyes, what in either scheme represents the fact that the objects a and b stand in the relation R (i.e., what says that aRb) is the (syntactic) fact that the signs ‘a’ and ‘b’ themselves stand in a relation — call it ‘r ’ — where ‘s1r s2’ is true just in case the signs s1 and s2 stand respectively to the left and the right of an ‘R’. Regarding (BT) and (BC) simply as collections of configurations of signs, then, we can ask how each collection stands vis-à-vis the r-relationships obtaining among the r-related signs occurring in it. Letting ‘u’ and ‘v’ range over the apposite signs, and noting that

(C)    (" u)(u=‘av u=‘b’)

is true of both (BT) and (BC), we can see that, whereas both

(S)    (" u)(" v)(urv º vru) and

(I)    (" u)~(uru)

are true of (BT), (S) but not (I) is true of (BC). On the contrary, since both ‘aRa’ and ‘bRb’ occur in the Carnapian state-description, viz, as constitutents (respectively) of ‘~aRa’ and ‘~bRb’, it plainly follows that

(R)    (" u)(uru)

is true of (BC). In short, the ex hypothesi symmetric and irreflexive (spatial) relation, R, that Black’s thought experiment posits as holding between the two spheres in World B, is represented by a (syntactic) relation, r , which itself occurs symmetrically and irreflexively in (BT) but symmetrically and reflexively in (BC). In this somewhat curious way, then, the disposition of primitive relational properties postulated for World B is indeed more perspicuously represented Tractarianly, by (BT), than Carnapianly, by (BC).

Can a would-be critic of Leibniz’s Principle (LP) defend the status of Black’s thought experiment as a counterexample in the face of Objection O? That is, is there any way sensibly to maintain, despite the objector’s argument to the contrary, that World B is not Leibnizian but indeed Black? One strategy plainly would be to deny the tacit assumption, now made explicit, that R is an irreflexive relation. That is, one strategy would be to hold that the correct complete formal description of World B is given neither by (BT) nor by (BC), but rather by

(BB) Ca, Cb, Da, Db, Ga, Gb, aRb, bRa, aRa, bRb

which, inter alia, answers equally well to both Tractarian and Carnapian specifications for perspicuity.

But can one make sense of this strategy in light of the intended interpretation of ‘R’, i.e., "… is at a distance of nine sphere diameters from …"? Johanna Seibt has argued that the answer is "Yes". The key to doing so is to recognize that the reflexivity or irreflexivity of a spatial relation depends upon formal or structural features of the space within which its relata are embedded. For instance, she proposes, we can consistently posit that World B’s two spheres occur in a "radial-symmetric space", i.e.,

that the total set of points constituting the space under consideration (call it S) is organized in the following way:
For any point p in S with coordinates <x*,y*,z*>, there is a [distinct] point q with coordinates <x*,y*,z*>.

Given this assumption, however,

one can consistently maintain that two spheres occur in the same place as much as one can say that 50 million American read the same newpaper article.

Nothing then stands in the way of interpreting the (quasi-) metric relation R in terms of places, i.e., xRy iff the place occupied by x is at a distance of nine sphere diameters from the place occupied by y. And, since the radial-symmetric space of World B, the place occupied by a is the same as the place occupied by b, it follows that    

(SBW)       (" x)(xRa º xRb)   

from which, given bRa, we may infer the truth of ‘bRb’, and correlatively, given aRb, the truth of ‘aRa’. Objection O, which presupposes the falsity of both ‘aRa’ and ‘bRb’, would then fail. To put it another way, on this reading, the property of being at a distance of nine sphere diameters from [the place occupied by] b, which the objector concedes is true of sphere a (but not of b), is arguably identical to the property of being at a distance of nine sphere diameters from [the place occupied by] a, which consistency demands the objector concede is true of sphere b. On this interpretation, in short, World B is indeed Black and a successful countersample to Leibniz’s Principle (LP).

This line of thought is hardly going to satisfy our objector, however. He is rather likely to find the notion of such a radial-symmetric space irredeemably problematic. For one thing, he might argue, the fact that the two spheres stand in the spatial relationship that they do (i.e., R) implies that they are located in different places. In general, at any given time, distinct (spatial) things necessarily occupy different places, and the notion of a place, in turn, is parasitic on this notion of something’s occupying a place. For another, one cannot really be as sanguine about the notion of spatial coordinates as this strategy seems to be. Space, even "absolute space" does not come coordinatized. The right to appeal to "spatial coordinates" must be suitably earned, at least by demonstrating that the space in question is in principle coordinatizable by reference (only) to the items in it. Given its radical symmetries at the level of elementary states of affairs, however, it seems unlikely that World B offers sufficient resources to get even an in principle coordinatization off the ground.

It is doubtful, then, that the considerations invoked by the strategy we have been considering will move our hypothetical objector to retreat from his critique of the claim that Black’s thought experiment describes a counterexample to Leibniz’s Principle. And one cannot help having considerable sympathy with his resoluteness. It is surely intuitively much more plausible to hold fast to our original tacit assumption that the relation R is (necessarily) irreflexive than to posit for World B a (somehow) pre-coordinatized "radial-symmetric space" structured in terms of a peculiar, non-standard distance metric. Objection O, I conclude, survives this first response. But is there any other, more plausible, way in which a defender of World B’s contra-Leibnizian Blackness might undertake to criticize the objection?

Well, one tempting strategy at this point is surely to deny that World B contains any asymmetrically-instantiated properties on the grounds that it contains no derived, extrinsic or relational properties at all. Strictly speaking, all its genuine monadic properties are basic (simple or elementary). Reference to a so-called "defined relational property" is merely a façon de parler, i.e., a way of abbreviating claims about properties and relations properly so called, claims that can be literally expressed only by combinations of elementary sentences. On this strategy, the schema (DfP) in particular will fail to introduce or pick out any property which could be instantiated by one sphere and not the other. Correlatively, (E2) will be interpreted as a necessary truth about World B’s properties, properly so called. Since either (BT) or (BC), taken together with both (E1) and (E2), indeed implies (BW), it will follow that World B is not Leibnizian but genuinely Black.

The problem with this strategy is that it appears to be entirely ad hoc. In ordinary circumstances, there is nothing wrong with defined relational properties nor even anything puzzling about them. Being born later than Socrates and being shorter than Mont Blanc strike us as completely unproblematic monadic properties, severally instantiated by multitudes of individuals (including the present author and most likely the present reader as well). The only motivation for excluding derivative extrinsic or relational properties from World B seems to be to avoid conceding the objector’s case. But even the most uncompromising defense of World B’s Blackness does not require ruling out defined relational properties per se, since many of them — e.g., being nine sphere diameters distant from something — occur entirely symmetrically. Yet if the schema

(DfQ) Qx =df ($ y)(xRy)

introduces or picks out a genuine monadic property, it is difficult to see why the schema (DfP) does not.

This last remark, however, suggests another, perhaps more plausible, strategy for responding to Objection O. A blanket rejection of defined relational properties indeed seems ad hoc and arbitrary. But perhaps there is some principled way to distinguish the objector’s putative property P — being at a distance of nine sphere diameters from sphere b — from such prima facie innocuous cases as Q — the property of being at a distance of nine sphere diameters from something. The fact that (DfQ), in contrast to (DfP), is completely general, i.e., contains neither ‘a’ nor ‘b’, immediately suggests itself, of course, but unless we can explain why that difference makes a difference here, we are liable to be met only with another charge of ad hoc and arbitrary stipulation. After all, didn’t we just implicitly concede that being born later than Socrates and being shorter than Mont Blanc are perfectly legitimate relational properties, despite the fact that the proper names ‘Socrates’ and ‘Mont Blanc’ would obviously need to occur in any schemata used, analogously to (DfP) and (DfQ), to introduce them or pick them out, e.g.,

(DfL) Lx =df ($ t1)($ t2)[Born(Socrates,t1) & Born (x,t2) & (t1 < t2)]

(DfM) Mx =df Shorter(x, Mont Blanc)

Nevertheless, I propose to argue that, although the schemata (DfQ), (DfL), and (DfM) are all acceptable ways of picking out or introducing derived relational properties, the objector’s scheme (DfP) is not. To understand why, we need to take a more careful look at ‘a’ and ‘b’.

I have deliberately been very cagey about ‘a’ and ‘b’. When necessary, I have referred to them as "signs", but I have carefully avoided such more specific and familiar labels as "singular terms", "individual constants", and "names". ‘Name’, of course, is the Tractarian category, but it in this connection it is crucial to remember that what Wittgenstein initially purports to characterize in the Tractatus is a logically-perspicuous notation (eine richtige Begriffsschrift). At least to begin with, names properly so-called — e.g., ‘Socrates’ and ‘Mont Blanc’ — live in languages (so-called "natural languages") — e.g., English and French — and, at least to begin with, rhetorical customs of philosophers notwithstanding, a logical calculus is not a language, not even an "artificial language". If we are going to speak of names and languages in this context, then we need to understand what is involved in extending such notions to logical calculi. The Tractatus is equally a locus classicus for this extended notion, the notion of a symbology (Zeichensprache), or, in a more familiar idiom, an "interpreted formal system".

In the Tractatus, Wittgenstein formulates his remarks, e.g.,

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’.

in an idiolect concocted out of bits of language (originally German, here English) and symbols drawn from the notation of a prima facie familiar calculus. The textual situation is complicated by the fact that it is unclear whether the symbolic notation there displayed is intended to belong to the Tractarian Begriffsschrift, or simply to assist Wittgenstein in describing a perspicuous Zeichensprache. That, in any event, the mixture is potentially philosophically explosive is suggested by the fact that its grammar is pied. The matrix "There is one and only one ___ …" demands a count noun (e.g., "There is one and only one dog who has starred in two television series"), whilst the substituends foreseen for ‘x’ are clearly (proper) names. Our present question, however, is what to make of the symbol ‘a’. The most useful way to think about Wittgenstein’s philosophical intent, I want to suggest, is to regard the Tractarian symbology as the schema a candidate language and, correlatively, such symbols as ‘a’ as candidate proper names.

On this reading, a language, properly so-called, is an applied notation, i.e., as 5.526 suggests, a representational system put to use in "describing the world", paradigmatically, in saying something (true or false) about something. A name, correlatively, picks out something in the world (an item or "object") about which something (true or false) is being said. Wittgenstein’s Tractarian conception of a name thus answers to the familiar notion of a "singular designator".

3.202 The simple signs applied in sentences are called names.

3.203 A name designates (bedeutet) an object. The object is its designation (Bedeutung). …

3.22        A name represents an object in a sentence.

Names are singular designators in that a name is properly understood as picking out one and only one item. An applied representational element will be a name, then, just in case it is in principle possible uniquely to fix its referent, i.e., to identify and specify the single item it purports to pick out. 5.526 in turn reflects this plausible requirement in the form of a global constraint on the applicability of a candidate language, viz., that its descriptive resources are adequate for fixing the referents of its singular designators. Let us call an applied representational system which satisfies this requirement Tractarian.

We restore contact with our preceding discussions as soon as we notice that Wittgenstein’s remark 5.526 appears to be false of Black worlds. Suppose, for instance, that World B is Black. In that case, the "fully generalized proposition"

(B1)        ($ x)($ y)[(x¹ y) & (" F)(Fx º Fy) & (" z)(z=x v z=y)]

surely bids fair to offer what is, at least in one sense of the phrase, a "complete description" of that world, but (B1) does not issue in a pair of uniquely individuating descriptions which might serve, treating the terms as candidate proper names, to fix distinct individual referents for ‘a’ and ‘b’. That is, there is no route from (B1) to any propositions of the form

(UI)        ($ 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 (BT) or (BC). In short, Black worlds are not Tractarian. It follows that, strictly speaking, individual items in a Black world are not nameable. If World B is Black, then, it further follows that ‘a’ and ‘b’ in (BT) and (BC) cannot be (candidate) names. What they then would be remains to be seen.

Objection O presupposes that ‘a’ and ‘b’ in (BT) and (BC) are (candidate) proper names. More generally, it presupposes that World B is Tractarian. That is how it goes wrong. In general, a schema of the form

                        (DfM)         M1x =df xR2n,
will succeed in introducing a defined (monadic) relational property, M1, only if ‘n’ is a (candidate) name, i.e., if the descriptive resources of the representational system within which the schema is formulated are adequate to fix referents for its singular designators, specifically, for ‘n’. The schemata (DfL) and (DfM), for instance, unproblematically succeed in introducing the relational properties being born later than Socrates and being shorter than Mont Blanc precisely because the expressions ‘Socrates’ and ‘Mont Blanc’ occurring essentially (non-vacuously) in them are actual proper names whose references can be fixed by suitable deployment of the deictic (demonstrative and indexical) descriptive resources of our language, and the complete generality of (DfQ) renders the question of singular designators (names) moot.

(DfP) simply presupposes that, in particular, the expression ‘b’ occurring in (BT) and (BC), is a (candidate) proper name. That is, it presupposes the in principle possibility of fixing a referent for ‘b’ by uniquely identifying and specifying the single item it purports to pick out — i.e., in this instance, by uniquely identifying and specifying one sphere in contrast to the other — and so to satisfy a necessary condition for introducing a defined (monadic) relational property of (other) items in terms of their relations to it. (DfP), in short, presupposes that World B is Tractarian or, equivalently, the nameability of the items (the two spheres) in it — but, since Black worlds are not Tractarian, making this presupposition is in fact equivalent to assuming that World B is not Black. Since that is precisely what Objection O was intended to demonstrate in the first place, I conclude that the objection simply begs the question.

There remains, however, the matter of what to make of the signs ‘a’ and ‘b’ in (BT) and (BC). If they are not (candidate) proper names, what are they? Well, I’m not sure we have a properly neutral, non-controversial label for them. It turns out to be easier to say what they’re not. It’s clear enough that they belong to a large family of representational devices that I’ll call proxy nominals. Proxy nominals do part of the representational work done by proper names, but not all of it. Among other things, they represent the distribution of properties and relations among items (objects) without presupposing the possibility of identifying or picking out individual exemplars of any property or individual relata of any relation.

We meet proxy nominals within logical calculi, in the first instance, as bookkeeping devices, counters or markers, in formal derivations e.g., the derivation by indirect proof of ‘($ x)[Fx É  (" y)Fy]’:

[P] 1. ~($ x)[Fx É (" y)Fy]     Hypothesis
        2. (
" x)~[Fx É (" y)Fy]    1, QE
        3. (
" x)~[~Fx v (" y)Fy]     2, É - Equiv
        4. (
" x)[ Fx & ~(" y)Fy]      3, DeMorgan
        5. Fa &
~(" y)Fy             4, UI
        6. (
$ y)~Fy                             5, Simp, QE       
~Fb                                    6, EI       
        8. Fb &
~(" y)Fy              4, UI
        9. Fb                                      8, Simp
       10. (
$ x)[Fx É (" y)Fy]           1, 9, 7, IP

In this capacity, ‘a’ and ‘b’ subserve some of the representational purposes of a common subspecies of proxy nominals, viz. temporary proper names, and that model is indeed sometimes invoked for "individual constants" (or even unbound variables) occurring in this bookkeeping role. Hypothetical contexts are analogously characteristic linguistic loci for (genuine) temporary proper names, e.g.,

[K] Let us imagine that Saul Kripke has an identical twin brother — call him ‘Paul’ —
         and let’s suppose further that, in sharp contrast to his brother, Paul Kripke is both
         logically and mathematically incompetent.

The expression ‘Paul Kripke’ functions in such a context as a vehicle of what Strawson called "story-relative identification". Like a genuine proper name, it is an accumulation point for claims, a peg upon which descriptive content can be anaphorically hung, and, like a genuine proper name, it carries a (contextual) presupposition of uniqueness. But precisely this shows why proxy nominals in their bookkeeping function are not quite candidate temporary proper names. The job of the ‘b’ recurring on lines 7-9 of the derivation [P], for instance, is certainly to indicate that the given hypothesis implies that there is one item which both has and lacks the property F, but not that there is exactly one such item. While the "identification question" ‘Who is Paul Kripke?’ makes sense in the hypothetical context generated by [K] — which context also yields a determinate "story-relative" answer to the question — the context generated by the derivation [P] is insufficient to confer an analogous sense on the parallel question ‘Which item is b?’, much less to yield even a "derivation-relative" answer to that question.

In this respect, the ‘a’ and ‘b’ occurring in derivation [P] more closely resemble individual pronouns than they do temporary proper names. Like genuine pronouns, they too occupy syntactic places where proper names could go, and, like genuine pronouns, they signal and serve as anaphoric vehicles of intra-contextual cross-reference.

But, literally and strictly speaking, genuine pronouns are, in the first instance, placeholders for proper name substituends, and there are no proper names in the representational system to which the proxy nominals ‘a’ and ‘b’ belong. In particular, if we are going to respect the distinction between a notation (Begriffsschrift) and a symbology (Zeichensprache), then we will need to distinguish the syntactic from the representational role of pronouns. Thus, describing World B’s spheres, we might observe that, e.g.,

[D] If one is made of iron, then so is the other.
        Each is at a distance of nine sphere diameters from the other.

While genuine proper names of course occur within the natural language used to formulate [D], the discursive context supplied by Black’s thought experiment precisely disqualifies them as potential pronominal substituends here. It is thus perhaps philosophically more perspicuous to say that, in the description [D], the syntacticpronouns ‘one’, ‘each’, and ‘the other’ are themselves, so to speak, representationally degenerate. The proxy nominals ‘a’ and ‘b’ occurring in the formal descriptions (BT) and (BC) could then be understood as candidate pronouns in a sense, but only syntactically, on the model of the pronouns ‘one’, ‘each’, and ‘the other’ occurring in the properly linguistic description [D].

I originally introduced the signs ‘a’ and ‘b’ into our candidate formal descriptions of World B with the proposal that we "call the two spheres in Black’s thought experiment ‘a’ and ‘b’". The contrast between such proxy nominals and the more familiar representational categories of proper names, temporary proper names, and individual pronouns comes clearly into view when we observe that the linguistic formulation here is irreducibly plural. The two signs ‘a’ and ‘b’, that is, can be introduced into discourse about Black’s world only together. I could instead, of course, have written

[D ] Call one of the spheres ‘a’ and the other ‘b

but, although the syntax of [D ] is distributive, its sense is not, for this conjunction is indissoluble. Despite appearances, [D ] does not describe a conjunction of distinguishable, independent, and hence separable cognitive or imaginative proceduresfirst calling one of the spheres ‘a’ and then calling the other ‘b’. For, if it did, there would then be a context, subsequent to consummating the first procedure but before having embarked upon the second, within which it would make sense to ask the identification question, "Which sphere have we called ‘a’?" It is the demonstrable lack of any possible answer to that question in connection with Black’s world that shows that not only the proxy nominals ‘a’ and ‘b’ but also the expressions ‘one’ and ‘the other’ used in [D ] to introduce them are, if pronouns at all, then merely syntactically so.

I shall close, finally, by noting that, in order to make this journey, it has been necessary, as it were, to sail part of the way under false colors. For if I am right about the proxy nominals ‘a’ and ‘b’, then, strictly speaking, (BT) cannot be a basic Tractarian world-story nor (BC) a Carnapian state-description. Rather both (BT) and (BC) will presuppose a completely general description of World B — prima facie something along the lines of

(B1)        ($ x)($ y)[(x¹ y) & (" F)(Fx º Fy) & (" z)(z=x v z=y)]

— to supply an interpretive context for the ‘a’s and ‘b’s occurring in (BT) and (BC), as the substitution and transformation rules governing the derivation [P] supply the (only) interpretive context for the ‘a’s and ‘b’s occurring in it. In the last analysis, a Black world could then be described only generally and as a whole. Its generality would be "irreducible" and its internal multiplicity or plurality "indecomposable". That is, the world ostensibly described in general by (B1) would not admit of any redescription as a conjunction of individually specifiable particularities. And what I want last to suggest is that it is not at all obvious that these last remarks indeed express a "logical possibility". Early on, I remarked that intuitively unproblematic informal descriptions can rest on unarticulated presuppositions that in fact embody deep-seated and well-concealed incoherencies. I will end by expressing my suspicion that the idea of "irreducible generality" or "indecomposable internal multiplicity or plurality" that these Tractarian reflections have unexpectedly unearthed within the dialectic of indiscernibility and identity in the end will turn out to be one such incoherency.