possibility and combinatorialism: wittgenstein versus armstrong

Canadian Journal of Philosophy

Possibility and Combinatorialism: Wittgenstein versus ArmstrongAuthor(s): Raymond BradleySource: Canadian Journal of Philosophy, Vol. 19, No. 1 (Mar., 1989), pp. 15-41Published by: Canadian Journal of PhilosophyStable URL: http://www.jstor.org/stable/40231631 .

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CANADIAN JOURNAL OF PHILOSOPHY 15 Volume 19, Number 1, March 1989, pp. 15-41

Possibility and Combinatorialism: Wittgenstein versus Armstrong*

RAYMOND BRADLEY Simon Fraser University Burnaby, BC Canada V5A 1S6

I Introduction

In his recently published paper, The Nature of Possibility/1 David Arm- strong presents an account of possibility which, he correctly claims, is partly an elaboration of the early Wittgenstein's.2 Both are combinatori- alists. That is to say, both hold that there is a fixed ontology of individuals, properties and relations whose combinations determine the range of all possible states of affairs, and therewith the range of all those totalities of states of affairs which they call possible worlds.

But Armstrong's account, I believe, is fatally flawed in ways that Witt- genstein's isn't. And this, I shall argue, is mainly because Armstrong is both an actualist, whose fixed ontology is one of actual individuals,

* This paper was delivered, in Armstrong's presence, at the annual conference of the Australasian Association of Philosophy, held in Brisbane from 23-28 August, 1987.

1 David Armstrong, "The Nature of Possibility/ Canadian Journal of Philosophy 16, 4 (1986) 575-94. Jaegwon Kim's critical commentary, 'Possible Worlds and Arm-

strong's Combinatorialism,' is published in the same issue (595-612). I am indebted to Kim's paper for helping me understand a good deal of what is going on in

Armstrong's. For the most part, however, the criticisms developed in my paper are independent of those in Kim's.

2 Ludwig Wittgenstein, Notebook 1914-1916, G.H. von Wright and G. E. M. An- scombe, eds. (Oxford: Basil Blackwell 1961); and Tractatus Logico-Philosophicus, trans. D. F. Pears and B. F. McGuiness (London: Routledge and Kegan Paul; New York: The Humanities Press 1961).



16 Raymond Bradley

properties and relations, and a reductionist, who tries to reduce the notion of possibility to that of 'all' their combinations. Armstrong seems to think that Wittgenstein at least shares his actualism, and perhaps even his reductionism. But in fact, Wittgenstein - as his Notebooks makes particularly clear and the Tractatus reflects to a lesser extent - is both a possibilist, whose fixed ontology embraces all possible individuals, properties and relations, and a nonreductionist, since he endows his simple objects with essential formal properties which determine the range of other properties they can have and the range of possible relations of combination into which they can enter. Moreover, for any given individual, a, Wittgenstein endows a with the Haecceitist essential property of being identical to a, whereas Armstrong denies that individuals have any such nonrepeatable properties. In short, Wittgenstein is a Haecceitist de re modal possibilist, whereas Armstrong is what might be called a weak anti-Haecceitist de re non-modal actualist.

In what follows, I shall be as much concerned to expound and de- fend Wittgenstein's version as I will to expound and criticise Armstrong's.

II Some Conceptual Geography

Before getting into the similarities and differences between them, I should say a bit about the terminology just used and the conceptual geography it marks out.3

Let's start with 'combinatorialist.' This term has a descriptive content which allows it to have a wider extension than it was originally intended to mark. William Lycan4 introduced the term to characterize the idea, suggested by Quine5 and elaborated by Max Cress well,6 of taking possible worlds to be 'set-theoretic combinatorial rearrangements

3 Although most of the terms derive from different authors, the way of organizing them which I adopt here is largely that proposed by Michael J. Loux in 'Modality and Metaphysics/ his editorial introduction to Michael J. Loux, ed., The Possible and the Actual (Ithaca and London: Cornell University Press 1979).

4 William Ly can's paper, The Trouble with Possible Worlds,' first appeared in The Possible and the Actual and, together with Loux's 'Modality and Metaphysics,' does a lot of valuable sorting out of positions. See that volume, 301.

5 Quine, 'Propositional Objects,' in Ontological Relativity and Other Essays (New York: Columbia University Press 1986)

6 Max Cress well, 'The World is Everything That is the Case,') Australasian Journal of Philosophy 50 (1972), 1-13



Possibility and Combinatorialism: Wittgenstein versus Armstrong 17

of the posited basic atoms of which our own world is composed/7 But there is no good reason why we should take its basic meaning to be constrained by Lycan's initial characterization. Even Armstrong doesn't. His combinatorial rearrangements aren't abstract set-theoretic entities, which is how Jaegwon Kim thinks he should construe them,8 but mere fictions. Taking a little more liberty still, we can affirm the central combinatorialist notion that possible worlds are determined by 'rearrangements' of the items in a fixed ontology while denying that those items are all ones of which our own world, the actual world, is composed.

This allows the distinction between actualist and possibilist versions of combinatorialism to get its grip. Actualists, quite generally, believe that all the materials for the construction of other possible worlds are at hand in the actual world. Possibilists believe that actualia do not suffice and that in some sense or other we are compelled to assert the existence9 of mere possibilia: entities which are neither identical with nor constructable from any that actually exist. The important point is that both actualism and possibilism leave open the precise mode of construction of possible worlds. Neither is committed to the combinatorial view of the matter.

Actualists like Adams, Plantinga and Stalnaker,10 for instance, are modal actualists who try to construct possible worlds out of suitably modalized entities: abstract but actually existing maximally consistent sets of propositions in Adams' case; nonobtaining but actually existing states of affairs in Plantinga's case; and uninstantiated but actual-

ly existing properties in Stalnaker's. Adams is a de dicto modal actualist since his primitive entities are propositional in character. Plantinga and Stalnaker are de re modal actualists, since their primitive entities have the de re modal properties of obtainability, in the one case, and in-

stantiability, in the other. Armstrong is not a modal actualist.

7 Lycan, 306

8 Kim, 611

9 I say 'in some sense or other' because there is a great deal of dispute over the

question whether all existence is actual existence. Against the tradition of those who say it is, David Lewis argues that we need to distinguish between two domains of quantification: the restricted domain of actual existence and the unrestricted domain of actual and possible existence. Neo-Meinongians, like Terence Parsons and Richard Routley (now Richard Sylvan) argue that even if all existence is actual existence, there is a good sense in which 'there are/ in addition to actualia, both mere possibilia and impossibilia.

10 Robert Merrihew Adams, Theories of Actuality/ Nous 8 (1984), 211-31; Alvin Plan-

tinga, 'Actualism and Possible Worlds,' Theoria 42 (1976), 139-60; Robert Stalnak- er, 'Possible Worlds,' Nous 10 (1976), 65-75. All three of these papers reappear in Loux.



18 Raymond Bradley

Actualists with a nominalist bent, like Carnap,11 propose to construct possible worlds out of actually existing sentences. Those with a con- ceptualist bent, like Rescher,12 try to construct them out of actual acts of human conceiving. Both of them are nonmodal actualists since the basic materials out of which possible worlds are to be constructed, and to which they are supposed to be reduced, are nonmodal in character. It is within this camp, I believe, that Armstrong is to be located, though - being of a realist bent - he pitches his tent at some distance from these others. Carnap, I suppose, might be called a de dicto nonmodal actualist since his primitives are sentential. I am not sure what to call Rescher. By way of contrast, Armstrong is clearly a de re nonmodal actualist. And his position is even more sharply demarcated from Car- nap's and Reseller's by virtue of the fact that he is a combinatorialist.

Wittgenstein, I have said, is also a combinatorialist. But whereas Arm- strong is an actualist - and a nonmodal one at that - Wittgenstein is a possibilist. In this respect he is in the same camp as David Lewis.13 Indeed, both of them are de re modal possibilists since they take non- propositional modal entities as the primitives of their ontologies. The main difference between them is that Lewis, unlike Wittgenstein, is not a combinatorialist. It is essential to any combinatorial theory that the basic entities with which one starts should be able to combine in different ways so as to constitute different possible worlds. Wittgen- stein, as we shall see, allows that. Lewis does not. For him, every possible world is stocked with its own entities. They can be counterparts of entities in other worlds. But they can never be identical to them. Wittgenstein, I shall argue, takes as his basic entities the set of all possible objects, and allows these to combine in all possible ways so as to produce all possible worlds. His basic entities, therefore, would include Lewisian counterparts, and the worlds constructed therefrom. But Witt- gensteinian worlds outrun Lewisian ones since they also allow for the inclusion of individuals which are genuinely identical with individuals inhabiting other worlds.

The fact that Wittgenstein allows one and the same object to appear in different possible worlds is connected with the fact that he is a Haec- ceitist. Haecceitism, as we shall understand it here (following Adams,14

11 Rudolph Carnap, Meaning and Necessity (Chicago: Chicago University Press 1947)

12 Nicholas Rescher, A Theory of Possibility (Oxford: Basil Blackwell 1975)

13 David K. Lewis, 'Counterpart Theory and Quantified Modal Logic/ Journal of Phi-

losophy 65 (1968), 113-26; Counter/actuals (Cambridge, MA: Harvard University Press

1973); and On the Plurality of Worlds (New York: Basil Blackwell 1986)

14 Robert Merrihew Adams, 'Primitive Thisness and Primitive Identity/ Journal of Philosophy 76 (1979), 5-26




Kaplan,15 and others), is the doctrine that not all the essential properties of an object a are repeatable ones, properties which other objects too can instantiate; at least one is nonrepeatable, the property of being identical with a, for instance. Anti-Haecceitists deny that objects have such uniquely individuating essences. Lewis is an anti-Haecceitist.16 So is Armstrong, who calls himself a 'weak' anti-Haecceitist, thus in-

voking a further distinction whose import will be described later.

Ill Similarities

1. The primacy of states of affairs

Although both Armstrong and Wittgenstein allow for individuals ('objects,' as Wittgenstein calls them), properties and relations (generical- ly referred to as 'universals' by Armstrong, 'attributes' by Kim), neither will allow that any of these entities can exist outside of those combinations which they call 'states of affairs': things like a's being F, b's

having R to c, and so on.17 There can be no unpropertied individuals, and no uninstantiated universals.

This is not quite what Wittgenstein says. What he does say is that 'It is essential to things that they should be possible [emphasis added] constituents of states of affairs' (2.011). But since being a possible constituent of a state of affairs is an essential property ('part of the nature'

[2.0123] of every object), there can be no objects which are entirely un-

propertied, no 'absolutely bare particulars' (to use Copi's phrase). And since there is never a suggestion, in either the Notebooks or the Tracta- tus, that properties or relations may exist in a world without being in- stantiated in that world, it seems likely that Wittgenstein would agree with Armstrong's claim that 'States of affairs rule' (578), in the sense that it is states of affairs from which (as Armstrong puts it) objects, properties and relations are 'abstractions' (577), 18 or (as Wittgenstein

15 David Kaplan, 'How to Russell a Frege-Church/ Journal of Philosophy 72 (1975), 716-29

16 David Lewis, On the Plurality of Worlds, esp. 220-48

17 Armstrong (575) thinks that Wittgenstein would say 'facts' where he says 'states of affairs.' But he is wrong. Facts (Tatsachen), for Wittgenstein, are a proper subset of states of affairs (Sachverhalten), namely those which are not just possible but actual. Their agreement, in this respect, is closer than Armstrong thinks.

18 Armstrong explains that by 'abstraction' he does not mean that they have mental existence but only that 'by an act of selective attention they may be considered

apart from the states of affairs in which they figure ...' (578).



20 Raymond Bradley

puts it) inferred 'as the end-product of analysis' (NB 50[5]). Neither thinks of this or any other world as what Armstrong calls a 'tinker-toy construction' (577) from three different parts.

2. The elements: simple individuals, properties and relations

Armstrong is quite explicit about the items from which states of affairs are combinatorially constructed. They are simple individuals, simple properties and simple relations.

It may be questioned, however, whether Wittgenstein holds properties to be as basic as objects and relations. Do not his elementary states of affairs, those which Armstrong calls 'atomic,' consist solely of objects in the relation of combination? Tractatus 2.01, after all, tells us that 'A state of affairs [a state of things] is a combination of objects [things]/ and here there is no mention of properties.

The answer is that, while simple objects in elementary states of affairs have no material properties - determinate properties like a particular colour or particular taste, for instance - since material properties are produced 'only by the configuration of objects' (2.0231), they certainly do not lack what Wittgenstein calls 'formal' properties. Indeed, the very next passage (2.011), as we have already seen, insists that no object can lack the essential property (which turns out to be a formal one)19 of being a possible constituent of states of affairs. And elsewhere, Wittgenstein lists other formal properties as well which are essential to every object that possesses them.20 It follows that some properties, namely formal ones, feature essentially in the constitution of elementary states of affairs, even though material ones do not and cannot.

The concept of form features prominently in Wittgenstein's account. Armstrong, as we shall see, smuggles it into his account in order to make his reductionism sound plausible; but otherwise he ignores it and leaves it unanalyzed.21

19 That it is a formal property is evident for two reasons. First, every essential property, according to Wittgenstein, is either formal or structural. But structural properties can belong only to objects which have structure and which, therefore, are

complex. In 2.011, however, Wittgenstein is clearly talking about simple objects. Second, 2.0141 tells us explicitly that 'The possibility of its occurring in states of affairs is the form [and hence a formal property] of an object/

20 2.0251 tells us that being in space, being in time and being coloured are all forms, and hence formal properties, of objects. 4.1272 extends the list to being an object, being a complex, being a fact, being a function, being a number, etc.

21 The effect of his invoking it as he does, I shall argue in section VI. 3, is to subvert his attempted reduction by making it circular.




3. Structural properties

By way of contrast, the related concept of structure plays an important role in the combinatorial theories of both philosophers.

For both of them, any complex whatever - whether it is a complex individual such as a rod, or a state of affairs such as a rod's being red, or a rod leaning against a wall - consists of a determinate connexion of simpler, and ultimately of absolutely simple, elements. In Wittgen- stein's terminology this determinate connexion of the elements of a complex is called the 'structure' of that complex (2.032); and the property of having that structure he calls a 'structural property' of that complex. Now since a complex would not be what it is unless it had the structural properties which it has, Wittgenstein says that structural properties are essential, or internal, to their possessors (4.122[2]). Sup- pose that complex individual or state of affairs, X, consists of simpler parts a and b standing in relation R to each other. Since, as we have seen, no individual can exist wholly unpropertied, we must suppose each part to have its own property or properties. Let a have F and b have G. Then the structure of X consists of having an F-part standing in Rto a G-part. And the property of having that structure, Wittgen- stein claims, is essential to X.

Now it is a consequence of Wittgenstein's general atomism that any determinate material property,22 such as red, is 'produced' (2.0231) by the combination of simpler elements which do not themselves possess any such properties. A determinate colour such as red, will be produced, of course, only when there is a sufficiently great degree of complexity of structure in the individual which has that colour. Presumably, since many complex individuals may have the same property of being red, their being red will consist in their having a certain determinate structure of simpler parts in common.23 Hence, one may

22 It is important to note that, for Wittgenstein, material properties are always determinate, never determinable. This follows from the fact that material properties, such as having a determinate colour, are paradigmatically 'external'; nonessential, properties of the objects that have them, whereas determinable properties such as having some colour are always essential to their possessors. As he puts it: 'A

speck in the visual field, though it need not be red, must have some colour . . . Tones must have some pitch, objects of the sense of touch some degree of hardness, and so on' (2.0131).

23 Needless to say, sameness of structure can be preserved even where sameness of parts structured is not. Suppose a ball and a rod both to be red. None of the

parts of one will be identical to parts of the other. But provided that some of the

parts of the ball are structured in the same way as are some of the parts of the rod, they will both have, at least in part, the structure in whose possession being red consists.



22 Raymond Bradley

regard the property red itself as consisting in a determinate connexion of simpler propertied elements standing in relation to one another, where the property of having that determinate structure will be essential to being red.24

Note that Wittgenstein's doctrine of the internality of all structural properties (and relations) precludes him from saying that the property of being red is itself a structural property. For being red is certainly not essential to any of its possessors.

Nevertheless, since that property consists in having a certain structure (a structure which is common to all things which are red) one might choose to speak, more loosely, of being red itself as a structural property. There will be nothing wrong with so speaking provided we recognize that, in this looser sense, being a structural property of an individual X does not entail being an internal property of X.

It is in this looser sense that Armstrong uses the expressions 'structural property' and (generalizing) 'structural universal.' Thus he writes: 'If a compound individual consists of an F-part having R to a G-part, with F, R and G universals, then the individual instantiates the structural universal an F having RtoaG' (577). And, departing from Witt- genstein's usage, he further suggests that the colours themselves 'are suitable structural properties' (593).

For ease of exposition hereafter, I shall abandon strict fidelity to Witt- genstein's usage and follow Armstrong's instead.25 In Armstrong's looser sense, we may refer to material properties such as red and green as

24 Consider the state of affairs of a rod's being red. This state of affairs, since it is a complex of simpler (ultimately simple) objects in configuration, will have its own overall essential structure. Part of that overall structure will consist of the structure which comprises being red. The structure which comprises being red will, of course, be essential to the state of affairs of the rod's being red. But this does not mean that it is essential to the rod. For the structure which is essential to the rod is different from that which is essential to the property of being red. The former structure may happen to accompany the latter, i.e., may happen to be combined with it. But if it does so, it does so accidentally. Hence, having the structure which comprises being red is accidental to the rod even though it is essential to the state of affairs of the rod's being red.

25 The difference in usage amounts to this. Suppose complex X to have structure S. Then obviously the property of having S may be called a 'structural property' of X. Wittgenstein and Armstrong both agree on this point. Suppose, further, that complex X is itself a property which has structure S. Then, Armstrong wants to call X, as well as S, a 'structural property,' whereas Wittgenstein does not. The

potential for terminological confusion might have been minimized had Armstrong called X a 'structured' rather than a 'structural' property. But since he did not, and since there is an obvious warrant for thus extending the meaning of the ex-

pression 'structural property,' I will use the expression with his wider sense.




themselves structural properties, properties which are 'produced' only when the structures in which their respective identities consist are parts of the structures of the complex objects which have these determinate colours. After all, what is important for both Wittgenstein and Arm- strong is that they hold it to be the task of science to reveal the structure of all material properties, and in so doing to reveal how such properties are produced combinatorially from simple objects. Wittgen- stein, who may be taken to be speaking for both of them, puts it like this: 'We see that there is a difference of structure between red and green. And then physics arranges them in a series. And then we see how here the true structure of the objects is brought to light' (NB 81[8-9]).26

Despite their disagreement over the extension of the expressions 'structural property' and 'structural relation,' Wittgenstein and Arm- strong agree that all complex properties and relations are structured combinations of simple elements.

They agree, too, in a number of things they say about simple individuals, or objects, at one end of the scale of combinatorial complexity, and possible worlds at the other.

4. Simple individuals

Both agree that individuals can be either simple or complex (in the sense of being combinations of simple, or at least simpler,27 individuals). Both agree that the simplicity of individuals lies not in their being properti- less but only in their having no parts which themselves are individuals. And both avow a modicum of ignorance as to exactly what the ultimately simple individuals will turn out to be, while at the same time suggesting that occupied point-instants are at least plausible candidates.28

5. Possible worlds

As to possible worlds, both agree that such worlds are generated out of simple individuals, properties and relations in a minimum of two

26 See also NB 82(12-13): 'That the colours are not [simple] properties is shewn by the analysis of physics, by the internal relations in which physics displays the colours. Apply this to sounds too/

27 Both suggest strategies for dealing with the seeming logical possibility of individuals being infinitely complex so that there are no ultimate simples after all. See

Armstrong, 586-8; and Wittgenstein, 4.2211.

28 Armstrong, 576; Wittgenstein, NB 67(8) and 69(5)



24 Raymond Bradley

stages: first, by combinations of these fixed elements so as to produce all elementary states of affairs, merely possible as well as actual; and second, by combinations of possible states of affairs so as to produce those totalities of possible states of affairs which they call possible worlds.29

Needless to say, neither of them would allow that there can be worlds consisting of no states of affairs whatever. As Armstrong puts it: The reason is that the empty world is not a construction from given elements' (586).

IV Differences

1. Haecceitism

A major point of disagreement concerns the truth of Haecceitism: the doctrine that simple individuals have individual essences, that is properties which are not only essential to, but unique to, the objects which possess them. The unrepeatable property to being identical to a is commonly held to be such a property.

The position a combinatorialist takes regarding Haecceitism determines the number of worlds which are possible relative to the initial stock of individuals and attributes. Suppose, for example, that we were given nothing but the simple individuals a and b along with the simple properties F and G. Then, among the seven possible worlds coun- tenanced by the combinatorial Haecceitist, we have (using Armstrong's numbering):

I: Fa & Gb

II: Ga&Fb

VII: Fa & Ga & Fb & Gb

As Armstrong points out, an anti-Haecceitist would deny that there are any properties other than F and G by virtue of which a and b are distinct, and so would conclude that worlds I and II are identical. By the same token, other pairs of worlds, recognized as distinct by the Haecceitist, are collapsed by the anti-Haecceitist so as to produce a total of only four worlds relative to our assumed base.

29 Kim (601), in his exposition of Armstrong, says that there are three stages. But while it is true that for some worlds there is an intermediate stage in which, by conjunction of elementary states of affairs, molecular states of affairs are produced, this is not an essential stage since both Wittgenstein (NB 11[7]) and Armstrong allow for possible worlds consisting of a single elementary state of affairs.




Anti-Haecceitism, Armstrong also points out, comes in two degrees of strength. Strong anti-Haecceitism, since it denies that individuals are

anything more than bundles of their properties, would not even allow our initial supposition that a and b are distinct individuals, and hence would collapse VII into a one-individual world comprising the individual which has both F and G. Weak anti-Haecceistism, since it allows for the possibility of two or more distinct individuals having the very same repeatable properties, would accept VII as a two-individual world in which each individual has both F and G.

Armstrong is inclined to be a weak anti-Haecceitist. He denies that worlds I and II are different worlds, on the grounds that there is no-

thing about either a or b which distinguishes one from the other ex-

cept the repeatable properties F and G. But, he allows that world VII is a genuinely possible world containing two individuals, on the

grounds that a and b are nevertheless 'merely, barely, numerically different from each other' (584).

Wittgenstein, by way of contrast, is a Haecceitist. The decisive evi- dence for this is 5.53. 'Identity of object I express by identity of sign. . . Difference of objects I express by difference of signs/ For him, world I is distinguished from world II by virtue of the fact that in I the object a has the property F, whereas in II that very same object lacks the property F and takes on the property G instead. Object a in I is identical with object a in II, as the identity of sign 'a' shows. The fact that Witt- genstein treats names as Kripkean rigid designators - signs which sim-

ply name (3.221) their bearers without describing them, and designate one and the same object (5.53) no matter in what possible states of affairs that object occurs - rams the point home.30 It shows that identity

30 That Wittgenstein treats names of complexes, not just of simples, as rigid designators is also evident from 3.3442, where he claims that the sign for a complex 'would not have a different resolution every time it was incorporated in a different proposition/ His position is made even clearer by the parallel passage in the Notebooks, viz., NB 50(2), where he considers the example of a book named 'A.' Consider the two propositions 'A is red' and 'A is green/ Then his point is that the name 'A' has the same analysis, i.e., stands for the same simple objects-in- configuration in both 'propositional formations/ It could do so, however, only if the complex object A could occur in both of the corresponding states of affairs - A's being red and A's being green - where only one of these, at most, can be actual though both, of course, are possible. In short, 'A' designates the same

object in each of the different propositional formations in which it occurs, independently of which of these propositional formations is true, and independently therefore of which corresponding possible state of affairs actually obtains. NB 50 (3) endorses this construal: 'And like the name of a thing in different propositions, the occurrence of the name of compounded objects shews that there is a form and a content in common/



26 Raymond Bradley

of object is not just a function of those of its essential properties which are repeatable (ones that would enter into its description), but of its nonrepeatable Haecceitist ones as well.

Obviously, if all else were equal, Wittgenstein's Haecceitism would allow him to recognize more worlds than Armstrong.

2. Essentialism and logical kinds

Further major points of disagreement concern the issues of essentialism and logical kinds: whether simple objects belong to different logical kinds by virtue of the different sets of essential properties which they possess.

Armstrong says nothing explicit about essential properties. But he does say that To be individuals, individuals must be an [sic] individual, must be one thing' (578). His use of 'must' here at least suggests that he regards the 'unit-making property' of being one thing as an essential property of every individual.31 There, however, his list of essential properties apparently stops. As already noted, he does not allow for Haecceitist essential properties. And so far as I can tell, he doesn't allow for any other essential properties either. Different individuals, qua individuals, are 'merely, barely, numerically different from one another' (584). His basic ontology therefore has no room, as I understand it, for differences of logical kind.

Wittgenstein's position is very different. Not only does he allow for nonrepeatable, Haecceitist, properties to count as essential to their possessors. He also allows for a wide range of repeatable essential properties, in addition to the unit-making property which Armstrong recognizes. And he holds that certain of these in effect sort out individuals into what, in the Notebooks, he calls different logical kinds.

The class of essential properties, according to Wittgenstein, divides into two proper sub-classes: structural and formal. Structural properties, since they consist in structured combinations of simpler properties and relations, can belong only to complex objects. Formal ones can belong to either simple or complex objects. The form of a simple object consists of 'the possibility of its occurring in states of affairs' (2.0141). The form of a complex consists of the 'possibility of its structure' (2-033); that is to say (given the account of structure presented

31 Strangely, he expresses uncertainly (587) as to whether the simplicity of an individual is a noncontingent matter, while confidently producing a reductio ad absurdum argument (587) for saying that universals, if simple, are necessarily so. But not only does his reductio seem to apply mutatis to individuals. His claim that a simple individual is necessarily 'one' thing suggests the same conclusion.




in 2.032) it consists in the possibility of determinate connections of the elements of that complex. Thus, the concept of form turns out to be a modal one since to speak of the form (or formal properties) of X is to speak of the possibilities of combination of X itself or of X's elements.32

Formal properties, for Wittgenstein, include both the nonrepeatable Haecceitist properties and a number of repeatable ones. Some of the

repeatable ones, such as the dispositional property of being a possible constituent of states of affairs and the nondispositional one of being self- identical, are possessed by all objects and hence do not provide any basis for dividing them into logical kinds. But others, such as the determinable properties of being a number, being a function, being spatially located, and being temporally located, for instance, are essential properties of whole classes of objects, simple as well as complex, while still others such as the relational properties of being able to lean against a wall or being able to be put into a hole, and nonrelational onces such as being coloured, having some hardness, and having some pitch, are essential

properties which only complex objects can possess. The idea that objects belong to different logical kinds according to

certain of the repeatable essential properties which they possess, properties which constitute what he calls their 'internal natures,' is

32 'Form is the possibility of structure' is a dark saying. Yet Wittgenstein makes this claim not only regarding states of affairs (2.033), but also regarding pictures, i.e., propositions (2.15[2]). And in both cases, he goes on to say that the structure concerned consists in the determinate way in which the elements are combined. What this amounts to, I take it, is that where X is a complex, the form of X determines a range of possible structures where each of these possible structures consists in a possible combination of the elements of X. To make this more

comprehensible, consider the case of a proposition having the form 'aRb.' This form, we may say, determines a range of possible structures such as 'John is a friend of Mary/ 'John is a friend of Mary and Bill/ and so on, where each of these

propositional structures consists in a possible combination of propositional elements, viz., names of objects and relations. The form of a simple element cannot, of course, be explicated in terms of the possible combinations of its elements. Nevertheless, Wittgenstein wants to say, the form of a simple element is to be understood in terms of the possible combinations into which the element itself can enter. Quite generally, therefore, to speak of form is to speak of possibilities of combination of elements. This is how Wittgenstein wants us to understand his talk of the forms, not only of objects, states of affairs, pictures, and propositions, but also of reality (2.18), language and the world (6.12), and all possible worlds (2.022). His fundamental claim, that a picture must have the same form as reality 'in order to be able to depict it - correctly or incorrectly - in any way at all' (2.18), amounts to the claim that representation is possible only if the elements of a picture can combine in the same ways (have the same possibilities of combination) as the elements of reality.



28 Raymond Bradley

implicit in the Tractatus but explicit in the Notebooks. There Wittgen- stein writes: 'If, e.g., I call some rod "A," and a ball "B," I can say that A is leaning against the wall, but not B. Here the internal nature of A and of B comes into view. A name designating an object thereby stands in a relation to it which is wholly determined by the logical kind of the object and which signalises that logical kind' (NB 70 [8-9]).

The differences between Wittgenstein and Armstrong in this regard give their respective combinatorialisms totally different characters, as we shall now see.

3. Impossible combinations and logical syntax

For Armstrong, all combinations of the simples of his ontology are corn- possible. As he puts it: 'I begin in Wittgensteinian fashion with a world of simple objects whose recombinations determine the possibilities' (576). Since his simple individuals are merely numerically different from one another, having no essential natures other than oneness, there are no limits to their possibilities of combination. Likewise for properties '... all distinct simple properties and, more generally, all wholly distinct properties, are compossible' (589).

Such unrestricted promiscuousness of combination is no part of Witt- genstein's program. Rather, for Wittgenstein, there are limits to the possible combinations of simples. And these limits are determined by the logical kinds to which the objects belong, where these logical kinds are in turn determined by the various sets of essential properties which the simples instantiate.

The Notebooks makes all this particularly clear. The doctrine of logical (and corresponding syntactical) kinds lies behind such claims as the following: that the internal natures of the rod A and ball B, respec- tively, 'come into view' in so far as 'I can say that A is leaning against the wall but not B' (NB 70[8]); that the sentence 'The watch is sitting on the table' is senseless (NB 70[ll]); and that 'An illogical language would be one in which, e.g., you could put an event into a hole' (NB 107[5]).33 Sentences such as 'The ball is leaning against the wall,' 'The watch is sitting on the table,' and 'I put an event into a hole' - to use one of Armstrong's phrases - 'do not respect the form of an atomic state of affairs' (578), where respecting such a form, on Wittgenstein's

33 To be sure, objects such as rods, balls, and watches are simple only in the sense of being 'simple for me' (NB 70[7]). But the context makes it clear that what he is saying about them applies also to genuinely simple objects.




account, consists in respecting the possibilities of combination of the given simple elements.34

The most general formulation of Wittgenstein's position also occurs in the Notebooks. Commenting on his tableau vivant, 'picture/ theory of language,35 he points out that 'In a proposition we - so to speak - arrange things experimentally, as they do not have to be a reality; but we cannot make any unlogical arrangement...' (NB 13[2]). Elaborat- ing, he explains that 'the logical connexion [between names] must, of course, be one that is possible as between the things that the names are representatives of ../ (NB 26[11]); and, continuing a little later: Then in order for a proposition to present a situation it is only necessary for its component parts to represent those of the situation and for the former [the names] to stand in a connexion which is possible for the latter [the objects]' (NB 27[1]).

Wittgenstein's talk, in the Tractatus, of the rules of logical syntax36 mirrors this doctrine of logical kinds. Such rules are not arbitrary mat- ters of convention, as are the particular modes of signifying;37 they are rules which permit some combinations of names and forbid others according as the internal natures, and consequent logical kinds, of the objects named allow or disallow.38 If, at the ontological level, all combinations were permissible - as they are for Armstrong - Wittgen- stein would be committed (by his 'picture' theory of language) to saying that all combinations of names are equally permissible. But he denies the conclusion: 'A proposition,' he insists, 'is not a medley of words' (3.141).

An obvious consequence of the restricted liaisons into which Wittgen- stein's simples can enter is that, if all else were equal - i.e., if we were to ignore Wittgenstein's Haecceitism and regard his simples as identical to (and hence as bare as) Armstrong's - there could not be as many Wittgenstein worlds as there are Armstrong worlds. But for reasons which we shall now examine, all else being equal, there are more.

34 'The logical form of the proposition/ he claims, 'must already be given by the

logical forms of its component parts' (see NB 23[1]).

35 This is first presented in NB 7(3), where Wittgenstein cites the case of a motor- car accident being represented in a Paris law-court by means of dolls.

36 See, for instance: 3.325; 3.33; 3.334; 3.344; 6.124.

37 See 3.342.

38 The possibility of a particular mode of signifying, i.e., the fact that the rules of

logical syntax allow for it, he goes on to say, 'discloses something about the essence of the world' (3.3421).



30 Raymond Bradley

4. Aleins and objed-heterogenous worlds

Armstrong, to be sure, thinks the converse is true. He assumes, without discussion, that what he calls 'Wittgenstein worlds' are ones which involve all and only the simples which exist in the actual world. And, since he finds such object-homogeneity counterintuitive, Armstrong suggests a strategy whereby so-called Wittgenstein worlds may be both expanded and contracted. His own worlds, he explains, differ from 'Wittgenstein worlds' in being object-heterogeneous. The ironic fact is, however, that genuine Wittgenstein worlds - as opposed to the mythical ones that Armstrong refers to by that label - are robustly heterogeneous in a sense in which Armstrong's own are not.

Ever since Ramsey wrote his 'Review of Tractatus/39 philosophical mythology has portrayed Wittgenstein's worlds in Armstrongian fashion. Reason? Mainly that Wittgenstein claims that all possible worlds have the same form in common (2.022) and that he goes on to say that this form is 'constituted' by objects (2.023). But these passages, I have argued elsewhere, do not sustain the Ramsey-Armstrong construal.40 And if they did, they would flagrantly contradict much else that Witt- genstein says about possible worlds.

In the first place, a number of passages make it clear that Wittgen- stein's worlds range all the way from single-object worlds (NB 11[7]), and two-object worlds (NB 14[4]), to ones with infinitely many objects (4.2211). The actual world, it seems, is to be conceived as lying in between, being both object-impoverished with respect to some and object- enriched with respect to others.

Second, arguing against Russell's treatment of the so-called 'axioms' of infinity and reducibility as logical truths, Wittgenstein points out (6.1233 and, more elaborately, NB 127) that it is a logically open question as to whether the actual world is as these so-called axioms claim it to be. As Russell, who came to accept Wittgenstein's correction on this point, says: 'We are left to empirical observation to determine whether there are as many as n individuals in the world. In the

39 F. P. Ramsey, 'Review of Tractatus/ originally published in Mind (1923); reprint- ed in I. M. Copi and R. W. Beard, eds., Essays on Wittgenstein's Tractatus (New York: Macmillan 1966).

40 Raymond D. Bradley, Tractatus 2.022-2.023/ Canadian Journal of Philosophy 17 (1987) 349-60. My conclusion is that the form which all worlds have in common consists in the set of all possibilities of combination, and that this set is 'constituted' by the set of all possible objects, not just the set of all actual ones.




Leibnizian sense, there will be worlds having one, two, three ... individuals.'41

Third, Wittgenstein not only believes (in Leibnizian fashion) that a

god 'could create anything except what would be contrary to the laws of logic' (3.031), but also that 'he could not create a world in which the proposition "p" was true without creating all its objects' (5.123).42 The conjunction of these two passages is especially telling. It entails that since there is nothing contrary to the laws of logic in the supposition that, e.g., Sherlock Holmes was a friend of Dr. Watson, a god could create a world in which this proposition is true, and in so doing would necessarily bring it about that both Sherlock Holmes and Dr. Watson exist in that world. Yet, since neither Sherlock Holmes nor Dr. Watson is an inhabitant of the actual world, it follows that they are merely possible individuals.43 It will not do, of course, to object that both are complex objects capable of being produced solely by recombinations of simple objects which populate the actual world. For the argument can be run through again regarding such suppositions as that there are n+1, n+2, n+3 ... simple ojects in the world, where n is the number of simple objects which actually exist. Wittgenstein, it is clear, admits what Armstrong, following Lewis, calls 'alien individuals' - objects like Sherlock Holmes, Pegasus, unicorns, 'extra' material particles or occupied point-instants - into his ontology. He thinks in terms of a fixed domain, alright. But this is not the actualist's domain of actualia. Rather it is the possibilist's domain of all possible objects.44

Armstrong seems unaware of Wittgenstein's real position on this issue. But he is moved by Wittgensteinian considerations to allow both

41 Bertrand Russell, Introduction to Mathematical Philosophy, 203. See also: '...a world

containing more than 3,000 things and a world containing fewer than 3,000 things are both possible, so that if it happens that there are exactly 3,000 things, that is what one might call an accident and is not a proposition of logic' ('Philosophy of Logical Atomism,' 240).

42 The reference of 'its' here may seem unclear. (The world's objects, or the proposition's?) But NB 98(2) settles the question: 'No world can be created in which a proposition is true, unless the consituents of the proposition are created also.'

43 My choice of Conan Doyle's characters is not intended to prejudge any issue as to whether or not fictional entities need their own special philosophical analysis. Those who are uncomfortable about treating Holmes and Watson as nonactual

possibles are free to substitute examples - such as extra children, extra planets, or extra electrons - of their own choosing.

44 Significantly, in NB 99(1), Wittgenstein uses the expression 'possible things' (and 'possible relations') in a way which accords with the above conclusion.



32 Raymond Bradley

for what he calls 'expanded' worlds (worlds in which the stock of objects that exist in the actual world is supplemented by so-called aliens) and for 'contracted' worlds (worlds in comparison with which the actual world contains at least some aliens). Thus, on the one hand: 'alien individuals seem straightforwardly possible' (584). And, on the other hand, 'Of any individual in the actual world, it seems true to say that it might not have existed' (584).

Armstrong's problem, of course, is to reconcile these intuitions with the fact that he calls himself 'an actual world chauvinist' (582), and that he therefore wants to be able to show how the contents of all possible worlds can be constructed combinatorially from the contents of the actual world. He points out, correctly, that 'there seems to be no particular difficulty about the contraction of individuals' (585). They can be constructed combinatorially out of a subset of the contents of the actual world. But how about expanded worlds and the alien individuals which they are posited as containing?

His proposed solution is to be somewhat less than strict in his construal of combinatorialism, by reviving Brian Skyrms's doctrine of 'analogy'45 and arguing that his abandonment of Haecceitism allows him to form 'a fully determinate concept of an indefinite number of alien individuals "by analogy'" (584). 46 This, he confesses, involves a 'qualification' of combinatorialism. But he hopes that it is 'only a little one' (584). About that we shall see in due course.

For now, note that even if Armstrong's analogical aliens were allowed to enrich his basic ontology, they do not enrich it to the same extent as Wittgenstein's more full-blooded aliens47 enrich his. For Armstrong's

45 Brian Skyrms, Tractarian Nominalism,' Philosophical Studies 40 (1981), 199-206. Skyrms, who observes that his title is an oxymoron, helps perpetuate the actualist myth that Wittgenstein's worlds are object-homogeneous. He also argues for a Haec- ceitist version of combinatorialism. Armstrong says that he was converted to his own combinatorialism by Skyrms' article. But, as noted, he rejects Skyrms' Haecceitism.

46 He needs to abandon Haecceitism because, if Haecceitism were true, each of those individuals in other worlds which are not identical with individuals in the actual world, would have to have its own individual essence. But individual essences which did not appear in the actual world could not be constructed from any which did appear in the actual world by any process of mere combination. Kim, follow-

ing Armstrong (582), claims that Haecceistism is 'inconsistent with nonactual simple individuals' (606). Strictly, however, this is not so. It is only when Haecceitism is

conjoined with both actualism and strict combinatorialism that alien individuals are precluded.

47 'Full-blooded' seems the right word here, not just because Armstrong's aliens turn out, by way of comparison, to be but pale shadows of actual objects, but also because at least some of Wittgenstein's aliens - e.g., Sherlock Holmes and Pegasus - arguably have the property of being warm-blooded among their essential properties.




aliens are merely, barely, analogies of whatever actual simples turn out to inhabit the real world: occupied point-instants, for example. But Witt- genstein's aliens encompass whatever objects would have to exist in order for God to make any self-consistent propositions true. And these may well include what Lycan has called 'irreducibly spiritual objects' such as 'ghosts, monads, or Cartesian egos.'48 Indeed, if F. H. Brad- ley's Absolute were to be - as Bradley claims it is - both self-consistent and a single indivisible unity, there is no reason why it too should not count as one of the simple objects within the domain of Wittgensteini- an possibilia, a simple object such that one possible world would consist sole of it.49 Armstrong comments that Wittgenstein's avowal of ignorance as to which objects, properties and relations are the Tmild- ing blocks' of the actual and other possible worlds is 'a stroke of genius' (579). But Armstrong seems not to realize just how many more worlds than his own that stroke of genius enables Wittgenstein to embrace.

5. S5, S4, and the accessibility relation

G. H. von Wright and Raymond D. Bradley have given independent arguments for saying that Wittgenstein is committed to the system of modal logic known as S5, a system in which each possible world is possible relative to all the others and, in this sense, is 'accessible' to all the others.50 The viewpoint of S5 is, in effect, a God's-eye point of view in so far as to say that p is possible is just to say that there is, somewhere in logical space, a world in which p is true. In the Note- books Wittgenstein writes: The thing seen sub specie aeternitatis is the thing seen together with the whole of logical space' (NB 83[11]). I think it no exaggeration to say that this is the point of view to which Witt- genstein aspires throughout his early work.51

48 Lycan, in Loux, 306

49 Lycan comments: 'I should think that nonatomistic worlds are also possible, such as ones which consist of an undifferentiated miasma of Pure Spirit. A nonatomistic world can hardly be regarded as an arrangement of '[the actual world's] stock of atoms' (Loux, 306). Lycan here overlooks the fact that combinatorial atomism can allow, as Wittgenstein does (NB 11[7]), for worlds consisting of just one object.

50 G. H. von Wright, Wittgenstein (Oxford: Basil Blackwell 1982), especially 200; and

Raymond D. Bradley, Tractatus 2.022-2.023/ Canadian Journal of Philosophy 17 (1987) 349-60

51 In the Preface to the Tractatus, Wittgenstein tells us that his aim is to portray the boundaries of what is thinkable; that is, of what is expressible; that is, of what is possible. But the boundaries of what is possible, it turns out, just are the boundaries of logical space.



34 Raymond Bradley

Armstrong points out - correctly, but for the wrong reason (namely, that he thinks Wittgenstein's worlds are all object-homogeneous) - that Wittgenstein's combinatorialism does indeed require an S5 modal logic, and that in this logic the relation of accessibility is reflexive, transitive and symmetrical. He also points out that this desired symmetry cannot be provided on his own account of the matter. As he puts it: 'we must content ourselves with an S4 modal logic, with accessibility reflexive and transitive, but not symmetrical' (585). His reasons? He wants to give a combinatorial account not just of the domain of the logically possible, something that Wittgenstein does, but also of the very nature of possibility itself. Now, if he is to do that, then he needs to give a purely combinatorial account of the nature of the accessibility relation. And that, he thinks, requires him to be able to show that one world is possible relative to another just when its contents are combinatorial constructs from the contents of the other.

How is this to be effected if some worlds contain individuals which are alien to others? No problem, Armstrong thinks, provided we adopt the 'little' qualification that allows for anti-Haecceitist analogical aliens. What does pose a real problem for symmetry of the accessibility relation, combinatorially construed, is the supposition (which he wants to allow) that some worlds contain fewer universals, i.e., fewer properties or fewer relations than others.52 For instance, suppose that the actual world, Ww, contains Fs which a contracted world, Wc, lacks. Then, he points out, 'given combinatorial theory, Ww is not accessible from Wc ... Symmetry of accessibility thus fails' (585). Hence the need to accept S4, rather than the preferred S5, as the logic of Armstrongian combinatorialism.

I think Armstrong is right about this, but only - it should be noted - because he gives a (qualified) combinatorialist account of the accessibility relation. Wittgenstein, in effect, adopts a different account: one according to which the question whether one world is accessible to, or possible relative to, another is to be determined not from the point of view of this world or that, or from the point of view of what is con- ceivable by the inhabitants of this world or that, or from the point of view of what is constructable from the materials in this world or that,

52 The supposition that some worlds might contain universals which don't exist in the actual world has already been ruled out by Armstrong on grounds similar to those which led him to reject Haecceitism regarding invidiauls. Each alien universal would have to have its own unique 'quiddity7 (its own distinctive nature), and such alien quiddities, he argues, cannot be constructed combinatorially from those

given in the actual world. For a criticism of his reasoning on this and related points, see Kim (604-5).




but rather from the God's-eye point of view of what worlds are possible in themselves. Sub specie aeternitatis, all possible worlds are possible in themselves. From this point of view, no possible world is shut off from any other. S5 reigns.

More generally, the point is worth emphasizing that other accounts can be given of the accessibility relation in modal logic than the combinatorial one proposed by Armstrong. Kripke's semantics for quantified S5, for instance, retains symmetry of accessibility ('relative possibility,' as he calls it) and yet allows for different domains in different worlds.53 Thus symmetry can be preserved even in cases where mutual combinatorial constructability is not.

The main reason why Armstrong gives a combinatorial account of accessibility is that he wants to reduce the notion of possibility to that of combination. Wittgenstein, by way of contrast, is no reductionist. For him, not all combinations are possible; hence the notions of possibility and combination are independent of one another. Wittgenstein is a combinatorialist, alright. But the realm of what is possible (logical space), for him, isn't determined just by what is combined with what, i.e., by what Armstrong calls 'all the combinations ... of given, actual, elements' (575). It is determined by what is combiiwWe with what, where what is combinable with what is a function of the de re (essential) properties of the elements in his ontology, viz., of members of the set of all possible objects.

This difference between their respective combinatorialisms, as we shall now see, affects the ways in which they are able to respond to a serious problem which troubles both of them.

V A major difficulty: colour-incompatibilities

The problem is that such determinable properties as being red and being green are not compossible even though they are, on the face of it, simple properties. Let's call this the problem of colour-incompatibility, though, as noted by Armstrong and Wittgenstein alike, it extends (at the very least) to all cases of different determinates falling under a single determinable.

Drawing upon some rather schematic remarks of Wittgenstein's in Tractatus 6.3751, ones elaborated on in Notebooks 81[8-11], Armstrong suggests a Wittgensteinian solution. Give up the supposition that

53 Saul Kripke, 'Semantical Considerations on Modal Logic/ Ada Philosophica Fen- nica (1963) 83-94. See esp. page 85.



36 Raymond Bradley

colour properties are simple, treat them as structural properties in the manner previously outlined, and the problem should go away. Thus

Armstrong writes: 'Such properties are structural properties ... As a result, these properties are not compossible. Individuals instantiating both of a pair of such properties will never appear in possible worlds, that is, in recombinations of wholly distinct "atoms'" (592-3).

The problem of colour-incompatibility manifests itself in a challenge to the doctrine of the independence of elementary propositions - what Armstrong calls 'atomic statements/ Armstrong does not explicitly state such a doctrine. But he is committed to it. For him an 'atomic statement' is one which attributes a simple property F to a simple object a, or asserts that a simple n-adic relation R holds between n simple objects. Hence his point about the compossibility of all such simples may be expressed by saying that all atomic statements are independent.

Now clearly, neither Wittgenstein nor Armstrong can allow that 'a is red' and 'a is green' (with temporal parameters understood) are atomic, for these statements, both would agree, are not independent. This is why both want to say that the colour properties turn out, on analysis, to be complexes with their own different structures. So far, so good. But the problem then is that it is difficult, if not impossible, to cite any examples of properties (or relations, for that matter) which would count as simple in the required sense; hence difficult, if not impossible, to cite examples of statements which would both count as atomic and be independent of one another.

The problem can be posed in a particularly acute form by consider- ing the name of a hypothetically simple relation R as it features, along with the names of hypothetically simple objects a and b, in an atomic statement of the form 'aRb.' If 'aRb' is to count as atomic, then l>Ra' must also count as atomic. Yet, if R is an asymmetrical relation, 'aRb' is not independent of l>Ra' but inconsistent with it.54 The proposed solution could work only if no simple relations were asymmetrical.55 Yet how could that be? Not only is it difficult to see how the preferred

54 Kim notes this, 609. But the point does not originate with him. Leonard God- dard and Brenda Judge, in The Metaphysics of Wittgenstein's Tractatus (Bundoora, Victoria: Australasian Journal of Philosophy, Monograph No. 1, June, 1982), 7-8, argue on this basis that the only simple relation admissible in Wittgenstein's ontology is the symmetrical one of combination. They too, however, fail to explain how

asymmetrical and nonsymmetrical relations can be generated out of symmetrical ones.

55 The idea that no simple relations are asymmetrical is something which Wittgen- stein would have to reject. In NB 49(13), he refers to the relation of naming as a 'simple relation/ Yet it is surely an asymmetrical one.




simples of Armstrong's ontology, occupied point-instants, can all be related to one another in space-time only by relations which are either symmetrical or nonsymmetrical. It is surely impossible for any of the complex asymmetrical relations which in fact obtain between objects in the actual world to be generated combinatorially (i.e., simply by conjunction) from simple relations which themselves are not asymmetrical.

This objection, if sound, is absolutely devastating for Armstrong's position. Not only does it mean that he must give up the doctrine of the independence of atomic statements. It means, as well, that his proposed combinatorialist reduction of the notion of possibility to that of 'all combinations' cannot succeed. It could work only if all combinations of simple objects, simple properties and simple relations were compossible - only if, that is, all such combinations generated possibilities. But some, like the combination of aRb and bRa where R is asymmetrical, generate impossibilities.

For Wittgenstein, however, the damage is more limited. He, too, must abandon the doctrine of the independence of elementary propositions.56 But his combinatorialism remains intact. Wittgenstein has said all along that not all combinations of simple objects, properties, and relations are possible. Some, he has emphasized, are impossible: those which are ruled out by the internal natures, and consequent differences in logical kind, of the simples involved.

Indeed, by appealing to this fact, Wittgenstein is able to offer his own sketchy solution to the colour-incompatibility problem. The first step, as we have seen, is to reject the colour properties of being red and being green as simples which could feature in elementary propositions.57 The second step is to treat both properties as complex properties whose structures are revealed by physics. The third step is to suggest that physics reveals the incompatibility as stemming from the fact that 'a particle cannot have two velocities at the same time; that is to say, it cannot be in two places at the same time; that is to say, particles that are in different places at the same time cannot be identical' (6.3751).

56 It can, of course, be partially rescued by importing the qualification 'entirely different' from his statement, in 5.135, of the corresponding thesis of the independence of different situations: 'There is no possible way of making an inference from the existence of one situation to the existence of another, entirely different [my em-

phasis] situation/ Were he to claim only that entirely different elementary propositions are independent, his claim would be obviously (though perhaps trivially) true.

57 NB 91(9): 'If the logical product of two propositions is a contradiction, and the

propositions appear to be elementary propositions, we can see that in this case the appearance is deceptive. (E.g.: A is red and A is green.)'



38 Raymond Bradley

The contradiction, Wittgenstein is claiming, stems from the identity conditions, the internal natures, of those presumed simples with which physics deals, viz., particles. Wittgenstein's combinatorialism can allow that, indeed requires that, certain combinations of simples should be impossible in this sort of way. Armstrong's can't.

VI Further difficulties for Armstrong's combinatorialism

1. Alien individuals and Armstrong's 'little' qualification

Armstrong, it will be remembered, thinks he can satisfy the intuitive demand for alien individuals by denying Haecceitism and allowing for analogically constructed alien individuals, simple individuals which are 'simply other' than any actual individuals. (584). How, one wonders, is their incursion to be reconciled with his actual world chauvinism? By virtue of the fact, he answers, that 'this concept of other- ness is derivable from actuals' (584).

This talk of conceptual derivability, however, involves much more than the little' qualification of combinatorialism which Armstrong had hoped for. On the one hand, such analogical aliens are not identical to, since they are other than, any actual individuals. On the other hand, they are not combinatorially derivable from actual individuals. If they were, they would all have to be complex.

But this opens the door to possibilism. For possibilism must surely be true provided only that there are individuals which are neither identical to, nor combinatorially derivable from, actual individuals. Maybe Armstrong's admission of analogically alien individuals opens the door to possibilism 'only a little way,' in as much as alien properties and alien relations are still excluded. But that little opening suffices to let in a flood of merely possible individuals: individuals which, though they start off as 'merely, barely, numerically different' from any actual ones, can then be clothed, in wholly respectable Armstrongian fashion, with whatever actual properties and relations we care to bestow on them. Extra children, extra particles, and perhaps even Sherlock Holmes and Dr. Watson, would seem to be cases in point.

In short, if Armstrong were to stick with strict combinatorialism, his account could not do justice to our, and his, intuitions that alien individuals are logically possible. But when qualified by the doctrine of analogy, his account can no longer do justice to his actual world chauvinism.

2. Alien universals and S4

Although Armstrong's concession to intuition extends to alien individuals, it does not extend to alien universals, i.e., to simple properties




and relations which fail to occur in the actual world. Tor universals/ he says, 'the way of analogy fails' (581). His reasoning seems to be that universals cannot be stripped of their natures, their 'quiddities' as he calls them, in the way that individuals can be stripped of their Haecceities.

I am not sure that I understand why. Why doesn't the way of analogy allow for weak anti-Quidditism, so to speak, on a par with weak anti-Haecceitism? Consider the case of relations, and suppose that no actual relation has more than n places. Then, according to Armstrong, no relation with n+1 places is possible.58 Yet if analogically alien individuals are to be allowed on the grounds that the concept of other- ness is derivable from actuals, why cannot analogically alien relations also be legitimated on similar grounds? An n + 1-place relation, one would think, would be conceptually derivable from actual relations simply by addition of one more place. And a similar strategy would seem to be possible for properties. If so, the way of analogy would seem to allow for simple properties and relations which, with respect to the actual world, are alien after all.

Be that as it may, Armstrong certainly does allow that there are properties and relations in the actual world which are alien with respect to certain universal-depleted worlds. This, it will be remembered, is what leads him to settle for S4 as the appropriate logic for his actualist version of combinatorialism. From the point of view of such universal-depleted worlds some actual properties and relations will, since they are combinatorially inaccessible, be strictly impossible. But surely this shows that something is wrong with the S4 conception of logical impossibility to which Armstrong's actualist version of combinatorialism leads. For a property or relation which is actual is ipso facto not logically impossible.

The dilemma for Armstrong this time is that, if he were to admit analogically alien universals along with analogically alien individuals, his capitulation to the possibilists would be virtually complete. But if he does not admit them, he is stuck with a modal logic which even he finds somewhat difficult to accept since it does less than justice to our ordinary notion of logical possibility.59

58 Unless, of course, a relation with n + 1 places is supposed to be combinatorially constructable from simple relations. How such a construction might be effective, however, is not at all obvious.

59 He writes: 'the contraction of universals does raise problems for combinatorialism/ viz., the problem of seeming to make S5 unavailable, and concludes that 'we must content [emphasis added] ourselves with an S4 modal logic' (585).



40 Raymond Bradley

3. Armstrong's reductionism

Armstrong is not only an actualist but a reductionist as well. Unlike modal actualists, he does not want any modal notions to feature in his ontology but instead wants to reduce such notions to nonmodal ones. For him it will not do to say that what states of affairs and worlds are possible is a function of what combinations of simple objects, simple properties and simple relations are possible. That would be to leave the notion of possibility unanalyzed. He wants to be able to explicate this notion itself in terms of the notion of the totality of combinations. The notion of possibility, he claims, just is the notion of 'all the combinations which respect a certain simple form - of given, actual elements' (575).

The problem is to see how the notion of possibility can altogether be avoided. For it certainly is not Armstrong's view that all these con- binations of actual elements - all states of affairs, that is - are themselves actual.60 Some are nonactual, merely possible, combinations. Hence the need for the qualifying phrase 'which respect a certain form.' It turns out, in fact, that it is not quite the case that all combinations of actual elements are possible. Rather, those combinations only which respect a certain form are possible.

What, then, is this form? It is, he tells us, 'the form of an atomic state of affairs' (578). Now I take it that paradigmatic forms of atomic states of affairs will include such combinations of elements as those described by the atomic statements 'Fa,' 'Ga,' 'aRb,' and the like. But what, we then need to know, would it be like for a combination of elements not to respect such forms as these? Presumably, it would be for the elements to combine in ways such as would be described by nonsensical medleys like 'Fab' (where F is monadic), 'aR' (where R is dyadic), and so on. Such combinations, clearly, would not respect the form (or forms) of atomic states of affairs. And the reason is, of course, that such combinations of elements are impossible.61 At any rate, as we have already seen, that is how Wittgenstein would understand this talk of respecting form.

60 If they were, we could join Kim in asking: 'where are these combinations and

arrangements? If they are legitimate in the eye of the actualist, they must be found somewhere in the actual world. But where are they?' (610)

61 As Wittgenstein explains: 'Whatever is possible in logic is also permitted. (The reason why "Socrates is identical" means nothing is that there is no property called "identical" ...)' (5.473). In order words, Wittgenstein holds that the combination of elements in this sentence is not possible in logic, since the sentence has the

impermissible form 'aR' (with R dyadic).




Understood in this way, however, the insertion of the qualifying phrase 'which respect a certain form' has the effect of smuggling the notion of possibility back into the analysans from which it was supposed to be absent.62 It means that the idea of possibility, for states of affairs and worlds, is being analyzed as 'all the combinations which are possible for given, actual elements.'

In itself, there is nothing wrong with such an analysis. Indeed, when Armstrong's qualifying phrase is construed in this way, it has the same import as Wittgenstein's insistence that 'the connexion of the propositional components [the components of a proposition with sense] must be possible for the represented things' (NB 28[13]). The trouble is that, when the qualification is so construed, Armstrong's proposed reduction turns out not to be a reduction at all.

The final, and fatal, dilemma for Armstrong is this: If by 'all combinations' he means 'all actual combinations,' his reduction of the idea of possibility falsely claims that the domain of the possible extends no further than the domain of the actual. But if, in order to avoid this, he qualifies 'all combinations' to mean 'all combinations which are possible for the given, actual, elements,' his account is circular. Either way, his reductionism fails.

Received November, 1987

62 This is exactly the effect which would result were Wittgenstein to use the expression 'which respect a certain form' in this way. For the form of a state of affairs, he tells us, is 'the possibility of its structure' (2.033). And, since the structure of a state of affairs is the 'determinate way in which objects are connected' within it (2.032), this means that the form of a state of affairs just is the possibility of determinate combinations of the given elements.



possibility and combinatorialism: wittgenstein versus armstrong

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