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

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 Lycan'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 Cresswell, ‘The World is Everything That is the Case,’) Australasian Journal of Philosophy 50 (1972), 1-13

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 Plantinga, ‘Actualism and Possible Worlds,’ Theoria 42 (1976), 139-60; Robert Stalnaker, ‘Possible Worlds,’ Nous 10 (1976), 65-75. All three of these papers reappear in Loux.

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 Philosophy 65 (1968), 113-26; Counterfactuals (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

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).

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.

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.

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 expression ‘structural property,’ I will use the expression with his wider sense.

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)

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.

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-inconfiguration 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.’

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.

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.

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.

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 motorcar 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).

39 F. P. Ramsey, ‘Review of Tractatus,’ originally published in Mind (1923); reprinted 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.

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.

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 Haecceitist 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, following 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.

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.

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 ‘quiddity’ (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).

53 Saul Kripke, ‘Semantical Considerations on Modal Logic,’ Acta Philosophica Fennica (1963) 83-94. See esp. page 85.

54 Kim notes this, 609. But the point does not originate with him. Leonard Goddard 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 Wittgenstein 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.

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 emphasis] 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.)’

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).

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).

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.