LEO K.C. CHEUNG
Logical atomism – a term first coined by Bertrand Russell in 1911 (see Russell, [1911] 2003, p.94) – in general consists of a methodological part and a metaphysical part. In the methodological part, there is a process of logical analysis of propositions. In the metaphysical part, there is the view that logical analysis reveals the metaphysics of reality. More precisely, logical analysis discloses that what correspond to true propositions are atomic facts, or facts as the obtaining of determinate combinations of logical atoms, of which the world is constituted (see Russell, 1918; 1924). In the Tractatus, Wittgenstein adopts a version of logical atomism (although he does not use the term “logical atomism”). In the course of the present chapter, I shall offer an exposition of the Tractarian version of logical atomism.
According to the Tractatus, “a proposition has one and only one complete analysis” (TLP 3.25). By this, Wittgenstein means that every proposition is completely and uniquely analyzable into a truth function, or a truth‐functional combination, of elementary propositions (TLP 5). An elementary proposition is an immediate combination of what he calls “simple signs” or “names” (TLP 3.202, 4.221). Names in propositions denote objects, and those objects are their meanings (TLP 3.202–3.203). A state of affairs is a determinate combination of objects (TLP 2.01). The obtaining of a state of affairs is a fact (TLP 2). The obtaining and non‐obtaining of states of affairs is reality (TLP 2.06). The sum total of reality, or the totality of the obtaining of states of affairs, or facts, is the world (TLP 2, 2.04, 2.063).
Objects are simple, or not composite (TLP 2.02, 2.021). They are devoid of internal complexity. So an object only possesses its combinatorial possibility, that is, its logical form (TLP 2.032–2.033), and its being a different object from all other objects (TLP 2.02, 2.0233–2.02331), that is, its individuality. Moreover, objects subsist, or necessarily exist (TLP 2.027–2.0271). The forms and contents of objects constitute the substance, including the unalterable form, of the world, which is also the substance of reality (TLP 2.021, 2.024–2.025, 2.05–2.06). Therefore, substance also subsists and is unalterable (TLP 2.024). Objects must be necessarily existent simple objects. Since objects are the end products (or, more precisely, the meanings of the constituents of the end products) of logical analysis, they are the logical atoms of the Tractarian world. In this way, logical analysis reveals the metaphysics of reality.
Wittgenstein argues for the possibility of complete analysis. By doing this, he also argues for the claims that the constituents of the end products of complete analysis are simple signs, and that there are necessarily existent objects. Their conjunction is the thesis that the world has substance, or that there are necessarily existent simple objects. His main argument is that if there are necessarily existent simple objects, then a proposition has a complete analysis (the first premise); and there are necessarily existent simple objects or, equivalently, the world has substance (the second premise); hence, a proposition has a complete analysis. What may be called “the substance argument” in TLP 2.0211–2.0212 is actually a sub‐argument for the second premise, that is, the thesis that the world has substance.
In the remaining three sections, I shall first explain Wittgenstein’s main argument for the possibility of complete analysis, assuming that its second premise is true. Then I shall comment on three recent interpretations of the substance argument; finally, I shall offer an exposition of the substance argument.
Wittgenstein holds that thinking is making to us a logical picture of facts, and that a logical picture of facts is a thought (TLP 2.1, 3–3.001, 4.021). A proposition is a perceptible expression, or symbol (TLP 3.31), of a thought, and is also a picture of reality (TLP 3.1, 4.01). A symbol, or an expression (TLP 3.31), is a sign used in accordance with a logical form (TLP 3.326–3.328), that is, a combinatorial possibility (TLP 2.01, 2.0123, 2.014–2.0141) or a possibility of structure (TLP 2.033). The sign is its sign, and the logical form its form. A sign is what is perceptible of a symbol (TLP 3.32). The perceptible sign of a proposition, as a symbol, is its propositional sign, and its logical form its propositional form (TLP 3.12, 4.5). A proposition is a propositional sign used, or applied, in accordance with its propositional form, via thinking, to represent a possible situation, that is, its sense (TLP 3.11, 3.5, 4.031). More precisely, when a propositional sign is used or applied, the objects of which its constituent signs are their representatives are thought to be connecting with one another in accordance with its propositional form such that a possible situation is represented (TLP 3.221, 4.031–4.0312). In this way, a thought, whose perceptible expression is a proposition, is a propositional sign applied and thought out (TLP 3.1, 3.5). It is also in this way that a proposition is a picture of reality. The fact that a proposition is a picture of reality is to be accounted for by the picture theory, which will be explained by means of the example of the proposition “fa” later.
The notions “completely analyzed proposition,” “simple sign,” and “name” are introduced as follows:
| 3.2 | In a proposition a thought can be expressed in such a way that elements of the propositional sign correspond to the objects of the thought. |
| 3.201 | I call such elements ‘simple signs’, and such a proposition ‘completely analyzed’. |
| 3.202 | The simple signs employed in propositions are called names. |
| 3.203 | A name means an object. The object is its meaning. (TLP 3.2–3.203) |
Since the propositional elements denote objects, the objects of the thought mentioned here are not the constituents of the thought, but the objects to which the constituents correspond. Moreover, reality and its sum total – the world – are always thinkable (TLP 3.001–3.01). If there were unthinkable reality or world, we could not think about it, nor could we talk about it (TLP 5.61). If the world has objects, they must be the objects of thought. The constituents of thought correspond to objects of the world in the same way as the constituent names of a relevant propositional sign.
The notion “completely analyzed proposition” can be characterized as follows:
A proposition is completely analyzed if and only if it has a propositional sign with simple signs (names) as its constituent elements.
However, Wittgenstein also seems to hold a different characterization of the notion: a proposition is completely analyzed if and only if it is analyzable into a truth function of immediate combinations of names, or elementary propositions (TLP 4.22–4.221, 5, 5.5). The truth function is produced by applying logical operations to elementary propositions. Logical operations are symbolized by logical constants (TLP 5.5, 5.51 s, 5.52 s). So, amongst the end products of logical analysis, there are also logical constants.
Logical operations are always definable in terms of a single fundamental operation (TLP 6–6.001). The fundamental operation in the Tractarian system is symbolized by the logical constant N (TLP 5.502; see also Geach, 1981; 1982; and Cheung, 2000). But the sign “N” is inessential here. In a completely analyzed proposition, in which N occurs, no sign is needed to symbolize N, although brackets indicating the applications of N may be needed (TLP 5.46–5.4611). There are no primitive signs of logical operations. For Wittgenstein, this entails the Grundgedanke that logical constants do not denote (TLP 4.0312; see also Cheung, 1999). Therefore, logical constants should not be seen as the constituent elements of propositions. This explains why, in TLP 3.2–3.203, none of the constituents of the completely analyzed propositional sign is the sign of a logical constant, even though the proposition still contains logical constants.
Logical analysis proceeds via the dissection of signs by means of definitions, and arrives at signs, which are not further dissectible – primitive signs (TLP 3.24–3.25). Yet, in TLP 3.2–3.203, Wittgenstein calls the constituents of a completely analyzed proposition “simple signs” or “names.” Thus, primitive signs are also simple signs. Moreover, he also says in TLP 3.26 that “a name cannot be dissected any further by means of a definition; it is a primitive sign.” Simple signs are then also primitive signs. Any acceptable interpretation of the Tractarian notion of logical analysis must explain why primitive signs are simple signs (or names), and vice versa.
To investigate in detail the idea of analysis via the dissection of signs by means of definitions, consider this:
| 3.24 | A proposition about a complex stands in an internal relation to a proposition about a constituent of the complex. |
| A complex can be given only by its description, which will be right or wrong. A proposition that mentions a complex will not be nonsensical, if the complex does not exist, but simply false. | |
| When a propositional element signifies a complex, this can be seen from an indeterminateness in the propositions in which it occurs. In such cases we know that the proposition leaves something undetermined. (TLP 3.24) | |
Let us call a sign that is further dissectible by means of definitions “a nonprimitive sign.” TLP 3.24 concerns, amongst other things, a special type of nonprimitive sign, namely, propositional elements signifying complexes. I shall employ the proposition “The broom is in the corner” – taken from section 60 of Philosophical Investigations – as an example to illustrate some of the points here. Let “a” be a symbol for the broom, and “f—” short for “—is in the corner.” The proposition “The broom is in the corner” can be expressed as “fa.” Since the broom is composite, it would be called “a complex” in the Tractatus. For the sake of illustration, let us take the property, or the attribute, signified by “—is in the corner” to be a complex, though admittedly this may sound odd. The proposition “fa” is then about the complexes f and a.
How does the proposition “fa” say anything about the world? The answer is to be given by the so‐called “picture theory (of proposition).” The picture theory consists of the introduction of the notion of a picture and the explanation of how a picture depicts reality, mainly in TLP 2.1–2.225 and 4.011–4.016, and the theses in TLP 3–3.001 that a thought is a logical picture of facts, and in TLP 4.01 that a proposition is a picture of reality. The latter thesis may be called “the picture thesis (of proposition).”
According to the picture theory, a proposition expresses a sense, or says something about the world, because it is a picture of reality (TLP 4.03). A picture is an already obtained determinate combination of things (physical signs, in the case of a proposition), and thus is a fact (TLP 2.14–2.141). The form of the fact, or the possibility of structure constituted by the forms of its constituents, is the pictorial form (TLP 2.0141, 2.032–2.034, 2.15). A fact becomes a picture, when its constituent elements are correlated with, or denote, objects in a way subject to this constraint:
| 2.1511 | That is how a picture is attached to reality; it reaches right out to it. |
| 2.1512 | It is laid against reality like a measure. |
| 2.15121 | Only the end‐points of the graduating lines actually touch the object that is to be measured. |
| 2.1513 | So a picture, conceived in this way, also includes the pictorial relationship, which makes it into a picture. (TLP 2.1511–2.1513) |
That is, only objects of the same forms (combinatorial possibilities) of the constituent elements can be correlated with the constituent elements. This guarantees that the picture and the situation it represents have the same form. In this way, the fact becomes a picture by representing a situation – the obtaining and non‐obtaining of states of affairs (TLP 2.11).
The picture theory, especially the account in the 2.15 s, is applicable to the case of a proposition in this way: the constituent signs of a proposition denote objects of the same forms, respectively, such that its propositional sign represents a situation whose objects are connected to one another in the same determinate way as the constituent signs. This is described vividly in TLP 4.031:
In a proposition a situation is, as it were, constructed by way of experiment.
Instead of, ‘This proposition has such and such a sense’, we can simply say, ‘This proposition represents such and such a situation’. (TLP 4.031)
The account in TLP 2.15 s, which is part of the picture theory, appears to be applicable to elementary propositional signs only, because the constituent elements of the picture appear to be simple, that is, not further analyzable. But, as suggested by David Pears (1987, p.78), it can also be taken to be applicable to the case of nonelementary propositional signs, as if “a short cut” is taken. So let us apply the picture theory to the case of the propositional sign “fa,” and let us answer the question how “fa” says anything about the world. In this case, the following holds:
- (1) If the complexes f and a exist, then “fa” makes sense, or represents a situation.
That is, the constituent signs “f” and “a” of “fa” can be symbolized to denote the complexes f and a, respectively, such that “fa” represents the situation that fa, provided the complexes f and a exist.
However, “fa” may still make sense, even if, say, the complex a does not exist. Recall that TLP 3.24 says, amongst other things, that “a complex can be given only by its description, which will be right or wrong.” With respect to the present example, the broom is given by one of its descriptions, say, “The broomstick is fixed in the brush.” Let “b” and “c” be the propositional elements signifying the broomstick and the brush, respectively, and “—R——” be short for “—is fixed in——.” The broomstick and the brush are, of course, complexes, and the relation —is fixed in——, or R, may be regarded as a complex. The complex a, or the broom, is then given by the description “bRc.” The complex a exists if and only if the description “bRc” is true.
If “bRc” is false, then the complex a does not exist; and yet the proposition “fa” may still make sense. For Wittgenstein would hold this:
- (2) If the complexes f, b, c, and R exist, then “fb.fc.bRc” makes sense, and is equivalent to the proposition “fa.”
(Here, “fa.fb.fc” means the conjunction of the propositions “fb”, “fc” and “bRc”.) In other words, if —is in the corner, the brush, the broomstick, and —is fixed in—— exist, then the proposition “The broomstick is in the corner, the brush is in the corner, and the broomstick is fixed in the brush” (i.e., “fb.fc.bRc”) makes sense, and is equivalent to “The broom is in the corner.” This is indeed suggested by an entry dated 5 September 1914 in the Notebooks, where there is the sentence, or formula, “φ(a).φ(b).aRb = Defφ[aRb]” (NB 4; see also Kenny, 1973, pp.79–80). Moreover, according to TLP 5.14–5.141, Wittgenstein accepts uncritically the view that two logically equivalent propositions are actually “one and the same proposition.” Thus, he would regard “fa” and “fb.fc.bRc” as one and the same proposition, provided the complexes f, b, c, and R exist. Hence, “fa” may still make sense, even if the complex a does not exist. For instance, when f, b, c, and R exist, but a doesn’t (or “bRc” is false), “fb.fc.bRc,” and thus “fa,” still makes sense.
Actually, (2) illustrates a step in the analysis of the proposition “fa.” The logical equivalence fa ≡ (fb.fc.bRc), under the assumption that those complexes exist, can be seen as involving a contextual definition of the sign “a” in terms of the signs “b”, “c,” and “R.” In this case, “a” is a nonprimitive sign. Thus, the process of analysis is one of the dissection of nonprimitive signs by means of definitions. This is why “a proposition about a complex stands in an internal relation to a proposition about a constituent of the complex” (TLP 3.24) – for example, if the relevant complexes exist, “fa” entails “bRc.”
However, the description “bRc” is not an instance of those complete descriptions mentioned in TLP 2.0201:
Every statement about complexes can be resolved into a statement about their constituents and into the propositions that describe the complexes completely. (TLP 2.0201)
This is because the complexes b, c, and R have internal complexities, which are also that of the complex a, and thus “bRc” cannot describe a completely. Nevertheless, this explains why TLP 3.24 says, amongst other things, this:
When a propositional element signifies a complex, this can be seen from an indeterminateness in the propositions in which it occurs. In such cases we know that the proposition leaves something undetermined. (TLP 3.24)
For example, the propositional sign “fa” has at least the constituent element “a” signifying the complex a. It does not reflect any internal complexities of the complex a, including the internal complexities shown by the description “bRc,” structurally. Therefore, the propositional sign “fa” has an indeterminateness, or has left something undetermined.
When Wittgenstein claims in TLP 2.0201 that “every statement about complexes can be resolved into a statement about their constituents and into the propositions that describe the complexes completely,” he is assuming the conclusion of the substance argument and its implication that complexes are constituted solely by necessarily existent simple objects. It is only when complexes are constituted solely by objects that there are complete descriptions of complexes. If there are objects, then every proposition (about complexes) can be expressed as a truth function of propositions about the constituent objects of the complexes it is about. Since objects exist necessarily, there is the logical equivalence between the proposition and the truth function, under no condition of existence. (Note that both the simplicity and necessary existence of objects are crucial here.) Let me explain this by considering the proposition “fa” again.
Suppose that the complexes f and a are constituted by necessarily existent simple objects, and, without loss of generality, that f, b, c, and R are objects (and thus “f”, “b”, “c,” and “R” are simple signs or names). Then Wittgenstein would hold this:
- (3) The propositional sign “fb.fc.bRc” expresses a sense, and produces a proposition logically equivalent to the proposition “fa.”
Here, the biconditional fa ≡ (fb.fc.bRc) is a logical equivalence, under no condition of existence. For, as objects exist necessarily, there is no chance of reference failure with respect to the propositional sign “fb.fc.bRc”; and thus the possibility of the biconditional’s not being a logical equivalence is ruled out. In this case, “bRc” is a complete description of the complex a. (The object f is also completely, but trivially, described by the sign “f” in the propositional signs “fa” and “fb.fc.bRc.”) Hence, it is only under the assumption that the world has substance that “every statement about complexes can be resolved into a statement about their constituents and into the propositions that describe the complexes completely” (TLP 2.0201), and that “a proposition that mentions a complex will not be nonsensical, if the complex does not exist, but simply false” (TLP 3.24).
Wittgenstein never gives any example of completely analyzed propositions. But he would accept that, given the assumption that f, b, c, and R are objects, fa ≡ (fb.fc.bRc) is a logical equivalence, indicating the final step of the complete analysis of “fa.” The signs “f”, “b”, “c,” and “R” in the relevant propositional sign cannot be further dissected by means of definitions. (Notational variants are, of course, possible.) They are primitive signs, as well as simple signs (TLP 3.26). This answers the previous question why primitive signs are simple signs, and vice versa.
The proposition “fa” therefore can be completely and uniquely – disregarding notational variants – analyzed into the proposition “fb.fc.bRc,” with “fb,” “fc,” and “bRc” being elementary propositions. The consideration here is supposed to be applicable to any proposition about the world. This shows that if there are necessarily existent simple objects, which Wittgenstein certainly thinks he has already proven in TLP 2.0211–2.0212, every proposition has a complete unique analysis.
To complete the exposition of the Tractarian logical atomism, it remains for me to explicate the substance argument in the following entries:
2.0211If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true.
2.0212In that case we could not sketch any picture of the world (true or false).
However, before offering my interpretation, I shall first comment on three recent interpretations of the substance argument by Ian Proops (2004), Michael Morris (2008), and José Zalabardo (2012), respectively.
Proops (2004) thinks that when the substance argument is put forward in TLP 2.0211–2.0212, both the possibility of complete analysis and the rejection of the possibility of the contingent existence of objects are assumed. The aim of the substance argument is then only to argue that there are objects, which exist necessarily. The simplicity of objects, however, is not assumed, but follows from the conclusion that there are necessarily existent objects.
Proops’s reconstruction of the substance argument starts with the first inference in TLP 2.0211. Suppose, for reductio, that the world has no substance, or that everything exists contingently. Then everything is complex, because contingently existent simples were ruled out already. Thus, since the possibility of complete analysis is assumed here, there must be names in fully analyzed propositions referring to complexes. Hence, the sense of a proposition about a complex depends on the truth of another proposition, whose truth constitutes the complex as existing. This is applicable to any proposition about any number of complexes. Therefore, in general, whether a proposition has sense depends on whether another proposition is true. Since Proops construes “having sense” as a matter of having truth value, this means that “every interpreted sentence lacks a truth‐value with respect to at least one possible world” (2004, p.116).
To see the second inference in TLP 2.0212, suppose that whether any sentence “has sense,” that is, is truth‐valued, depends on whether another is true. Then every sentence will have an “indeterminate sense” such that it will lack a truth value with respect to at least one possible world. But an indeterminate sense is no sense at all, because, according to TLP 3.42, a proposition is truth‐valued with respect to every possible world. Hence, no sentence has a determinate sense; and so no sentence has sense. In this case, we cannot frame propositions (viz., sentences that have a sense), or “draw up pictures of the world (true or false).” This is an unacceptable consequence, as we can frame propositions. The final conclusion is then that the world has substance.
Proops’s interpretation has at least the following two difficulties, however. First, the view that both the possibility of complete analysis and the rejection of the contingent existence of objects are assumed in the substance argument does not seem to be right. In fact, as I have already argued in the previous discussion, the presence of necessarily existent simple objects guarantees that every proposition has a complete analysis. Therefore, they should not only be assumed there, but they are entailed by the conclusion of the substance argument.
Second, Proops’s interpretation fails to explain the employment of the picture theory in the substance argument. He explains that the reason why “we could not sketch any picture of the world (true or false)” is that it would follow that every proposition had an indeterminate sense, and thus that we would be unable to frame any propositions at all. But this kind of reasoning fails to employ the picture theory. This failure is probably due to Proops’s construing “having sense” as a matter of having a truth value, which leaves out the formal, or structural, aspects of sense. But it is exactly the formal aspects of sense that identify sense with a possible situation, and thus support the picture theory.
Morris (2008, pp.39–50) holds that the claim that the world has substance is equivalent to the claim that there must be a fixed form which is common to all possible worlds, and that the latter, in turn, is equivalent to the claim that whatever is possible is necessarily possible. Thus, the substance argument aims to argue for the claim that every possibility is a necessary possibility. Morris’s reconstruction of the substance argument is as follows. For any combination of names, say, “abcde,” since we can sketch pictures of the world, “abcde” has sense if and only if “abcde” is a possible combination (that is, the corresponding objects can be combined in the same way). So the picture theory supports the following claim:
- (A) The sentence “abcde” has sense if and only if “It is possible that abcde” is true.
Thus, whether the sentence “abcde” has sense depends on whether “It is possible that abcde” is true. The proposition “It is possible that abcde” is the “other proposition” mentioned in TLP 2.0211, with respect to the case of the arbitrarily chosen sentence “abcde.” Moreover, Wittgenstein would hold this:
- (B) The sentence “abcde” has sense if and only if “It is possible that abcde” has sense.
It follows from (A) and (B) that:
- (C) “It is possible that abcde” has sense if and only if “It is possible that abcde” is true.
But (C) implies that “It is possible that abcde” is necessarily true. Thus, what is asserted by “abcde” is necessarily possible. Since “abcde” is arbitrarily chosen, the conclusion that every possibility is a necessary possibility, that is, that the world has substance, is derived. Finally, Morris seems to think that one can see that his interpretation explains the text, if one sees that the argument he presents is basically the reverse of what Morris takes to be the reductio ad absurdum in TLP 2.0211–2.0212.
Morris’s interpretation apparently has the merits that the “another proposition” in TLP 2.0211 is specified, and that the picture theory is employed in the argument. But, like Proops’s, Morrris’s interpretation has some serious difficulties. First, in his reconstruction, the picture theory entails the statement that whether a proposition has sense depends on whether another proposition is true. So, according to him, Wittgenstein would hold this statement in the Tractatus. This cannot be right. In TLP 2.0211–2.0212, the statement, or, rather, the proposition that “whether a proposition had sense would depend on whether another proposition was true,” is only a step in a reductio ad absurdum. Moreover, the fact that this step is leading to the unacceptable conclusion that we could not sketch any picture of the world (true or false) proves that Wittgenstein would not hold the statement.
Second, the preceding point also shows that Morris’s reconstructed argument cannot be the substance argument. For, if his reconstructed argument were the reverse of the reductio ad absurdum in TLP 2.0211–2.0212, then the picture theory should entail the denial of the statement that whether a proposition has sense depends on whether another proposition is true, and not, as Morris mistakenly thinks, the statement itself.
Third, according to the Tractatus, propositions show the logical form of reality, which is constituted by the logical forms of states of affairs (TLP 2.031–2.033, 2.06, 4.121). The logical forms of states of affairs cannot be said; that is, cannot be represented by propositions (TLP 4.121, 4.1212). Hence, Wittgenstein would never hold that “It is possible that abcde” is a proposition, not to mention its having a truth value. What is supposed, by Morris, to be said by “It is possible that abcde” is actually unsayable, but is shown by the relevant proposition’s having sense. That is, that “abcde” makes sense shows that it is possible that abcde. (Of course, Wittgenstein would say that, strictly speaking, the “it is possible that abcde” in the preceding sentence is nonsensical.)
In order to explain the substance argument, Zalabardo (2012) appeals to an entry dated 21 October 1914 in the Notebooks:
I thought that the possibility of the truth of the proposition φa was tied up with the fact (∃x,φ).φx. But it is impossible to see why φa should only be possible if there is another proposition of the same form. φa surely does not need any precedent. (For suppose there existed only the two elementary propositions ‘φa’ and ‘ψa’ and that ‘φa’ were false: Why should this proposition make sense only if ‘ψa’ is true?) (NB 17)
According to Zalabardo, Wittgenstein takes (∃x,φ).φx to be a logical form, which is also a constituent (complex) of the understanding complex U[S, P, a, (∃x,φ).φx] which was introduced by Russell in order to account for the possibility of false representations in his 1913 manuscript Theory of Knowledge (Russell, 1913/84, pp.115–17; Zalabardo, 2012, p.139). Moreover, a Russellian logical form, like (∃x,φ).φx, exists only if one of its substitution instances obtains.
Zalabardo’s interpretation of the substance argument goes as follows. First, substance consists in possibilities of combination. The claim that the world has substance means that “the way in which certain objects are combined in an actually obtaining state of affairs can be a possible mode of combination for other objects” (2012, p.141). Second, if the world had no substance, then, in order for a false representation to be possible, there would have to be a Russellian logical form, which was a constituent of a representational state. But the logical form would not exist, unless one of its substitution instances obtains. Thus, the meaningfulness of a false representation would depend on the truth of a proposition expressing a substitution instance of the existentially generalized proposition representing the Russellian logical form. The latter is the “another proposition” in TLP 2.0211. Third, it follows that an infinite regress would be generated in this way: “the particular fact that would make the logical form exist would have to be understood, and for this another logical form would be required and so on” (2012, p.143). The unacceptable consequence is then the impossibility of false representations.
Zalabardo’s interpretation has the merits that the “another proposition” in TLP 2.0211 is specified as well as that an elegant reductio ad absurdum is reconstructed for the substance argument. But, like Proops’s and Morris’s, Zalabardo’s interpretation also suffers from serious difficulties. The first is this. The substance argument belongs to the group of comments on the simplicity of objects (TLP 2.02). The simplicity of objects, however, plays no role in Zalabardo’s reconstructed argument. The second difficulty is that the picture theory is not employed in the reconstructed argument at all. Instead, Zalabardo makes use of Russell’s accounts of understanding complex and false representations. But these are very different from the picture theory. The third difficulty is that whereas Wittgenstein says in TLP 2.0212 that we could not sketch any picture of the world (true or false), the unacceptable conclusion drawn by Zalabardo’s reconstructed argument only states that false representations are impossible.
I am now going to offer my own interpretation of the substance argument. In my reconstructed argument, both the possibility of complete analysis and the necessary existence of objects are not assumed, but derived; the “another proposition” mentioned in TLP 2.0211 is specified, and the picture theory is employed in a nontrivial, crucial manner.
Before proceeding further, a remark should be made here. Probably, because the antecedent of the counterfactual in TLP 2.0211, or the first premise of the substance argument, refers to the world, Wittgenstein uses the term “any picture of the world (true or false)” in TLP 2.0212, or the second premise of the substance argument. Actually, “any picture of the world (true or false)” means the same as “any picture of reality.” To see this, first, note that the totality of existing states of affairs is the world (TLP 2.04), and that the existence and nonexistence of states of affairs is reality (TLP 2.06). “A proposition shows how things stand if it is true. And it says that they do so stand” (TLP 4.022). Moreover, “a proposition can be true or false only in virtue of being a picture of reality” (TLP 4.06). It follows that, in general, a picture of the world (true or false) is actually a picture of reality. For if a picture of the world is true, it depicts the existence of states of affairs and thus is a picture of reality; and if it is false, it depicts the nonexistence of states of affairs and is still a picture of reality.
The substance argument is a reductio ad absurdum. The reductio is the premise that the world had no substance, which is equivalent to the claim that there were no necessarily existent simple objects. It also implies that all complexes are constituted by complexes only, or that complexes are gunky. The unacceptable conclusion to be deduced is that we could not sketch any picture of the world (true or false) (true or false). It is unacceptable, because Wittgenstein takes it to be a plain truth that we can sketch any picture of the world (true or false).
To produce a thought, or a proposition, is to sketch a picture of the world (true or false), that is, a picture of reality. That a proposition is a picture of reality can be seen from the fact that “we understand the sense of a propositional sign without its having been explained to us” (TLP 4.02; see also 4.01, 4.021). Wittgenstein even thinks it follows that we “can actually see from the proposition how everything stands logically if it is true” (TLP 4.023). For we understand the sense of a proposition when we understand how, as depicted by the proposition, things stand, if it is true (TLP 4.022). The reason why we can see the sense of a proposition is that a proposition shows or displays its sense (TLP 4.022, 4.121). This leads to what may be called “the doctrine of showing,” which includes at least the thesis that a proposition “shows its sense,” or “how things stand if it is true” (TLP 4.022).
Sense, as the content of a proposition, is determinate (TLP 3.23). Because a proposition, as a picture, shows its sense, it can always be expressed in a determinate manner that its sense can be “set out” clearly:
What a proposition expresses it expresses in a determinate manner, which can be set out clearly: a proposition is articulate. (TLP 3.251)
Remember that a proposition is a picture of reality, because it shows its sense. Therefore, “it is only in so far as a proposition is logically articulated that it is a picture of a situation” (TLP 4.032).
TLP 3.251 is the immediate comment on the thesis that “a proposition has one and only one complete analysis” (TLP 3.25). Thus, the possibility of unique complete analysis is guaranteed by the fact that a proposition shows its determinate sense. Moreover, Wittgenstein also says in TLP 3.23 that “the requirement that simple signs be possible is the requirement that sense be determinate.” This, together with TLP 3.25–3.251, also suggests that the thesis that there are simple objects (or that the world has substance) is supported by the thesis that a proposition shows its determinate sense. It is precisely because of the thesis that a proposition shows its determinate sense that the substance argument provides support for both the possibility of complete analysis and the existence of necessarily existent simple objects.
Owing to factors like “the outward form of clothing” and “the tacit conventions” of language (TLP 4.002), there may be expressions of the proposition from which one cannot see its sense. Thus, not all propositional signs show its sense. Nevertheless, the thesis that a proposition shows its sense can be formulated as follows:
- (i) A proposition must have a propositional sign showing its sense.
What plays a crucial role in the substance argument is actually this conditional:
- (ii) If we can produce a proposition as a picture of reality (that is, if we can sketch a picture of the world (true or false)), then it must have a propositional sign showing its determinate sense.
The substance argument would deduce from the reductio that the world had no substance the proposition that a proposition could not have any propositional sign showing its sense. Form this, and employing (ii), the unacceptable conclusion that we could not sketch any picture of reality would be derived.
Let me now turn to the substance argument, and begin by considering the first step:
If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true. (TLP 2.0211)
The reductio is that the world had no substance, or that there were no necessarily existent simple objects. It follows that there would only be complexes, and that:
A complex must be constituted by complexes, and those constituent complexes are constituted by other complexes, and so on and so forth.
If there were no objects, complexes had to be gunky. Moreover, since, of course, there are propositions about the world, they could only be propositions about complexes.
Wittgenstein would hold that, if there were no objects, a proposition (say) “gb”, viz. about the complexes g and b, had sense if and only if one of the following would hold:
- (a) The complexes g and b existed;
- (b) g and some constituent complexes of b existed;
- (c) b and some constituent complexes of g existed;
- (d) Some constituent complexes of g existed or some constituent complexes of b existed.
If the convention of regarding a complex as a constituent complex of itself is adopted, the general point here can be expressed as follows:
- (iii) If there were no objects, then a proposition (about the world) had sense if and only if some relevant constituent complexes of the complexes the proposition was about existed (and there were true descriptions of those complexes).
Thus, if there were no objects, then a proposition had sense if and only if the descriptions of some relevant constituent complexes of the complexes the proposition was about were true. It might not be known which descriptions were true, but there had to be those true descriptions.
This also leads to another general point, which, if the aforementioned convention is adopted, can be expressed as follows:
- (iv) If there were no objects, then whether a proposition (about the world) had sense would depend on whether some descriptions of the constituent complexes of the complexes it was about were true.
The answer to the controversial exegetical question “What is the ‘another proposition’ mentioned in TLP 2.0211?” is then this: the conjunction of some descriptions of the constituent complexes of the complexes the proposition was about. With this, TLP 2.0211 is explained.
Let me now turn to the second, and final, step of the substance argument:
In that case we could not sketch any picture of the world (true or false). (TLP 2.012)
To begin, note that (iv), together with the adoption of the aforementioned convention, implies this:
- (vi) Under the assumption that there were no objects, if a proposition (about the world) had sense, then there had to be a true description of a constituent complex of one of the complexes it was about.
But then whenever a proposition had sense, its propositional sign in that case could not have a propositional part that was a propositional sign of the relevant true description of the relevant complex. For, otherwise, the propositional sign might still express a sense, even if the description were false. This, in turn, entails that:
- (vii) Under the assumption that there were no objects, for any propositional sign of a proposition (about the world), there had to be a true description of a constituent complex of the complexes it was about such that the propositional sign did not have a propositional part that was a propositional sign of the description.
It follows from (vi) that no propositional sign of a proposition (about the world) could reflect the internal complexities of the complexes it was about completely. Thus, every propositional sign of the proposition had an indeterminateness. Hence, a proposition could not be articulate, and also could not have a propositional sign showing its sense completely. But a picture must show its sense. Therefore, if there were no objects, the proposition could not be a picture of the world (true or false). This applies to any proposition about the world. In this case, every proposition about the world had intrinsic indeterminateness, and thus could not show its sense. Consequently, if there were no objects, one could not sketch any picture of the world (true or false). Since we can sketch any picture of the world (true or false) by producing propositions or thoughts, those complexes must be constituted by necessarily existent simple objects solely; and, thus, there must be objects. The final conclusion is then that the world has substance constituted by necessarily existent simple objects.