The Representation of Logical Form: A Dilemma

The Representation of Logical Form: A Dilemma Benjamin Ferguson 1 Introduction Throughout the Tractatus Logico-Philosophicus and especially in the 2.17 s and 4.1 s Wittgenstein asserts that propositions cannot represent logical form. Specifically, he states: 2.172 The picture [or proposition], however, cannot depict its form of depiction; it shows it forth. And later he argues that 4.12 Propositions can represent the whole reality, but they cannot represent what they must have in common with reality in order to be able to represent it the logical form. To be able to represent the logical form, we should have to be able to put ourselves with the propositions outside logic, that is outside the world. These passages are puzzling since, prima facia, it does seem possible to represent the logical form of a proposition. As McGuiness points out, the logical form of the statement John loves Mary can be expressed as xry and there does not seem to be any difficulty in stating this. 1 We have a dilemma: on one hand we have Wittgenstein s statement that propositions cannot represent the logical form, and on the other we have the seemingly unproblematic statement that xry represents the logical form of John loves Mary. In the following section, I will examine both horns of this dilemma and argue that there is some reason to believe we can construct pseudo-propositions that represent the logical form of other propositions. Therefore, we must find a way of dissolving the horn of the dilemma claiming that such representation is impossible. I will argue that we can reconcile the representation of logical form with what Wittgenstein says in the 4.1 s and 2.17 s by drawing a distinction between type and token logical form, in the case of the 4.1 s, and drawing a distinction between intra- and inter-propositional representation in the case of the 2.17 s. In a short final section, I will examine the consequences this interpretation has for the statements Wittgenstein makes about the project of the Tractatus as a whole in 6.54. 1 McGuiness (1966, p.144) 1

2 Two Approaches 2.1 The First Approach As noted above, there are two general approaches that would allow us to resolve the dilemma. First, we can take what Wittgenstein says as a statement that it is impossible for a picture to represent its form. In this case, we would need to show that it is a mistake to say that xry represents the logical form of John loves Mary. The second approach is to argue for an interpretation of passages like 4.12 and 2.172 that both fits with the global argument of the Tractatus and allows us to say that xry does represent the logical form of John loves Mary. Let s consider the first approach. If we accept that it is impossible for a proposition to represent its form, then we must show that it is not the case that xry is really a representation of the logical form of John loves Mary. In order to do this, we need to understand how Wittgenstein defines proposition, representation, and logical form. For Wittgenstein, propositions are special kinds of pictures. They are pictures we construct in our minds that represent reality (4.01) and, since propositions are subsets of pictures, what is true of pictures is also true of propositions. In section two of the Tractatus, Wittgenstein explains some features of pictures that will be important for understanding the way in which he conceives of propositions. Pictures are models of reality (2.12) and the elements of a picture correspond to a particular arrangement of objects in reality (what Wittgenstein calls a state of affairs (2.01, 2.13)). Propositions are true or false, depending on whether or not they correspond to an existing state of affairs. As he says, if we want to check to see whether or not a picture [or proposition] is true or false we must compare it to reality (2.223). Pictures can represent reality because of the correspondence of their elements with states of affairs. So, we can think of representation as an isomorphism between reality and a picture of reality a picture represents a state of affairs if it is isomorphic to that state of affairs. The logical form of a proposition is the possibility of structure (2.033) and the structure is the determinate way in which objects are connected (2.032). So the logical form is a possible way in which objects are connected. As McGuiness, points out a fact and its picture may have the same form (must have, indeed) but cannot have the same structure. 2 The logical form of a proposition is like a system that has placeholders to be filled in with particular objects; a metaphysical version of mad-libs. Propositions are realised once some configuration of objects is placed in the placeholder of logical form. As an example, consider the proposition if it is raining and I am in my home I will be dry but if I am outside I will be wet. We could symbolize this proposition as P and get an idea of its logical form by considering F. P [(R H) D] [(R O) W ] F [( ) ] [( ) ] 2 McGuiness (1966, p.144) 2

The letter symbols stand in for the objects 3 that we ve used to populate the proposition, but these are interchangeable. That is, the form of the proposition does not change if we swap out R for S, where S denotes Saturday. Swapping R for S may change the truth value of the sentence, but it does not alter the logical form. From these definitions we see that a picture that represents logical form will be a model that is isomorphic to the logical form of a picture. Further, it should be clear that it is impossible for a picture to represent its own logical form. Take for example the following proposition: ( x)( y)(xly) (1) This proposition says that there is some x and there is some y such that x and y are in a particular relation (following the above examples, let s use the love relation). However, (1) says nothing about its own form, rather it says something about x and y and their relation. If we tried to construct a proposition that did speak about the form of (1) this second proposition would still not represent its own form; rather, it would represent the form of (1). If it is impossible not just for a proposition to represent its own form, but also for any proposition to represent the form of another, then the construction of a proposition that speaks about the form of (1) would be impossible. But consider this proposition: ( x)(((p x F 1 x) F 2 x) F 3 x) (2) The above proposition says something like there exists some x such that x is a proposition and x has some features (arbitrarily F1, F2 and F3). If (2) represents the logical form of (1), then the elements of (2) must be isomorphic not to (1), but to the logical form of (1). Is there reason to think that (2) (or a more accurate proposition in the spirit of (2)) is isomorphic to the logical form of (1)? Consider the following argument for the fact that (2) represents logical form. Let s label (1) s logical form (L). Either (2) is isomorphic to (L), or it is not. Suppose we want to argue that (2) is not isomorphic to (L). In order to do this, we must compare (L) and (2) and if they are not isomorphic, then (2) does not represent (L). Since (2) is proposition, it is a model of a state of affairs. It does not make sense to argue that (2) is not isomorphic to (L) if (L) is not a state of affairs by which we can judge whether or not an isomorphism obtains. If (L) is a state of affairs, then although (2) may not be isomorphic to (L), it must be possible to construct some proposition that is isomorphic to (L). To put it more intuitively, if we compare two things and we say one is not like the other we must already have a notion of what it would mean for the two to be alike. By appealing to that notion we could describe a third object that was comparable to the first. It seems that no matter how the problem is set out we can show 3 Objects for Wittgenstein are not everyday objects like in (P), rather they are the elementary entities of his system. All propositions can be reduced to propositions about these elementary entities. However, for illustrative purposes it is useful to think of objects as everyday objects. 3

that it is possible to represent the logical form of (1). However, on reflection we might be a bit sceptical about the above argument. By constructing (L) we are smuggling in a proposition that is already supposed to be a representation of the logical form of (1). The above argument begs the question. Consider this argument against (2) s ability to represent logical form. If (2) does not represent the logical form of (1), then it is not because (2) is not isomorphic to (L). It is because it does not make sense to even talk about representing the logical form of (1). We cannot test the adequacy of (2) by comparing it to (L), because in this case we tacitly assume our conclusion. It is never possible to show that a proposition represents logical form without begging the question; therefore, we must reject the idea that any proposition, (2) included, can represent logical form. This second argument is odd as well. It too seems to beg the question. While it may be reasonable to accept the claim that assuming (L) begs the question, the motivation for this claim is that it is simply impossible to represent the logical form of a proposition. But this brings us back to where we began. We have no way of evaluating (2) and we are still left with our dilemma. On one hand we have Wittgenstein s assertion that it is impossible to represent the logical form of a proposition and on the other we have (2), which seems to be a perfectly good proposition. Perhaps we can make sense of a third option. We could think of (2) as a pseudo-proposition. That is, we may wonder if (2) as a proposition really pictures any state of affairs in reality. This option compromises between the view that (2) does represent the logical form of (1) and the view that it does not. On one hand, the argument that (2) is a pseudo-proposition presupposes that construction of (2) is possible, that it does represent something. On the other hand, as a pseudo-proposition, (2) will be vacuously true. Propositions that are not pseudo-propositions are either true or false depending on whether or not the state of affairs they describe obtains. Pseudopropositions are statements that seem to be proper propositions, but do not make use of this correspondence notion of truth. They may, for example, have indeterminate truth values or they may be tautologically true that is, their truth does not depend on a state of affairs. In order to see whether or not (2) is a pseudo-proposition, we must determine those conditions under which we can determine its truth value. All of the information needed to construct (2) we find in (1), or more accurately, we are shown that (1) has the features (2) describes when we come to understand what (1) says. In order to check to see if what (2) describes is true we must ascertain whether (1) does or does not have the features described in (2). First, suppose (1) does not have these features, then whatever it is that (2) represents (if anything at all), it is not the logical form of (1). On the other hand, if (1) does have the features described, then (2) does represent the form of (1), but in this case, since (2) is a restatement of the features of (1), (2) is not saying anything that is not already shown when we come to understand (1). As McGuiness argues, since, in order to understand any proposition P n about a proposition p, you must already understand p,... the 4

proposition P k ascribing a certain logical form to p is bound to be otiose. 4 If (2) does represent the logical form of (1), then (2) will be true. So any representation of the logical form of a proposition is tautological. Since (2) is either nonsense in the first case or vacuously true in the second, we can conclude that (2) is a kind of pseudo-proposition. So far we have found that it is impossible for a proposition to represent its own form because any attempt to describe the form results in a second sentence that is not identical to the first. Arguments for and against the ability of (2) to represent the logical form of (1) proved problematic and the best approach is to classify (2) as a pseudo-proposition. By constructing a pseudo-proposition, (2) that represents the form of (1) we don t gain anything that is not already contained in (1). Returning to our dilemma, we have not shown that xry does not represent the logical form of John loves Mary ; rather, we have shown that we must already understand what it means to say John loves Mary if we are to construct xry. 2.2 The Second Approach After an analysis of the first horn of the dilemma we have concluded that (2) can be constructed and is a pseudo-proposition. This conclusion is still at odds with a strictly literal interpretation of what Wittgenstein says in sections 2.17 and 4.1. Now we must test our conclusion against a careful reading of these sections if we are to dissolve the dilemma. It seems that xry does, with some qualification, represent the logical form of John loves Mary. How can we square this with what Wittgenstein says about the representation of logical form? First let s take a look at 2.172. Here he does not say that it is impossible for a picture (or proposition) to have its form represented; rather, he argues that a picture cannot depict its form of depiction. That is, a picture cannot represent its own form. We have already seen that this is indeed the case and Wittgenstein s statements in 2.17 do not rule out the possibility of a second proposition representing the form of a first. What about section 4.1? The first half of 4.12 might plausibly be read similarly to 2.172, but this seems difficult in light of the second half of 4.12 where Wittgenstein states, to be able to represent the logical form, we should have to be able to put ourselves with the propositions outside logic, that is outside the world. It seems puzzling that Wittgenstein thinks we should have to put ourselves outside of logic to represent logical form since we didn t have trouble constructing our xry from John loves Mary. Addtionally, 4.121 contains a direct statement about the impossibility of representing logical form: 4.121 Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means of language. Propositions show the logical form of reality. They display it. 4 McGuiness (1966, p.144) 5

In order to understand Wittgenstein s meaning in these passages we should closely examine the work being done in the fourth section and the way this section differs from the second. In the second section of the Tractatus, Wittgenstein sets up his picture theory of meaning and the way in which these pictures connect with the basic ontology he outlined in section one. Section two deals with the nature of these pictures and the way they represent states of affairs. The statements Wittgenstein makes about propositions in section two deal with token propositions. Wittgenstein is trying to help us see how this type of entity works in his ontology by giving us examples of what can and cannot be represented by a particular proposition. References to logical form in section two are references to token logical forms that are found in token propositions. In the fourth section the goal is a bit different. Here Wittgenstein sets out to show the most essential features of propositions and the way they hook up with the logical form of reality. When Wittgenstein describes the general form of propositions his statements about propositions are statements about propositions as a class. Once we understand his goal in section four, we can see that his statements at 4.12 and 4.121 about the representation of logical form apply not to token logical form found in token propositions, but rather to logical form writ large. What 4.121 asserts is that no proposition can represent logical form itself that is, no proposition can represent the possible ways in which objects can be connected. In (1) we have a proposition and in (2) we have a representation of the logical form of (1), but (2) does not tell us anything about the logical form in (1), it merely represents it. (2) is a statement that says here is a proposition (1) and (1) has this logical form. If we want to represent logical form in general, we would need a statement (3) that says, something like here is (2), a representation of the logical form of (1) and this type of thing, logical form, is such and such. It is the last step of (3) that Wittgenstein claims is impossible. When we represent the logical form of (1) in (2) we are merely re-presenting explicitly that which we saw in (1), but when we try to speak about the type of thing we saw in (1), to characterize it as such and such by constructing (3), we try to go beyond the limits of what can be meaningfully said. In the context of section four we can see that it would indeed be impossible to represent logical form in general without stepping outside of the world, outside of logic, because any attempt to represent this form by another proposition would be a more general statement than the general form of the proposition. We began with a dilemma: it seemed that Wittgenstein was claiming that it was impossible to represent the form of a proposition or picture, and yet, it also seemed we could straightforwardly represent the logical form of a proposition John loves Mary with a second proposition xry. We have resolved this dilemma by clarifying what Wittgenstein meant in those phrases that are seemingly at odds with the construction of xry. First, it is clear that it is impossible for a proposition to represent its own logical form. Propositions that represent the logical form of another proposition are, although possible to construct, pseudo-propositions because they tell us nothing that is not contained in 6

the first proposition. Second, this conclusion is consistent with the statements Wittgenstein makes in 2.17 and 4.1 when we draw a distinction between type and token logical form, in the case of the 4.1 s and a distinction between intraand inter-propositional representation in the case of the 2.17 s. One proposition may represent the logical form of another proposition, but no proposition can represent the logical form of reality. No proposition can make characterising statements about logical form. 3 Senlessness and Nonsense While the dilemma has been resolved, a puzzle still remains. The final sentence of the preceding paragraph is slightly odd: No proposition can make characterising statements about logical form. As I noted above, this is exactly what Wittgenstein is trying do in section four, and in fact, the general project of the Tractatus is to examine the limits of language. Wittgenstein is attempting to do exactly what we cannot do according to 4.1212: 4.1212 What can be shown cannot be said. Wittgenstein s a realisation of the impossibility of his task motivates the cryptic final statements in the Tractatus: 6.54 My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (he must so to speak throw away the ladder, after he has climbed up on it.) He must surmount these propositions; then he sees the world rightly. 7 Whereof one cannot speak, thereof one must be silent. Much has been written about the meaning of this final section. While it is clear that Wittgenstein is addressing the self-referential nature of the book in the last section, the impact of these later statements on what he said in the earlier sections has been hotly debated. In the short section that follows I will to argue for an interpretation of 6.54 and 7 that mixes elements of Russell s interpretation (which Wittgenstein believes missed the point) and a traditional reading. White 5 points out that in his introduction to the Tractatus, Russell notices that it is a lack of reflexivity that makes Wittgenstein s arguments about language odd. Russell s solution to this problem was to argue that propositions that cannot be said within one language can be expressed in a meta-language. He suggests that every language has a structure concerning which, in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit. 6 Wittgenstein rejected 5 White (2006, p.122) 6 Wittgenstein (1999, p.22) 7

this approach. The Tractatus is not about what can be expressed in a particular language; rather, it is a generalized statement about what can be expressed in any language. The conclusion we reached on analysis of 4.121 that it is impossible to construct a statement about logical form as a type of entity applies to Russell s attempted solution. At the level of language it is impossible to talk about logical form. However, at the propositional level, it does make sense to speak of representing logical form, provided we recognize such statements as pseudopropositions. Although the approach is not applicable at the general level of language, an analogy between the propositional and linguistic levels can help us understand Wittgenstein s statements at 6.54 and 7 and will also show how the Tractatus can still be useful despite problems of reflexivity. If we consider the statement John loves Mary and a representation of its form xry, xry says nothing not already contained in John loves Mary. However, detailing the features of the logical form of propositions can still be an epistemically meaningful exercise. For example, we may want to talk about the properties of the love relation is it transitive, symmetrical, reflexive etc. In order to answer these questions we must think about the original statement. If a teacher asks a class whether or not the love relation is transitive they will appeal to what they already know about love to answer the question. While they already knew that love was not transitive by the way they used the term, the epistemic discovery that it is not transitive is still meaningful to them. The features of the language they used were not clear to them, but by considering the logical form of their propositions they discovered something new that was presupposed in the way they were already using the terms. As Wittgenstein points out: 4.002... Colloquial language is a part of the human organism and is not less complicated than it. From it it is humanly impossible to gather immediately the logic of language. Language disguises the thought; so that from the external form of the clothes one cannot infer the form of the thought they clothe, because the external form of the clothes is constructed with quite another object than to let the form of the body be recognized... Note that, for Wittgenstein, the propositions of mathematics are, in fact, pseudopropositions (6.2), but nevertheless, mathematic inquiry is epistemically (and practically) fruitful. In the same way, inquiry into the logical form of propositions is also epistemically fruitful. But is inquiry into the general features of propositions and logical form similarly fruitful in light of the impossibility of constructing propositions about these entities? I think we must conclude that while such inquiry is not strictly meaningful it is not nonsense. Rather it is something in between. It is senseless. We can distinguish between three levels of meaning: nonsense (N), senselessness (L) and sense (S) as shown in the following sentences: 8

N asdfasq34t2349-edfvn24-0uxsfkln34r0-sdufgnl. L The logical form of (1) is ( x)(((p x F 1 x) F 2 x) F 3 x). S It is raining outside. Sentence (N) contains no words, conveys nothing, and does not even trick us into a belief that it is meaningful. It offers no insight into any aspect of the world. Sentence (L) is a tautological pseudo-proposition, but (L) can give us insight into some features of the world, despite being meaningless. (S) is a meaningful sentence that is a true (or false) picture of a state of affairs. The propositions of the Tractatus fall into the second category. As White indicates we need to keep firmly in mind the distinction between the meaning of a sentence and the use we make of it. The issue is not : Does a [senseless] sentence have a surreptitious meaning? but Can we use a sentence that is confessedly nonsense to communicate something? The bald answer to the second question is undoubtedly Yes. 7 The propositions of the tracatus are useful for helping us understand the way our language works and the way our world must be if we are to use language in this way. By elucidating features of our language that are obscured the Tractatus enables us to better evaluate our propostions, gives us some ideas about the characteristics of a logically perfect language and classifies many philosophical problems as pseudo-problems. 4 Conclusion In this paper I ve analyzed a puzzle about the representation of logical form that turns out to be central to the entire project of the Tractatus. The fact that it seemed easy enough to construct propositions that represented logical form contradicted statements in sections two and four of the Tractatus that seemed to suggest otherwise. By examining both the validity of the claim that we can construct propositions that represent logical form and Wittgenstein s comments in sections two and four I have shown that it is indeed possible to construct propositions about logical form; however, any proposition representing logical form is a pseudo-proposition. Despite first appearances, this is not at odds with what Wittgenstein states. The apparent contractions in section two can be dissolved when we draw a distinction between inter- and intra- propositional representation. The apparent contractions in section four disappear when we make a distinction between type and token logical form. Finally, we saw that pseudo-propositions about logical form may still be epistemically meaningful and, applying this insight to the Tractatus we see that Wittgenstein s own pseudo-propositions enable us to understand some deep truths about the nature of the world and language. 7 White (2006, p.132) 9

References Black, M. (1964). A Companion to Wittgensteins Tractatus. Cambridge University Press, Cambridge, UK. Copi, I. and Beard, R. (1966). Essays on Wittgensteins Tractatus. Routledge, London, UK. McGuiness, B. F. (1966). Pictures and Form in Wittgenstein s Tractatus. In Copi, I. and Beard, R., editors, Essays on Wittgenstein s Tractatus. Cambridge University Press, Cambridge, UK. Morris, M. (2008). Wittgenstein and the Tracatus Logico-Philosophicus. Routledge, ew York, NY. White, R. (2006). Wittgensteins Tractatus Logico-Philosophicus. Continuum, London, UK. Wittgenstein, L. (1999). Tractatus Logico-Philosophicus. Dover, Mineola, NY. 10