Just for a change of pace, here is the paper that I’m planning on using for my PhD application as my writing. Enjoy, but a warning, this paper is not for the philosophically faint of heart.
Wittgenstein’s Argument for Why a Picture cannot Depict its Logical Form
In this paper I will be reconstructing Wittgenstein’s argument for his claim that a picture cannot depict it’s logical from (Tractatus, 2.172). I start off by presenting the direct argument he offers for the claim, which uses quasi-mystical sounding language about “standing outside of our picture,” which of course is something he thinks we cannot do. After the initial reconstruction I attempt to explicate the main principles and lines of reasoning Wittgenstein develops leading up to this argument and immediately the argument in an attempt shed some light on the argument itself. Then I will present a reconstructed version of the original argument. Wittgenstein’s argument hinges on the crucial distinction with between form and structure, and what is required to depict something. Structure is the determinate way that things are and form is the possibility of structure. To depict something means to show not only its structure but its form. However, it is impossible to depict the logical form of our picture because to do so would require what I will later call a higher order logico-pictorial form capable of depicting not only the structure of our logico-pictorial form but also the possibilities of the structure of our logico-pictorial form, which we do not have.
Part I – Why a picture cannot depict its pictorial form
In proposition 2.172 of the Tractatus Logico-Philosophicus Wittgenstein makes his famous claim (C) that a picture cannot depict its pictorial form but can only show it. Although this sounds somewhat quasi-mystical Wittgenstein does offer an argument as to why it is he thinks it is that a picture cannot depict its pictorial form. Here is the claim and the propositions that immediately follow it that contain his argument for (C):
2.172: A picture cannot, however, depict its pictorial form: it displays it.
2.173: A picture represents its subject from a position outside it. (Its standpoint is its representational form.) That is why a picture represents its subject correctly or incorrectly.
2.174: A picture cannot, however, place itself outside its representational form.
All the pieces of the argument are not right there but there is enough, I think, to get a clear idea of the argument that he has in mind, especially given the propositions that have lead up to this argument. Although Wittgenstein starts with the conclusion here is how the argument for (C) is supposed to run:
Initial formulation of the Argument for C (AI):
- If it is possible to depict the pictorial form of our picture then it must be possible to depict the pictorial form of our picture from a position outside of the pictorial form of our picture.
- It is not possible to depict the pictorial form of our picture from a position outside of the pictorial form of our picture.
- Therefore, it is not possible to depict the pictorial form of our picture.
The first premise is specific statement what would be required to depict the pictorial form of our picture given the an important principle that Wittgenstein holds, which he alludes to in 2.173, which states the necessary condition for the possibility of depiction, which is:
Wittgenstein’s Depiction Principle (WDP): If it is possible to depict x then it must be possible to depict x from a position outside of x.
The second premise of the argument is a pretty straightforward reconstruction of what Wittgenstein says at 2.174. And the conclusion is merely the original claim (C). The above argument seems to be a pretty accurate reconstruction of the argument contained in propositions 2.172-4 but much more needs to said to fill out the picture of why Wittgenstein holds the first two premises to be true. Most importantly we need to know what it means to be able to depict x from outside of x, and why it is that we can’t “get outside” of our pictorial representation.
The language Wittgenstein uses is not the most perspicuous as it is stated in the somewhat metaphorical terms of the picture analogy but the principle itself rests on earlier principles regarding what it is to know an object as well as a second claim about the structure and limits of our thought. So, let’s work our way to (C) from earlier in the text.
For Wittgenstein, to know an object means to know all of its possibilities in states of affairs, which requires knowing all of the its logical possibilities. He says this most explicitly at 2.0123, “If I know an object I also know all its possible occurrences in states of affairs. (Every one of these possibilities must be part of the nature of the object).”
And leading up to this claim he says:
2.011 It is essential to things that they should be possible constituents of states of affairs.
2.012 In logic nothing is accidental: if a thing can occur in a state of affairs, the possibility of the state of affairs must be written into the structure of the thing itself.
2.0121 … If things can occur in states of affairs, this possibility must be in them from the beginning.
(Nothing in the province of logic can be merely possible. Logic deals with every possibility and all possibilities are facts.)
Just as we are quite unable to imagine spatial objects outside space or temporal objects outside time, so too there is no object that we can imagine excluded from the possibility of combining with others.
If I can imagine objects combined in states of affairs, I cannot imagine them excluded from the possibility of such combinations.
So, from this can assume Wittgenstein holds the following principle:
Wittgenstein’s Possibility Principle (WPP): If I know an object then I also know all its possible occurrences in states of affairs.
And, in explicating what it would be to know all the possible states of affairs of an object he assumes a further principle, which he states at 2.0131:
Wittgenstein’s Logical Property Principle (WLPP): If I know an object then I must know all of its internal (logical) properties.
And the internal properties of an object constitute the form of an object. He says, “The possibility of its occurring in state of affairs is the form of an object” (2.0141). And form, for Wittgenstein, is the possibility of structure (2.033). And structure, for Wittgenstein, is “The determinate way in which objects are connected in a state of affairs . . .” (2.031). So, for any given state of affairs x, the way that things actually are, their determinate way, is the structure of x, and the form is all the possible ways in which x could’ve been, the possibility of x’s structure. So, to know an object is to know all of the logically possible ways in which it can connect to other objects. He uses the metaphor of a chain (2.03) to describe the way that objects fit together. Logical possibility determines exactly what objects can “link up” with other objects, which is the form of an object, and the structure of an object is contained in the state of affairs of which it is a part and in which it is “linked up” with other objects. With these basic principles laid out he begins to work towards (C).
He first introduces the picture metaphor in 2.1 saying, “We picture facts to ourselves.” But what is a picture? “A picture,” Wittgenstein says, “presents a situation in logical space . . . ” (2.11). And, “A picture is a model of reality” (2.12). The picture is model of reality in the sense that “In a picture the elements of the picture are the representatives of objects” (2.131). But this is only one half of the model theory. The picture models reality in two ways. It is a model of the structure of reality but also of the form of reality. It models the structure of reality by representing elements of the picture in a certain way, the determinate way that things actually are, and it models the form of reality by containing all the logically possible ways in which objects could be combined in all possible states of affairs.
And this brings us to Wittgenstein’s introduction of “pictorial form” in 2.15. There he says:
The fact that elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way.
Let us call this connexion of its elements the structure of the picture, and let us call the possibility of this structure the pictorial form of the picture.
And in 2.16 he says, “Pictorial form is the possibility that things are related to one another in the same way as the elements of the picture.” And, pictorial form is what a picture must have in common with reality in order to depict reality (2.17). It is the fact that our picture has the same pictorial form as reality that we are able to depict reality, because a picture can only depict a reality whose form it has (2.171). And voila we arrive at the original claim (C) that “A picture cannot, however, depict its pictorial form; it displays it” (2.172). What is interesting is that in the argument for (C) Wittgenstein doesn’t appear to use any of the terminology he has been developing but seems to shift to talking about how we can’t place ourselves outside our representational form. Can this new way of talking be connected to the previous arguments he has developed? I think it can but there is one further principle that needs to be clarified to show exactly how.
Wittgenstein goes on to clarify that the pictorial form that a picture has in common with reality is really logical form. He says, “What any picture, of whatever from, must have in common with reality, in order to be able to depict it . . . is logical form, i.e. the form of reality” (2.18). Not all pictures are spatial, but all pictures are logical pictures. There is a logical structure in our pictures that mirrors the logical structure of reality. The pictures that Wittgenstein has in mind are clearly not images because he says that not all our pictures are spatial but all our pictures are logical (2.182). More properly speaking a picture does not merely have a pictorial form but a logico-pictorial form (2.19). This becomes crucial because logical structure is what determines the form of our pictures and the possibility of depiction of states of affairs. And this is why, for Wittgenstein, we cannot think illogically (3.03). The rules of logic are the rules that determine the possible states of affairs that can be represented in our picture. Really we cannot even think or say what it would mean for something to be illogical. He says, “It used to be said that God could create anything except what would be contrary to the laws of logic. – The truth is that we could not say what an ‘illogical’ world would look like” (3.031). And in 3.032 “It is as impossible to represent in language anything ‘contradicts logic’ as it is in geometry to represent by it coordinates a figure that contradicts the laws of space, or to give the co-ordinates of a point that does not exist.” When speaking about things that are ‘illogical’ and ‘contradict logic’ Wittgenstein uses quotation marks to indicate that he is not really saying anything by those terms because there is nothing that can be said about something that contradicts logic because logic dictates the rules regarding what we can think and say. To say something illogical is like trying to give the co-ordinates for a point that does not exist. And with that we are now in a position to see exactly how Wittgenstein’s argument is supposed to work.
At 2.173 Wittgenstein says that “A picture represents its subject from a position outside it” and “Its standpoint is its representational form.” What I take him to mean is that the logico-pictorial form of reality, and our pictures of reality, is a kind structure i.e it is determinate. It is confusing because he calls the structure of our picture the ‘the logico-pictorial form’, or sometimes just the “pictorial form’, of our picture. It is a kind of form in that it represents the possibility of the structure of objects but it is a structure in that it is a determinate set of possibilities. So what he means when he says that “A picture represents its subject from a position outside it” is that to depict our logico-pictorial form we would have to depict the form of our logico-pictorial form i.e the possibilities of the structure our logico-pictorial form. A picture shows the structure of its pictorial form but it does not show its form i.e it does not depict the structure of its pictorial form. A picture depicts the form of objects but cannot depict the structure of its logico-pictorial form. What is implied by Wittgenstein’s account, which is the one thing he is not explicit about, is that there are levels of form and structure, the form and structure of objects and the form and structure of our logico-pictorial form:
First Order Form and Structure:
(S1): The structure of objects i.e. The determinate way that things are in reality and in our picture of reality.
(F1): The form of objects i.e. All the logically possible states of affairs of the objects in our pictures of reality
Second Order Form and Structure:
(S2): The structure of the logico-pictorial form of our picture i.e the determinate structure of our picture that determines what possibilities (F1) it can represent.
(F2): The form of the logico-pictorial form of our picture i.e. all the possible structures of logico-pictorial form.
Third Order Form and Structure:
(S3): The Structure of the logico-pictorial form that would be required to depict (S2).
So, what Wittgenstein means when he says that “a picture represents its subject from a position outside it” is that a higher order structure is required to depict the possibilities of form. To depict the possibilities of objects (F1)we must use a picture capable of depicting objects as they are and all the logically possible states of affairs of those objects (S2). The terminology is slightly misleading because the ‘form’ in ‘logico-pictorial form’ refers to F1, the form of objects, so although it represents the form, or the possibilities, of objects it is a structure, the structure of our ‘logico-pictorial’ picture (S2). The key point here is that the total set of the possibilities of objects (F1) form a determinate set of possibilities of representation (S2). So, to depict objects we must depict their possibilities, which requires a position “outside” of the objects, a position that can represent the determinate way the objects are as well as all the possible ways that objects could be. This is so, for Wittgenstein, because of what we earlier introduce as Wittgenstein’s Possibility Principle (If I know an object then I also know all its possible occurrences in states of affairs.)
Now, we are in a position to see what Wittgenstein meant when he said that “A picture cannot, however, place itself outside its representational form” and why it is a picture cannot depict its logico-pictorial form. To depict F2, all the possible structures of logico-pictorial form, we would require a higher order picture with an S3 –logico-pictorial form. But, the determinate S2-logico-pictorial form of our pictures limits our thought, for we cannot think ‘illogically.’ Or, in other words we cannot get outside of the S2-logical-pictorial form of our picture, which is what Wittgenstein is intending to argue for at 3.031-2 when he talks about the impossibility of ‘illogical’ thought.
So now what need is a fully reconstructed of the original argument for why it is that a picture cannot depict its form, but merely shows it:
Revised formulation of the Argument for C (AR):
- If it is possible to depict x then it must be possible to depict the form of x.
- If it is possible to depict the form of x then it must be possible to depict the form of x from a position outside of x.
- If it is possible to depict the form of the logico-pictorial form of our picture, (F2) then it must be possible to depict the form of the logico-pictorial form of our picture (F2) from a position outside of the form of the logico-pictorial form of our picture (S3).
- If it is possible to depict the form of the logico-pictorial form of our picture, (F2), from a position outside of the logico-pictorial form of our picture, (S3), then it would require use of a higher order logico-pictorial form, (S3), than the logico-pictorial form of our picture, (S2).
- We cannot depict anything that requires use of a higher order logico-pictorial form than the logico-pictorial form of our picture i.e. we cannot think illogically.
- (MT 2 &3) It is not possible to depict the form of the logico-pictorial form of our picture from a position outside of the (S2)-logico-pictorial form of our picture).
- (MT 1 & 4) Therefore, it is not possible to depict (F2), the form of the (S2)-logico-pictorial form of our picture.
I believe the above is an accurate reconstruction of Wittgenstein’s original argument (AI) for why a picture cannot show its logical form. The crucial claim was that “A picture cannot, however, place itself outside its representational form” (2.174) and what I have argued is that by “place itself outside its representational form” Wittgenstein means that to depict its logico-pictorial form a picture would have to use a higher order picture capable of depicting the form of its logico-pictorial form, which is of course impossible because we cannot use any other logico-pictorial form than the one we have. The following illustrates the above argument pictorially:
Our (S2)-logico-pictorial form is capable of depicting the determinate way that objects are (S1) and all the possible ways that objects could be in our picture (F1), which are represented as (S1a, S1b, S1c, . . . S1n). What our picture cannot represent is all the possible structures of logico-pictorial form (F2), represented as (S2a, S2b, S2c, . . . S2n). However, that is exactly what it would need to do to depict the logico-pictorial form of our picture. To depict the logico-pictorial form of our picture would require an (S3)-logico-pictorial form, but alas, we are limited to our (S2)-logico-pictorial form.
It is important to note that nothing Wittgenstein explicitly says commits him to the belief in other possible logico-pictorial forms, either at the (F2) or (S3) level. Nothing he says commits him to alternative logico-pictorial forms or to higher order logico-pictorial forms capable of depicting the various possible structures of logico-pictorial form. His argument is merely intended to show what would be required to depict our logico-pictorial form and that we cannot do such a thing. And certainly by his own lights he is going to want to pass over such a question in silence.
In this paper I have argued that Wittgestein’s argument for his claim that a picture cannot depict its logical form coheres nicely with the rest of his arguments in the Tractatus. I offered a reformulated version of Wittgestein’s original argument that cached out what it would mean for a picture to “stand outside its representational form” as meaning that for a picture to depict its logico-pictorial form it would require the use of a higher order logico-pictorial form that can depict the structure of its logico-pictorial form as well as the form of its logico-pictorial form. No picture, however, can do such a thing.
The upshot of this argument is quite significant for Wittgenstein. The above view that I’ve been developing undergirds his later claims about logic and his view that “ . . . the propositions of logic say the same thing, to wit nothing” (5.43). They are meaningless or trivial because we cannot know anything about them except that they are the structure of our logico-pictorial form. For them to be meaningful we would have to know what it would for them to be false, but we cannot even intelligibly ask that question because we have no idea what it means. This is a quite a radical departure from the traditional view regarding the propositions of logic and logical possibility. The way Wittgenstein illustrates this departure by raising the question of whether God could do something that contradicted the laws of logic. He says, “It used to be said that God could do anything except what would be contrary to the laws of logic . . .” (3.031), with the implication that logical possibility is something even God is subject to. However, he goes on to say that, “The truth is we cannot say what an ‘illogical’ world would look like,” which I take to mean that for Wittgenstein logical possibility and impossibility do not carry the same kind of necessity as has traditionally been afforded to them by philosophers.
Wittgenstein, Ludgwig. Tractatus Logico-Philosophicus. Trans. D.F. Pears and B.F. McGuiness. (New York: Routledge, 2009).
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