Logical Space and Objects

One may have the impression that Wittgenstein presents simple objects as a metaphysical postulate, claiming their form is displays in logical space. The opposite is the case: elementary logical spaces are revealed by physical analysis and simple objects that contain the form of these spaces are established as a matter of reductio

argumentation. An example of the kind of analysis Wittgenstein has in mind is revealed in a recorded conversation with Waismann in 1929:170

Although Waismann’s notes are not all that clear, their gist is that the colour signified by the line intersecting the white-black axis is a shade of the elementary colour yellow. The

point of intersection stands for an elementary picture in colour-space that is the configuration of two elements of representation: the coordinate value “yellow” and the coordinate value of the particular shade. In this conversation each coordinate value is simply taken to be correlated to a simple object, the configuration of which is represented by the elementary picture.

That the two elements of representation are indeed correlated to two simple objects is established through the following set of propositions:

2.02 Objects are simple.

Sign for a colour:

white

blue

black red

Every statement about colours can be represented by means of such symbols. If we say that four elementary colours [i.e., red, yellow, green, blue] would suffice [to generate all colours], I call such symbols of equal status

elements of representation. These elements of representation are the ‘objects’.

The following question now has no sense: Are objects something thing-like, something that stands in subject- position, or something property-like, or are they relations, and so forth? It is simply where we have elements of representation of equal status that we speak of objects.

2.021 Objects make up the substance of the world. That is why they cannot be composite.

2.0211 If the world had no substance, then whether a proposition had sense would

depend on whether another proposition was true.

2.0212 In that case we could not sketch any picture of the world (true or false).

2.022 It is obvious that an imagined world, however different it may be from the real

one, must have something – a form – in common with it.

2.023 Objects are just what constitute this unalterable form.

The argument found in these propositions is actually three sub-arguments that build upon one another.

The first establishes substance and occurs in propositions 2.0211 and 2.0212. It can be reconstructed as follows:

P1: The whole of logical space is a multi-dimensional form of all possible worlds. P2: A picture presents a situation in the whole of logical space.

P3: A picture represents a possible situation.

P4: A picture has sense only if the situation it represents is structure that 1) is isomorphic; and 2) can make the picture true.

P5: Assume there is no substance that could produce a situation in the world.

P6: It follows that a picture can only represent an isomorphic situation presented by another picture, which can only represent an isomorphic situation presented by another picture, and so on in infinite regress.

P7: But since no situation can make a picture true, no picture has sense.

P8: Yet we routinely sketch pictures in multi-dimensional space that convey information. C: P5 is false.

What this first argument establishes is that there must be isomorphic situations in the world that can make a picture true and a supporting substrate that produces them.

The second argument found in propositions 2.022 and 2.023 builds upon the first and concerns the form of substance. We picture all situations of the real world as located in a form of all possible worlds, i.e., the whole of logical space. Given this form, there is a possible world that agrees with a particular picture. Since a picture represents a possible situation in the real world that can make it true, the real world must share the form of this possible world. Since substance is what produces a situation in the world, substance must also have the same form of the possible world.

The third argument concerns ontology and is mentioned in 2.021 and 2.023. That the form of substance must be a constitution of simple objects follows straightforwardly. If, as physical analysis shows, one elementary logical space is independent of another (e.g., the logical space for touch is independent of the logical space for sound), then it follows that the form of substance must be divided into kinds of substance. And if, as physical analysis also shows, there are elementary logical spaces that have more than one coordinate dimension, then substance must be divided (at most) into objects corresponding to individual coordinate values of a certain axis with an essence that can be represented by another a coordinate axis.

Note that nothing in the forgoing arguments establishes that there are simple objects that are bachelors. They merely establish simple objects insofar as they configure with other simple objects to produce material properties that can make a picture true. This is probably why Wittgenstein is uncommitted one way or the other to their existence or non-existence apart from these configurations. Furthermore, nothing establishes that elementary logical spaces we imagine simple objects to be situated in are real, only that reality shares its form. Finally note that if we combine these arguments with the argument that if elementary propositions are given we are given all elementary propositions or, equivalently, that if names with different meanings are given we are given all names with different meanings, then it follows that “[i]f all objects are given, then at the same time all possible states of affairs are also given” (2.0124) or, equivalently, that “[i]f objects are given, then at the same time we are given all objects” (5.524).