In propositions of the form "A says that p" or "A believes that p," it would seem that the proposition "p" is being used in a larger proposition without having any bearing on that larger propositions truth or falsity. For instance, the proposition "John hopes that it will rain tomorrow" can true or false regardless of whether or not it does actually rain tomorrow. This would seem to problematize Wittgenstein's claim that all propositions are truth-functions of elementary propositions. If "John hopes that it will rain tomorrow" is a composite of, among other things, the proposition "p," then p would have to have some bearing on the truth or falsity of the overall proposition.
Wittgenstein answers that a proposition of the form "A believes that p" does not actually involve a relationship between A and the proposition "p." "'A believes that p,' 'A has the thought p,' and 'A says p' are of the form '"p" says p'" (5.542). For A to think, believe, or say that p is the case, the words that constitute the verbal utterance of p must occur to A. It is then not A but these words that are related to p, and the internal similarity between the words and the proposition is obvious. Wittgenstein further infers that there is no such thing as a "soul" where thoughts and beliefs reside (5.5421).
We cannot learn a priori what kinds of objects or elementary propositions there are. Logic is prior to any particular experience, but not prior to the fact of experience itself: it is the shape experience takes. Logic can teach us that there are objects and elementary propositions, but it is a matter of applying logic that we come to learn about the varieties of objects and elementary propositions that there are. There is no such thing as "logical experience" that we can consult regarding the various forms of elementary propositions (5.552).
"The limits of my language mean the limits of my world" (5.6): the limits of language are determined by the totality of elementary propositions, and the limits of the world are determined by the totality of facts. There is a one-to-one correspondence between facts and elementary propositions, so we cannot say what lies outside the limits of the world (5.61).
This observation leads Wittgenstein to reflect on the limited truth of solipsism. Where, within the limits of my world, do I fit? Wittgenstein draws the analogy between the relationship between the metaphysical subject and the world on one hand, and the relationship between the eye and the visual field on the other (5.633). I cannot see my eye anywhere in my visual field, but the existence of a visual field presupposes the existence of the eye. Similarly, my self is not something I encounter in the world, but the existence of the world (my experience of the world) presupposes that there is a self to experience it. However, I cannot talk about this self because it is outside the limits of the world, and hence outside the limits of language. The only way the self appears in philosophy is in the fact that "the world is my world" (5.641).
The term "solipsism" defines a number of related philosophical positions, all of which claim that the objects and people in the world only exist as objects of my awareness, that only I, as a thinking consciousness, truly exist. No philosopher has seriously defended this position (although there is a story about a woman who wrote to Bertrand Russell, claiming she was a solipsist and wondering why there weren't more people like her), but it has fascinated philosophers as a doctrine that is very difficult to refute. How can we convince a solipsist that we, or the objects around him, exist? What proof can we give him?
Wittgenstein's discussion of solipsism, strongly influenced by Schopenhauer, is one of the most difficult sections of the book, and there are a great number of differences in interpretation. On one hand, Wittgenstein feels that there is some kind of truth in the solipsist's position, but that the solipsist misfires in his attempt to express that position in language (5.62).
The truth of solipsism rests on the fact that my only knowledge of the world comes from my own consciousness of it. I only know of the existence of objects and other people because I am conscious of them. The solipsist would make the argument, "as far as I'm concerned, these things and people exist only as objects of my consciousness." The problem comes with the conclusion that only I exist. What is this "I" that I am referring to? Wittgenstein picks up a theme first introduced by Hume: that I cannot find my own consciousness anywhere in my conscious experience. The solipsist runs into trouble when faced with the question of what is this "I" that is the only thing that exists.
What this amounts to in the language of the Tractatus is that there are no objects or elementary propositions that correspond to this "I": there are no propositions with sense, true or false, relating to it. The "I" is not a part of the world. Rather, this "I" is the limit of the world, much as the eye is the limit of the visual field. The metaphysical subject is effectively the same thing as the world, logic, and language: it is all there is. This is the truth the solipsist wants to express, but cannot anymore than we can make general statements about the nature of the world, logic, or language.
At this point, however, there is no real distinction between solipsism and pure realism, the doctrine that there are in fact objects and people in the world in just the way that common intuition tells us there is. The solipsist's idea of a self is not something that can be expressed, nor does it rule out any factual statements that can be made about the world. The solipsist and the realist might think that they are disagreeing, but any disagreement they can state will be in the form of pseudo-propositions that try to make unsayable claims about the nature of the self or the world. Wittgenstein is not trying to show that the solipsist is wrong so much as he is trying to show that the distinction between solipsism and realism is an artificial one: to the extent that either position can be stated without nonsense, they are the same.
Wittgenstein defines the metaphysical subject as the "philosophical self" and distinguishes it from the human body and the soul as treated by psychology (5.641). This psychological soul is what he deals with in his earlier discussion of propositions like "A thinks that p." He is reacting here primarily against the claim that the subject A is held in a relationship with a proposition p. According to Wittgenstein, no such unified "self" exists such that it can hold the place of an object in a proposition. Rather, the soul is a composite made up of the numerous different thoughts, beliefs, and attitudes that it entertains. Thus, when we talk about a person's belief, we should not analyze this proposition as existing between the belief and a unified consciousness. Rather, we should analyze it as existing between the belief as it is expressed and the belief as it appears in this composite consciousness. Effectively, Wittgenstein is denying that there is a self that is somehow distinct from, and more essential than, the thoughts, ideas, and beliefs that constitute it.