The totality of all propositions is the sum total of what we are capable of expressing in language (4.001). Everyday language, however, does not break down neatly into distinct propositions. It has a logical structure, but this structure is disguised by the complicated conventions of ordinary speech, so much so that the logical structure beneath ordinary speech is difficult to ascertain. (4.002). Most of the problems of philosophy result from a misunderstanding of the logic of language. We can only answer such philosophical questions by pointing out that they are nonsensical (4.003).
A proposition is a picture of reality in the same way that notes in a score form a picture of a piece of music. And just as there is a general rule for translating notes on a page into music, there is also a general rule for translating written propositions into pictures of reality (4.0141). It is because of this correspondence between the form of a proposition and the form of reality that we can understand the sense of a proposition from the proposition itself: we don't need a further proposition to explain what one proposition means. "A proposition shows its sense. A proposition shows how things stand if it is true. And it says that they do so stand" (4.022). This is like saying a picture of a person shows what the person looks like. We don't need further explanations to tell us how we are supposed to correspond the marks on a piece of paper to a human face.
The logical connections between objects are not represented in a proposition. Propositions depict facts, not the logical structure of these facts, as expressed in connectives like "and" or "not," as well as more general logical concepts like those of class or relation. "My fundamental idea," Wittgenstein claims, "is that the 'logical constants' are not representatives; that there can be no representatives of the logic of facts" (4.0312).
Wittgenstein emphasizes that the truth-value of a proposition has no bearing on its sense. True or false, it still makes a picture of the world, and we can still draw logical inferences from that picture. The propositions p and ~p depict the same possible situation, only they have opposite sense (4.0621): one says that the picture presented is the case, and the other says that it is not the case.
"The totality of true propositions is the whole of natural science" (4.11). In this way, Wittgenstein sets philosophy apart as distinct from natural science. Natural science describes the world, whereas philosophy's aim is the logical clarification of thoughts (4.112). Philosophy itself is not a body of propositions; rather, it is the activity of clarifying the propositions of natural science. In clarifying the propositions of natural science, philosophy will not only render clear what can be said, but also show what cannot be said (4.114 and 4.115). Because it treats all the propositions of natural science equally, no branch of science (such as psychology or evolutionary theory) is more closely related to philosophy than any other branch (4.1121 and 4.1122).
The sense of a proposition is internal to the proposition, while the meaning of a name is external to the name. The meaning of a name is the object it denotes, and there is nothing in the name itself (as a written or spoken sign) that can tell us what object it denotes. Rather, we learn the meaning of a name by observing how and in reference to what it is used. The meaning of a name lies outside the name, and this meaning must be made clear by means of elucidations.
The point Wittgenstein makes with his picture theory of propositions is that the sense of a proposition does not need to be made clear by means of elucidations. A proposition and the reality that it depicts share a logical form, and that is enough for the one to depict the other. There is nothing external to the proposition that can make the connection between proposition and what it depicts any clearer than it already is. Wittgenstein likens this connection to the one that holds between sheet music and a symphony: if you can read music, nothing else is necessary to help you (and indeed, nothing else can help you) in translating written notes into sounds.
In saying that the sense of a proposition is internal to it, Wittgenstein is simply saying that a proposition like "the tree is in the garden" stands in no need of further clarification. Reflect on the absurdity of replying to that proposition with, "I understand English perfectly, but I need some further clarification as to how I'm supposed to connect the words you just uttered to the objects you're pointing at." If you understand English perfectly, the sense of the proposition should be clear from the words alone.
A concern that subtly underlies this discussion, and which will become more evident in Wittgenstein's later philosophy, is the concern with rule following. Suppose there were certain rules established to instruct us on how to understand the sense of a proposition. How, then, are we to know how to interpret these rules? Is there a further set of rules telling us how to interpret this first set of rules? And if there isn't, what is it about this set of rules that makes them so unambiguously clear? Wittgenstein's answer to this question in the Tractatus (his later answer in the Philosophical Investigations is far deeper and more complicated) is that elucidations (rules, interpretations) cease to be necessary when there is a common logical form between two things.
This point stands as further support for Wittgenstein's fundamental distinction between saying and showing. We do not need to (and cannot) say what the sense of a proposition is because this sense shows itself by means of sharing a logical form with the reality it depicts. This commonality of logical form stands in place of speech, it expresses what cannot be said, and it is only because of this commonality that speech can be understood. Logical form itself cannot be spoken about. We cannot (as Frege and Russell had done) talk about logical inference or logical relations. The workings of logic are shown in the way the world is held together, and we can say nothing that will make these workings clearer.
Wittgenstein calls his conclusion that logical constants cannot be represented in propositions his "fundamental idea" (4.0132). This idea underlies the distinction between saying and showing, and the significance of this distinction will become increasingly clear as we progress through this book. It is worth noting also that Wittgenstein's "fundamental idea" relates to logic, not to language or the world. Though his discussion of logic comes later in the Tractatus than his discussion of language or the world, it was logical concerns that first motivated his construction of the book.
One of the upshots of the distinction between saying and showing is that it places a limit on what can be said. Specifically, Wittgenstein limits propositions to making claims, true or false, about how things stand in the world, which is the business of natural science. To think of philosophy as made up of propositions is a common error that Wittgenstein suggests is the source of a great deal of philosophical confusion (4.003). In referring to philosophy as an "activity" (4.112), Wittgenstein suggests that the business of philosophy is not of saying, but of showing: philosophy clarifies the logical structure of our propositions that is clouded by everyday language.