"We picture facts to ourselves." (2.1)
This quotation introduces Wittgenstein's picture theory of propositions. According to this view, propositions can represent reality by making a logical picture of the facts they represent. If something is to be considered a picture of something else, the picture and the thing depicted must share something, a form, in common. According to Wittgenstein, propositions and the world share a common logical form. Accordingly, a proposition has the same logical form as the fact it represents. Though the words in a proposition do not in any way resemble facts in the world, their common logical form makes it possible for us to recognize one as a picture of the other.
"My fundamental idea is that the 'logical constants' are not representatives; that there can be no representatives of the logic of facts." (4.0312)
"Logical constants" are the objects used in the logical schematization of propositions to tie the elements of those propositions together. There are truth-functional constants like "and" or "not" as well as constants to express generality ("for every " and "there exists a ") and constants to express sets and membership of sets. Wittgenstein argues that these constants do not signify anything (e.g. there is no such thing as negation, represented by the symbol "~"), and shows that they are unnecessary to logical schematization. This argument contradicts Frege and Russell, who introduced logical constants as essential parts of their axiomatic systems. This idea is significant because it reflects Wittgenstein's view of logic as being structural and without content. There are no logical propositions, and there is no body of objects, ideas, or thoughts that we can define as "logical."
"Philosophy aims at the logical clarification of thoughts. Philosophy is not a body of doctrine but an activity." (4.112)
There is no subject matter for philosophy, so there are no propositions that can be called "philosophical." As a result, any propositions that purport to be philosophical are nonsense. Rather than conceive of philosophy as a body of knowledge like any other, Wittgenstein suggests that we conceive of philosophy as an activity directed toward all other bodies of knowledge. The role of philosophy is not to state truths about the universe, but to clarify the truths stated by the other sciences. Language can often confuse meaning, and it is the business of philosophy to clarify the logical structure of language.
"Logic must look after itself." (5.473)
Frege and Russell both conceived of logic as being a fundamental set of laws that then rested on self-evident axioms. Wittgenstein found this conception distasteful for a number of reasons, not least of which was the fact the axioms themselves were given no justification. According to Wittgenstein, logic should not stand in need of justification, nor should it need laws to say what can and cannot be the case. Logic defines the boundaries of sense: any proposition that has sense has a logical form, and any proposition that is nonsense lacks logical form. Thus, we do not need laws, axioms, or anything else to tell us what is and is not allowed in logic.
"My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them—as steps—to climb up beyond them. (He must, so to speak, throw away the ladder after he has climbed up it.)" (6.54)
A central theme of the Tractatus is that we cannot talk about the nature of philosophy, the world, metaphysics, or anything "transcendental" without descending into nonsense. However, the propositions of the Tractatus deal with precisely these sorts of things, and on reflection, we realize that Wittgenstein could not have proscribed this kind of talk without descending into it himself. On the face of it, it would seem that his work is self-defeating. We must be careful to understand the purpose of the Tractatus, however: Wittgenstein is not trying to tell us a number of things that we did not already know; he is trying to instruct us in a way of thinking that will help us out of philosophical muddles. While the propositions of the Tractatus may themselves be nonsense, Wittgenstein hopes that they have served their instructive purpose. We are expected to put down this book not with a knowledge that the world is made up of objects and states of affairs and that propositions depict facts, but with an understanding of why it is impossible to say these sorts of things. The goal of the Tractatus, as Wittgenstein claims in his preface, is "to draw a limit to the expression of thoughts."
just to add another insight to a fine summary
have a look at