The opening pages of the Tractatus (sections 1–2.063) deal with ontology—what the world is fundamentally made up of. The basic building blocks of reality are simple objects combined to form states of affairs. Any possible state of affairs can either be the case or not be the case, independent of all other states of affairs. The world is the totality of all states of affairs that are the case. States of affairs can be combined together to form complex facts.
States of affairs are combinations of objects. Objects are utterly simple and unanalyzable, and they can exist only in the context of states of affairs. They have a logical form that determines the ways in which they can be combined into states of affairs, and they fit into these states of affairs "like links in a chain" (2.03). That is, they fit together by virtue of their logical form alone, and do not need something extra (like a relational object) to hold them together.
From 2.1 to 4.128, the discussion deals with the question of how language works so that it can describe the world accurately. According to Wittgenstein, language consists of propositions that are complexes built from simple, elementary propositions. Elementary propositions are unanalyzable and consist solely of names. Language mirrors reality by sharing its logical form. Thus, names mirror objects, elementary propositions mirror as states of affairs, and propositions mirror facts. The totality of true propositions is the totality of language just as the totality of facts is the world. A proposition is a logical picture of reality: the elements of a proposition are arranged in such a way that they resemble the reality they represent, just as the elements of a portrait are arranged in such a way that they resemble the person they represent.
Signs are given meaning through their use in propositions, so it follows that if a sign is used in two different ways we are actually dealing with two different signs. For instance, the "is" in "John is tall" is different from the "is" in "John is the captain of the guard."
While a picture can represent a fact by means of sharing its logical form, this logical form itself cannot be depicted. We cannot say what the logical form of a proposition or fact is, but this form shows itself in the way the proposition or fact is held together. Similarly, the logical connections between states of affairs and between elementary propositions show themselves, so that there is no need for logical objects (like "and" and "not") to hold them together. Wittgenstein calls the observation that logical objects do not represent anything his "fundamental idea" (4.0312).
Most of the problems of philosophy arise when people try to talk about things that can only be shown, such as the logical structure of the world or language. Wittgenstein distinguishes between formal concepts (e.g. "x is a number"), which cannot be spoken about, and concepts proper (e.g. "x is a horse"), which are the legitimate constituents of propositions. Philosophy, unlike science, is not a body of propositions. It should be thought of as the activity of clarifying the often obscure logical structure of language and thought.
Starting at 4.2, Wittgenstein discusses logic. At 4.31, he introduces truth tables, a notation that makes clear that we can represent propositions and their truth-conditions without making use of logical connectives. There are three kinds of propositions: tautologies, which are always true, contradictions, which are always false, and propositions with a sense, which can be true or false depending on what is or is not the case in the world. One proposition follows from another if that proposition is true whenever the other proposition is true. We do not need laws of inference to tell us what follows from what, as this is clear from the structure of the propositions themselves. Wittgenstein also shows how logical form can explain probability.
We can generate new propositions out of old ones by means of operations. The successive application of an operation produces a series of new propositions. Given the elementary propositions, we can generate all other propositions by successive application of the operation that negates all the propositions it is applied to.
The propositions of logic are all tautologies, and so are all equivalent. We do not need axioms or laws of inference to tell us how to proceed in logic, since this should make itself manifest. "Logic must look after itself" (5.473): we should not need external laws to tell us how proceed with logic since there is nothing external to logic. Wittgenstein also shows how signs for generality and identity are unnecessary to logic.
Propositions of the form "A believes that p" do not relate a proposition, p, to a person, A. Rather, they relate p to the verbal expression of p, so that what we are really saying is "'p' says that p."
That both language and the world share the same limits leads to the reflection that solipsism is correct in the claim that "the world is my world" (5.62). However, the thesis of solipsism cannot be put into language, but can only show itself. With regard to everything that can be said, there is no difference between solipsism and pure realism, so Wittgenstein suggests the distinction between the two is an artificial development of muddled philosophy.
Mathematics is a logical method derived from the repeated application of operations. The number 2, for instance, is the exponent given to an operation that is applied twice. Thus, the propositions of mathematics do not say anything about the world, but only reflect the method in which propositions are constructed.
The laws of science are not logical laws, nor are they empirical observations. Rather, they constitute an interpretive method, by means of which we can more accurately describe reality. Science is ultimately descriptive, not explanatory.
There is no perspective external to the world from which we can talk about the world or its contents generally. Thus, statements of value (as we find in ethics or aesthetics) are nonsense, since they evaluate the world as a whole. The feeling of life as a limited whole is what Wittgenstein calls "the mystical."
The only correct method in philosophy is to remain silent about philosophical questions, and to point out to anyone who tries to talk philosophy that he or she is talking nonsense. The propositions of the Tractatus themselves make general statements about the nature of the world, and so they too are nonsense. They should serve only as a ladder to be climbed and then discarded. "What we cannot speak about we must pass over in silence" (7).