The Tractatus consists of a series of terse propositions numbered in a decimal form from 1 to 7. It divides roughly into three parts: propositions 1 to 2.063 deal with the nature of the world; 2.1 to 4.128 deal with the nature of language; and 4.2 to 7 deal with the nature of logic and its implications for mathematics, science, philosophy, and the meaning of life.
Proposition 1.1 announces, “The world is the totality of facts and not things.” A complete description of the world is not a list of all the objects in the world but a list of all the facts that are true of the world. In other words, facts are metaphysically prior to objects: an object only has being insofar as it is a constituent of a fact. Facts can be logically analyzed into constituent parts. Fundamental, atomic facts that cannot be further analyzed are called states of affairs, and they are all logically independent of one another. Any given state of affairs can be true or false regardless of the truth or falsity of any other state of affairs. Objects link together to form facts by virtue of their logical form, much as pieces of a jigsaw puzzle link together by virtue of their shape.
Language depicts reality by virtue of sharing a logical form in common with reality. We know that a picture of a sunset represents a sunset because both the picture and the sunset share a similar “pictorial form.” Similarly, a proposition and what it represents share a similar “logical form”: a proposition depicts a fact, and just as a fact can be analyzed into independent states of affairs, a proposition can be analyzed into independent elementary propositions.
Wittgenstein draws an important distinction between saying and showing: while a proposition says that such-and-such fact is the case, it shows the logical form by virtue of which this fact is the case. The upshot of this distinction is that we can only say things about facts in the world; logical form cannot be spoken about, only shown. Because logical form shows itself and cannot be spoken about, there is no need for the so-called logical objects, the connecting glue between different propositions that plays a central role in the logic of Frege and Russell. Wittgenstein asserts that most philosophical confusion arises from trying to speak about things that can only be shown.
At proposition 4.31, Wittgenstein introduces his method of truth tables, which show how logical form makes itself apparent without the need for logical relations or objects. One consequence of this view is that all the propositions of logic are tautologies—they are the set of propositions that are true no matter what. As such, they tell us nothing about the world, and they are all equivalent.
The foregoing reflections on the nature of the world, language, and logic lead Wittgenstein to address a series of long-standing philosophical problems. He suggests that solipsism, the belief that we have no knowledge of a world outside of our own minds, is technically valid but that there is no distinction between solipsism and realism that can properly be expressed in language. He claims that mathematics can be derived from the successive application of logical operations and that the laws of science are neither logical laws nor empirical observations but rather an interpretive method. Because language can speak about only facts in the world, we can say nothing about the world as a whole (metaphysics) or about the value of things in the world (ethics and aesthetics).