Before outlining the philosophical project of the Tractatus a cursory note about the style and structure of the text should be made. It consists of short supposedly self-evident aphorisms in the form 7 general statements as well as many supplementary sentences that explain or reveal the deeper meaning of the more general statement above, e.g. 7.1 is taken to be an explanatory proposition of 7, 7.1.1 supplements 7.1, and so on. There are no arguments per se in the text. This does not mean that the propositions are unreasoned, but the responsibility lies with the student in teasing the arguments out of Wittgenstein's subtle pointers. It is of necessity to point out that the Tractatus is not a work accessible to laymen or beginners: one does need some understanding of contemporary formal logic as well as the logical atomism of Russell and to a lesser extent the philosophy of Immanuel Kant.
Wittgenstein's aim with the Tractatus was to demystify philosophy through the conceptual clarification. Wittgenstein did not believe philosophical problems existed in the traditional sense, but resulted from confusions caused by a fundamental misuse and misunderstanding about the form and meaning of language. In this sense, the Tractatus, according to Wittgenstein, was to put an end to philosophizing in the ordinary sense and instead see philosophy as a process of clarification of fundamental concepts that would aid those subjects that seek genuine answers in terms of facts, e.g. the sciences.
Central to this new definition of philosophy is the picture theory, for which the Tractatus is most famed. In the preface Wittgenstein alludes to the Kantian principle that the limits of language (thought) are the limits of the world. This link between language and reality lies at the heart of the metaphysics of the early Wittgenstein's philosophy. The picture theory borrows heavily from Russellian referentialism and can be briefly stated as the view that language represents reality. The first statement of the Tractatus is that the world is everything that is the case - or the totality of facts as the supplementary statements explain. Facts can be broken down into constituent parts (objects), which can be further broken down. Facts are expressed in language by propositions, which too are divisible. In the Tractatus Wittgenstein adopted the logical atomism of Russell and claimed that the chain of divisibility must end somewhere - he postulates the existence of simple ideas underpinning reality. Furthermore, it is by denoting real world objects that words gain meaning. So the words `chair', `table', `computer' all have meaning only because they denote objects in the world. So if words have meaning because they denote objects, propositions too only have meaning if they denote possible states of affairs (which are constituted of actual objects in a possible configuration). The result is that only propositions that can be subjected to tests of truth and falsehood have any meaning (or sense), everything else can be cast aside as nonsense. Wittgenstein's claim is that the sorts of propositions philosophers have used throughout history in the fields of metaphysics, epistemology and ethics & religion are nonsense since they use words and concepts such as `God', `Justice', `Knowledge' that do not denote objects in reality. They do fit within the boundaries of our language, according to Wittgenstein's model, therefore cannot form part of our meaningful reality and are nonsense.
The rest of the Tractatus can be separated into two parts. The first consists of the statements between 4 and 6.241 and outlines Wittgenstein's theory of an ideal language of logic. His analysis is done mostly through the use of truth-tables (a standard model of semantics in modern sententional logic). There is also a theory on the essential form of sentences, as well Wittgenstein expressing the view that mathematical and logical sentences are tautological and transcendental and therefore are themselves nonsense (and also re-iterating the Russellian thesis that all mathematic propositions are derivative of logical propositions). The purely logical language reflects the world and can only derive meaning from this and is in possession of none in itself (My lack of detail here does not reflect the opinion that I consider these parts of the Tractatus of lesser importance, but is due to my inability to render the technical features lucidly and with the justice they deserve).
The second part is of particularly interesting in light of what has gone before it and has sparked much intrigue and debate since (see the New Wittgensteinians). Beginning at around 6.3 it consists of statements that are taken to confront religion, ethics and the mystical. There is also to be found the claim by Wittgenstein that what one has just read (the Tractatus) is itself nonsense, expressed mischievously by the metaphor of the ladder (6.54). I leave the intrigued student to seek and figure this out for themselves.
The Tractatus is an incredible work in its scope considering its short length; its conclusions are profound, groundbreaking and contain serious implications for philosophers. However, as with any work of philosophy there are flaws. Most of these were ruthlessly exposed and criticised by Wittgenstein himself in his posthumously published Philosophical Investigations. The picture theory is now taken as an ultimately incorrect account of the relationship between language and reality - it is too narrow and the `use theory' of the mature Wittgenstein appears more versatile and far more robust to challenges. There is the work itself: Wittgenstein aimed to redefine philosophy as a therapeutic process of dispelling the myths of metaphysics that were caused by the muddyness of language; he rejected metaphysical doctrines as nonsense, yet a large chunk of the Tractatus is devoted to a positive metaphysical theory.
Nevertheless, the Tractatus constitutes a very noble attempt at establishing its theories in their own right, even if it fails to deliver on its promise to reduce philosophy to a process of conceptual clarification in aid of the sciences. Its influence remains strong and there are many scholars who consider the Tractatus the pinnacle of Wittgenstein's philosophical output.