Wittgenstein's Tractatus is a series of short sections that I will refer to as "remarks." The key remarks for our purposes are numbered n, n.m, or n.0m; these numbers are boxed in the handout. It has been suggested for remarks with such numbers that n.1, n.2, etc. lead up to n+1 while n.01, n.02, etc. fall under n as further elaborations. It is okay to limit your reading to the remarks with boxed numbers but other remarks are included to provide some context and enable you to pursue ideas a little further if you are interested. Some of the further remarks provide especially good examples of some of the ideas in the work; I'll refer you to these below.
The text can be divided roughly into the remarks before 4.1 and those from 4.1 on.
• For our purposes, the key idea in first group is what is known as the "picture theory of meaning" (see, for example, 2.15) as an alternative to the account of meaning implicit in Russell's ch. 12. Wittgenstein's "objects" play a role analogous to Russell's "constituents" with the important difference that an object brings with it all the possibilities of combining it with other objects to form a state of affairs. (Recall that Russell says, on the contrary, that we can be acquainted with a constituent without knowing anything about its relations.) This array of possible combinations of objects is one of the things Wittgenstein has in mind when he speaks of "logical space."
• From 4.1 on, Wittgenstein can be understood to develop an alternative to Russell's platonistic account of a priori knowledge. Remarks 4.46, 6.1, and 6.3 present the basic statement of this, but Wittgenstein's characterization of Newtonian mechanics as a conventional framework for describing nature (6.341 and 6.35) provides a more concrete way into the idea. Another way of making the point that a priori truths have no content is Wittgenstein's idea that they say nothing although they may show the structure of a language (see, for example, 6.12 and 6.13). Notice also the conception of philosophy sketched in the last few remarks and think how it is related to this conception of the a priori.
A guide to Wittgenstein's notation in the Tractatus
• Wittgenstein's notation for logical operations:
• negation ('not') is expressed by the tilde (~)
• conjunction ('and') is expressed by a period
• disjunction ('or') is expressed by the letter v
• the conditional ('if ... then') is expressed by the right pointing horseshoe (⊃)
• the existential quantifier ('some' or 'there exists') is expressed by an inverted E (∃)
• The period is also used as symbolic punctuation whose function is roughly to mark the beginning of the parenthesized group (so the first period in 5.47 it indicates that the existential quantifier is applied to the full conjunction 'fx.x=a' rather than 'fx' alone).
• A right single quote is used to mark the application of an operation (something Wittgenstein distinguishes from a function--in these terms, addition and multiplication are operations but physical quantities varying with time or position would be examples of functions).
• Notation of the form '[a, x, O'x]' indicates the collection of things that may be formed by beginning with a and repeatedly applying the operation O--i.e., moving from x to O'x. Thus '[0, x, x+1]' would indicate the accumulated results of beginning with 0 and repeatedly adding 1.
• Wittgenstein uses a bar over a variable is used to indicate the set of all values for a variable. This notation is analogous to a common notation for vectors except that it stands for a set rather than an ordered series of components.
• The summations in 4.27 and 4.42 use notation for the number of sets of a given size that can be drawn from a set of a given size (the number of "combinations of x things taken y at a time") to write an expression for the number of ways of dividing a set into two parts. In 4.27 this gives the number of ways a collection of n states of affairs can be divided into those which exist and those which do not exist (or, equivalently, the number of "truth-possibilities" for n elementary propositions), and in 4.42, it gives the number of pattern of agreement and disagreement between a given proposition and the truth-possibilities of a group of n elementary propositions. (Simpler expressions can be given, 2n--i.e., 2 to the nth power--for the former and 22n for the latter. For example, 23 (= 8) and 28 (= 256), respectively, for 3 elementary propositions.)