Identity in Tractatus

In Tractatus, identity of object Wittgenstein expresses by identity of sign, and not by using a sign for identity (5.53). He speaks roughly that (5.5303):

To say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all.

Then he also says at 5.535 that Russell's "Axiom of Infinity", which says that there are infinitely many objects (at 4.1272, Wittgenstein says one even cannot say so), would express itself in language through the existence of infinitely many names with different meanings. In fact, as Russell says in his introduction to Tractatus, the rejection of identity removes one method of speaking of the totality of things.

To Wittgenstein, tautologies also say nothing at all (4.461), although they are not nonsensical (4.46211). Of course, everyone absolutely agrees that tautologies like "if it is raining, then it is raining" really say nothing at all. However, the "problem" of human beings (or of the world) is that the world is so complicated that we cannot determine many tautologies at first sight.

Identity is a relation in Mathematics. There are occasions at which one object may have more than one name / sign - signs are used to refer to the descriptions of some objects, and eventually it is found (proved) that these descriptions have the same reference. In the real world, we might call the murderer of some case of murder X, and eventually we found that he was the man called A: A = X. The introduction of the identity-sign simplifies our deduction.

Negative & Positive Propositions in Tractatus

As Wittgenstein says at the very beginning of Tractatus that the world is all that is the case. What is the case is a fact. For the totality of facts determines what is the case, and also whatever is not the case (1.12) - if we have the collection of all facts, then we can tell what is not in this collection. Maybe we can call what is not the case a negative fact. A negative fact is just a fact that tells you something that is not the case.

If p is the sign of a proposition that asserts something is the case, then the sign of the corresponding negative proposition can be constructed (by truth-operations) as ~p. However (5.5151), it is also possible to express the negative proposition by means of a negative fact: p is not the case. Functions f(x) like "x is the case", "x is not the case", "x is true", and "x is false" are not truth-functions (in the sense of Tractatus, e.g., 5). But really even in this case the negative proposition is constructed by an indirect use of the positive (5.5151).

Consider a simple example with the notation of Mathematics: 0 > 1. We can simply say that "0 > 1" is not the case. In fact, this negative proposition can also be expressed by 0 ≦ 1. We could define first the sign > for all real numbers, and then just define for any real numbers that a ≦ b whenever "a > b" is not the case if we want such a sign ≦ for the sake of simplicity. Such a sign ≦ is in fact not necessary. Nevertheless, it is clear that "a > b" presupposes the existence of "a ≦ b" and vice versa - if we know when "a" stands to "b" in a certain relation, then we can tell when "a" does not stand to "b" in that relation.

As Wittgenstein concludes at 5.5151, the positive proposition necessarily presupposes the existence of the negative proposition and vice versa.