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.