| p | q | p→q |
| T | T | T |
| F | T | T |
| T | F | F |
| F | F | T |
It is in fact the proposition p→q. Here the order or the truth-possibilities (of p and q) in a scheme is fixed once and for all by a combinatory rule (4.442). Wittgenstein does not state the order explicitly, while the sequences of the truth-possibilities of no more than three propositions (terms of the truth-functions) are shown at 4.31. The order is not stated explicitly because it is really not important in the discussion in Tractatus, except that the truth-possibilities are all F in the last row in the truth table. Thus, (---T)(p,q) is the same as p↓q, where ↓ is the Peirce arrow, and has the following truth table:
| p | q | p↓q |
| T | T | F |
| F | T | F |
| T | F | F |
| F | F | T |
For propositions p, q, r, ..., a variable ξ having them as its values (the terms of the truth-function) is used, and the sign becomes (-...-T)(ξ, ...) at 5.5. At 5.501 and 5.502, the sign is further simplified to (-...-T)(`ξ) and N(`ξ) respectively, while the order of the terms (the propositions) is indifferent. The sign is called (5.5) the negation of the propositions. Thus (5.51), if ξ has only one value p, then N(`ξ) is ¬p (not p); if ξ has two values p and q, then N(`ξ) is ¬p∧¬q (neither p nor q). Furthermore (5.52), if ξ has as its values all the values of a function f(x) for all values of x, then N(`ξ) is ¬∃xf(x).
The importance of the Peirce arrow in logic (and therefore the importance of the operation N in Tractatus) is that any logical operation can be expressed in terms of it (completeness). For example,
- ¬p is equivalent to p↓p;
- p∧q is equivalent to (p↓p)↓(q↓q);
- p∨q is equivalent to (p↓q)↓(p↓q);
- p→q is equivalent to ((p↓q)↓q)↓((p↓q)↓q).
A truth-function can therefore be obtained by successive negation of the propositions. In Wittgenstein's notation, for example, if ξ has values p and q, then N( `ξ) is p↓q. Furthermore, if ξ has value p↓q, then N(`ξ) is p∨q. Any truth-function and any proposition can be obtained by successive application of such operations. Thereby for a series of successive application of an operation O' on a variable a
a, O'a, O'O'a, ...,
Wittgenstein introduces the sign for the general form at 5.2522:
[a, x, O'x]
where the first term of the bracketed expression is the beginning of the series of forms, the second is the form of a term x arbitrarily selected from the series, and the third is the form of the term that immediately follows x in the series. For example if x is the third term O'O'a of the series, then O'x is the fourth term O'O'O'a. Nevertheless, while each of these variables a and x only has one values, the situation of truth-functions or operations on propositions is a bit more complicated, at least in the sense of mathematical rigorousness. Let us consider an expression
[`a, `x, O'`x]
where both a and x have more than one value, and O'`x is a family of operations of different numbers of variables (for a fixed number n, there is a well-defined operation of n variables in this family). The words in Russell's introduction to Tractatus can be used for the explanation of this expression. The symbol means whatever can be obtained by taking any selection of values of a, taking the result of the operation of O' `x on them, then taking any selection of the set of values now obtained, together with any of the originals - and so on indefinitely. Obviously, we cannot interpret it simply with a series as in 5.2522. We cannot regard Wittgenstein's bracket as a symbol in Mathematics or Mathematical Logic.
Hence in Wittgenstein's notation, the general form of truth-function and of proposition (6 of Tractatus) is given by:
[`p, `ξ, N(`ξ)]
where p has all elementary propositions (or just the elementary propositions that appear in the expression) as its values, and ξ has some selection of propositions obtained in the previous operations (together with any of the original elementary propositions) as its values, and N( `ξ) is the negation of all the propositional values of ξ.
