Tautology and Contradiction.

We can use a mathematical function to calculate that n elemental propositions produce L(n) groups of truth values. These can be ordered in a row (or added as a column to the truth table).

Among all the possible groups of truth values there are two extreme cases. The first case agrees with all combinations of truth values and we call the proposition a tautology. The second case agrees with none and we call it a contradiction.

A proposition shows what it has to say; tautologies and contradictions show that they have nothing to say. Because a tautology is unconditionally true, it has no truth-conditions; and a contradiction has none because it is never true.

Tautologies and contradictions have no sense; just as a point from which arrows go out in two directions. (For example, I know nothing about the weather when I know that it is either raining or not raining.) Tautologies and contradictions are not, however, absurd. They belong to symbolism; much as zero to the symbolism of arithmetic.

Tautologies and contradictions are not images of reality. They do not represent possible situations, for the former admit all situations, and latter none. In a tautology the conditions of agreement with the world - the embodying relations - cancel one another, so that it does not express reality.

The truth-conditions of a proposition determine the range of the facts. (Interpreted in the negative sense, a proposition, a picture, or a model is like a solid body that restricts the freedom of movement of others. Interpreted in the positive sense, it is like a space bounded by solid substance in which there is room for a body.) A tautology leaves the infinite whole of logical space open to reality. A contradiction fills it, leaving no point of it for reality. Thus neither of them can determine reality in any way.

A tautology is certainly true, a proposition possibly, and a contradiction certainly not. (We have the scale that we need in the theory of probability.) The logical product of a tautology and a proposition says the same thing as the proposition and is therefore identical with the proposition because one cannot change the essence of a symbol without changing its sense.

A particular logical combination of signs corresponds to a particular logical combination of their meanings. Absolutely any combination corresponds to uncombined signs, but only to them. In other words, propositions that are true for every situation cannot be sign combinations at all, for if they were, only particular combinations of objects could correspond to them. (And what is not a logical combination has no combination of objects corresponding to it.)

Tautology and contradiction are the limiting cases of sign combinations - their dissolution. Admittedly the signs are still combined with one another even in tautologies and contradictions. That is, they relate to one another. But these relations have no meaning, they are not essential to the symbol.