If I know the meaning of an English and a synonymous German word, it is impossible for me not to know that they are synonymous, it is impossible for me not to be able to translate them into one another.
Expressions like “a=a”, or expressions deduced from these are neither elementary propositions nor otherwise significant signs. (This will be shown later.)
4.25
If the elementary proposition is true, the atomic fact exists; if it is false the atomic fact does not exist.
4.26
The specification of all true elementary propositions describes the world completely. The world is completely described by the specification of all elementary propositions plus the specification, which of them are true and which false.
4.27
With regard to the existence of n atomic facts there are Kn=∑ν=0n(nν) possibilities.
It is possible for all combinations of atomic facts to exist, and the others not to exist.
4.28
To these combinations correspond the same number of possibilities of the truth—and falsehood—of n elementary propositions.
4.3
The truth-possibilities of the elementary propositions mean the possibilities of the existence and nonexistence of the atomic facts.
4.31
The truth-possibilities can be presented by schemata of the following kind (“T” means “true,” “F” “false.” The rows of T’s and F’s under the row of the elementary propositions mean their truth-possibilities in an easily intelligible symbolism).
| p | q | r |
| T | T | T |
| F | T | T |
| T | F | T |
| T | T | F |
| F | F | T |
| F | T | F |
| T | F | F |
| F | F | F |
| p | q |
| T | T |
| F | T |
| T | F |
| T | F |
| F | F |
| p |
| T |
| F |
4.4
A proposition is the expression of agreement and disagreement with the truth-possibilities of the elementary propositions.
4.41
The truth-possibilities of the elementary propositions are the conditions of the truth and falsehood of the propositions.
4.411
It seems probable even at first sight that the introduction of the elementary propositions is fundamental for the comprehension of the other kinds of propositions. Indeed the comprehension of the general propositions depends palpably on that of the elementary propositions.
4.42
With regard to the agreement and disagreement of a proposition with the truth-possibilities of n elementary propositions there are ∑K=0Kn(KnK)=Ln possibilities.
4.43
Agreement with the truth-possibilities can be expressed by coordinating with them in the schema the mark “T” (true).
Absence of this mark means disagreement.
4.431
The expression of the agreement and disagreement with the truth-possibilities of the elementary propositions expresses the truth-conditions of the proposition.
The proposition is the expression of its truth-conditions.
(Frege has therefore quite rightly put them at the beginning, as explaining the signs of his logical symbolism. Only Frege’s explanation of the truth-concept is false: if “the true” and “the false” were real objects and the arguments in ~p, etc., then the sense of ~p would by no means be determined by Frege’s determination.)
4.44
The sign which arises from the coordination of that mark “T” with the truth-possibilities is a propositional sign.
4.441
It is clear that to the complex of the signs “F” and “T” no object (or complex of objects) corresponds; any more than to horizontal and vertical lines or to brackets. There are no “logical objects.”
Something analogous holds of course for all signs, which express the same as the schemata of “T” and “F”.
4.442
Thus e.g.
| p | q | |
| T | T | T |
| F | T | T |
| T | F | |
| F | F | T |
is a propositional sign.
(Frege’s assertion sign “⊢” is logically altogether meaningless; in Frege (and Russell) it only shows that these authors hold as true the propositions marked in this way. “⊢” belongs therefore to the propositions no more than does the number of the proposition. A proposition cannot possibly assert of itself that it is true.)
If the sequence of the truth-possibilities in the schema is once for all determined by a rule of combination, then the last column is by itself an expression of the truth-conditions. If we write this column as a row the propositional sign becomes: “(TT—T)(p,q),” or more plainly, “(TTFT)(p,q)”.
(The number of places in the left-hand bracket is determined by the number of terms in the right-hand bracket.)
4.45
For n elementary propositions there are Ln possible groups of truth-conditions.
The groups of truth-conditions which belong to the truth-possibilities of a number of elementary propositions can be ordered in a series.
4.46
Among the possible groups of truth-conditions there are two extreme cases.
In the one case the proposition is true for all the truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological.
In the second case the proposition is false for all the truth-possibilities. The truth-conditions are self-contradictory.
In the first case we call the proposition a tautology, in the second case a contradiction.
4.461
The proposition shows what it says, the tautology and the contradiction that they say nothing.
The tautology has no truth-conditions, for it is unconditionally true; and the contradiction is on no condition true.
Tautology and contradiction are without sense.
(Like the point from which two arrows go out in opposite directions.)
(I know, e.g. nothing about the weather, when I know that it rains or does not rain.)
4.4611
Tautology and contradiction are, however, not nonsensical; they are part of the symbolism, in the same way that “0” is part of the symbolism of Arithmetic.
4.462
Tautology and contradiction are not pictures of the reality. They present no possible state of affairs. For the one allows every possible state of affairs, the other none.
In the tautology the conditions of agreement with the world—the presenting relations—cancel one another, so that it stands in no presenting relation to reality.
4.463
The truth-conditions determine the range, which is left to the facts by the proposition.
(The proposition, the picture, the model, are in a negative sense like a solid body, which restricts the free movement of another: in a positive sense, like the space limited by solid substance, in which a body may be placed.)
Tautology leaves to reality the whole infinite logical space; contradiction fills the whole
