4.27
With regard to the existence of n atomic facts there are 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 non-existence 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		p	q		p
T	T	T		T	T		T
F	T	T		F	T		F
T	F	T		T	F
T	T	F		F	F
F	F	T
F	T	F
T	F	F
F	F	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 general propositions.

4.42
With regard to the agreement and disagreement of a proposition with the truth-possibilities of n elementary propositions there are possibilities.

4.43
Agreement with the truth-possibilities can be expressed by co-ordinating with them in the scheme 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 co-ordination 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	F

'' 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 possible assert of itself that it is true.)

If the sequence of the truth-possibilities in the scheme 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: "(T T - T)(p, q)", or more plainly: "(T T F T)(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 L_n 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 modem, 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 logical space and leaves no point to reality. Neither of them, therefore, can in any way determine reality.

4.464
The truth of tautology is certain, of propositions possible, of contradiction impossible.

(Certain, possible, impossible: here we have an indication of that gradation which we need in the theory of probability.)

4.465
The logical product of a tautology and a proposition says the same as the proposition. Therefore that product is identical with the proposition. For the essence of the symbol cannot be altered without altering its sense.

4.666
To a definite logical combination of signs corresponds a definite logical combination of their meanings; every arbitrary combination only corresponds to the unconnected signs.

That is, propositions which are true for every stat of affairs cannot be combinations of signs at all, for otherwise there could only correspond to them definite combinations of objects.

(And to no logical combination corresponds no combination of the objects.)

Tautology and contradiction are the limiting cases of the combination of symbols, namely their dissolution.

4.4661
Of course the signs are also combined with one another in the tautology and contradiction, i.e. they stand in relations to one another, but these relations are meaningless, unessential to the symbol.

4.5
Now it appears to be possible to give the most general form of proposition; i.e. to give a description of the propositions of some one sign language, so that every possible sense can be expressed by a symbol, which falls under the description, and so that every symbol which falls under the description can express a sense, if the meanings of the names are chosen accordingly.

It is clear that in the description of the most general form of proposition only what is essential to it may be described -- otherwise it would not be the most general form.

That there is a general form is proved by the fact that there cannot be a proposition whose form could not have been foreseen (i.e. constructed).

The general form of proposition is: Such and such is the case.

4.51
Suppose all elementary propositions were given me: then we can simply ask: what propositions I can build out of them. And these are all propositions and so are they limited.

4.52
The propositions are everything which follows from the totality of all elementary propositions (of course also from the fact that it is the totality of them all). (So, in some sense, one could say, that all propositions are generalizations of the elementary propositions.)

4.53
The general proposition form is a variable.