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.)
The sign which arises from the coordination of that mark “T” with the truth-possibilities is a propositional sign.
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”.
Thus e.g.
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.)
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.
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.