5
Propositions are truth-functions of elementary propositions.
(An elementary proposition is a truth-function of itself.)
5.0
The elementary propositions are the truth-arguments of propositions.
5.02
It is natural to confuse the arguments of functions with the indices of names. For I recognize the meaning of the sign containing it from the argument just as much as from the index.
In Russell's "+c", for example, "c" is an index which indicates that the whole sign is the addition sign for cardinal numbers. But this way of symbolizing depends on arbitrary agreement, and could choose a simple sign instead of "+c": but in "~p" "p" is not an index but an argument; the sense of "~p" cannot be understood, unless the sense of "p" has previously been understood. (In the name Julius Caesar, Julius is an index. The index is always part of a description of the object to whose name we attach it, e.g. The Caesar of the Julian gens.)
The confusion of argument and index is, if I am not mistaken, at the root of Frege's theory of the meaning of propositions and functions. For Frege the propositions of logical were names and their arguments the indices of these names.
5.1
The truth-functions can be ordered in series.
That is the foundation of the theory of probability.
5.101
The truth-functions of every number of elementary propositions can be written in a scheme of the following kind:
| (T T T T)(p, q) | Tautology (if p then p, and if q then q) [p |
| (F T T T)(p, q) | in words: Not both p and q. [~(p . q)] |
| (T F T T)(p, q) | '' '' If q then p. [q |
| (T T F T)(p, q) | '' '' If p then q. [p |
| (T T T F)(p, q) | '' '' p or q. [p v q] |
| (F F T T )(p, q) | '' '' Not q. [~q] |
| (F T F T)(p, q) | '' '' Not p. [~p] |
| (F T T F)(p, q) | '' '' p or q, but not both. [p . ~q :v: q . ~p] |
| (T F F T)(p, q) | '' '' If p, then q; and if q, then p. [p |
| (T F T F)(p, q) | '' '' p |
| (T T F F)(p, q) | '' '' q |
| (F F F T)(p, q) | '' '' Neither p nor q. [p . ~q or p | q] |
| (F F T F)(p, q) | '' '' p and not q. [p . ~q] |
| (F T F F)(p, q) | '' '' q and not p. [q . ~p] |
| (T F F F)(p, q) | '' '' p and q. [p . q] |
| (F F F F)(p, q) | Contradiction (p and not p; and q and not q.) [p . ~p . q . ~q] |
p . q 
q]Those truth-possibilities of its truth-arguments, which verify the proposition, I shall call its truth-grounds.
5.11
If the truth-grounds which are common to a number of propositions are all also truth-grounds of some one proposition, we say that the truth of this proposition follows from the truth of those propositions.
In particular the truth of a proposition p follows from that of a proposition q, if all the truth-grounds of the second are truth-grounds of the first.
5.121
The truth-grounds of q are contained in those of p; p follows from q.
5.122
If p follows from q, the sense of "p" is contained in that of "q".
5.123
If a god creates a world in which certain propositions are true, he creates thereby also a world in which all propositions consequent on them are true. And similarly he could not create a world in which the proposition "p" is true without creating all its objects.
5.124
A proposition asserts every proposition which follows from it.
5.1241
"p . q" is one of the propositions which assert "p" and at the same time one of the propositions which assert "q".
Two propositions are opposed to one another if there is no significant proposition which asserts them both.
