4.53 The general propositional form is a variable.

5 A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.)

5.01 Elementary propositions are the truth-arguments of propositions.

5.02 The arguments of functions are readily confused with the affixes of names.  For both arguments and affixes enable me to recognize the meaning of the signs containing them.  For example, when Russell writes ‘+c’, the ‘c’ is an affix which indicates that the sign as a whole is the addition-sign for cardinal numbers.  But the use of this sign is the result of arbitrary convention and it would be quite possible to choose a simple sign instead of ‘+c’; in ‘Pp’ however, ‘p’ is not an affix but an argument:  the sense of ‘Pp’ cannot be understood unless the sense of ‘p’ has been understood already. (In the name Julius Caesar ‘Julius’ is an affix.  An affix is always part of a description of the object to whose name we attach it:  e.g. the Caesar of the Julian gens.) If I am not mistaken, Frege’s theory about the meaning of propositions and functions is based on the confusion between an argument and an affix.  Frege regarded the propositions of logic as names, and their arguments as the affixes of those names.

5.1 Truth-functions can be arranged in series.  That is the foundation of the theory of probability.

5.101 The truth-functions of a given number of elementary propositions can always be set out in a schema of the following kind:  (TTTT) (p, q) Tautology (If p then p, and if q then q.) (p z p . q z q) (FTTT) (p, q) In words :  Not both p and q. (P(p . q)) (TFTT) (p, q) " :  If q then p. (q z p) (TTFT) (p, q) " :  If p then q. (p z q) (TTTF) (p, q) " :  p or q. (p C q) (FFTT) (p, q) " :  Not g. (Pq) (FTFT) (p, q) " :  Not p. (Pp) (FTTF) (p, q) " : p or q, but not both. (p .  Pq :  C :  q .  Pp) (TFFT) (p, q) " :  If p then p, and if q then p. (p + q) (TFTF) (p, q) " :  p (TTFF) (p, q) " :  q (FFFT) (p, q) " :  Neither p nor q. (Pp .  Pq or p | q) (FFTF) (p, q) " :  p and not q. (p .  Pq) (FTFF) (p, q) " :  q and not p. (q .  Pp) (TFFF) (p,q) " :  q and p. (q . p) (FFFF) (p, q) Contradiction (p and not p, and q and not q.) (p .  Pp . q .  Pq) I will give the name truth-grounds of a proposition to those truth-possibilities of its truth-arguments that make it true.

5.11 If all the truth-grounds that are common to a number of propositions are at the same time truth-grounds of a certain proposition, then we say that the truth of that proposition follows from the truth of the others.

5.12 In particular, the truth of a proposition ‘p’ follows from the truth of another proposition ‘q’ is all the truth-grounds of the latter are truth-grounds of the former.

5.121 The truth-grounds of the one are contained in those of the other:  p follows from q.

5.122 If p follows from q, the sense of ‘p’ is contained in the sense of ‘q’.