A Rule of the propositional function
If is a propositional function of variable letters which is identically true then the result of replacing each by any statement is a valid statement.

B Rule of Inference
If is a valid statement and is a valid statement then is a valid statement.

C Rule of Equality
First equality
The statements , and , & are valid statements where are any three constant symbols.
Second equality
If is a statement and are constant symbols and if represents with every occurence of replaced by , then is a valid statement.

D Rule of Change of Variables
If be a statement and are variable symbols and results from replacing every occurence of in by then is a valid statement.

Let denote a statement with one free variable and in which every occurence of is free, and let for any constant symbol denote with every occurence of replaced by , then

E Rule of Specialization
is a valid statement, where is any constant symbol.

F Rule of existence
Let be any statement not involving or . If for any constant symbol , is a valid statement then, is a valid statement.

G Rule of validity under conditions
Let be a statement with only one free variable and in which every occurence of is free . Let be a statement which does not contain .
Then the following are valid statements.
~ ~
& &
& &


Notations