 
Summary: 1. Axioms and rules of inference for propositional logic.
Suppose T = (L, A, R) is a formal theory. Whenever H is a finite subset of L
and C L it is evident that
(H, C) R H C.
Fix a set X of propositional variables. We work with the language
p(X).
1.1. The standard setup (or so I think). This is, essentially, what you see in
the coursepack on page 78.
We axiomatize propositional logic by using following rules of inference.
Suppose A, B, C are statements. Then
EM , A A
Ass {(A (B C))}, ((A B) C)
Ex {A}, (B A)
Contr {(A A)}, A
Cut {(A B), ( A C))}, (B C)
are rules of inference.
EM stands for "excluded middle"; Ass stands for "associative"; Ex stands for
"expansion"; Contr stands for "contraction"; and Cut stands for "cut".
An axiom is a rule of inference where the set of hypotheses is empty; thus EM
is an axiom.
