Proof Systems

An argument is inferable, if and only if there exists no interpretation in which the premises are True and the conclusion False.

Justified Argument

From a set of n premises follow the conclusion C:

{P1, P2, …, Pn} ⊢ C

To be justified the argument must be either derived from axioms or from other justified arguments by means of inference rules.

In this way the definition of a justified argument is recursive:

  • Base case: An argument is instance of axiom
  • Recursive: A justified argument can be an argument that is derived from other justified arguments

In propositional logic, any argument where the conclusion is a member of the set of premises is an axiom e.g.

{P} ⊢ P
{Q} ∪ {P} ⊢ P
{P, Q} ⊢ P

Axiom schema

Γ ∪ {A} ⊢ A

Where Gamma is a set of premises and A a specific proposition.

Inference Rules

If on the basis of some set of premises Γ, the conclusion A is justified and on the set of the same premises B is also justified, the conclusion A /\ B is justified.

Γ ⊢ A       Γ ⊢ B
−−−−−−−−−−
    Γ ⊢ A /\ B

to be continued...