# 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:

{P_{1}, P_{2}, …, P_{n}} ⊢ 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...*