# Proof Theory

Proof theories (also called deductive systems) provide means of ”calculating” things about the logic. Each proof theory is a means of doing proofs in that logic.

- Proof theories are methods that perform mechanical manipulations on strings of symbols.
- Proof theories are based on pattern matching.
- For a particular proof theory, there may be multiple ways to prove a formula.

$P_{1},P_{2},…,P_{n}∣−Q$ means that from $P_{1},P_{2},…P_{n}$ we can prove $Q$ using a proof theory.

- $∣−$ defines what is a proof
- We say that $P_{1},P_{2},…,P_{n}$ “proves” $Q$

Also see Semantics.