# Definition:Deductive Apparatus

## Definition

Let $\mathcal L$ be a formal language.

A **deductive apparatus for $\mathcal L$** is a formally specified system for deriving conclusions about the well-formed formulas of $\mathcal L$.

In mathematics and logic, **deductive apparatuses** can by and large be divided into proof systems and formal semantics.

### Proof System

A **proof system** $\mathscr P$ for $\mathcal L$ comprises:

**Axioms**and/or**axiom schemata**;**Rules of inference**for deriving theorems.

It is usual that a **proof system** does this by declaring certain arguments concerning $\mathcal L$ to be valid.

Informally, a **proof system** amounts to a precise account of what constitutes a **(formal) proof**.

### Formal Semantics

A **formal semantics** for $\mathcal L$ comprises:

- A collection of objects called
**structures**; - A notion of
**validity**of $\mathcal L$-WFFs in these structures.

Often, a **formal semantics** provides these by using a lot of auxiliary definitions.