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.