Definition:Deductive Apparatus

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Let $\LL$ be a formal language.

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

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 $\LL$ 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 $\LL$ to be valid.

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

Formal Semantics

A formal semantics for $\LL$ comprises:

  • A collection of objects called structures;
  • A notion of validity of $\LL$-WFFs in these structures.

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

Also see