# Derivation (logic)

A **derivation** , **derivation** , or deduction in logic is the extraction of statements from other statements. In doing so, final rules are applied to premises in order to arrive at conclusions . Which final rules are allowed is determined by the calculation used .

The derivation, together with the semantic inference, is one of the two logical methods of arriving at the conclusion.

## Example: propositional and predicate logic

The sequence calculus deals with the derivation of sequences of the shape with the help of the sequence rules . As an illustration, we take the derivation of the sentence about the excluded third party . The rules used are described in the rules of the sequential calculus of the first-order predicate logic .

The following new sequence rule was thus derived:

It can now be used just like the basic rules of calculus.

## The derivability relation and the derivability operator

### definition

To formalize the derivability is often **derivative operator** (also inference) is used which on the *derivation relationship* (also *Inferenzrelation* ) is defined.

If - according to the rules of a concrete calculus - the expression (the conclusion or the consequence) can be derived from the set (the premises) in finitely many steps, one writes for it
; here is the *derivative relation* .

This *derivability relation* (also *inferential **relation* ) is a relation between a set of statements, the premises, and a single statement, the conclusion. is to be read as: " can be derived from ".

If one adds *all* expressions *which can be* derived from to a given set of expressions (one says that one forms the deductive closure), then the derivative operator (also inference operation ) is defined:

Different logics each define a different concept of derivability. There is a propositional concept of derivability, a predicate logic , an intuitionistic , a modal logic , etc.

### Properties of derivative operators

There are a number of properties that most of the derivability relations (at least those mentioned above) have in common

- Inclusion: (Every assumption is also a consequence).
- Idempotence: if and , then (adding inferences to the assumptions does not lead to new conclusions.)
- Monotony : If , then (adding assumptions preserves the consequences possible so far.)
- Compactness ; If so , then there is a finite set with such that . (Every conclusion from an infinite set of assumptions can already be reached from a finite subset .)

From the first three of these properties it can be deduced that an envelope operator is; H. an extensive , monotonous , idempotent mapping .

## References and comments

- ↑ An example of a definition is given by Kruse and Borgelt (2008) on p. 8.

## literature

- R. Kruse, C. Borgelt: Basic concepts of predicate logic . Computational Intelligence . Otto von Guericke University, Magdeburg 2008, p. 14 .