Provable

A derivation, or derivation or deduction is in the logic of the collection of statements from other statements. This conclusion rules are applied to premises in order to arrive at conclusions. Which rules of inference are allowed here, is determined by the calculus used.

Example: propositional and predicate logic

The sequent calculus is concerned with the derivation of sequences of the form using the sequences of rules. For illustration, we take the derivation of the law of excluded middle. The rules used are described in.

Thus, the following new rule was derived sequences:

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

The Ableitbarkeitsrelation and Ableitbarkeitsoperator

Definition

To formalize the derivability of the derivative operator is often ( and inference ) is used, the derivation of the relation ( also Inferenzrelation ) is defined.

If - according to the rules of a specific calculus - the expression ( the conclusion or consequence ) of the amount ( the premises ) can be derived in a finite number of steps, to write it; Here, the derivation relationship.

In this Ableitbarkeitsrelation (also Inferenzrelation ) is a relation between a set of statements, the premises, and a single statement, the conclusion. is to read it as: " is derived from ".

Adds one ( say, we form the deductive conclusion ) of a given set of expressions from all derivable added expressions accepted, the derivative operator is defined ( and inference ):

Different logics each define a different Ableitbarkeitsbegriff. So there is a propositional Ableitbarkeitsbegriff, a predicate logic, an intuitionistic, modal logic one, etc.

Properties of derivative operators

There are a number of properties, the most Ableitbarkeitsrelationen (at least the above ) are together

• Inclusion: ( Any acceptance is also an inference ).
• Idempotency: If and, then ( by addition of inferences about the assumptions we obtain no new conclusions. )
• Monotony: If, then (adding assumptions given the previously possible inferences. )
• Compactness; If, then there is with a finite set, so that. (Any inference from an infinite amount of acceptance is already accessible from a finite subset. )

From the first three of these properties can be concluded that a closure operator, ie extensive, monotone, idempotent mapping.

Swell

• Mathematical Logic