Game semantics

The dialogical logic (English: game semantics ) is a standard developed by the German logicians and philosophers Kuno Lorenz and Paul Lorenzen game-theoretic, semantics near approach to logic. The motivation is compared to derive in logic calculi detailed orientation on human reasoning.

Through the framework of the dialogical logic of the claim is made that the parties require no external referees in conversation, but check the validity of statements in freedom itself.

The rules for the connectives and quantifiers are conceived as a dialogue game. The dialogue is generally determined by the framework and rules in detail by attack and defense rules for the logical operators. True is called a composite of logical signs statement if they can be in dialogue always win. Formal true such a statement is called, if it can be won always, without entering into a dialogue on the Primaussagen ( elementary sets).

Is assumed in the conventional calculus of elementary formulas and then derived by calculus rules to the final result, so you go into the dialogical logic before the other way around: It is started with a composite assertion and this reduces in compliance with the rules of the game on elementary propositions.

Frame rules

Effective frame rule

The effective frame rule is particularly important for the interpretation of the subjunction ( if A then B ) are relevant: No player has to defend an attack, not before this attack was in turn defended finitely many attacks. Before an attack, the attacker respective indicates itself to a maximum number of attacks.

If the effective frame rule, the dialogical logic is a model of intuitionistic logic. This statements for dialogue are admitted whose truth value is not fixed: some unsolved problems in mathematics, statements about future events or infinite.

The classical two-valued logic can be obtained by further liberalization of competition rules in that each statement can be defended at any stage of the dialogue.

Attack and defense rules for the logical operators

Here are the offensive and defensive rules of dialogical logic are listed below:

The latter connective operation if - then here is subjunction, otherwise mostly called implication.

Quantorzeichen: ( Einsquantor (even existential ): " for ( at least) one " ) or ( universal quantifier " for all " )

Examples

Here a simple example of a dialog to enter. The statement is formally logically true:

Dialogue can always win, because he can take over.

The following additional examples, first for the classical and intuitionistic set true, then the only classic true sentence.

Here, it is stated in your defenses against attack which they are directed. " 1 " so called " defend itself against the attack under 1" and "1? " Means "takes the statement below 1 at ." Brackets indicate features which are not possible in accordance with the effective frame rule.

In step 3 a Primaussage, namely, which has been maintained in step 2. According to the rules of the dialogue has been gained for.

Situation is quite different for this:

In the last step defended the statement below 1, which has attacked in step 2. There has to step 2 still attacked statements by the defense would only be possible if the effective frame rule would not apply. Also another gameplay does not help:

Engages in step 3 to the Primaussage. Although these Primaussage admits in step 4 itself may no longer defend against this attack, since the meantime, another attack took place.

Since the proponent can not force a gameplay where he wins in compliance with the effective frame rule, the statement in the intuitionistic logic is not to prove. In classical logic, however, it is true, as the examples show.

Applications

Interesting are the special effects, which occur in the ( intuitionistic ) interpretation of Subjunktors (): During the dialogue are not wahrheitsdefinite ( a statement is either true or false) statements allowed. The truth value of the statements can be left in a limbo. In the framework of effective rule of the law of excluded middle is not required. Only at the end of the dialogue is clear, the truth value of the overall presentation.

Performs a one frame rule, in which a statement is no longer available later in the dialog, so you can develop a temporal logic of the dialogical logic. Carl Friedrich von Weizsäcker and Peter Mittelstaedt added this rule for the interpretation of quantum physics by temporal logic. Here is an example: While we consider whether the moon goes down or not, he goes under.

Other applications arise for the argumentation theory, as the dialogical logic shows in the course of the dialogue, who, when the burden of proof for allegations of fact in the form of elementary propositions takes over.