Dialogic logic

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The dialogical logic ( Engl. : Game semantics ) is one of the German logicians and philosophers Kuno Lorenz and Paul Lorenzen developed game theory , semantics near approach to logic . In comparison to deductions in logic calculations, motivation is closer to human reasoning.

Through the framework of the dialogic logic, the claim is made that those involved in the conversation do not need an external referee, but rather freely check the validity of statements themselves.

The rules for the joiners and quantifiers are designed as a dialogue game. The dialogue is generally determined by framework rules and in detail by attack and defense rules for the logical operators. A statement composed of logical signs is called true if it can always be won in dialogue. Such a statement is called formally true if it can always be obtained without entering into a dialogue about the prime statements ( elementary sentences ).

If elementary formulas are used as the starting point in conventional calculi and then derived according to calculus rules up to the end result, then in dialogical logic one proceeds exactly the other way around: It starts with a compound assertion and this is reduced to elementary sentences while observing the rules of the game.

Framework rules

  1. The proponent (right column noted as P) begins the dialogue by making a statement linked to logical signs.
  2. It is the turn of the dialogue partners to take turns.
  3. The further course of action consists of attacks and defenses.
  4. An attack represents a right to attack a statement made by the opponent that is still vulnerable.
  5. A defense is the duty to defend yourself on an attacked statement, at the latest when you are no longer allowed to attack yourself.
  6. The attacks and defenses are standardized in the particle rules.
  7. The proponent has won if he has defended an attacked elementary statement (prime or atomic statement) or if the opponent (noted with O in the left column) does not defend an attacked elementary statement.

Effective framework rule

The effective framework rule is particularly relevant for the interpretation of the subjunction ( if A then B): No player has to defend himself against an attack until this attack has not been defended against a finite number of attacks. Before starting an attack, the attacker sets himself a maximum number of attacks.

If the effective framework holds, dialogic logic is a model of intuitionist logic . This allows statements for the dialogue whose truth value is not fixed: for example in the case of unsolved mathematical problems , statements about future events or about infinity.

The classic two-valued logic can be maintained through a further liberalization of the framework rules, so that every statement can be defended at any point in the dialogue.

Rules of attack and defense for the logical operators

The attack and defense rules of dialogical logic are listed here, whereby the question mark should be read as a defense request:

Junctures attack defense
(and)
(and)
/ (or)
... (Not)
(if then)

The last-mentioned join operation if-then is called a subjunction here , otherwise it is usually called an implication .

Quantifiers attack defense

Quantifier: (one quantifier (also existential quantifier ): "for (at least) one") or (universal quantifier: "for all")

Examples

Here as a simple example a dialog around . The statement is formally logically true:

(The subjunction assertion is attacked according to the subjunction rule: for this the preceding prime statement is asserted.)
(The following prime statement is given as a defense, this is also an adoption of the previous line.)

can always win the dialogue because he can take over.

In the following further examples, first for the classically and intuitionistically true sentence , then for the only classically true sentence .

In the case of defenses, it is also indicated here which attack they are directed against. 1! means “defends against the attack under 1”, and 1? means "attacks the statement under 1". Brackets denote moves that are not possible if the effective framework rule is observed .

1.
2.
3
4th

makes a prime statement in step 3, namely , which was already asserted in step 2. According to the rules, the dialogue is won.

It looks completely different for :

1.
2.
3. ( )

In the last step, defend the statement under 1, which attacked in step 2. Since statements from attacked after step 2 , the defense would only be possible if the effective general rule did not apply. A different course of the game does not help either:

1.
2.
3.
4th ( )

attacks the prime statement in step 3 . Although this prime statement in step 4 itself admits, you are no longer allowed to defend yourself against this attack, since another attack has now taken place.

Since the proponent cannot force the course of the game where he wins while observing the effective framework, the statement cannot be proven in intuitionistic logic. In classical logic, however, it applies, as the examples show.

Applications

The special effects that occur with the ( intuitionist ) interpretation of the subjunctor ( ) are interesting : During the dialogue, statements that are not truthful (a statement is either true or false) are allowed. The truth value of the statements can remain in a limbo state. In the effective framework, the excluded third party is not required. The truth value of the overall statement is only determined when the dialogue is concluded.

If you introduce a general rule in which a statement is no longer available later in the dialog, you can develop a temporal logic from the dialogic logic. Carl Friedrich von Weizsäcker and Peter Mittelstaedt took up this rule for the interpretation of quantum physics through temporal logic . Here's an example: as we ponder whether the moon is setting or not, it is setting.

There are further applications for argumentation theory , since dialogical logic shows in the course of the dialogue who takes on when the burden of proof for factual assertions in the form of elementary statements.

literature

  • Kuno Lorenz: Logic, dialogical. In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. Volume 2, Metzler, Stuttgart / Weimar 1995, pp. 643ff.
  • Jaakko Hintikka , Esa Saarinen: Game-Theoretical Semantics. Springer, 1979, ISBN 90-277-0918-1 .
  • Kuno Lorenz , Paul Lorenzen : Dialogical Logic. WBG, Darmstadt 1978
  • Mathieu Marion: Why Play Logical Games? In: Unifying Logic, Language, and Philosophy. Springer, 2009, ISBN 978-1-4020-9373-9 .
  • S. Rahman, L. Keiff: On how to be a dialogician . In: Daniel Vanderken (Ed.): Logic Thought and Action. Springer, 2005, ISBN 1-4020-2616-1 , pp. 359-408.
  • Rüdiger Inhetveen : Logic: A dialogue-oriented introduction. 2003, ISBN 3-937219-02-1 .
  • J. van Benthem: Logic in Games . Elsevier, 2006.
  • L. Keiff: Introduction à la logique modale et hybride. In: M. Rebusqui, T. Tulenheimo (eds.): Logique et théorie de jeux. Kimé, 2004, pp. 89-102. ISSN  1281-2463 .
  • S. Rahman: Non-Normal Dialogics for a Wonderful World and More. In: J. van Benthem, G. Heinzmann, M. Rebuschi, H. Visser (eds.): The Age of Alternative Logics . Springer, 2006, ISBN 1-4020-5011-9 .
  • S. Rahman and N. Clerbout: Linking Games and Constructive Type Theory: Dialogical Strategies, CTT-Demonstrations and the Axiom of Choice . Springer Briefs (2015)
https://www.springer.com/gp/book/9783319190624
  • S. Rahman, Z. McConaughey, A. Klev, N. Clerbout: Immanent Reasoning or Equality in Action. A plaid for the play level . Springer (2018).
https://www.springer.com/gp/book/9783319911489
  • H. Rückert: Logiques dialogiques multivalentes. In: M. Rebusqui, T. Tulenheimo (eds.): Logique et théorie de jeux. Kimé, 2004, pp. 59-88. ISSN  1281-2463 .
  • J. Ehrensberger, C. Zinn: DiaLog - A System for Dialogue Logic. In: William McCune (Ed.): Proceedings of the 14th. Conference on Automated Deduction - CADE-14. (= Lecture Notes in Artificial Intelligence. Volume 1249). Springer, 1997, pp. 446-460.
  • C. Zinn: Colosseum - An Automated Theorem Prover for Intuitionistic Predicate Logic based on Dialogue Games. In: Position Papers of the International Conference on Analytic Tableaux and Related Methods (Tableaux-99). Technical Report, Saratoga Springs 1999, pp. 133-147.

Web links

Individual evidence

  1. Kuno Lorenz: The dialogical justification of effective logic. 1973 In: Paul Lorenzen, Kuno Lorenz: Dialogical Logic. WBG, Darmstadt 1978, p. 184; Rüdiger Inhetveen : Logic. A dialogue-oriented introduction. (= EAGLE 002). Edition at Gutenbergplatz, Leipzig 2003, p. 40.