Dialogical Logic

Origin. Lorenzen, Lorenz (1960s). Erlangen school. Game semantics. Dialogue games. Foundation of constructive semantics.

Models. Logic as dialogue game. Proponent defends thesis. Opponent attacks. Winning strategy = validity. Constructive and classical variants.

Formalism.

Players: P (Proponent): defends thesis. O (Opponent): attacks thesis. Alternating moves. Dialogue.

Particle rules (attacks/defenses): A ∧ B: O chooses, P defends chosen conjunct. A ∨ B: P chooses, defends chosen disjunct. A → B: O asserts A, P must defend B. ¬A: O asserts A, P must show contradiction. ∀x.A: O chooses term, P defends instance. ∃x.A: P chooses term, defends instance.

Structural rules: Turn-taking. When P can use O's assertions. Winning conditions. Different rule sets → different logics.

Intuitionistic vs classical: Intuitionistic: P may use only last O statement. Classical: P may use any O statement. Structural rule difference. Game semantics.

Winning: P wins if dialogue ends with O unable to respond. O wins if P unable to respond. Validity = P has winning strategy. Regardless of O's moves.

Formal vs material: Formal dialogue: only logical structure. Material dialogue: non-logical content. Formal: P wins by form alone. Logic proper.

Extensions: Modal dialogues. Relevance dialogues. Linear logic dialogues. Flexible framework.

Symbols.

Symbol	Unicode	Meaning
P	—	Proponent
O	—	Opponent
!	—	assertion
?	—	attack

Metatheory. Game semantics. Winning strategy. Constructive validity. Dialogue.

Applies to. Proof theory. Game semantics. Constructive logic. Argumentation.

Limitations. Rule set variations. Learning curve. Implementation. Philosophical debates.