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Nixon diamond

"Nixon is anti-pacifist since he is a Republican." / "Nixon is a pacifist since he is a Quaker." — the canonical example for frameworks with multiple extensions, introduced in Dung (1995).

The setup​

Two arguments, each attacking the other. Neither attacks anything else.

Nixon diamond
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Symmetric mutual attack. Grounded semantics leaves both undecided; preferred semantics picks one or the other — the framework alone cannot decide between them.

Which sets stand?​

  • Grounded extension: ∅. Neither A nor B is unconditionally defensible.
  • Preferred extensions: {A}, {B}. Two maximal admissible sets.
  • Stable extensions: {A}, {B}. Each stably defeats the other.

Credulously accepted: both A and B (each is in some preferred). Skeptically accepted: neither (neither is in every preferred).

In code​

use argumentation::dung::{Framework, grounded};

let mut fw = Framework::new();
fw.add_argument("A");
fw.add_argument("B");
fw.add_attack("A", "B");
fw.add_attack("B", "A");

let g = grounded(&fw);
assert!(g.is_empty());  // grounded extension is empty

let prefs = fw.preferred_extensions();
assert_eq!(prefs.len(), 2);  // two preferred extensions

Why it matters​

The diamond is the smallest case where semantics disagree. It's the argument-theoretic equivalent of "should I believe he flies or doesn't?" — a forced choice. Scene AI designers should notice that diamonds create indecision: either branch is equally defensible, so the scene resolution depends on which preferred extension you pick. In practice, you'd break the tie by adding a weighted attack, a preference ordering, or β — turning the diamond into a one-sided framework.

Further reading​