eyeprolog

The Nixon diamond

What should a reasoner conclude when two good defaults disagree?

Quakers are usually pacifists. Republicans are usually not. Nixon was both. This classic puzzle from non-monotonic reasoning has no single right default conclusion, so the example does not force one: it reports the conflict explicitly.

Run it in the playground · Source

The question

A default is a rule that holds unless there is reason to think otherwise. Two defaults apply to Nixon and point in opposite directions. The name comes from the diamond shape of the inheritance diagram: Nixon at the bottom, Quaker and Republican in the middle, pacifist and hawk at the top.

Should the reasoner pick one side, refuse to conclude anything, or say that the case is conflicted?

The facts

kind(nixon, quaker).
kind(nixon, republican).

supports_default(Person, pacifist) :-
  kind(Person, quaker).
supports_default(Person, hawk) :-
  kind(Person, republican).

contrary(pacifist, hawk).
contrary(hawk, pacifist).

Each default is a rule that supports a conclusion rather than asserting it, and the two conclusions are declared contrary.

The rules

conflicted(Person, A, B) :-
  supports_default(Person, A),
  supports_default(Person, B),
  contrary(A, B).

status(Person, conflicted_default_case) :-
  conflicted(Person, _a, _b).

A person is conflicted when two supported conclusions are contrary. No priority is invented between the defaults.

The answer

eyeprolog examples/nixon-diamond.pl

defaultSupports(nixon, pacifist).
defaultSupports(nixon, hawk).
conflict(nixon, conflict(pacifist, hawk)).
conflict(nixon, conflict(hawk, pacifist)).
status(nixon, conflicted_default_case).

Both defaults apply, the conflict is reported in both orders, and Nixon’s case is marked conflicted, not settled.

The reason

step(conflicted(nixon, pacifist, hawk),
     rule(7),
     ['Person' = nixon, 'A' = pacifist, 'B' = hawk],
     [supports_default(nixon, pacifist),
      supports_default(nixon, hawk),
      contrary(pacifist, hawk)]).

The conflict rests on exactly three premises: each default’s support, and the fact that the two conclusions are contrary.

The check

condition('C1', resolution, ok, 13).
condition('C4', coverage, ok, 18).
steps(13).
verified(13).
trusted(0).
verdict(checked).

All 13 steps replay against their clauses. There is no negation, arithmetic, or aggregation, so nothing needs to be recomputed or trusted.

Try it

Add a priority fact such as stronger(hawk, pacifist) and a rule that resolves the conflict with it. What is lost by doing so?
Add a second person who is only a Quaker. Which answers does that person get?
Rewrite the defaults with \+ (“pacifist unless known to be a hawk”). What happens when each default blocks the other?

Learn more in The Art of EyeProlog, Chapter 7.

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