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In nonmonotonic reasoning, the Nixon diamond is a scenario in which default assumptions lead to mutually inconsistent conclusions. The scenario is:
A non-monotonic logic is a formal logic whose consequence relation is not monotonic. In other words, non-monotonic logics are devised to capture and represent defeasible inferences, i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences. Intuitively, monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known. A monotonic logic cannot handle various reasoning tasks such as reasoning by default, abductive reasoning, some important approaches to reasoning about knowledge, and similarly, belief revision.
Contents
- usually, Quakers are pacifist
- usually, Republicans are not pacifist
- Richard Nixon is both a Quaker and a Republican
Quakers, also called Friends, are a historically Christian group of religious movements formally known as the Religious Society of Friends, Society of Friends or Friends Church. Members of the various Quaker movements are all generally united in a belief in the ability of each human being to experientially access "the light within", or "that of God in every one".
The Republican Party, also referred to as the GOP, is one of the two major political parties in the United States; the other is its historic rival, the Democratic Party.
Richard Milhous Nixon was an American politician who served as the 37th president of the United States from 1969 until 1974, when he resigned from office, the only U.S. president to do so. He had previously served as the 36th vice president of the United States from 1953 to 1961, and prior to that as both a U.S. representative and senator from California.
Since Nixon is a Quaker, one could assume that he is a pacifist; since he is Republican, however, one could also assume he is not a pacifist. The problem is how a formal logic of nonmonotonic reasoning should deal with such cases. Two approaches can be adopted:
- skeptical
- since Nixon can neither be proved to be a pacifist nor the contrary, no conclusion is drawn;
- credulous
- since Nixon can be proved to be a pacifist in at least one case, he is believed to be a pacifist; however, since he can also be proved not to be a pacifist, he is also believed not to be a pacifist.
The credulous approach can allow proving both something and its contrary. For this reason, the skeptical approach is often preferred. Another solution to this problem is to attach priorities to default assumptions; for example, the fact that “usually, Republicans are not pacifist” can be assumed more likely than “usually, Quakers are pacifist”, leading to the conclusion that Nixon is not pacifist.
The name diamond comes from the fact that such a scenario, when expressed in inheritance networks, is a diamond shape. This example is mentioned for the first time by Reiter and Criscuolo in a slightly different form where the person that is both a Republican and a Quaker is a John instead of Richard Nixon.
In plane Euclidean geometry, a rhombus is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.
