Nixon diamond
Encyclopedia
In nonmonotonic reasoning
, the Nixon diamond is a scenario in which default assumptions can lead to mutually inconsistent conclusions. This scenario is:
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 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.
Non-monotonic logic
A non-monotonic logic is a formal logic whose consequence relation is not monotonic. 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...
, the Nixon diamond is a scenario in which default assumptions can lead to mutually inconsistent conclusions. This scenario is:
- usually, QuakersReligious Society of FriendsThe Religious Society of Friends, or Friends Church, is a Christian movement which stresses the doctrine of the priesthood of all believers. Members are known as Friends, or popularly as Quakers. It is made of independent organisations, which have split from one another due to doctrinal differences...
are pacifist - usually, RepublicansRepublican Party (United States)The Republican Party is one of the two major contemporary political parties in the United States, along with the Democratic Party. Founded by anti-slavery expansion activists in 1854, it is often called the GOP . The party's platform generally reflects American conservatism in the U.S...
are not pacifist - Richard NixonRichard NixonRichard Milhous Nixon was the 37th President of the United States, serving from 1969 to 1974. The only president to resign the office, Nixon had previously served as a US representative and senator from California and as the 36th Vice President of the United States from 1953 to 1961 under...
is both a Quaker and a Republican
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
Formal logic
Classical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...
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 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
Rhombus
In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...
. 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.