In Predicate Logic Without Equality
The axiom given above assumes that equality is a primitive symbol in predicate logic. Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a definition of equality. Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol. Most of the axioms of equality still follow from the definition; the remaining one is
and it becomes this axiom that is referred to as the axiom of extensionality in this context.
Read more about this topic: Axiom Of Extensionality
Famous quotes containing the words predicate, logic and/or equality:
“The only thing that one really knows about human nature is that it changes. Change is the one quality we can predicate of it. The systems that fail are those that rely on the permanency of human nature, and not on its growth and development. The error of Louis XIV was that he thought human nature would always be the same. The result of his error was the French Revolution. It was an admirable result.”
—Oscar Wilde (1854–1900)
“...some sort of false logic has crept into our schools, for the people whom I have seen doing housework or cooking know nothing of botany or chemistry, and the people who know botany and chemistry do not cook or sweep. The conclusion seems to be, if one knows chemistry she must not cook or do housework.”
—Ellen Henrietta Swallow Richards (1842–1911)
“I have no purpose to introduce political and social equality between the white and black races. There is a physical difference between the two, which, in my judgement, will probably for ever forbid their living together upon the footing of perfect equality; and inasmuch as it becomes a necessity that there must be a difference, I ... am in favour of the race to which I belong having the superior position.”
—Abraham Lincoln (1809–1865)