Axiom of Extensionality - in Predicate Logic Without Equality

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 predicate of truth-value of a proposition, therefore, is a mere fictive quality; its place is in an ideal world of science only, whereas actual science cannot make use of it. Actual science instead employs throughout the predicate of weight.
    Hans Reichenbach (1891–1953)

    The logic of worldly success rests on a fallacy: the strange error that our perfection depends on the thoughts and opinions and applause of other men! A weird life it is, indeed, to be living always in somebody else’s imagination, as if that were the only place in which one could at last become real!
    Thomas Merton (1915–1968)

    Proclaim human equality as loudly as you like, Witless will serve his brother.
    Thomas Henry Huxley (1825–95)