**Equality and Its Axioms**

There are several different conventions for using equality (or identity) in first-order logic. The most common convention, known as **first-order logic with equality**, includes the equality symbol as a primitive logical symbol which is always interpreted as the real equality relation between members of the domain of discourse, such that the "two" given members are the same member. This approach also adds certain axioms about equality to the deductive system employed. These equality axioms are:

**Reflexivity**. For each variable*x*,*x*=*x*.**Substitution for functions.**For all variables*x*and*y*, and any function symbol*f*,*x*=*y*→*f*(...,*x*,...) =*f*(...,*y*,...).

**Substitution for formulas**. For any variables*x*and*y*and any formula φ(*x*), if φ' is obtained by replacing any number of free occurrences of*x*in φ with*y*, such that these remain free occurrences of*y*, then*x*=*y*→ (φ → φ').

These are axiom schemes, each of which specifies an infinite set of axioms. The third scheme is known as **Leibniz's law**, "the principle of substitutivity", "the indiscernibility of identicals", or "the replacement property". The second scheme, involving the function symbol *f*, is (equivalent to) a special case of the third scheme, using the formula

*x*=*y*→ (*f*(...,*x*,...) = z →*f*(...,*y*,...) = z).

Many other properties of equality are consequences of the axioms above, for example:

**Symmetry.**If*x*=*y*then*y*=*x*.**Transitivity.**If*x*=*y*and*y*=*z*then*x*=*z*.

Read more about this topic: First-order Logic

### Famous quotes containing the words equality and, equality and/or axioms:

“I’m tired of earning my own living, paying my own bills, raising my own child. I’m tired of the sound of my own voice crying out in the wilderness, raving on about *equality and* justice and a new social order.... Self-sufficiency is exhausting. Autonomy is lonely. It’s so hard to be a feminist if you are a woman.”

—Jane O’Reilly, U.S. feminist and humorist. The Girl I Left Behind, ch. 7 (1980)

“Trying to love your children equally is a losing battle. Your children’s scorecards will never match your own. No matter how meticulously you measure and mete out your love and attention, and material gifts, it will never feel truly equal to your children. . . . Your children will need different things at different times, and true *equality* won’t really serve their different needs very well, anyway.”

—Marianne E. Neifert (20th century)

““I tell you the solemn truth that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the *axioms* of physics.””

—Henry Brooks Adams (1838–1918)