# Axiom:Leibniz's Law

## Axiom

Let $=$ represent the relation of equality and let $P$ be an arbitrary property.

Then:

$x = y \dashv \vdash \map P x \iff \map P y$

for all $P$ in the universe of discourse.

That is, two objects $x$ and $y$ are equal if and only if $x$ has every property $y$ has, and $y$ has every property $x$ has.

### Application to Equality of Sets

Let $S$ be an arbitrary set.

Then:

$x = y \dashv \vdash x \in S \iff y \in S$

for all $S$ in the universe of discourse.

This is therefore the justification behind the notion of the definition of set equality.

## Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.

## Historical Note

Leibniz used this law as the definition of equality.

However, Alfred Tarski notes:

To regard Leibniz's law here as a definition would make sense only if the meaning of the symbol "$=$" seemed to us less evident than that of expressions [such as 'every property $x$ has, $y$ has'].

Hence Leibniz's law can also be adopted as an axiom, or not adopted at all.