# Axiom:Leibniz's Law

## Contents

## 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.

From Set Definition by Predicate, the above formulation can be expressed as:

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

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.

## Also see

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.): $\S 3.16-17, \ \S 4.21$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Equality: $\text{(d)}$