Axiom:Leibniz's Law

Definition

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

Then:

$x = y \dashv \vdash P \left({x}\right) \iff P \left({y}\right)$

for all $P$ in the Universe of Discourse.

That is, two objects $x$ and $y$ are equal iff $x$ 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 this definition is equivalent to:

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

for all $S$ in the Universe of Discourse.

This is therefore the justification behind the notion of Equality of Sets.

Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.

He used this law as the definition of equality.

However, Alfred Tarski points out:

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

• Axioms of Equality
• Equals

Sources

• Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (1936) $\S 3.16-17, \ \S 4.21$