Axiom:Leibniz's Law

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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.


Also see


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.


Sources