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 it can be argued that Leibniz's law should either be adopted as an axiom, or not adopted at all.
Sources
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.): $\S 3.17$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Equality: $\text{(d)}$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Leibniz's law: 1.