What is the principle of the identity of indiscernibles in Gottfried Wilhelm Leibniz’s philosophy?

The principle of the identity of indiscernibles is a philosophical doctrine enunciated by G.W. Leibniz which asserts that no two objects can be numerically distinct while sharing all their properties in common.1

Formally, the principle states that if two objects, x and y, are not identical, then there must be at least one property P that holds for one but not the other. Conversely, if x and y share all their properties, they must be identical.12

It is important to distinguish this from the principle of the indiscernibility of identicals (also known as Leibniz’s Law), which states that if x is identical to y, then every property of x is a property of y. While the latter is a standard axiom of formal logic, the identity of indiscernibles is a metaphysical claim that many philosophers argue is not a necessary truth, as it is possible to conceive of a universe where distinct objects share all properties. Leibniz utilized this principle to support various metaphysical doctrines, including his arguments against the existence of Newtonian absolute space.12