Leibnitz's law

Leibnitz's law is generally understood to say that two entities are identical if and only if they share all and only the same properties. It was shown by Quine to have two parts, though it's not clear that Leibnitz actually knew about the first:

1. The indiscernibility of identicals
   • For any x and y, if x is identical to y, then x and y have all the same properties.

     <math>\forall x \forall y[x=y \rightarrow \forall P(Px \leftrightarrow Py)]</math>

   • For any x and y, if x and y differ with respect to some property, then x is non-identical to y.

     <math>\forall x \forall y[\neg \forall P(Px \leftrightarrow Py) \rightarrow x \neq y]</math>

2. The identity of indiscernibles
   • For any x and y, if x and y have all the same properties, then x is identical to y.

     <math>\forall x \forall y[\forall P(Px \leftrightarrow Py) \rightarrow x=y]</math>

   • For any x and y, if x is non-identical to y, then x and y differ with respect to some property.

     <math>\forall x \forall y [x \neq y \rightarrow \neg \forall P(Px \leftrightarrow Py)]</math>

Leibnitz's law is an important fundamental ontological principle.