How does the principle of identity work in logic?

In the philosophy of logic, the principle of identity is one of the three fundamental laws of thought. It asserts that a thing is identical with itself, symbolically represented as $(\forall x)(x=x)$, or simply that $x$ is $x$.1

In formal logic, the properties of numerical identity are codified by two primary axioms:23

• Reflexivity: Any object $x$ is identical with itself ($x=x$).23
• Leibniz’s Law (Indiscernibility of Identicals): If $x$ is identical to $y$, then any property true of $x$ must also be true of $y$ ($(x=y)\supset (Fx\equiv Fy)$).23

These axioms imply that identity is symmetrical (if $x=y$, then $y=x$) and transitive (if $x=y$ and $y=z$, then $x=z$). Furthermore, the principle supports the substitution of identicals, meaning that if two terms refer to the same entity, they can be interchanged in any statement without altering its truth value. While some philosophers have explored theories of relative identity to address complex problems like personal identity or theological doctrines, the standard logic of identity remains a cornerstone of metaphysical and logical analysis.23