How does first-order logic with identity differ from first-order logic without identity?

In first-order logic, the primary difference between systems with and without identity lies in the inclusion of the identity symbol ("=") as a logical constant.1

Logic Without Identity

In first-order logic without identity, the system focuses on quantification over individuals and predication (the attribution of properties or relations to individuals). While this system is sufficient for many logical expressions, it lacks the formal mechanism to assert that two terms refer to the same object.12

Logic With Identity

First-order logic with identity incorporates the "=" symbol, which is governed by two fundamental axioms:31
* Reflexivity: $x=x$, stating that every object is identical to itself.3
* Leibniz’s Law (Indiscernibility of Identicals): $(x=y)\supset (Fx\equiv Fy)$, which asserts that if $x$ and $y$ are identical, they must share all the same properties.3

These axioms allow for the formal expression of numerical identity, such as "Clark Kent is identical to Superman". While identity can be treated as an arbitrary relation in systems without it, including it as a logical constant provides a standard, rigorous framework for reasoning about equality and substitution.432