 Thus, formulas such as ‘\(x = y\)’ are part of the second-order predicate calculus with identity. Frege, too, had primitive identity statements; for him, identity is a binary function that maps a pair of objects to The True whenever those objects are the same object. So whereas we shall suppose that statements like ‘\(2^{2} = 4\)’ are simply true assertions and statements like ‘\(2^{2} = 3\)’ are simply false ones, Frege took ‘\(2^{2} = 4\)’ to be a name of The True and took ‘\(2^{2} = 3\)’ to be a name of The False. The statement form ‘\(f(x) = y\)’ plays an important role in Frege’s axioms and definitions, but we shall not need to assert claims of this form in order to derive Frege’s Theorem. Instead, we shall assume (a) that identity is simply a 2-place relation and (b) that a unary function \(f\) is really a relation \(R\) that has the following property: \(Rxy \amp Rxz \to y = z\) (i.e., that functions are relations that always relate their first argument to at most one second argument). We may call such relations functional relations. In other words, when Frege asserts \( f(x) = y\), we may represent this as asserting that \(f\) is a functional relation \(R\) such that \(Rxy\). This generalizes to \( n\)-place relations for \(n\geq2\). For example, where \(+\) is the binary addition function of arithmetic, we may represent the arithmetic statement \(2+3 = 5\) in our language as a claim of the form \(+(2,3,5)\), where \(+\) is taken to be a 3-place functional relation that obeys the condition: \(+(x,y,z) \amp +(x,y,w) \to z = w\).