Function and Concept

In general, a concept is a function whose value is always a truth value (139). A relation is a two place function whose value is always a truth value (146).

Frege draws an important distinction between concepts on the basis of their level. Frege tells us that a first-level concept is a one-place function that correlates objects with truth-values (147). First level concepts have the value of true or false depending on whether the object falls under the concept. So, the concept ${\displaystyle F}$ has the value the True with the argument the object named by 'Jamie' if and only if Jamie falls under the concept ${\displaystyle F}$ (or is in the extension of F).

Second order concepts correlate concepts and relations with truth values. So, if we take the relation of identity to be the argument ${\displaystyle f}$, the concept expressed by the sentence:

${\displaystyle \forall x\forall yf(x,y)\rightarrow \forall z(f(x,z)\rightarrow y=z)}$

correlates the relation of identity with the True.

The conceptual range (Begriffsumfang in Frege 1891, p. 16) follows the truth value of the function:

${\displaystyle x^{2}=1}$ and ${\displaystyle (x+1)^{2}=2(x+1)}$ have the same conceptual range.

• "On Function and Concept" in Michael Beaney, ed., The Frege Reader, Blackwell, 1997, pp. 130–148