 Frege originally conceived of the quantifier \(\forall\) as a monadic predicate that is true of a first-level concept \(F\) under which only objects fall if, and only if, all objects fall under \(F\); likewise, Frege assimilated \(\exists\) to a monadic predicate that is true of a first-level concept if, and only if, at least one object falls under \(F\). In modern terms, we may say that the truth conditions of \(\forall x A\) are given in terms of a certain condition on the extension of the formula \(A\). Given a model \(\langle D, I\rangle\) and a formula \(A(x)\) with a single free variable \(x\), it will be convenient to use \(A(x)[d]\) to abbreviate: \(A\) is satisfied in \(\langle D, I\rangle\) by variable assignments on which \(d\) is the value assigned to the variable \(x\). The extension of a formula \(A(x)\) with a single free variable \(x\) in a model \(\langle D, I\rangle\) is just \(\{ d \in D: A(x)[d] \}\), which we may abbreviate \(I(A)\).