Deriving a binary logistic classifier from a multi class logistic classifier

Given a multi class logisitic classifier $f(x)=argmax(softmax(Ax + \beta))$, and a specific class of interest $y$, is it possible to construct a binary logistic classifier $g(x)=(\sigma(\alpha^T x + b) 0.5)$ such that $g(x)=y$ if and only if $f(x)=y$?