 The exclamation mark “\(\bang\)” is used in PM to indicate predicative functions. In Church’s system of r-types this is expressed by saying that the variable x is of r-type \(\tau\) and \(\phi\) is of r-type \((\tau )/1\) and \(\psi\) is of r-type \((\tau) /n\). In other words, \(\phi\) is of the lowest order compatible with its arguments. This notion of predicative functions is taken from the Introduction. In ∗12 Whitehead and Russell propose a narrower conception of predicative function, by which \(\phi\) must be a matrix, or function in the definition of which no quantifiers at all appear. See the accompanying entry on   the notation in Principia Mathematica. 