  Formulas are constructed more-or-less in the obvious way, with two kinds of quantified variables instead of one: extensional and intensional. But there is one really important addition to the syntactic machinery, and it requires some discussion. Suppose we have an intension, \(f\), that picks out an object in each state. For example, the states might be various ways the universe could have been constituted, and at each state \(f\) picks out the number of the planets which could, of course, be 0. Suppose \(P\) is a one-place relation symbol—what should be meant by \(P(f)\)? On the one hand, it could mean that the intension \(f\) has the property \(P\), on the other hand it could mean that the object designated by \(f\) has the property \(P\). Both versions are useful and correspond to things we say every day. We will allow for both, but the second version requires some cleaning up. Suppose \(P(f)\) is intended to mean that the object designated by \(f\) (at a state) has property \(P\). Then how do we read \(\Diamond P(f)\)? Under what circumstances should we take it to be true at state \(\Gamma\)? It could be understood as asserting the thing designated by \(f\) at \(\Gamma\) (call it \(f_{\Gamma })\) has the ‘possible-\(P\)’ property, and so at some alternative state \(\Delta\) we have that \(f_{\Gamma }\) has property \(P\). This is the de re reading, in which a possible property is ascribed to a thing. Another way of understanding \(\Diamond P(f)\) takes the possibility operator as primary: to say the formula is true at \(\Gamma\) means that at some alternative state, \(\Delta\), we have \(P(f)\), and so at \(\Delta\) the object designated by \(f\) (call it \(f_{\Delta })\) has property \(P\). This is the de dicto reading, possibility applies to a sentence. Of course there is no particular reason why \(f_{\Gamma }\) and \(f_{\Delta }\) should be identical. The de re and de dicto readings are different, both need representation, and we cannot manage this with the customary syntax.