Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is typable. For example, suppose there is a derivation in S ending with the judgement A |- M : tau meaning that M has result type tau when assuming the types of free variables are given by A. Then (A, tau) is a typing for M.
A principal typing in S for a term M is a typing for M which somehow represents all other possible typings in S for M. It is important not to confuse this notion with the weaker notion of principal type often mentioned in connection with the Hindley/Milner type system. Previous definitions of principal typings for specific type systems have involved various syntactic operations on typings such as substitution of types for type variables, expansion, lifting, etc.
This talk presents a new general definition of principal typings which does not depend on the details of any particular type system. This talk shows that the new general definition correctly generalizes previous system-dependent definitions. This talk explains why the new definition is the right one. Furthermore, the new definition is used to prove that certain polymorphic type systems using "for all" quantifiers, namely System F and the Hindley/Milner system, do not have principal typings.