  There is no straightforward way in which the truth-functions and the quantifiers (over any type of objects) are identifiable with notions in the sense of Tarski. According to John Corcoran (see footnote 6 in Tarski 1986a, p. 150), in one version of the lecture Tarski indicated that the truth-functions and classical quantifiers can be constructed as certain objects in the type hierarchy that are invariant under all permutations, and, in this sense, are logical notions. (Clearly, Tarski took the logicality of these notions almost for granted, and was concerned with a definition which gave some account of the status of other more substantive notions in various mathematical languages.) For example, the truth-values “true” and “false” can be identified with the universe of individuals and the null set, respectively, and the truth-functions in turn with functions having (tuples of) these classes as arguments and values; and the classical universal and existential quantifiers over a type of objects t can be identified with certain functions from the class of sets of objects of type t into the class of truth-values—identifying “true” with the universal set of objects of type t and “false” with the empty set of that type. (A universal quantifier will assign “true” to the set of all objects of type t, and “false” to all other subsets of t; and the existential quantifier will assign “true” to the non-empty subsets, and “false” to the empty subset.)