 Let us return to the propositional languages with which we began. A valuation for such an L is a function from L to the two-element set {T, F} (of truth-values). As in the case of first-order languages touched on above, we put the set of formulas L in the source position (i.e. on the left) of our relational connections ⟨L, V, R⟩, with the set V of all valuations for L as the target (on the right), and R (the “is true on” relation) relates φ ∈ L to v ∈ V just in case v(φ) = T. In view of the one-to-correspondence between sets and characteristic functions, this relational connection is just a mild variant of the ∈-based relational connection considered above, in particular sharing with it the feature that it has conjunctive and disjunctive combinations on the right but not the left, and the feature that no two elements from the left bear the relation R to exactly the same elements on the right, and no two elements on the right have have exactly the same elements from the left bearing the relation R to them. To get some interesting logic under way, some of these features needed to be disturbed, though as we shall see, one of them—the existence of conjunctive combinations on the right—will prove surprisingly tenacious. If our interest lies in classical logic, we will be especially keen to discard most of (the above) V and concentrate only on Boolean valuations, in the sense to be explained now. A function from {T, F}n to {T, F} is called an n-ary truth-function, and we say that an n-ary truth-function f is associated with the n-ary connective # of L on the valuation v (for L)—or alternatively, under these same conditions, that # is associated with f on v—just in case for all φ1,…,φn ∈ L: