 We describe the connective # as truth-functional with respect to a class U of valuations when there is some f such that for each v ∈ U, f is associated with # on v, and as pseudo-truth-functional with respect to U when U is a union of classes of valuations with respect to each of which classes # is  truth-functional.[4]  A valuation v is #-Boolean, speaking informally, when the expected truth-function is associated with # on v. More specifically, v, assumed to be a valuation for a language supporting the connectives to be mentioned here, is ∧-Boolean when the binary truth-function f∧ with f∧(x, y) = T iff x = y = T is associated with ∧ on v, ∨-Boolean when a similar association holds for ∨ and f∨ defined by f∨(x, y) = F iff x = y = F. We assume that the idea is now clear without providing separate definitions of “→-Boolean”, “↔-Boolean”, “¬-Boolean”, “⊥-Boolean”, etc. A Boolean valuation for L is just a valuation for L which is #-Boolean for each primitive connective # of L for which a notion of #-Boolean valuation has been defined. (This is just a way of registering an intended truth-functional interpretation for the connectives # in question.) Similarly we call any connective # for which a notion of #-Boolean valuation has been defined, a Boolean connective, even when discussing its behaviour according to a logic not determined by a class of Boolean valuations. (This terminology will be explained in due course.)