 One may wonder which truth-functions are like that associated (on ∧-Boolean valuations) with ∧ in this respect. More precisely, let ⊢ be the consequence relation determined by the class of valuations on each of which an n-ary connective #f is associated with the n-ary truth-function f. Our question is: for which choices of f does ⊢ enforce the interpretation of #f as f in the sense that on every v ∈ Val(⊢), #f is associated with f. This question was answered in Dov Gabbay (1978) and (1981) (Chapter 1, Section 3) as follows. Say that f (as above) is a projection-conjunction truth-function just in case for some J ⊆ {1,…,n} we have, for all x1,…,xn (xi ∈ {T, F}):