 The rule on the left is usually called ∧-Introduction while the two on the right are collectively known as ∧-Elimination—thus “(∧I)” and “(∧E)”—because they take us, respectively, to and from a formula with ∧ as main connective, and they enable us to define a consequence relation, ⊢∧ in purely syntactic terms thus: Γ ⊢∧ φ just in case starting with formulas all of which lie in Γ, successive application of the above rules allows us to pass eventually to the formula φ. And then one can show, not only that this is the consequence relation determined by the class of ∧-Boolean valuations—so that if one began with that class and sought a syntactic codification of the logic it gave rise to, one’s search would be at an end—but further that the valuations consistent with ⊢∧ are precisely the ∧-Boolean valuations. Here we calling v consistent with ⊢ when for no Γ, ψ, for which Γ ⊢ ψ, do we have v(φ) = T for all φ ∈ Γ while v(ψ) = F; the “but further” in the preceding sentence is explained by the fact that every consequence relation ⊢ is determined by the class Val(⊢) of all valuations consistent with ⊢, which in the case of ⊢∧ are precisely the ∧-Boolean valuations. (See the discussion of Gabbay’s result on the projection-conjunction truth-functions below for the extent of this phenomenon, which, for example, none of ∨, →, ¬ exhibits.) The significance of this is that if one started not with the class of ∧-Boolean valuations (or any other kind of valuational semantics), but with the rules (∧I) and (∧E) themselves, then that class of valuations would force itself on one’s attention anyway as the only class of valuations, preservation of truth on each of which coincided with derivability using those rules. (We return to this theme below, in connection with Carnap, by which time we will have traded in formula-to-formula rules for sequent-to-sequent rules, allowing for the convenient representation of inferences in which assumptions are discharged.) Moreover, the rules have a considerable claim to the status of rules in accordance with which one must reason if one is understanding ∧ as “and”, as providing a codification of the way in which conjunctive premisses and conclusions are reasoned from and to in natural day-to-day reasoning.