 In saying that a consequence relation ⊢ is determined by a class V of valuations what is meant that Γ ⊢ ψ if and only if for all v ∈ V, whenever v(φ) = T for each φ ∈ Γ, then v(ψ)= T. When ⊢ has been described by means of some proof system (some collection of rules), then the “only if” (“if”) direction of this claim amounts to the claim that ⊢ is sound (respectively, complete) with respect to V. Take the language whose sole primitive connective is ∧ and the class of ∧-Boolean valuations. A simple proof system is provided by the rules indicated in the obvious way by the following figures: