 Let us forget about → and ¬ and the rules governing them, and consider the formulas constructed with the aid only of ∧ and ∨. On sequents of Set-Fmla in this fragment, classical and intuitionistic logic agree, and the consequence relation concerned—⊢∧,∨, we may call it—is the consequence relation determined by the class of all ∧- and ∨-Boolean valuations. Carnap’s point is one anticipated earlier with the remark that every v ∈ Val(⊢∧) is ∧-Boolean, which extends to the case of ⊢∧,∨: all valuations in Val(⊢∧,∨) are ∧-Boolean, but they are not all ∨-Boolean, for the simple reason that for any consequence relation ⊢, Val(⊢) is always closed under conjunctive combinations, and it is easy to find ∨-Boolean u, v, with u ⋅ v not  ∨-Boolean.[6]  It is interesting to see this as an appeal to the considerations of Section  1  as follows. Consider the binary relational connection between the set of formulas in a language one of whose connectives is ∨ and the class of Boolean valuations, with the relation is true on. We have disjunctive combinations on the left, with φ + ψ as φ ∨ ψ, since the valuations on the right are all ∨-Boolean; therefore we cannot in general have conjunctive combinations on the left, since the crossover condition is not satisfied for this connection. The closure claim applies quite generally, with ∏ V ∈ Val(⊢) whenever V ⊆ Val(⊢). This fact appears in the philosophical literature in the fact that supervaluations over classes of bivalent valuations deliver the same consequence relation as the original valuations, which is accordingly classical when the valuations are Boolean. (Supervaluations are supposed to be non-bivalent, in verifying those formulas verified by all the underlying valuations and falsifying those formulas false on all the underlying valuations, with those in neither category being neither true nor false on the supervaluation. But everything in this definition after the words “verified by all the underlying valuations” is mere rhetoric, since all that matters—for a sequent’s holding on a valuation or a valuation’s being consistent with a consequence relation—is truth-preservation; setting it to one side, we can identify the supervaluation over V with ∏ V. Likewise in the case of ‘subvaluations’—as in Hyde (2005)—where it is ∑ V that would be relevant.) A special case arises with V = ∅, for which ∏ V is the unique valuation vT assigning T to every formula (of the language in question, to which the claim of uniqueness is understood as relativized).