 A Set-Set sequent Γ≻ Δ holds on a valuation v if and only if it is not the case that v verifies every formula in Γ while falsifying every formula in Δ; note that this coincides with the definition of “holding on v” given for Set-Fmla in the case in which Δ = {ψ} for some ψ. Similarly, we have the notion of a generalized consequence relation in which suitably two-sided versions of the conditions definitive of consequence relations are given, which can be found in Shoesmith and Smiley (1978), Chapter 2, for example (under the name ‘multiple-conclusion consequence relation’). If ⊩ is such a relation on a language L then an L-valuation v is said to be consistent with ⊩ when there do not exist Γ, Δ, with v(φ) = T for all φ ∈ Γ and v(ψ) = F for all ψ ∈ Δ. The class of all such valuations is denoted, much as in the consequence relation case, by Val(⊩). Sometimes formulations appear below with the phrase “(generalized) consequence relation”, meaning that the claim being made or considered applies both in the case of consequence relations and in the case of generalized consequence relations.