 In classical logic we can take φ #1 ψ and φ #2 ψ as (equivalent to) ¬φ and ¬ψ respectively, and φ #3 ψ as ⊥. There are no three candidates playing the desired role in any conservative extension of intuitionistic logic, however, because the inset equivalences render arbitrary φ and ψ head-linked in the sense of Section  3,  with φ #3 ψ as the head formula.