“…The non-modal fragment of the logics we are going to study is the multi-valued logic of Arieli and Avron [1] (henceforth referred to as bilattice logic or Arieli-Avron logic), which arose from Belnap's famous proposal for a "useful four-valued logic" [3]. Belnap saw that his four truth values can be arranged as lattices in two distinct but equally meaningful ways, ordering them either by information content (the knowledge order ≤ ) or by logical strength (the truth order ≤ ), see Generally, a (bounded) bilattice is an algebra ⟨ , ∧, ∨, ⊗, ⊕, ¬, f, t, ⊥, ⊤⟩ such that ⟨ , ∧, ∨, f, t⟩ and ⟨ , ⊗, ⊕, ⊥, ⊤⟩ are both bounded lattices.…”