Linear logic is a refinement of classical and intuitionistic logic. Instead of emphasizing truth, as in classical logic, or proof, as in intuitionistic logic, linear logic emphasizes the role of formulas as resources.To achieve this focus, linear logic does not allow the usual structural rules of contraction and weakening to apply to all formulas but only those formulas marked with certain modals. Transformations. If a formula A logically entails a formula B and B also entails A , written A ⊣⊢ B , then we say that A and B are logically equivalent. In propositional logic, what this amounts to is that they have the same truth table, or express the same truth function. It is a fundamental feature of logic that if two formulae are.