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We define and study a notion of , which is a broadening of the familiar notion of classical consequence. The concept is studied first in its simplest form, and then in a syntax-independent version which we call .

Beyond the specific results obtained, a general message emerging from the investigation is that even in such an elementary and well-trodden area as classical propositional logic, there are intriguing concepts that remain little explored, and non-trivial problems concerning them. Once they are well-understood in a classical setting, the concepts can also be formulated in a more general manner to be applicable to a wide variety of non-classical logics.

We discuss conditions under which the following ‘truism’ does indeed express a truth: the weaker a logic is in terms of what it proves, the stronger it is as a tool for registering distinctions amongst the formulas in its language.