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The distinction between language and metalanguage is in my opinion a fundamental fact of life. It is necessary to explain how language acquires meaning.

It is interesting that this can be made precise in logic. In fact, one can see that each logical constant is the reflection at the level of propositions (language) of a link between assertions (metalanguage). This is expressed by requiring an equivalence to hold, which defines that logical constant implicitly. It is a virtuous circle, since one can obtain an explicit solution, that is the inference rules; they explain the meaning of the logical constant inductively.

This I call the principle of reflection. The least logical system in which all logical constants are obtained in this way is called basic logic.

The role of basic logic is that of reaching unification by giving a conceptual structure to the space of logic: each of the most common logics (linear, intuitionistic, classical, ortho,...) is obtained as a simple and natural extension of basic logic, by adding assumptions on the structure of consequence (structural rules). The main reference is Basic logic: Reflection, symmetry, visibility, written with Giulia Battilotti and Claudia Faggian; section 5 of Steps gives a short introduction.

The novelty of basic logic also from a technical point of view is shown for instance by the fact that it yields a simple cut-free formulation for orthologic.

Monoids with an arbitrary relation provide with a mathematical interpretation and a completeness theorem for basic logic (except implication, which has to be undertstood better). Phase spaces coincide with monoids with a strongly symmetric relation. See Relational semantics, written with Damiano Macedonio.

Logic, the magic of reflection