We introduce a sequent calculus B for a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic. quantum and non-modal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate three properties, which characterize B positively: reflection, symmetry and visibility.A logical constant obeys to the principle of reflection if it is characterized semantically by an equation binding it with a metalinguistic link between assertions, and if its syntactic inference rules are obtained by solving that equation. All connectives of basic logic satisfy reflection.To the control of weakening and contraction of linear logic, basic logic adds a strict control of contexts, by requiring that all active formulae in all rules are isolated, that is visible. From visibility, cut-elimination follows. The full, geometric symmetry of basic logic induces known symmetries of its extensions, and adds a symmetry among them, producing the structure of a cube.

We introduce the notion of infinitary preorder and use it to obtain a predicative presentation of sup-lattices by generators and relations. The method is uniform in that it extends in a modular way to obtain a presentation of quantales, as “sup-lattices on monoids”, by using the notion of pretopology. Our presentation is then applied to frames, the link with Johnstone’s presentation of frames is spelled out, and his theorem on freely generated frames becomes a special case of our results on quantales. The main motivation of this paper is to contribute to the development of formal topology. That is why all our definitions and proofs can be expressed within an intuitionistic and predicative foundation, like constructive type theory

In a predicative framework from basic logic, defined for a model of quantum parallelism by sequents, we characterize a class of first order domains, termed virtual singletons, which allows a generalization of the notion of duality, termed symmetry. Although consistent with the classical notion of duality, symmetry creates an environment where negation has fixed points, for which the direction of logical consequence is irrelevant. Symmetry can model Bell's states. So, despite its nonsense in a traditional logical setting, symmetry can hide the peculiar advantage for the treatment of information, that is proper of quantum mechanics.

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We introduce an interpretation of quantum superposition in predicative sequent calculus, in the framework of basic logic. Then we introduce a new predicative connective for the entanglement. Our aim is to represent quantum parallelism in terms of logical proofs.