Giulia Battilotti

Basic Logic: Reflection, Symmetry, Visibility

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.

Pretopologies and a uniform presentation of sup-lattices, quantales and frames

Abstract 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.

BASIC LOGIC AND QUANTUM COMPUTING: LOGICAL JUDGEMENTS BY AN INSIDER OBSERVER

We consider the logical assertions of a hypothetical observer who is inside a quantum computer and performs a reversible quantum measurement, obtaining a symmetric couple of new axioms, valid only inside the quantum computer. The result is that, in this logical framework, symmetry and paraconsistency hold.

Quantum States as Virtual Singletons: Converting Duality into Symmetry

In a predicative framework from basic logic, defined for a model of quantum parallelism by sequents, we characterize a class of first order domains, called 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 states. So, despite not having sense in a traditional logical setting, symmetry can hide the peculiar advantage of quantum mechanics for the treatment of information.

Characterization of Quantum States in Predicative Logic

We develop a characterization of quantum states by means of first order variables and random variables, within a predicative logic with equality, in the framework of basic logic and its definitory equations.We introduce the notion of random first order domain and find a characterization of pure states in predicative logic and mixed states in propositional logic, due to a focusing condition. We discuss the role of first order variables and the related contextuality, in terms of sequents.

Symmetry in sequent calculus and Matte Blanco's bi-logic

We discuss the problem of symmetry, namely of the orientation of the logical consequence, represented by the sequent sign in sequent calculus. We show that the problem is surprisingly entangled with the problem of “being infinite”. We present a model based on quantum states and we show that the requirements of Matte Blancos symmetric mode are satisfied. We briefly discuss the model for symmetry to include correlations, in order to obtain a possible approach to displacement. In this setting, we find a possible reading of the structural rules of sequent calculus, whose role in computation, on one side, and in the representation of human reasoning, on the other, has been debated for a long time.