Basic logic: Reflection, symmetry, visibility

Journal of Symbolic Logic 65 (3):979-1013 (2000)
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Abstract

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.

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Author's Profile

Giulia Battilotti
University of Florence (PhD)

References found in this work

Elements of Intuitionism.Michael Dummett - 1980 - British Journal for the Philosophy of Science 31 (3):299-301.
Proof Theory.Gaisi Takeuti - 1990 - Studia Logica 49 (1):160-161.
Display logic.Nuel D. Belnap - 1982 - Journal of Philosophical Logic 11 (4):375-417.
Logical constants as punctuation marks.Kosta Došen - 1989 - Notre Dame Journal of Formal Logic 30 (3):362-381.
On the unity of logic.Jean-Yves Girard - 1993 - Annals of Pure and Applied Logic 59 (3):201-217.

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