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Basic logic: reflection, symmetry, visibility

Published online by Cambridge University Press:  12 March 2014

Giovanni Sambin ,
Giulia Battilotti  and
Claudia Faggian
Giovanni Sambin
Affiliation:
Dipartimento di Matematica Pura Ed Applicata, Università di Padova, Via Belzoni 7, I-35131 Padova, Italy E-mail: sambin@math.unipd.it
Giulia Battilotti
Affiliation:
Dipartimento di Matematica Pura Ed Applicata, Università di Padova, Via Belzoni 7, I-35131 Padova, Italy E-mail: giulia@math.unipd.it
Claudia Faggian
Affiliation:
Dipartimento di Matematica Pura Ed Applicata, Università di Padova, Via Belzoni 7, I-35131 Padova, Italy E-mail: claudia@math.unipd.it
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 979 - 1013
Copyright
Copyright © Association for Symbolic Logic 2000

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