by Giovanni Sambin, Giulia Battilotti, Claudia Faggian
Journal of Symbolic Logic
http://www.math.unipd.it/~claudia/download/B2main.ps
Add To MetaCart
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,
Citations
|
479
|
On the unity of logic
– Girard
- 1993
|
|
378
|
Untersuchungen über das logische Schliessen
– Gentzen
- 1934
|
|
173
|
Labelled Deductive Systems
– Gabbay
- 1996
|
|
94
|
1977] Elements of Intuitionism
– Dummett
|
|
77
|
Display logic
– Belnap
- 1982
|
|
13
|
1908], De onbetrouwbaarheid der logische principes, Tijdschrift voor Wijsbegeerte 2, 152-158. English trans. The unreliability of the logical principles
– Brouwer
- 1975
|
|
9
|
Basic logic and the cube of its extensions
– Battilotti, Sambin
- 1996
|
|
8
|
From basic logic to quantum logics with cut-elimination
– Faggian, Sambin
- 1997
|
|
8
|
A new and elementary method to represent every complete Boolean algebra
– Sambin
- 1996
|
|
7
|
Pretopologies and a uniform presentation of suplattices, quantales and frames
– Battilotti, Sambin
- 2006
|
|
6
|
On the meanings of the logical constants and the justifications of the logical laws
– of
- 1996
|
|
2
|
Basic logic through the reflection principle
– Battilotti
- 1997
|
|
2
|
Per una dinamica nei fondamenti (italian), in Atti del Congresso "Nuovi problemi della logica e della filosofia della scienza
– Sambin
- 1990
|
|
2
|
Basic logic, a structure in the space of logic
– Sambin
- 1998
|
|
1
|
From basic logic to full intuitionistic linear logic
– Battilotti
- 1997
|
|
1
|
Logica di base attraverso il principio di riflessione
– Battilotti
- 1997
|
|
1
|
Quantum logic and the cube of logics
– Battilotti, Faggian
- 1997
|
|
1
|
de Bruijn, Type-theoretical checking and the philosophy of mathematics, in Twenty-five years of constructive type
– G
- 1997
|
|
1
|
Logical consequence: a turn in style
– sen
- 1997
|
|
1
|
Teorema di eliminazione del taglio in basic logic e nelle logiche quantistiche. Tesi di Laurea in Matematica, Universit`a di Padova
– Faggian
- 1996
|
|
1
|
proofs via basic logic
– Classical
- 1997
|
|
1
|
e, A uniform display system for intuitionistic and dual intuitionistic logic
– Gor'
- 1995
|
|
1
|
Intuitionistic type theory", notes by Giovanni Sambin of a series of lectures given
– of
- 1980
|
|
1
|
Dipartimento di Matematica Pura ed Applicata Universit`a di Padova Via Belzoni 7, I--35131 Padova, Italy e-mail: sambin,giulia,claudia@math.unipd.it
– Takeuti, Theory, et al.
- 1975
|
