Research areas
Basic logic
Basic logic offers a unified and simple approach to the treatment of the
various weakenings of classical logic (mainly: intuitionistic, linear
and quantum logics).
The first formulation of basic logic consisted of a sequent calculus
which can prove theorems which hold in the three logics at the same
time.
It is contained in
G. Battilotti, G. Sambin, Basic logic and the cube of its extensions,
Proceedings of LMPS, Florence 1995, A. Cantini, E. Casari, and P. Minari,
eds., Kluwer.
Then, focusing on the proof-theoretical aspects of the problem, a radically
new formulation of basic logic has been obtained, which is symmetric and
which enjoys cut-elimination, due to
visibility (i.e.,
in every operational rule, all active formulae have no context).
The new calculus is justified by means of the reflection principle,
introduced by G. Sambin, which states the fact that logical constants
are the result of importing into a formal system a pre-existing metalinguistic
link between assertions.
The new formulation of basic logic together with the general formulation
of the reflection principle is contained in
G. Sambin, G. Battilotti and C. Faggian, Basic logic: Reflection, symmetry,
visibility, Journal of Symbolic Logic, 65 (2000) pp. 979-1013.
Basic logic can be characterized as the logic of connectives. The other
logics are then obtained as extensions, in a uniform framework.
Our aim is to obtain each extension by the addition of structural rules
(on contexts), while keeping fixed the rules on connectives. We already
have achieved this for the "symmetric" logics, including orthologic,
linear and classical logic, while it seems within reach for the asymmetric
logics, including intuitionistic logic.
Other published papers in the field of basic logic:
- C.
Faggian, G. Sambin, From basic logic to quantum logics with cut-elimination,
Proceedings of the International Quantum Structures Association (Berlin,
96). International Journal of Theorethical Physics, 12.97.
Orthologic is obtained as an extension of basic logic yielding a cut-free
formulation for such a logic and a wide range of new quantum-like
logics, including linear orthologic.
-
C. Faggian, Classical proofs via basic logic, Proceedings of CSL'97
(Aahrus, 97). LNCS, Springer-Verlag, in press.
A new calculus for classical logic is introduced as basic logic plus
structural rules. The features of the calculus benefit both the proof
search and the cut-elimination process.
-
G. Battilotti, Embedding classical logic into basic orthologic with
a primitive modality, Logic Journal of the IGPL, special issue
on Generalized Sequent Systems, H. Wansing ed., Oxford Univ. Press,
pp. 383-402.
The first proof-theoretical embedding of classical logic into a quantum-like
logic is given. The logic so obtained is interpretable as the coexistence
of classical and quantum logic.
- G. Battilotti,
C. Faggian, Quantum logic and the cube of logics, in Handbook
of Philosophical Logic (new edition), D. Gabbay, F. Guenthner (eds.),
Kluwer, Vol. VII, Ch. Quantum Logic by M.L. Dalla Chiara and R. Giuntini.
A survey about the treatment of quantum logic in the framework of
basic logic.
Other available papers can be found in the homepages of Giovanni
Sambin, Giulia
Battilotti, Claudia
Faggian, Damiano Macedonio.