Piero Pagliani

Scientific interests and related researches

My current research interests are in Logico-Algebraic models for Incomplete Information Systems, Knowledge Discovery in Databases and Data Mining, Formal Logic and Philosophy of Language (1).

This section displays my complete publication record after a short philosophical introduction (in the near future, I will add some of my papers to download. All the listed papers are in English, if not otherwise stated).

My Net-friend Gregg Restall maintains that he is a logical pluralist. I am a pluralist too, because, from a neo-conceptualistic interpretation of my work, I can claim that "any specific object has a specific logic" (as already observed by K. Marx). Therefore, I agree that "Logic is in the business of analysing and evaluating arguments, and there are many different standards of validity you can rightly use to do the job".
However, I conjecture that this issue has to be put into the framework of a complex and problematic unifying setting. Of course, if one agrees on the following points:

At least in the Propositional case, Classical Logic is the horizon of any system.

What we need is a unified framework in which several logical systems can be combined.

This ultimate system cannot be a mono-logical system (classical, or intuitionistic or whatever), but a pluralistic system, a system in which several codes can communicate.

As far as I know, to this end we have two major directions nowadays:

The "Unity of Logic" program, derived from Linear Logic, and the present development of the forgotten (or betrayed) Intuitionistic "Proof semantic" program (2).

The "Labelled Deductive Systems" program, as developed by D. Gabbay’s school, together with the connected "Fibred Semantics" (3).

In my view, the first approach proposes a monadic system with a plurality of logical contexts inside, whereas the second goes towards a plurality of monadic systems linked by a metacontext.
In accordance with the second program we have different systems interacting by means of a logical middleware. In accordance with the first, the target system itself must be a meta-system, at the same time.
The LDS program is, in a sense, more comfortable and indeed an impressive number of different systems can be described in an LDS fashion together with their functional interpretations (4).
The "Unity of logic" program is, perhaps, more radical and ambitious and the aforementioned notion of a system as a meta-system has a difficult conceptual coherent implementation.
Anyway, it is worthwhile to compare this approach with U. Eco's understanding of a semiotic code as a meta-code: ''When one speaks of a 'language' as a 'code', one has to think of a wide series of small semantic systems (or fields), which couple with the unities of the meaning system, in different ways. At this point the code starts appearing as [...] the system of the semantic systems and of the rules of semantic combination of the different unities [...]'' (U. Eco, "La struttura assente (The absent structure)", Milano, Bompiani, 1968).

As a matter of fact, a unifying logical framework is an obligatory target for my work, as one may see below.

My first researches had been on logics with strong (or constructive) negations which I applied to understand some algebraic structures of incomplete defined objects, in the sense of Pawlak’s Approximation Spaces (5).
Here we say "incomplete defined", not because for some property we have an information gap (a value gap), but because the parameters (or attributes) at hand are not able to single out each element of the universe of discourse:
(example of Approximation Space [please, click just once])

P. Pagliani, "Some remarks on Special Lattices and related constructive logics with Strong Negation". Notre Dame Journal of Formal Logic, 31 (4), 1990, pp. 515-528.

P. Pagliani, "From Concept Lattices to Approximation Spaces: Algebraic Structures of some Spaces of Partial Objects". Fundamenta Informaticae, 18 (1), 1993, pp. 1-25.

P. Pagliani, "A pure logico-algebraic analysis on rough top and rough bottom equalities". In W. Ziarko (ed.): Rough Sets, Fuzzy Sets and Knowledge Discovery '93. Springer-Verlag, 1994, pp. 227-236.

In my work about information systems, I found that we hardly deal with absolutely incomplete or absolutely precise systems. Usually we have precise and imprecise pieces of information mixed together. This leads to polymorphic systems fulfilling different logical behaviours at the same time.
More precisely, these systems have two extreme logical interpretations:

1) Classical, when there is no imprecise information.

2) Three-valued with an intermediate value (that represents a completely unknown situation), when there is no precise information.

Between these two extreme situations, there is the most general one:
3) When we have mixed (precise and imprecise) information, we do not have a Classical behaviour, of course, but we cannot have an intermediate value, since we cannot have a complete unknown situation (indeed, some information is precise anyway). In this case, since we can distinguish global and local logical behaviours (that collapse in the extreme cases), the system has a number of readings at different levels. Nelson Algebras, three-valued Lukasiewicz algebras, Heyting and co-Heyting algebras, double Stone Algebras and Chain based Lattices are the possible interpretations, endowed with local Boolean environments:
(example [please, click just once] )

P. Pagliani, "Rough Sets and Nelson Algebras", Fundamenta Informaticae, 27 (2-3), 1996, pp. 205-219.

P. Pagliani, ''Rough Set Systems and Logico-algebraic Structures''. In E. Orlowska (ed.): Incomplete Information: Rough Set Analysis, Physica-Verlag, 1997, pp. 109-190.

Moreover, the intermediate value of case 2) above, splits into a local top value and a local bottom value . The local top value validates Excluded Middle in the Heyting algebra reading (as usual, "A or not A" is not uniformly valid in this case). The local bottom value invalidates any contradictions in the co-Heyting reading ("A and not A" does not uniformly equal 0, in co-Heyting algebras, since it individuates the topological boundary of A) (6).
( click once to see a picture).
The local top element and the absolute top element collapse in case 1) above where, dually, the local bottom element collapses into the absolute bottom element.
In case 2) the two local intermediate values collapse into the autodual intermediate value of a Post algebra of order three:
(example [please, click just once] )

P. Pagliani, ''Intrinsic co-Heyting boundaries and information incompleteness in Rough Set Analysis''. In L. Polkowski & A. Skowron (eds.): Rough Sets and Current Trends in Computing. Lecture Notes in Artificial Intelligence 1424, 1998, pp. 123-130.

P. Pagliani, "Local and global logical behaviours: re-visiting Many-Valued Logics through Rough Set Systems". In Proceedings of the 5th European Congress on Intelligent Techniques and Soft Computing, EUFIT '97, Aachen, Germany, 1997, pp. 1592-1596.

P. Pagliani, "Local classical behaviours in three-valued logics and connected systems. An information-oriented analysis" Part 1 and 2. Journal of Multiple Valued Logics (in print).

This approach can be applied to somewhat different situations, such as information systems with information gaps:

P. Pagliani, ''From information gaps to communication needs: a new semantic foundation for some non-classical logics''. Journal of Logic, Language and Information, Vol. 6 (1), 1997, pp. 63-99.

The syntactic counterpart of the systems studied in the above papers, is interesting and presents some degree of difficulty. Here you can find a couple of suggestions:

P. Pagliani, "Towards a logic of rough set systems". In T. Y. Lin & A. M. Wildberger (eds.): Soft Computing, The Society for Comp. Simulation, 1995, pp. 59-62.

P. Pagliani, "Algebraic Models and Proof Analysis: A Simple Case-study". In E. Orlowska (ed.): Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, Physica-Verlag, 1998.

As a matter of fact. I have not developed a logical calculus enjoying a completeness theorem, since I am more interested in directly algebraic calculi for Rough Set Systems. Following this approach, on my personal computer I have implemented several routines that make it possible to compute all the functions provided by Rough Set Theory, using the functional programming language called APL. This implementation is based on a translation of Approximation Spaces into logico-relational structures, by means of a modal relational language (7) . APL provides a number of powerful primitive functions over relations, so that the implementation of this translation is straightforward and correctness is simple to prove. For these topics see:

P. Pagliani, "A modal relation algebra for generalized approximation spaces". In S. Tsumoto, S. Kobayashi, T. Yokomori, H. Tanaka & A. Nakamura (eds.): Proceed. Of the Fourth Int. Workshop on Rough Sets, Fuzzy Sets, and Machine Discovery. November 6-8, 1996, The University of Tokyo, Invited Section "Logic and Algebra", pp. 89-96.

P. Pagliani, "A practical introduction to the modal relational approach to Approximation Spaces". In A. Skowron (ed.): Rough Sets in Knowledge Discovery. Physica-Verlag, 1998, pp. 209-232.

P. Pagliani, "Modalizing Relations by means of Relations: a general framework for two basic approaches to Knowledge Discovery in Database". In Proc. of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU ’98, July, 6-10, 1998. "La Sorbonne", Paris, France, pp. 1175-1182.

In the Conclusions of the last paper above, I introduced the possibility to see modal relation operations for KDD as particular semantic categories, within categorical grammars. In my opinion, this reading could connect KDD with Description Logics, on one side, and with Cognitive Semantics, on the other side.

Moreover, one of the most interesting outcomes of all the above investigations, is the connection of these new mathematical approaches with classical philosophy, expecially with the gnoseological and epistemological problems faced by Locke, Berkeley, Kant, up to Husserl. Something is illustrated in the following paper:

P. Pagliani, "On the philosophical principles underlying some formal concept representation systems" (in Italian). In C. Cellucci, M. C. Di Maio & G. Roncaglia (eds.): Atti del Congresso Triennale 'Logica e Filosofia della Scienza, problemi e prospettive. Lucca 1992, Edizioni ETS, Pisa, pp. 543-561.

Presently, the major project I am involved in is a book about what could be termed a "geometry of approximation systems", in a broad sense.
I am writing the book together with Mihir Chakraborty (Mathematical Department of the University of Calcutta, India).
This work is intended for advanced undergraduates and postgraduates as well as other researchers in the area. When publication details are finalised, this page will be updated.

Professional interests as related to my scientific research

About Knowledge Management: I have accomplished a long study for an important telecommunication company, for which I have defined an Enterprise Ontology.
Based on that experience, but intended for a more philosophically oriented audience, I have written a short paper:

P. Pagliani, "Knowledge and Knowledge Management" (In Italian). Prometheus (in print).

Moreover, the following papers are connected to both my professional and scientific interests:

C: Zucchermaglio, M. Fantasia & P. Pagliani, "Skill Needs in Expert Systems". COMETT Conference Proceedings: Towards new models of university-industry cooperation: the example of COMETT, November 13-15, 1991, Amsterdam, The Netherlands, pp.52-54.

C. Zucchermaglio & P. Pagliani, "Organizational and cognitive design of technological learning environment". In: NATO Advanced Workshop on Organizational Learning and Technological Change. September 22-26, 1992, Certosa di Pontignano, Siena, Italy, pp. 74-80.

P. Pagliani, "Purposes and motivations of Formal Concept Analysis" (in Italian). In G. Negrini (ed.): Conference proc. 'Models and Modeling', May 17,1996, Roma, Italy. CNR-Roma, Note di bibliografia e di documentazione scientifica-LXIV, pp. 27-38.

If you are interested in any of this material please mail me or read the material I've written on these things. I welcome comments on it.

Piero Pagliani Research Group on Knowledge and Communication Models Via Imperia, 6. 00161 Rome, Italy p.pagliani@agora.stm.it

Notes

(1) For these topics, I suggest this crazy link.

(2) See: J-Y. Girard, ''On the Unity of Logics''. Annals of Pure and Applied Logic, 59, North-Holland, 1993, pp. 201-217.

(3) See D. M. Gabbay, "Labelled Deductive Systems". Oxford University Press, 1997 and D. M. Gabbay, ''Fibred semantics and the weaving of logics. Part I: Modal and Intuitionistic Logics''. Journal. Of Symbolic Logic, 61-4, 1996, pp. 1057-1120.

(4) See: D. M. Gabbay & R. J. G. B. De Queiroz, ''Extending the Curry-Howard interpretation to Linear, Relevant and other Resource Logics''. Journal. of Symbolic Logic, 57-4, 1992, pp. 1319-1365.

(5) See Z. Pawlak, " Rough Sets: A Theoretical Approach to Reasoning about Data". Kluwer, 1991.

(6) See F. W. Lawvere, Introduction to F. W. Lawvere & S. Schanuel (eds.): Categories in Continuum Physics, Lecture Notes in Mathematics, 1174, Springer-Verlag, 1986.

(7) For this notion see E. Orlowska, "Relational Interpretation of Modal Logics". In H. Andreka, J. D. Monk & I. Nemeti (eds.): Algebraic Logic, North-Holland, 1988, pp. 443-471.