The glorious past! A grim future? Petr Cintula 1 and Carles Noguera 2 - - PowerPoint PPT Presentation
The glorious past! A grim future? Petr Cintula 1 and Carles Noguera 2 1 Institute of Computer Science, Czech Academy of Sciences, Prague, Czech Republic 2 Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech
Petr Cintula1 and Carles Noguera2
1Institute of Computer Science,
Czech Academy of Sciences, Prague, Czech Republic
2Institute of Information Theory and Automation,
Czech Academy of Sciences, Prague, Czech Republic
Petr Cintula and Carles Noguera (CAS) The future of MFL I 1 / 27
On a foggy autumn day nearly 800 years ago a traveller happened upon a large group of workers adjacent to the River Avon. Curiosity convinced him to inquire about their work.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 2 / 27
On a foggy autumn day nearly 800 years ago a traveller happened upon a large group of workers adjacent to the River Avon. Curiosity convinced him to inquire about their work. With a slight detour he moved toward the first of the three tradesmen and said “my dear fellow what is it that you are doing?”
Petr Cintula and Carles Noguera (CAS) The future of MFL I 2 / 27
On a foggy autumn day nearly 800 years ago a traveller happened upon a large group of workers adjacent to the River Avon. Curiosity convinced him to inquire about their work. With a slight detour he moved toward the first of the three tradesmen and said “my dear fellow what is it that you are doing?” The man continued his work and grumbled, “I am cutting stones.”
Petr Cintula and Carles Noguera (CAS) The future of MFL I 2 / 27
On a foggy autumn day nearly 800 years ago a traveller happened upon a large group of workers adjacent to the River Avon. Curiosity convinced him to inquire about their work. The traveller moved toward the second of the three and repeated the
Petr Cintula and Carles Noguera (CAS) The future of MFL I 2 / 27
On a foggy autumn day nearly 800 years ago a traveller happened upon a large group of workers adjacent to the River Avon. Curiosity convinced him to inquire about their work. The traveller moved toward the second of the three and repeated the
“I am stonecutter, I came to Salisbury from the north to work but as soon as I earn ten quid I will return home.”
Petr Cintula and Carles Noguera (CAS) The future of MFL I 2 / 27
On a foggy autumn day nearly 800 years ago a traveller happened upon a large group of workers adjacent to the River Avon. Curiosity convinced him to inquire about their work. When he reached the third worker he once again asked the original
Petr Cintula and Carles Noguera (CAS) The future of MFL I 2 / 27
On a foggy autumn day nearly 800 years ago a traveller happened upon a large group of workers adjacent to the River Avon. Curiosity convinced him to inquire about their work. When he reached the third worker he once again asked the original
“I am a mason and I am building a cathedral.”
Petr Cintula and Carles Noguera (CAS) The future of MFL I 2 / 27
The worker continued, I have journeyed many miles to be part of the team that is constructing this magnificent cathedral. I have spent many months away from my family and I miss them dearly. However, I know how important Salisbury Cathedral will be
solace here. I know this because the Bishop once told me his vision for this great place. He described how people would come from all parts to worship here. He also told that the Cathedral would not be completed in our days but that the future depends on our hard work.
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Petr Cintula and Carles Noguera (CAS) The future of MFL I 4 / 27
Petr Cintula and Carles Noguera (CAS) The future of MFL I 4 / 27
Problem:
Petr Cintula and Carles Noguera (CAS) The future of MFL I 5 / 27
Problem: There are heaps of sand
Petr Cintula and Carles Noguera (CAS) The future of MFL I 5 / 27
Problem: There are heaps of sand If I remove one grain from a heap of sand, I still have a heap of sand.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 5 / 27
Problem: There are heaps of sand If I remove one grain from a heap of sand, I still have a heap of sand. Therefore, one grain of sand is a heap of sand.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 5 / 27
Problem:
Petr Cintula and Carles Noguera (CAS) The future of MFL I 6 / 27
Problem: Classical logic is boring http://www.xkcd.com/74/
Petr Cintula and Carles Noguera (CAS) The future of MFL I 6 / 27
Problem: Classical logic is boring http://www.xkcd.com/74/ Having more truth degrees to play with is fun even for such noble logicians as: Łukasiewicz, Tarski, Gödel, Skolem, Horn, . . .
Petr Cintula and Carles Noguera (CAS) The future of MFL I 6 / 27
Problem:
Petr Cintula and Carles Noguera (CAS) The future of MFL I 7 / 27
Problem: to deal with too much precision in control theory
Petr Cintula and Carles Noguera (CAS) The future of MFL I 7 / 27
Pavelka 1979 Pultr 1984 Takeuti, Titani 1984, 1992 Mundici 1987, 1993 Novák 1990 Gottwald 1993 Höhle 1995 Baaz 1996 Hájek–Esteva–Godo 1996 Hájek, 1998 Novák, Perfilileva, Moˇ ckoˇ r, 1999 Cignoli, d’Ottaviano, Mundici, 2000
Petr Cintula and Carles Noguera (CAS) The future of MFL I 8 / 27
It was established as a respectable branch of mathematical logic by Petr Hájek in Metamathematics of fuzzy logic Kluwer,1998. Citations: 1400+ (WOS) and 3500+ (Google Scholar)
Petr Cintula and Carles Noguera (CAS) The future of MFL I 9 / 27
This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of real-valued propositional and predicate calculus are defined and investigated.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 10 / 27
This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of real-valued propositional and predicate calculus are defined and investigated. The aim is to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named ‘fuzzy inference’ can be naturally understood as logical deduction.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 10 / 27
This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of real-valued propositional and predicate calculus are defined and investigated. The aim is to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named ‘fuzzy inference’ can be naturally understood as logical deduction. There are two main groups of intended readers.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 10 / 27
This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of real-valued propositional and predicate calculus are defined and investigated. The aim is to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named ‘fuzzy inference’ can be naturally understood as logical deduction. There are two main groups of intended readers. First, logicians: they can see that fuzzy logic is indeed a branch of logic and may find several very interesting open problems.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 10 / 27
This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of real-valued propositional and predicate calculus are defined and investigated. The aim is to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named ‘fuzzy inference’ can be naturally understood as logical deduction. There are two main groups of intended readers. Second, equally important, researchers involved in fuzzy logic applica- tions and soft computing.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 10 / 27
As a matter of fact, most of these are not professional logicians so that it can easily happen that an application, clever and successful as it may be, is presented in a way which is logically not entirely correct or may appear simple-minded. This fact would not be very important if only the bon ton of logicians were harmed; but it is the opinion of the author that a better understanding of the strictly logical basis of fuzzy logic (in the usual broad sense) is very useful for fuzzy logic appliers since if they know better what they are doing, they may hope to do it better. In addition, a better mutual understanding between (classical) logicians and researchers in fuzzy logic, promises to lead to deeper cooperation and new results.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 11 / 27
Mathematical Principles of Fuzzy Logic provides a systematic study of the formal theory of fuzzy logic. The book is based on logical formalism demonstrating that fuzzy logic is a well-developed logical theory. It includes the theory of functional systems in fuzzy logic, providing an explanation of what can be represented, and how, by formulas of fuzzy logic calculi. It also presents a more general interpretation of fuzzy logic within the environment of other proper categories of fuzzy sets stemming either from the topos theory, or even generalizing the latter. This book presents fuzzy logic as the mathematical theory of vagueness as well as the theory of commonsense human reasoning, based on the use of natural language, the distinguishing feature of which is the vagueness of its semantics.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 12 / 27
This unique textbook states and proves all the major theorems of many- valued propositional logic and provides the reader with the most re- cent developments and trends, including applications to adaptive error- correcting binary search.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 13 / 27
This unique textbook states and proves all the major theorems of many- valued propositional logic and provides the reader with the most re- cent developments and trends, including applications to adaptive error- correcting binary search. The book is suitable for self-study, making the basic tools of many- valued logic accessible to students and scientists with a basic math- ematical knowledge who are interested in the mathematical treatment
Petr Cintula and Carles Noguera (CAS) The future of MFL I 13 / 27
Stressing the interplay between algebra and logic, the book contains material never before published, such as a simple proof of the comple- teness theorem and of the equivalence between Chang’s MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 13 / 27
Stressing the interplay between algebra and logic, the book contains material never before published, such as a simple proof of the comple- teness theorem and of the equivalence between Chang’s MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic. Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean par- titions can be defined and computed in classical logic. Detailed biblio- graphic remarks at the end of each chapter and an extensive bibliogra- phy lead the reader on to further specialised topics.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 13 / 27
The following disciplines
were, and still are, developed in MFL proof theory model theory set theory recursion theory complexity theory
Metcalfe, Olivetti, Gabbay: Proof Theory for Fuzzy Logics. Springer, 2009. Hájek, Cintula: On theories and models in fuzzy predicate logics. Journal of Symbolic Logic, 2006. Hájek, Haniková: A Development of Set Theory in Fuzzy Logic. In Beyond Two, 2003. Hájek: Arithmetical Complexity of Fuzzy Predicate Logics. Annals of Pure and Applied Logic, 2009. Baaz, Hájek, Montagna, Veith: Complexity of t-tautologies. Annals of Pure and Applied Logic, 2002.
Petr Cintula and Carles Noguera (CAS) The future of MFL I 14 / 27
The connections of MFL with the following areas
and still are, explored: lattice theory group/field theory geometry game theory topology category theory measure theory
Mundici, Cignoli: An equivalence between MV-algebras and ℓ-groups with strong unit. Studia Logica. 1998. Mundici, Manara, Marra: Lattice-ordered Abelian groups and Schauder bases of unimodular fans. Transactions of the American Mathematical Society. 2007 Fermüller. Dialogue Games for Many- Valued Logics. Studia Logica, 2008.
set theory with applications to algebra and
Correspond to Probabilities of Rational
Approximate Reasoning, 2012.
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P . Cintula, C. Fermüller, P . Hájek, C. Noguera (editors).
College Publications, 2011, 2015.
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