Author: Nikolaos Galatos
Publisher: Elsevier
ISBN: 0080489648
Category : Mathematics
Languages : en
Pages : 532
Book Description
The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double negation is a classical one. We generalise it to the substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where we begin investigating lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones. Within the second, more technical part of the book another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric.
Residuated Lattices: An Algebraic Glimpse at Substructural Logics
Author: Nikolaos Galatos
Publisher: Elsevier
ISBN: 0080489648
Category : Mathematics
Languages : en
Pages : 532
Book Description
The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double negation is a classical one. We generalise it to the substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where we begin investigating lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones. Within the second, more technical part of the book another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric.
Publisher: Elsevier
ISBN: 0080489648
Category : Mathematics
Languages : en
Pages : 532
Book Description
The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double negation is a classical one. We generalise it to the substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where we begin investigating lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones. Within the second, more technical part of the book another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric.
Algebraic Perspectives on Substructural Logics
Author: Davide Fazio
Publisher: Springer Nature
ISBN: 303052163X
Category : Philosophy
Languages : en
Pages : 200
Book Description
This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra & Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over the past two decades by logicians of various persuasions. These researchers include mathematicians, philosophers, linguists, and computer scientists. Substructural logics are applicable to the mathematical investigation of such processes as resource-conscious reasoning, approximate reasoning, type-theoretical grammar, and other focal notions in computer science. They also apply to epistemology, economics, and linguistics. The recourse to algebraic methods -- or, better, the fecund interplay of algebra and proof theory -- has proved useful in providing a unifying framework for these investigations. The AsubL series of conferences, in particular, has played an important role in these developments. This collection will appeal to students and researchers with an interest in substructural logics, abstract algebraic logic, residuated lattices, proof theory, universal algebra, and logical semantics.
Publisher: Springer Nature
ISBN: 303052163X
Category : Philosophy
Languages : en
Pages : 200
Book Description
This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra & Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over the past two decades by logicians of various persuasions. These researchers include mathematicians, philosophers, linguists, and computer scientists. Substructural logics are applicable to the mathematical investigation of such processes as resource-conscious reasoning, approximate reasoning, type-theoretical grammar, and other focal notions in computer science. They also apply to epistemology, economics, and linguistics. The recourse to algebraic methods -- or, better, the fecund interplay of algebra and proof theory -- has proved useful in providing a unifying framework for these investigations. The AsubL series of conferences, in particular, has played an important role in these developments. This collection will appeal to students and researchers with an interest in substructural logics, abstract algebraic logic, residuated lattices, proof theory, universal algebra, and logical semantics.
Hiroakira Ono on Substructural Logics
Author: Nikolaos Galatos
Publisher: Springer Nature
ISBN: 3030769208
Category : Philosophy
Languages : en
Pages : 382
Book Description
This volume is dedicated to Hiroakira Ono life’s work on substructural logics. Chapters, written by well-established academics, cover topics related to universal algebra, algebraic logic and the Full Lambek calculus; the book includes a short biography about Hiroakira Ono. The book starts with detailed surveys on universal algebra, abstract algebraic logic, topological dualities, and connections to computer science. It further contains specialised contributions on connections to formal languages (recognizability in residuated lattices and connections to the finite embedding property), covering systems for modal substructural logics, results on the existence and disjunction properties and finally a study of conservativity of expansions. This book will be primarily of interest to researchers working in algebraic and non-classical logic.
Publisher: Springer Nature
ISBN: 3030769208
Category : Philosophy
Languages : en
Pages : 382
Book Description
This volume is dedicated to Hiroakira Ono life’s work on substructural logics. Chapters, written by well-established academics, cover topics related to universal algebra, algebraic logic and the Full Lambek calculus; the book includes a short biography about Hiroakira Ono. The book starts with detailed surveys on universal algebra, abstract algebraic logic, topological dualities, and connections to computer science. It further contains specialised contributions on connections to formal languages (recognizability in residuated lattices and connections to the finite embedding property), covering systems for modal substructural logics, results on the existence and disjunction properties and finally a study of conservativity of expansions. This book will be primarily of interest to researchers working in algebraic and non-classical logic.
Janusz Czelakowski on Logical Consequence
Author: Jacek Malinowski
Publisher: Springer Nature
ISBN: 3031444906
Category :
Languages : en
Pages : 473
Book Description
Publisher: Springer Nature
ISBN: 3031444906
Category :
Languages : en
Pages : 473
Book Description
Logical Aspects of Computational Linguistics
Author: Nicholas Asher
Publisher: Springer
ISBN: 3662437422
Category : Computers
Languages : en
Pages : 202
Book Description
Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this book constitutes the refereed proceedings of the 8th International Conference on Logical Aspects of Computational Linguistics (LACL 2014) held in Toulouse, France, in June 2014. On the broadly syntactic side, there are papers on the logical and computational foundations of context free grammars, pregroup grammars, on the Lambek calculus and on formalizations of aspects of minimalism. There is also a paper on Abstract Categorical Grammar, as well as papers on issues at the syntax/semantics interface. On the semantic side, the volume's papers address monotonicity reasoning and the semantics of adverbs in type theory, proof theoretical semantics and predicate and argument invariance.
Publisher: Springer
ISBN: 3662437422
Category : Computers
Languages : en
Pages : 202
Book Description
Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this book constitutes the refereed proceedings of the 8th International Conference on Logical Aspects of Computational Linguistics (LACL 2014) held in Toulouse, France, in June 2014. On the broadly syntactic side, there are papers on the logical and computational foundations of context free grammars, pregroup grammars, on the Lambek calculus and on formalizations of aspects of minimalism. There is also a paper on Abstract Categorical Grammar, as well as papers on issues at the syntax/semantics interface. On the semantic side, the volume's papers address monotonicity reasoning and the semantics of adverbs in type theory, proof theoretical semantics and predicate and argument invariance.
Information Processing and Management of Uncertainty in Knowledge-Based Systems
Author: Marie-Jeanne Lesot
Publisher: Springer Nature
ISBN: 3030501531
Category : Computers
Languages : en
Pages : 839
Book Description
This three volume set (CCIS 1237-1239) constitutes the proceedings of the 18th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2020, in June 2020. The conference was scheduled to take place in Lisbon, Portugal, at University of Lisbon, but due to COVID-19 pandemic it was held virtually. The 173 papers were carefully reviewed and selected from 213 submissions. The papers are organized in topical sections: homage to Enrique Ruspini; invited talks; foundations and mathematics; decision making, preferences and votes; optimization and uncertainty; games; real world applications; knowledge processing and creation; machine learning I; machine learning II; XAI; image processing; temporal data processing; text analysis and processing; fuzzy interval analysis; theoretical and applied aspects of imprecise probabilities; similarities in artificial intelligence; belief function theory and its applications; aggregation: theory and practice; aggregation: pre-aggregation functions and other generalizations of monotonicity; aggregation: aggregation of different data structures; fuzzy methods in data mining and knowledge discovery; computational intelligence for logistics and transportation problems; fuzzy implication functions; soft methods in statistics and data analysis; image understanding and explainable AI; fuzzy and generalized quantifier theory; mathematical methods towards dealing with uncertainty in applied sciences; statistical image processing and analysis, with applications in neuroimaging; interval uncertainty; discrete models and computational intelligence; current techniques to model, process and describe time series; mathematical fuzzy logic and graded reasoning models; formal concept analysis, rough sets, general operators and related topics; computational intelligence methods in information modelling, representation and processing.
Publisher: Springer Nature
ISBN: 3030501531
Category : Computers
Languages : en
Pages : 839
Book Description
This three volume set (CCIS 1237-1239) constitutes the proceedings of the 18th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2020, in June 2020. The conference was scheduled to take place in Lisbon, Portugal, at University of Lisbon, but due to COVID-19 pandemic it was held virtually. The 173 papers were carefully reviewed and selected from 213 submissions. The papers are organized in topical sections: homage to Enrique Ruspini; invited talks; foundations and mathematics; decision making, preferences and votes; optimization and uncertainty; games; real world applications; knowledge processing and creation; machine learning I; machine learning II; XAI; image processing; temporal data processing; text analysis and processing; fuzzy interval analysis; theoretical and applied aspects of imprecise probabilities; similarities in artificial intelligence; belief function theory and its applications; aggregation: theory and practice; aggregation: pre-aggregation functions and other generalizations of monotonicity; aggregation: aggregation of different data structures; fuzzy methods in data mining and knowledge discovery; computational intelligence for logistics and transportation problems; fuzzy implication functions; soft methods in statistics and data analysis; image understanding and explainable AI; fuzzy and generalized quantifier theory; mathematical methods towards dealing with uncertainty in applied sciences; statistical image processing and analysis, with applications in neuroimaging; interval uncertainty; discrete models and computational intelligence; current techniques to model, process and describe time series; mathematical fuzzy logic and graded reasoning models; formal concept analysis, rough sets, general operators and related topics; computational intelligence methods in information modelling, representation and processing.
VERY TRUE PSEUDO-BCK ALGEBRAS
Author: LAVINIA CORINA CIUNGU
Publisher: Infinite Study
ISBN:
Category :
Languages : en
Pages : 21
Book Description
In this paper we introduce the very true operators on pseudo-BCK algebras and we study their properties. We prove that the composition of two very true operators is a very true operator if and only if they commute.
Publisher: Infinite Study
ISBN:
Category :
Languages : en
Pages : 21
Book Description
In this paper we introduce the very true operators on pseudo-BCK algebras and we study their properties. We prove that the composition of two very true operators is a very true operator if and only if they commute.
Relational and Algebraic Methods in Computer Science
Author: Jules Desharnais
Publisher: Springer
ISBN: 3030021491
Category : Mathematics
Languages : en
Pages : 394
Book Description
This book constitutes the proceedings of the 17th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2018, held in Groningen, The Netherlands, in October/November 2018. The 21 full papers and 1 invited paper presented together with 2 invited abstracts and 1 abstract of a tutorial were carefully selected from 31 submissions. The papers are organized in the following topics: Theoretical foundations; reasoning about computations and programs; and applications and tools.
Publisher: Springer
ISBN: 3030021491
Category : Mathematics
Languages : en
Pages : 394
Book Description
This book constitutes the proceedings of the 17th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2018, held in Groningen, The Netherlands, in October/November 2018. The 21 full papers and 1 invited paper presented together with 2 invited abstracts and 1 abstract of a tutorial were carefully selected from 31 submissions. The papers are organized in the following topics: Theoretical foundations; reasoning about computations and programs; and applications and tools.
Petr Hájek on Mathematical Fuzzy Logic
Author: Franco Montagna
Publisher: Springer
ISBN: 3319062336
Category : Mathematics
Languages : en
Pages : 324
Book Description
This volume celebrates the work of Petr Hájek on mathematical fuzzy logic and presents how his efforts have influenced prominent logicians who are continuing his work. The book opens with a discussion on Hájek's contribution to mathematical fuzzy logic and with a scientific biography of him, progresses to include two articles with a foundation flavour, that demonstrate some important aspects of Hájek's production, namely, a paper on the development of fuzzy sets and another paper on some fuzzy versions of set theory and arithmetic. Articles in the volume also focus on the treatment of vagueness, building connections between Hájek's favorite fuzzy logic and linguistic models of vagueness. Other articles introduce alternative notions of consequence relation, namely, the preservation of truth degrees, which is discussed in a general context, and the differential semantics. For the latter, a surprisingly strong standard completeness theorem is proved. Another contribution also looks at two principles valid in classical logic and characterize the three main t-norm logics in terms of these principles. Other articles, with an algebraic flavour, offer a summary of the applications of lattice ordered-groups to many-valued logic and to quantum logic, as well as an investigation of prelinearity in varieties of pointed lattice ordered algebras that satisfy a weak form of distributivity and have a very weak implication. The last part of the volume contains an article on possibilistic modal logics defined over MTL chains, a topic that Hájek discussed in his celebrated work, Metamathematics of Fuzzy Logic, and another one where the authors, besides offering unexpected premises such as proposing to call Hájek's basic fuzzy logic HL, instead of BL, propose a very weak system, called SL as a candidate for the role of the really basic fuzzy logic. The paper also provides a generalization of the prelinearity axiom, which was investigated by Hájek in the context of fuzzy logic.
Publisher: Springer
ISBN: 3319062336
Category : Mathematics
Languages : en
Pages : 324
Book Description
This volume celebrates the work of Petr Hájek on mathematical fuzzy logic and presents how his efforts have influenced prominent logicians who are continuing his work. The book opens with a discussion on Hájek's contribution to mathematical fuzzy logic and with a scientific biography of him, progresses to include two articles with a foundation flavour, that demonstrate some important aspects of Hájek's production, namely, a paper on the development of fuzzy sets and another paper on some fuzzy versions of set theory and arithmetic. Articles in the volume also focus on the treatment of vagueness, building connections between Hájek's favorite fuzzy logic and linguistic models of vagueness. Other articles introduce alternative notions of consequence relation, namely, the preservation of truth degrees, which is discussed in a general context, and the differential semantics. For the latter, a surprisingly strong standard completeness theorem is proved. Another contribution also looks at two principles valid in classical logic and characterize the three main t-norm logics in terms of these principles. Other articles, with an algebraic flavour, offer a summary of the applications of lattice ordered-groups to many-valued logic and to quantum logic, as well as an investigation of prelinearity in varieties of pointed lattice ordered algebras that satisfy a weak form of distributivity and have a very weak implication. The last part of the volume contains an article on possibilistic modal logics defined over MTL chains, a topic that Hájek discussed in his celebrated work, Metamathematics of Fuzzy Logic, and another one where the authors, besides offering unexpected premises such as proposing to call Hájek's basic fuzzy logic HL, instead of BL, propose a very weak system, called SL as a candidate for the role of the really basic fuzzy logic. The paper also provides a generalization of the prelinearity axiom, which was investigated by Hájek in the context of fuzzy logic.
Fuzzy Logic and Technology, and Aggregation Operators
Author: Sebastia Massanet
Publisher: Springer Nature
ISBN: 303139965X
Category : Computers
Languages : en
Pages : 755
Book Description
This book constitutes the proceedings of the 13th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2023, and 12th International Summer School on Aggregation Operators, AGOP 2023, jointly held in Palma de Mallorca, Spain, during September 4–8, 2023. The 71 full papers presented in this book were carefully reviewed and selected from 161 submissions. The papers are divided into special sessions on: Interval uncertainty; information fusion techniques based on aggregation functions, preaggregation functions and their generalizations; evaluative linguistic expressions, generalized quantifiers and applications; neural networks under uncertainty and imperfect information; imprecision modeling and management in XAI systems; recent trends in mathematical fuzzy logics; fuzzy graph-based models: theory and application; new frontiers of computational intelligence for pervasive healthcare systems; fuzzy implication functions; and new challenges and ideas in statistical inference and data analysis.
Publisher: Springer Nature
ISBN: 303139965X
Category : Computers
Languages : en
Pages : 755
Book Description
This book constitutes the proceedings of the 13th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2023, and 12th International Summer School on Aggregation Operators, AGOP 2023, jointly held in Palma de Mallorca, Spain, during September 4–8, 2023. The 71 full papers presented in this book were carefully reviewed and selected from 161 submissions. The papers are divided into special sessions on: Interval uncertainty; information fusion techniques based on aggregation functions, preaggregation functions and their generalizations; evaluative linguistic expressions, generalized quantifiers and applications; neural networks under uncertainty and imperfect information; imprecision modeling and management in XAI systems; recent trends in mathematical fuzzy logics; fuzzy graph-based models: theory and application; new frontiers of computational intelligence for pervasive healthcare systems; fuzzy implication functions; and new challenges and ideas in statistical inference and data analysis.