The consequence relation in the logic of commutative GBL-algebras is PSPACE-complete
Theoretical Computer Science
The radical of a perfect residuated structure
Information Sciences: an International Journal
A note on intervals of residuated ℓ-groupoids
Fuzzy Sets and Systems
Nonassociative Lambek Calculus with Additives and Context-Free Languages
Languages: From Formal to Natural
Soft Constraints Processing over Divisible Residuated Lattices
ECSQARU '09 Proceedings of the 10th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty
Fuzzy Sets and Systems
Fuzzy logics from substructural perspective
Fuzzy Sets and Systems
Triangular norm based predicate fuzzy logics
Fuzzy Sets and Systems
MTL-algebras arising from partially ordered groups
Fuzzy Sets and Systems
Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics
Information Sciences: an International Journal
Continuation semantics for the Lambek--Grishin calculus
Information and Computation
On v-filters of residuated lattices with hedges
FSKD'09 Proceedings of the 6th international conference on Fuzzy systems and knowledge discovery - Volume 6
Expanding the realm of systematic proof theory
CSL'09/EACSL'09 Proceedings of the 23rd CSL international conference and 18th EACSL Annual conference on Computer science logic
Confluence and termination of fuzzy relations
Information Sciences: an International Journal
Grammar Induction by Unification of Type-logical Lexicons
Journal of Logic, Language and Information
Pseudo-BCK algebras as partial algebras
Information Sciences: an International Journal
Relational semantics for the Lambek-Grishin calculus
MOL'07/09 Proceedings of the 10th and 11th Biennial conference on The mathematics of language
Untyping typed algebraic structures and colouring proof nets of cyclic linear logic
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
Completeness with respect to a chain and universal models in fuzzy logic
Archive for Mathematical Logic
Extending conceptualisation modes for generalised Formal Concept Analysis
Information Sciences: an International Journal
Nuclei and conuclei on residuated lattices
Fuzzy Sets and Systems
Archive for Mathematical Logic
Fuzzy Sets and Systems
Strict core fuzzy logics and quasi-witnessed models
Archive for Mathematical Logic
Leibniz-linked Pairs of Deductive Systems
Studia Logica
Some extensions of the logic psUL
AICI'11 Proceedings of the Third international conference on Artificial intelligence and computational intelligence - Volume Part I
Generalized Bosbach and Riečan states based on relative negations in residuated lattices
Fuzzy Sets and Systems
Reasoning about mathematical fuzzy logic and its future
Fuzzy Sets and Systems
Amalgamation through quantifier elimination for varieties of commutative residuated lattices
Archive for Mathematical Logic
Logical grammars, logical theories
LACL'12 Proceedings of the 7th international conference on Logical Aspects of Computational Linguistics
Glivenko theorems and negative translations in substructural predicate logics
Archive for Mathematical Logic
Archive for Mathematical Logic
Polynomial space hardness without disjunction property
Theoretical Computer Science
On Birkhoff's Common Abstraction Problem
Studia Logica
Studia Logica
Metacompleteness of Substructural Logics
Studia Logica
On the Ordering Property and Law of Importation in Fuzzy Logic
International Journal of Artificial Life Research
Formal concept analysis and lattice-valued Chu systems
Fuzzy Sets and Systems
Institutional semantics for many-valued logics
Fuzzy Sets and Systems
Łukasiewicz logic: an introduction
TbiLLC'11 Proceedings of the 9th international conference on Logic, Language, and Computation
Fuzzy Sets and Systems
Residual implications on the set of discrete fuzzy numbers
Information Sciences: an International Journal
Archive for Mathematical Logic
Generalized Bosbach and Riečan states on nucleus-based-Glivenko residuated lattices
Archive for Mathematical Logic
Generalized Bosbach states: Part II
Archive for Mathematical Logic
Fuzzy Sets and Systems
Kripke Semantics for Modal Bilattice Logic
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
Derivation digraphs for dependencies in ordinal and similarity-based data
Information Sciences: an International Journal
Hi-index | 0.00 |
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. - Considers both the algebraic and logical perspective within a common framework. - Written by experts in the area. - Easily accessible to graduate students and researchers from other fields. - Results summarized in tables and diagrams to provide an overview of the area. - Useful as a textbook for a course in algebraic logic, with exercises and suggested research directions. - Provides a concise introduction to the subject and leads directly to research topics. - The ideas from algebra and logic are developed hand-in-hand and the connections are shown in every level.