Modal operators on bounded commutative residuated ℓ-monoids

2007 ◽  
Vol 57 (4) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Fuzzy Logic ◽  
Residuated Lattice ◽  
Heyting Algebras ◽  
Modal Operator ◽  
Modal Operators ◽  
Special Cases ◽  
Basic Fuzzy Logic ◽  
Commutative Residuated Lattice ◽  
Derived Algebras ◽  
Mv Algebras

AbstractBounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., Heyting algebras and BL-algebras, i.e., algebras of intuitionistic logic and basic fuzzy logic, respectively. Modal operators (special cases of closure operators) on Heyting algebras were studied in [MacNAB, D. S.: Modal operators on Heyting algebras, Algebra Universalis 12 (1981), 5–29] and on MV-algebras in [HARLENDEROVÁ,M.—RACHŮNEK, J.: Modal operators on MV-algebras, Math. Bohem. 131 (2006), 39–48]. In the paper we generalize the notion of a modal operator for general bounded commutative Rℓ-monoids and investigate their properties also for certain derived algebras.

Perfect residuated lattice ordered monoids

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Probability Measure ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Heyting Algebras ◽  
Effect Algebras ◽  
Quantum Logics ◽  
Mv Algebras ◽  
Intuitionistic Logics

AbstractBounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).

scholarly journals Residuated Structures and Orthomodular Lattices

Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Algebraic Logic ◽  
Residuated Lattices ◽  
Quantum Structures ◽  
Heyting Algebras ◽  
Quantum Algebras ◽  
Orthomodular Lattices ◽  
Lattice Of Subvarieties ◽  
Residuated Structures ◽  
Common Framework ◽  
Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

MEDUSA: A Fuzzy Expert System for Medical Diagnosis of Acute Abdominal Pain

1994 ◽  
Vol 33 (05) ◽  
pp. 522-529 ◽  
Author(s):  
M. Fathi-Torbaghan ◽  
D. Meyer
Keyword(s):  
Fuzzy Logic ◽  
Abdominal Pain ◽  
Expert System ◽  
Acute Abdominal Pain ◽  
Medical Knowledge ◽  
Clinical Problem ◽  
Fuzzy Relations ◽  
Diagnostic Decision ◽  
Special Cases ◽  
Hybrid Concept

Abstract:Even today, the diagnosis of acute abdominal pain represents a serious clinical problem. The medical knowledge in this field is characterized by uncertainty, imprecision and vagueness. This situation lends itself especially to be solved by the application of fuzzy logic. A fuzzy logic-based expert system for diagnostic decision support is presented (MEDUSA). The representation and application of uncertain and imprecise knowledge is realized by fuzzy sets and fuzzy relations. The hybrid concept of the system enables the integration of rulebased, heuristic and casebased reasoning on the basis of imprecise information. The central idea of the integration is to use casebased reasoning for the management of special cases, and rulebased reasoning for the representation of normal cases. The heuristic principle is ideally suited for making uncertain, hypothetical inferences on the basis of fuzzy data and fuzzy relations.

scholarly journals On the Decidability of Intuitionistic Tense Logic without Disjunction

2020 ◽  
Author(s):  
Fei Liang ◽  
Zhe Lin
Keyword(s):  
Proof Theory ◽  
Tense Logic ◽  
Heyting Algebras ◽  
Finite Model ◽  
Algebraic Proof ◽  
Model Property ◽  
Algebraic Models ◽  
Finite Axiomatizability ◽  
Finite Model Property ◽  
Modal Operators

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.

scholarly journals Different Arguments, Same Problems

2018 ◽  
Vol 13 (2) ◽  
pp. 5-22
Author(s):  
Rafal Urbaniak
Keyword(s):  
Propositional Variable ◽  
Modal Operator ◽  
Modal Operators

I illustrate with three classical examples the mistakes arising from using a modal operator admitting multiple interpretations in the same argument; the flaws arise especially easily if no attention is paid to the range of propositional variables. Premisses taken separately might seem convincing and a substitution for a propositional variable in a modal context might seem legitimate. But there is no single interpretation of the modal operators involved under which all the premisses are plausible and the substitution successful.

scholarly journals Residuation in non-associative MV-algebras

2018 ◽  
Vol 68 (6) ◽  
pp. 1313-1320
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Binary Operation ◽  
Residuated Lattice ◽  
Natural Question ◽  
Natural Conditions ◽  
Double Negation ◽  
Mv Algebra ◽  
Residuated Poset ◽  
Double Negation Law ◽  
Mv Algebras

Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

Categories of semigroups in quantum computational structures

2009 ◽  
Vol 59 (4) ◽  
Author(s):  
Hector Freytes ◽  
Antonio Ledda
Keyword(s):  
Fuzzy Logic ◽  
Quantum Computation ◽  
Variety Of Algebras ◽  
Reflective Subcategory ◽  
Mv Algebras ◽  
Computational Structures

AbstractWe investigate a categorial duality between quasi MV-algebras (a variety of algebras arising from quantum computation and tightly connected with fuzzy logic) and a reflective subcategory of l-groups with strong units.

Basic Fuzzy Logic is the logic of continuous t-norms and their residua

2000 ◽  
Vol 4 (2) ◽  
pp. 106-112 ◽  
Author(s):  
R. Cignoli ◽  
F. Esteva ◽  
L. Godo ◽  
A. Torrens
Keyword(s):  
Fuzzy Logic ◽  
Basic Fuzzy Logic

Probability as a modal operator: the possibilities of its combination with other modalities

2020 ◽  
Author(s):  
Nino Guallart
Keyword(s):  
Subjective Probability ◽  
Modal Operator ◽  
Relational Semantics ◽  
Modal Operators ◽  
Probabilistic Semantics ◽  
Probability Operator

Abstract In this work we examine some of the possibilities of combining a simple probability operator with other modal operators, in particular with a belief operator. We will examine the semantics of two possible situations for expressing probabilistic belief or the lack of it, a simple subjective probability operator (SPO) versus the composition of a belief operator, plus an objective modal operator (BOP). We will study their interpretations in two probabilistic semantics: a relational Kripkean one and a variation of neighbourhood semantics, showing that the latter is able to represent the lack of probabilistic belief more directly, just with the SPO, whereas relational semantics needs the combination of BOP probability to represent lack of belief.

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