237 research outputs found
Interval-valued algebras and fuzzy logics
In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter
Expanding FLew with a Boolean connective
We expand FLew with a unary connective whose algebraic counterpart is the
operation that gives the greatest complemented element below a given argument.
We prove that the expanded logic is conservative and has the Finite Model
Property. We also prove that the corresponding expansion of the class of
residuated lattices is an equational class.Comment: 15 pages, 4 figures in Soft Computing, published online 23 July 201
Conjuntos construibles en modelos valuados en retículos
We investigate different set-theoretic constructions in Residuated Logic based on Fitting’s
work on Intuitionistic Kripke models of Set Theory.
Firstly, we consider constructable sets within valued models of Set Theory. We present
two distinct constructions of the constructable universe: L
B and L
B
, and prove that the
they are isomorphic to V (von Neumann universe) and L (Gödel’s constructible universe),
respectively.
Secondly, we generalize Fitting’s work on Intuitionistic Kripke models of Set Theory using
Ono and Komori’s Residuated Kripke models. Based on these models, we provide a general-
ization of the von Neumann hierarchy in the context of Modal Residuated Logic and prove
a translation of formulas between it and a suited Heyting valued model. We also propose a
notion of universe of constructable sets in Modal Residuated Logic and discuss some aspects
of it.Investigamos diferentes construcciones de la teoría de conjuntos en Lógica Residual basados
en el trabajo de Fitting sobre los modelos intuicionistas de Kripke de la Teoría de Conjuntos.
En primer lugar, consideramos conjuntos construibles dentro de modelos valuados de la
Teoría de Conjuntos. Presentamos dos construcciones distintas del universo construible:
L
B y L
B
, y demostramos que son isomorfos a V (universo von Neumann) y L (universo
construible de Gödel), respectivamente.
En segundo lugar, generalizamos el trabajo de Fitting sobre los modelos intuicionistas de
Kripke de la teoría de conjuntos utilizando los modelos residuados de Kripke de Ono y
Komori. Con base en estos modelos, proporcionamos una generalización de la jerarquía de
von Neumann en el contexto de la Lógica Modal Residuada y demostramos una traducción de
fórmulas entre ella y un modelo Heyting valuado adecuado. También proponemos una noción
de universo de conjuntos construibles en Lógica Modal Residuada y discutimos algunos
aspectos de la misma. (Texto tomado de la fuente)MaestríaMagíster en Ciencias - MatemáticasLógica matemática, teoría de conjunto
A note on drastic product logic
The drastic product is known to be the smallest -norm, since whenever . This -norm is not left-continuous, and hence it
does not admit a residuum. So, there are no drastic product -norm based
many-valued logics, in the sense of [EG01]. However, if we renounce standard
completeness, we can study the logic whose semantics is provided by those MTL
chains whose monoidal operation is the drastic product. This logic is called
in [NOG06]. In this note we justify the study of this
logic, which we rechristen DP (for drastic product), by means of some
interesting properties relating DP and its algebraic semantics to a weakened
law of excluded middle, to the projection operator and to
discriminator varieties. We shall show that the category of finite DP-algebras
is dually equivalent to a category whose objects are multisets of finite
chains. This duality allows us to classify all axiomatic extensions of DP, and
to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure
Propositional calculus for adjointness lattices
Recently, Morsi has developed a complete syntax for the class of all
adjointness algebras . There, is a partially ordered set with top element , is a
conjunction on for which is a left identity
element, and the two implication-like binary operations and on
are adjoints of .
In this paper, we extend that formal system to one for the class of
all 9-tuples , called \emph{%
adjointness lattices}; in each of which is a bounded lattice, and is an
adjointness algebra. We call it \emph{Propositional Calculus for Adjointness
Lattices}, abbreviated . Our axiom scheme for features four
inference rules and thirteen axioms. We deduce enough theorems and
inferences in to establish its completeness for ; by means of
a quotient-algebra structure (a Lindenbaum type of algebra). We study two
negation-like unary operations in an adjointness lattice, defined by means
of together with and . We end by developing complete syntax for
all adjointness lattices whose implications are -type implications
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