250 research outputs found
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Logics without the contraction rule and residuated lattices
In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework
Fuzzy Sets and Formal Logics
The paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics. Our focus is on the historical side of these developments. © 2015 Elsevier B.V. All rights reserved.partial support by the Spanish projects EdeTRI (TIN2012-39348- C02-01) and 2014 SGR 118.Peer reviewe
Szubstrukturális logikák algebrai és bizonyításelméleti vizsgálata = Algebraic and Proof Theoretic Investigations of Substructural Logics
A kutatás fő eredményei: - Az ""Equality"" algebrák bevezetése (Studia Logica). - A ""strongly involutive uninorm"" algebrák bevezetése és osztályozása, valamint a vonatkozó logika komplexitásának vizsgálata (J Logic and Computation). - Az ""involutive FLe-monoid""-ok algebrai vizsgálata; kúp-reprezentáció és egyes véges láncok osztályozása (Archive for Mathematical Logic). - Az involutív uninormák egy osztályának osztályozása, az itt bevezetett ferde-szimmetrizáció segítségével (J Logic and Computation). - A forgatás konstrukció és az Ábel csoportok kapcsolatának vizsgálata (Fuzzy Sets and Systems). - A reziduált hálók geometriai jellegű vizsgálata (Annals of Pure and Applied Logic). Lektorálás alatt: az ""Elnyelő-folytonos, éles, szubreál láncon értelmezett FLe-algebrák osztályozása"" és a ""Pseudo Equality Algebras"" cikkek. | The scientific achievements of the project are the following: - The introducing of equality algebras (Studia Logica). - The introducing and classifying of strongly involutive uninorm algebras along with complexity issues of the related logic (J Logic and Computation). - Algebraic investigation of involutive FLe-monoids, in particular, conic representation and classification of certain finite chains (Archive for Mathematical Logic). - Structural description of a class of involutive uninorms via inrtoducing skew symmetrization (J Logic and Computation). - An investigation of the link between the rotation-construction and Abelian groups (Fuzzy Sets and Systems). - A geometric flavour study of residuated lattices (Annals of Pure and Applied Logic). - ""Classification of absorbent-continuous sharp FLe-algebras over subreal chains"" and Pseudo Equality Algebras"" (two articles under review
Poset products as relational models
We introduce a relational semantics based on poset products, and provide
sufficient conditions guaranteeing its soundness and completeness for various
substructural logics. We also demonstrate that our relational semantics unifies
and generalizes two semantics already appearing in the literature: Aguzzoli,
Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and
Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz
logic. As a consequence of our general theory, we recover the soundness and
completeness results of these prior studies in a uniform fashion, and extend
them to infinitely-many other substructural logics
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
The ubiquity of conservative translations
We study the notion of conservative translation between logics introduced by
Feitosa and D'Ottaviano. We show that classical propositional logic (CPC) is
universal in the sense that every finitary consequence relation over a
countable set of formulas can be conservatively translated into CPC. The
translation is computable if the consequence relation is decidable. More
generally, we show that one can take instead of CPC a broad class of logics
(extensions of a certain fragment of full Lambek calculus FL) including most
nonclassical logics studied in the literature, hence in a sense, (almost) any
two reasonable deductive systems can be conservatively translated into each
other. We also provide some counterexamples, in particular the paraconsistent
logic LP is not universal.Comment: 15 pages; to appear in Review of Symbolic Logi
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