129 research outputs found
Homomorphism Preservation Theorems for Many-Valued Structures
A canonical result in model theory is the homomorphism preservation theorem
which states that a first-order formula is preserved under homomorphisms on all
structures if and only if it is equivalent to an existential-positive formula,
standardly proved compactness. Rossman (2008) established that the h.p.t.
remains valid when restricted to finite structures. This is a significant
result in the field of finite model theory, standing in contrast to other
preservation theorems and as an theorem which remains true in the finite but
whose proof uses entirely different methods. It also has importance to the
field of constraint satisfaction due to the equivalence of existential-positive
formulas and unions of conjunctive queries. Adjacently, Dellunde and Vidal
(2019) established that a version of the h.p.t. holds for a collection of
first-order many-valued logics, those whose (possibly infinite) structures are
defined over a fixed finite MTL-chain.
In this paper we unite these two strands, showing how one can extend
Rossman's proof of a finite h.p.t. to a very wide collection of many-valued
predicate logics and simultaneously establishing a finite variant to Dellunde
and Vidal's result, one which not only applies to structures defined over
algebras more general than MTL-chains but also where we allow for those algebra
to vary between models. This investigation provides a starting point in a wider
development of finite model theory for many-valued logics and, just as the
classical finite h.p.t. has implications for constraint satisfaction, the
many-valued finite h.p.t. has implications for valued constraint satisfaction
problems.Comment: 22 page
Syntactic characterizations of classes of first-order structures in mathematical fuzzy logic
This paper is a contribution to graded model theory, in the context of
mathematical fuzzy logic. We study characterizations of classes of graded
structures in terms of the syntactic form of their first-order axiomatization.
We focus on classes given by universal and universal-existential sentences. In
particular, we prove two amalgamation results using the technique of diagrams
in the setting of structures valued on a finite MTL-algebra, from which
analogues of the Los--Tarski and the Chang--Los--Suszko preservation theorems
follow
Paraconsistency properties in degree-preserving fuzzy logics
Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. Aiming to find logics that can handle inconsistency and graded truth at once, in this paper we explore the notion of paraconsistent fuzzy logic. We show that degree-preserving fuzzy logics have paraconsistency features and study them as logics of formal inconsistency. We also consider their expansions with additional negation connectives and first-order formalisms and study their paraconsistency properties. Finally, we compare our approach to other paraconsistent logics in the literature. © 2014, Springer-Verlag Berlin Heidelberg.All the authors have been partially supported by the FP7 PIRSES-GA-2009-247584 project MaToMUVI. Besides, Ertola was supported by FAPESP LOGCONS Project, Esteva and Godo were supported by the Spanish project TIN2012-39348-C02-01, Flaminio was supported by the Italian project FIRB 2010 (RBFR10DGUA_02) and Noguera was suported by the grant P202/10/1826 of the Czech Science Foundation.Peer reviewe
Logics of formal inconsistency arising from systems of fuzzy logic
This article proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in
this article we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency
in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition
of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics,
expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs.The authors have been partially supported by the FP7-PEOPLE-2009-IRSES project MaToMUVI (PIRSES-GA-2009-247584). Coniglio was also supported by FAPESP (Thematic Project LogCons 2010/51038-0), and by a research grant from
CNPq (PQ 305237/2011-0). Esteva and Godo also acknowledge partial support by the MINECO project TIN2012-39348-C02-01Peer Reviewe
Structural and universal completeness in algebra and logic
In this work we study the notions of structural and universal completeness
both from the algebraic and logical point of view. In particular, we provide
new algebraic characterizations of quasivarieties that are actively and
passively universally complete, and passively structurally complete. We apply
these general results to varieties of bounded lattices and to quasivarieties
related to substructural logics. In particular we show that a substructural
logic satisfying weakening is passively structurally complete if and only if
every classical contradiction is explosive in it. Moreover, we fully
characterize the passively structurally complete varieties of MTL-algebras,
i.e., bounded commutative integral residuated lattices generated by chains.Comment: This is a preprin
Preservation theorems for algebraic and relational models of logic
A thesis submitted to the School of Computer Science,
Faculty of Science,
University of the Witwatersrand, Johannesburg
in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 15 May 2013In this thesis a number of different constructions on ordered algebraic structures
are studied. In particular, two types of constructions are considered: completions
and finite embeddability property constructions.
A main theme of this thesis is to determine, for each construction under
consideration, whether or not a class of ordered algebraic structures is closed
under the construction. Another main focus of this thesis is, for a particular
construction, to give a syntactical description of properties preserved by the
construction. A property is said to be preserved by a construction if, whenever
an ordered algebraic structure satisfies it, then the structure obtained through
the construction also satisfies the property.
The first four constructions investigated in this thesis are types of completions.
A completion of an ordered algebraic structure consists of a completely
lattice ordered algebraic structure and an embedding that embeds the former
into the latter. Firstly, different types of filters (dually, ideals) of partially ordered
sets are investigated. These are then used to form the filter (dually, ideal)
completions of partially ordered sets. The other completions of ordered algebraic
structures studied here include the MacNeille completion, the canonical
extension (also called the completion with respect to a polarization) and finally
a prime filter completion.
A class of algebras has the finite embeddability property if every finite partial
subalgebra of some algebra in the class can be embedded into some finite
algebra in the class. Firstly, two constructions that establish the finite embeddability
property for residuated ordered structures are investigated. Both of
these constructions are based on completion constructions: the first on the Mac-
Neille completion and the second on the canonical extension. Finally, algebraic
filtrations on modal algebras are considered and a duality between algebraic and
relational versions of filtrations is established
Gluing residuated lattices
We introduce and characterize various gluing constructions for residuated
lattices that intersect on a common subreduct, and which are subalgebras, or
appropriate subreducts, of the resulting structure. Starting from the 1-sum
construction (also known as ordinal sum for residuated structures), where
algebras that intersect only in the top element are glued together, we first
consider the gluing on a congruence filter, and then add a lattice ideal as
well. We characterize such constructions in terms of (possibly partial)
operators acting on (possibly partial) residuated structures. As particular
examples of gluing constructions, we obtain the non-commutative version of some
rotation constructions, and an interesting variety of semilinear residuated
lattices that are 2-potent. This study also serves as a first attempt toward
the study of amalgamation of non-commutative residuated lattices, by
constructing an amalgam in the special case where the common subalgebra in the
V-formation is either a special (congruence) filter or the union of a filter
and an ideal.Comment: This is a preprint. The final version of this work appears in Orde
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