5 research outputs found
Tense logic based on finite orthomodular posets
It is widely accepted that the logic of quantum mechanics is based on
orthomodular posets. However, such a logic is not dynamic in the sense that it
does not incorporate time dimension. To fill this gap, we introduce certain
tense operators on such a logic in an inexact way, but still satisfying
requirements asked on tense operators in the classical logic based on Boolean
algebras or in various non-classical logics. Our construction of tense
operators works perfectly when the orthomodular poset in question is finite. We
investigate the behaviour of these tense operators, e.g. we show that some of
them form a dynamic pair. Moreover, we prove that if the tense operators
preserve one of the inexact connectives conjunction or implication as defined
by the authors recently in another paper, then they also preserve the other
one. Finally, we show how to construct the binary relation of time preference
on a given time set provided the tense operators are given, up to equivalence
induced by natural quasiorders
Tense distributive lattices: algebra, logic and topology
Tense logic was introduced by Arthur Prior in the late 1950s as a result of
his interest in the relationship between tense and modality. Prior's idea was
to add four primitive modal-like unary connectives to the base language today
widely known as Prior's tense operators. Since then, Prior's operators have
been considered in many contexts by different authors, in particular, in the
context of algebraic logic.
Here, we consider the category tdlat of bounded distributive lattices
equipped with Prior's tense operators. We establish categorical dualities for
tdlat in terms of certain categories of Kripke frames and Priestley spaces,
respectively. As an application, we characterize the congruence lattice of any
tense distributive lattice as well as the subdirectly irreducible members of
this category. Finally, we define the logic that preserves degrees of truth
with respect to tdlat-algebras and precise the relation between particular
sub-classes of tdlat and know tense logics found in the literature
A Topological Approach to Tense LMn×m-Algebras
In 2015, tense n × m-valued Lukasiewicz–Moisil algebras (or tense LMn×m-algebras) were introduced by A. V. Figallo and G. Pelaitay as an generalization of tense n-valued Łukasiewicz–Moisil algebras. In this paper we continue the study of tense LMn×m-algebras. More precisely, we determine a Priestley-style duality for these algebras. This duality enables us not only to describe the tense LMn×m-congruences on a tense LMn×m-algebra, but also to characterize the simple and subdirectly irreducible tense LMn×m-algebras
Tense operators on De Morgan algebras
The purpose of this article is to investigate the variety of algebras, which we call tense De Morgan algebras, as a natural generalization of tense algebras developed in Burges (1984, Handbook of Philosophical Logic, vol. II, pp. 89–139) (see also, Kowalski (1998, Rep. Math. Logic, 32, 53–95)). It is worth mentioning that the latter is a proper subvariety of the first one, as it is shown in a simple example. Our main interest is the representation theory for these classes of algebras.Fil: Figallo, Aldo Victorio. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Instituto de Ciencias Básicas; ArgentinaFil: Pelaitay, Gustavo Andrés. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Instituto de Ciencias Básicas; Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Juan; Argentin