50 research outputs found
Pavelka-style completeness in expansions of \L ukasiewicz logic
An algebraic setting for the validity of Pavelka style completeness for some
natural expansions of \L ukasiewicz logic by new connectives and rational
constants is given. This algebraic approach is based on the fact that the
standard MV-algebra on the real segment is an injective MV-algebra. In
particular the logics associated with MV-algebras with product and with
divisible MV-algebras are considered
Super-\L ukasiewicz logics expanded by
Baaz's operator was introduced (by Baaz) in order to extend G\"odel
logics, after that this operator was used to expand fuzzy logics by H\'ajek in
his celebrated book. These logics were called -fuzzy logics. On the
other hand, possibility operators were studied in the setting of \L
ukasiewicz-Moisil algebras; curiously, one of these operators coincide with the
Baaz's one. In this paper, we study the operator in the context of
(-valued) Super-\L ukasiewicz logics. An algebraic study of these logics is
presented and the cardinality of Lindembaun-Tarski algebra with a finite number
of variables is given. Finally, as a by-product, we present an alternative
axiomatization of H\'ajek's \L ukasiwicz logic expanded with
Metabolic, Replication and Genomic Category of Systems in Biology, Bioinformatics and Medicine
Metabolic-repair models, or (M,R)-systems were introduced in Relational Biology by Robert Rosen. Subsequently, Rosen represented such (M,R)-systems (or simply MRs)in terms of categories of sets, deliberately selected without any structure other than the discrete topology of sets. Theoreticians of life’s origins postulated that Life on Earth has begun with the simplest possible organism, called the primordial. Mathematicians interested in biology attempted to answer this important question of the minimal living organism by defining the functional relations that would have made life possible in such a minimal system- a grandad and grandma of all living organisms on Earth. Genomic systems are also considered as molecular realizations of (M,R)-system subcatgeories
An algebraic approach to general aggregation theory: Propositional-attitude aggregators as MV-homomorphisms
This paper continues Dietrich and List's [2010] work on propositional-attitude aggregation theory, which is a generalised unification of the judgment-aggregation and probabilistic opinion-pooling literatures. We first propose an algebraic framework for an analysis of (many-valued) propositional-attitude aggregation problems. Then we shall show that systematic propositional-attitude aggregators can be viewed as homomorphisms in the category of C.C. Chang's [1958] MV-algebras. Since the 2-element Boolean algebra as well as the real unit interval can be endowed with an MV-algebra structure, we obtain as natural corollaries two famous theorems: Arrow's theorem for judgment aggregation as well as McConway's [1981] characterisation of linear opinion pools.propositional attitude aggregation, judgment aggregation, linear opinion pooling, Arrow's impossibility theorem, many-valued logic, MV-algebra, homomorphism, Arrow's impossibility theorem, functional equation
Towards understanding the Pierce-Birkhoff conjecture via MV-algebras
Our main issue was to understand the connection between \L ukasiewicz logic
with product and the Pierce-Birkhoff conjecture, and to express it in a
mathematical way. To do this we define the class of \textit{f}MV-algebras,
which are MV-algebras endowed with both an internal binary product and a scalar
product with scalars from . The proper quasi-variety generated by
, with both products interpreted as the real product, provides the
desired framework: the normal form theorem of its corresponding logical system
can be seen as a local version of the Pierce-Birkhoff conjecture