2,126 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
A temporal semantics for Nilpotent Minimum logic
In [Ban97] a connection among rough sets (in particular, pre-rough algebras)
and three-valued {\L}ukasiewicz logic {\L}3 is pointed out. In this paper we
present a temporal like semantics for Nilpotent Minimum logic NM ([Fod95,
EG01]), in which the logic of every instant is given by {\L}3: a completeness
theorem will be shown. This is the prosecution of the work initiated in [AGM08]
and [ABM09], in which the authors construct a temporal semantics for the
many-valued logics of G\"odel ([G\"od32], [Dum59]) and Basic Logic ([H\'aj98]).Comment: 19 pages, 2 table
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
First-order Nilpotent Minimum Logics: first steps
Following the lines of the analysis done in [BPZ07, BCF07] for first-order
G\"odel logics, we present an analogous investigation for Nilpotent Minimum
logic NM. We study decidability and reciprocal inclusion of various sets of
first-order tautologies of some subalgebras of the standard Nilpotent Minimum
algebra. We establish a connection between the validity in an NM-chain of
certain first-order formulas and its order type. Furthermore, we analyze
axiomatizability, undecidability and the monadic fragments.Comment: In this version of the paper the presentation has been improved. The
introduction section has been rewritten, and many modifications have been
done to improve the readability; moreover, numerous references have been
added. Concerning the technical side, some proofs has been shortened or made
more clear, but the mathematical content is substantially the same of the
previous versio
ON SOME AXIOMATIC EXTENSIONS OF THE MONOIDAL T-NORM BASED LOGIC MTL: AN ANALYSIS IN THE PROPOSITIONAL AND IN THE FIRST-ORDER CASE
The scientific area this thesis belongs to are many-valued logics: in particular, the logic MTL and some of its extensions, in the propositional and in the first-order case (see [8],[9],[6],[7]). The thesis is divided in two parts: in the first one the necessary background about these logics,
with some minor new results, are presented. The second part is devoted to more specific topics: there are five chapters, each one about a different problem. In chapter 6 a temporal semantics for Basic Logic BL is
presented. In chapter 7 we move to first-order logics, by studying the supersoundness property: we have improved some previous works about this theme, by expanding the analysis to many extensions of the first-order version of MTL. Chapter 8 is dedicated to four different families of n-contractive axiomatic extensions of BL, analyzed in the propositional and in the first-order case: completeness, computational and arithmetical complexity, amalgamation and interpolation properties are studied. Finally, chapters 9 and 10 are about Nilpotent Minimum logic (NM, see [8]): in chapter 9 the sets of tautologies of some NM-chains (subalgebras
of [0,1]_NM) are studied, compared and the problems of axiomatization and undecidability are tackled. Chapter 10, instead, concerns some logical and algebraic properties of (propositional) Nilpotent Minimum logic. The results (or an extended version of them) of these last chapters have
been also presented in papers [1, 4, 5, 2, 3]. ---------------------------------References---------------------------------------------
[1] S. Aguzzoli, M. Bianchi, and V. Marra. A temporal semantics for Basic
Logic. Studia Logica, 92(2), 147-162, 2009. doi:10.1007/s11225-009-9192-3.
[2] M. Bianchi. First-order Nilpotent Minimum Logics: first steps. Submitted
for publication,2010.
[3] M. Bianchi. On some logical and algebraic properties of Nilpotent Minimum
logic and its relation with G\uf6del logic. Submitted for publication, 2010. [4] M. Bianchi and F. Montagna. Supersound many-valued logics and
Dedekind-MacNeille completions. Arch. Math. Log., 48(8), 719-736, 2009.
doi:10.1007/s00153-009-0145-3.
[5] M. Bianchi and F. Montagna. n-contractive BL-logics. Arch. Math. Log.,
2010. doi:10.1007/s00153-010-0213-8.
[6] P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna, and C. Noguera.
Distinguished algebraic semantics for t-norm based fuzzy logics: methods
and algebraic equivalencies. Ann. Pure Appl. Log., 160(1), 53-81, 2009.
doi:10.1016/j.apal.2009.01.012.
[7] P. Cintula and P. H\ue1jek. Triangular norm predicate fuzzy logics. Fuzzy Sets
Syst., 161(3), 311-346, 2010. doi:10.1016/j.fss.2009.09.006.
[8] F. Esteva and L. Godo. Monoidal t-norm based logic: Towards a
logic for left-continuous t-norms. Fuzzy sets Syst., 124(3), 271-288, 2001.
doi:10.1016/S0165-0114(01)00098-7.
[9] P. H\ue1jek. Metamathematics of Fuzzy Logic, volume 4 of Trends
in Logic. Kluwer Academic Publishers, paperback edition, 1998.
ISBN:9781402003707
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