3,634 research outputs found
Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics
We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer
and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We
give new analytic proof systems for A and use the embeddings to derive
corresponding systems for L. These include: hypersequent calculi for A and L
and terminating versions of these calculi; labelled single sequent calculi for
A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur
Towards a Proof Theory of G\"odel Modal Logics
Analytic proof calculi are introduced for box and diamond fragments of basic
modal fuzzy logics that combine the Kripke semantics of modal logic K with the
many-valued semantics of G\"odel logic. The calculi are used to establish
completeness and complexity results for these fragments
Completeness of a Hypersequent Calculus for Some First-order Gödel Logics with Delta
All first-order Gödel logics G_V with globalization operator based on truth value sets V C [0,1] where 0 and 1 lie in the perfect kernel of V are axiomatized by Ciabattoni’s hypersequent calculus HGIF
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
Hypersequents and the Proof Theory of Intuitionistic Fuzzy Logic
Takeuti and Titani have introduced and investigated a logic they called
intuitionistic fuzzy logic. This logic is characterized as the first-order
Goedel logic based on the truth value set [0,1]. The logic is known to be
axiomatizable, but no deduction system amenable to proof-theoretic, and hence,
computational treatment, has been known. Such a system is presented here, based
on previous work on hypersequent calculi for propositional Goedel logics by
Avron. It is shown that the system is sound and complete, and allows
cut-elimination. A question by Takano regarding the eliminability of the
Takeuti-Titani density rule is answered affirmatively.Comment: v.2: 15 pages. Final version. (v.1: 15 pages. To appear in Computer
Science Logic 2000 Proceedings.
Admissibility in Finitely Generated Quasivarieties
Checking the admissibility of quasiequations in a finitely generated (i.e.,
generated by a finite set of finite algebras) quasivariety Q amounts to
checking validity in a suitable finite free algebra of the quasivariety, and is
therefore decidable. However, since free algebras may be large even for small
sets of small algebras and very few generators, this naive method for checking
admissibility in \Q is not computationally feasible. In this paper,
algorithms are introduced that generate a minimal (with respect to a multiset
well-ordering on their cardinalities) finite set of algebras such that the
validity of a quasiequation in this set corresponds to admissibility of the
quasiequation in Q. In particular, structural completeness (validity and
admissibility coincide) and almost structural completeness (validity and
admissibility coincide for quasiequations with unifiable premises) can be
checked. The algorithms are illustrated with a selection of well-known finitely
generated quasivarieties, and adapted to handle also admissibility of rules in
finite-valued logics
Neutrality and Many-Valued Logics
In this book, we consider various many-valued logics: standard, linear,
hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We
survey also results which show the tree different proof-theoretic frameworks
for many-valued logics, e.g. frameworks of the following deductive calculi:
Hilbert's style, sequent, and hypersequent. We present a general way that
allows to construct systematically analytic calculi for a large family of
non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and
p-adic valued logics characterized by a special format of semantics with an
appropriate rejection of Archimedes' axiom. These logics are built as different
extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's,
Product, and Post's logics). The informal sense of Archimedes' axiom is that
anything can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and it
is not well-founded and well-ordered. On the base of non-Archimedean valued
logics, we construct non-Archimedean valued interval neutrosophic logic INL by
which we can describe neutrality phenomena.Comment: 119 page
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
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
- …