115 research outputs found
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
Cyclic theories
We describe a geometric theory classified by Connes-Consani's epicylic topos
and two related theories respectively classified by the cyclic topos and by the
topos .Comment: 25 page
A Point-Free Look at Ostrowski's Theorem and Absolute Values
This paper investigates the absolute values on valued in the
upper reals (i.e. reals for which only a right Dedekind section is given).
These necessarily include multiplicative seminorms corresponding to the finite
prime fields . As an Ostrowski-type Theorem, the space of such
absolute values is homeomorphic to a space of prime ideals (with co-Zariski
topology) suitably paired with upper reals in the range , and
from this is recovered the standard Ostrowski's Theorem for absolute values on
.
Our approach is fully constructive, using, in the topos-theoretic sense,
geometric reasoning with point-free spaces, and that calls for a careful
distinction between Dedekinds vs. upper reals. This forces attention on
topological subtleties that are obscured in the classical treatment. In
particular, the admission of multiplicative seminorms points to connections
with Berkovich and adic spectra. The results are also intended to contribute to
characterising a (point-free) space of places of
Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective
We establish, generalizing Di Nola and Lettieri's categorical equivalence, a
Morita-equivalence between the theory of lattice-ordered abelian groups and
that of perfect MV-algebras. Further, after observing that the two theories are
not bi-interpretable in the classical sense, we identify, by considering
appropriate topos-theoretic invariants on their common classifying topos, three
levels of bi-intepretability holding for particular classes of formulas:
irreducible formulas, geometric sentences and imaginaries. Lastly, by
investigating the classifying topos of the theory of perfect MV-algebras, we
obtain various results on its syntax and semantics also in relation to the
cartesian theory of the variety generated by Chang's MV-algebra, including a
concrete representation for the finitely presentable models of the latter
theory as finite products of finitely presentable perfect MV-algebras. Among
the results established on the way, we mention a Morita-equivalence between the
theory of lattice-ordered abelian groups and that of cancellative
lattice-ordered abelian monoids with bottom element.Comment: 54 page
Probabilistic logics based on Riesz spaces
We introduce a novel real-valued endogenous logic for expressing properties
of probabilistic transition systems called Riesz modal logic. The design of the
syntax and semantics of this logic is directly inspired by the theory of Riesz
spaces, a mature field of mathematics at the intersection of universal algebra
and functional analysis. By using powerful results from this theory, we develop
the duality theory of the Riesz modal logic in the form of an
algebra-to-coalgebra correspondence. This has a number of consequences
including: a sound and complete axiomatization, the proof that the logic
characterizes probabilistic bisimulation and other convenient results such as
completion theorems. This work is intended to be the basis for subsequent
research on extensions of Riesz modal logic with fixed-point operators
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