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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
Generic substitutions
Up to equivalence, a substitution in propositional logic is an endomorphism
of its free algebra. On the dual space, this results in a continuous function,
and whenever the space carries a natural measure one may ask about the
stochastic properties of the action. In classical logic there is a strong
dichotomy: while over finitely many propositional variables everything is
trivial, the study of the continuous transformations of the Cantor space is the
subject of an extensive literature, and is far from being a completed task. In
many-valued logic this dichotomy disappears: already in the finite-variable
case many interesting phenomena occur, and the present paper aims at displaying
some of these.Comment: 22 pages, 2 figures. Revised version according to the referee's
suggestions. To appear in the J. of Symbolic Logi
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