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

Periodic pp-adic Gibbs measures of qq-states Potts model on Cayley tree: The chaos implies the vastness of pp-adic Gibbs measures

We study the set of $p$-adic Gibbs measures of the $q$-states Potts model on the Cayley tree of order three. We prove the vastness of the periodic $p$-adic Gibbs measures for such model by showing the chaotic behavior of the correspondence Potts--Bethe mapping over $\mathbb{Q}\_p$ for $p\equiv 1 \ (\rm{mod} \ 3)$. In fact, for$0 < |\theta-1|\_p < |q|\_p^2 < 1$, there exists a subsystem that isometrically conjugate to the full shift on three symbols. Meanwhile, for$0 < |q|\_p^2 \leq |\theta-1|\_p < |q|\_p < 1$, there exists a subsystem that isometrically conjugate to a subshift of finite type on$r$symbols where$r \geq 4$. However, these subshifts on$r$symbols are all topologically conjugate to the full shift on three symbols. The$p$-adic Gibbs measures of the same model for the cases$p=2,3$and the corresponding Potts--Bethe mapping are also discussed.Furthermore, for$0 < |\theta-1|\_p < |q|\_p < 1,$we remark that the Potts--Bethe mapping is not chaotic when$p=2,\ p=3$and$p\equiv 2 \ (\rm{mod} \ 3)$and we could not conclude the vastness of the periodic$p$-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case$0 < |q|\_p \leq |\theta-1|\_p < 1$for all$p$

Renormalization method in pp-adic λ\lambda-model on the Cayley tree

In this present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in $p$-adic statistical mechanics. We mainly study $p$-adic \l-model on the Cayley tree of order two. We consider generalized $p$-adic quasi Gibbs measures depending on parameter \r\in\bq_p, for the \l-model. Such measures are constructed by means of certain recurrence equations. These equations define a dynamical system. We study two regimes with respect to parameters. In the first regime we establish that the dynamical system has one attractive and two repelling fixed points, which predicts the existence of a phase transition. In the second regime the system has two attractive and one neutral fixed points, which predicts the existence of a quasi phase transition. A main point of this paper is to verify (i.e. rigorously prove) and confirm that the indicated predictions (via dynamical systems point of view) are indeed true. To establish the main result, we employ the methods of $p$-adic analysis, and therefore, our results are not valid in the real setting.Comment: 18 page