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Andrew Schumann, Florentin Smarandache, "Neutrality and Multi-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. Recall that hypersequents are a natural generalization of Gentzen’s style sequents that was introduced independently by Avron and Pottinger. In particular, we consider Hilbert’s style, sequent, and hypersequent calculi for infinite-valued logics based on the three fundamental continuous t-norms: Lukasiewicz’s, Gödel s, and Product logics. 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, Gödel 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. We consider two cases of non-Archimedean multi-valued logics: the first with many-validity in the interval [0,1] of hypernumbers and the second with many-validity in the ring of p-adic integers. Notice that in the second case we set discrete infinite-valued logics. The following logics are investigated: 1. hyperrational valued Lukasiewicz’s, Gödel s, and Product logics, 2. hyperreal valued Lukasiewicz’s, Gödel s, and Product logics, 3. p-adic valued Lukasiewicz’s, Gödel s, and Post’s logics. Hajek proposes basic fuzzy logic BL which has validity in all logics based on continuous t-norms. In this book, for the first time we survey hypervalued and p-adic valued extensions of basic fuzzy logic BL. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logic INL by which we can describe neutrality phenomena. This logic is obtained by adding to the truth valuation a truth triple t, i, f instead of one truth value t, where t is a truth-degree, i is an indeterminacy-degree, and f is a falsity-degree. Each parameter of this triple runs either the unit interval [0,1] of hypernumbers or the ring of p-adic integers.

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