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Schumann, Andrew and Smarandache, Florentin (2007) Neutrality and Multi-Valued Logics. ARP, USA.

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Abstract

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.

Item Type:Book
Subjects:B Philosophy. Psychology. Religion > BC Logic
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ID Code:86
Deposited By:Dr. Florentin Smarandache
Deposited On:10 Mar 2010 08:38
Last Modified:10 Mar 2010 09:27

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