Neutrality and Many-Valued Logics

arXiv:0707.3205v1 [cs.LO] 21 Jul 2007 Andrew Schumann & Florentin Smarandache Neutrality and Many-Valued Logics July, 2007 2 Preamble This book written by A. Schumann & F. Smarandache is devoted to advances of non-Archimedean multiple-validity idea and its applications to logical reason- ing. Leibnitz was the first who proposed Archimedes’ axiom to be rejected. He postulated infinitesimals (infinitely small numbers) of the unit interval [0, 1] which are larger than zero, but smaller than each positive real number. Robin- son applied this idea into modern mathematics in [117] and developed so-called non-standard analysis. In the framework of non-standard analysis there were obtained many interesting results examined in [37], [38], [74], [117]. There exists also a different version of mathematical analysis in that Archi- medes’ axiom is rejected, namely, p-adic analysis (e.g., see: [20], [86], [91], [116]). In this analysis, one investigates the properties of the completion of the field Q of rational numbers with respect to the metric ρp(x, y) = |x−y|p, where the norm | · |p called p-adic is defined as follows: • |y|p = 0 ↔y = 0, • |x · y|p = |x|p · |y|p, • |x + y|p ⩽max(|x|p, |y|p) (non-Archimedean triangular inequality). That metric over the field Q is non-Archimedean, because |n · 1|p ⩽1 for all n ∈Z. This completion of the field Q is called the field Qp of p-adic numbers. In Qp there are infinitely large integers. Nowadays there exist various many-valued logical systems (e.g., see Mali- nowski’s book [92]). However, non-Archimedean and p-adic logical multiple- validities were not yet systematically regarded. In this book, Schumann & Smarandache define such multiple-validities and describe the basic proper- ties of non-Archimedean and p-adic valued logical systems proposed by them in [122], [123], [124], [125], [128], [132], [133]. At the same time, non-Archimedean valued logics are constructed on the base of t-norm approach as fuzzy ones and p-adic valued logics as discrete multi-valued systems. Let us remember that the first logical multiple-valued system is proposed by the Polish logician Jan Lukasiewicz in [90]. For the first time he spoke 3 4 about the idea of logical many-validity at Warsaw University on 7 March 1918 (Wyk lad po˙zegnalny wyg loszony w auli Uniwersytetu Warszawskiego w dniu 7 marca 1918 r., page 2). However Lukasiewicz thought already about such a logic and rejection of the Aristotelian principle of contradiction in 1910 (O za- sadzie sprzeczno`sci u Arystotelesa, Krak´ow 1910). Creating many-valued logic, Lukasiewicz was inspired philosophically. In the meantime, Post designed his many-valued logic in 1921 in [105] independently and for combinatorial reasons as a generalization of Boolean algebra. The logical multi-validity that runs the unit interval [0, 1] with infinitely small numbers for the first time was proposed by Smarandache in [132], [133], [134], [135], [136]. The neutrosophic logic, as he named it, is conceived for a philosophical explication of the neutrality concept. In this book, it is shown that neutrosophic logic is a generalization of non-Archimedean and p-adic val- ued logical systems. In this book non-Archimedean and p-adic multiple-validities idea is regarded as one of possible approaches to explicate the neutrality concept. K. Trz¸esicki Bia lystok, Poland Preface In this book, we consider various many-valued logics: standard, linear, hy- perbolic, 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 gen- eralization of Gentzen’s style sequents that was introduced independently by Avron and Pottinger. In particular, we examine Hilbert’s style, sequent, and hypersequent calculi for infinite-valued logics based on the three fundamen- tal continuous t-norms: Lukasiewicz’s, G¨odel’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 for- mat of semantics with an appropriate rejection of Archimedes’ axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz’s, G¨odel’s, Product, and Post’s logics). The informal sense of Archimedes’ axiom is that anything can be mea- sured 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 Zp of p-adic integers. Notice that in the second case we set discrete infinite-v

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