Neutrality and Many-Valued Logics

Preamble This book written by A. Schumann & F. Smarandache is devoted to advances of non-Archimedean multiple-validity idea and its applications to logical reasoning. 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 ρ p (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 Q p of p-adic numbers. In Q p 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 properties of non-Archimedean and p-adic valued logical systems proposed by them in 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-sadziesprzecznò sci u Arystotelesa, Kraków 1910). Creating many-valued logic, Lukasiewicz was inspired philosophically. …