ThisbookwrittenbyA. Schumann &F. Smarandache isdevotedtoadvances 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. Robinson applied this idea into modern mathematics in  and developed so-called non-standard analysis. In the framework of non-standard analysis there were obtained many interesting results examined in , , , .
There exists also a diﬀerent version of mathematical analysis in that Archimedes’ axiom is rejected, namely, p-adic analysis (e.g., see: , , , ).
Let us remember that the ﬁrst logical multiple-valued system is proposed by the Polish logician Jan L ukasiewicz in . For the ﬁrst time he spoke about the idea of logical many-validity at Warsaw University on 7 March 1918 (Wyk lad poz˙egnalny wyg loszony w auli Uniwersytetu Warszawskiego w dniu 7 marca 1918 r., page 2). However L ukasiewicz thought already about such a logic and rejection of the Aristotelian principle of contradiction in 1910 (O zasadzie sprzeczno`sci u Arystotelesa, Krak´ow 1910). Creating many-valued logic, L ukasiewicz was inspired philosophically. In the meantime, Post designed his many-valued logic in 1921 in  independently and for combinatorial reasons as a generalization of Boolean algebra.
American Research Press
neutrality, many-valued logic, Jan Lukasiewicz
Smarandache, Florentin and Andrew Schumann. "Neutrality and Many-Valued Logics." (2007). https://digitalrepository.unm.edu/math_fsp/295
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