Improved Definition of Non Standard Neutrosophic Logic and Introduction to Neutrosophic Hyperreals (Fourth version)

Abstract

On the third version of this response-paper to Imamura’s criticism, we recall that Non Standard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century oldneutrosophic operators (1995-1998) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the Non Standard Analysis hyperreals, monads and binads to tiny intervals with the desired accuracy – otherwise they would be inapplicable. We point out several errors and false statements by Imamura [21] with respect to the inf/sup of nonstandard subsets, also Imamura’s “rigorous definition of neutrosophic logic” is wrong and the same for his definition of nonstandard unit interval, and we prove that there is not a total order on the set of hyperreals (because of the newly introduced Neutrosophic Hyperreals that are indeterminate), whence the transfer principle is questionable. After his criticism, several response publications on theoretical nonstandard neutrosophics followed in the period 2018-2022. As such, Iextended the Non Standard Analysis by adding the left monad closed to the right, right monad closed to the left,pierced  binad  (we  introduced  in  1998),  and  unpierced  binad-  all  these  in  order  to  close  the  newly extendednonstandard   space   (R^*)   under   nonstandard   addition,   nonstandard   subtraction,   nonstandard   multiplication,nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of Interval and Non Standard Neutrosophic Logic are presented.