Axioms of Soft Logic

Abstract

In this paper, we develop the foundation of a new mathematical language, which we term “Soft Logic”. This language enables us to present an extension of the number 0 from a singular point to a continuous line. We create a distinction between −0 and +0 and generate a new type of numbers, which we call ‘Bridge Numbers’ (BN):

\({\boldsymbol{a}}\overline {\bf{0}} \bot {\boldsymbol{b}}\overline {\bf{1}} ,\)

where a, b are real numbers, “a” is the value on the \(\overline {\bf{0}} \) axis, and “b” is the value on the \(\overline {\bf{1}} \) axis. We proceed by defining arithmetic and algebraic operations on the Bridge Numbers, investigate their properties, and conclude by defining goals for further research. In the Attachment, we continue comparing our results with existing mathematical work on Infinitesimals, Dual numbers, and Nonstandard analysis. The research is a part of “Digital living 2030” project with Stanford University.