Fundamentals of Soft Logic

In this paper, we develop a new theory termed Soft Logic. This theory addresses the need to combine real processes and cognitive ones in the same framework. At the same time, we develop a new concept of modeling and dealing with uncertainty: the uncertainty of time and space. We develop a language that can talk in two reference frames, and also suggest a way to combine them. This constitutes a fundamental new mathematics. In this paper, we present the theory from its foundation.In Soft Logic, we refer to situations of uncertainty occurring in the present. Ambiguity is created from the possible point of view of the observer of the phenomenon. Sometimes reality contains contradictory or opposite situations at the same time, which can be interpreted in different ways by the observer. Such is the case of the Necker cube or the Fechner color effect, in which a texture of black and white colors moving in a certain pattern creates a rainbow of colors. Another example is the Mobius strip, which from a local point of view seems to have two sides, but from a global point of view has only one.The mathematical solution for expressing uncertainty in Soft Logic is the blow-up of the number zero by distinguishing between multiples of this number. In this manner, we create a zero-axis and define a new type of number called Soft Numbers, which have the following form: a0¯+̇b1¯. The zero-axis expresses the inner world of the observer and is the component of a subjective interpretation and a certain type of uncertainty that occurs in the present.Soft Numbers are similar to complex numbers, but the specific mathematical definitions are different and therefore Soft Numbers are a different world. This paper presents five axioms of Soft Logic, which form a new system of coordinates that differs from the Cartesian system and Gauss’s complex plane. We define the algebra of Soft Numbers, show the connection to differential and integral calculus, expand analytical functions to soft functions, and describe continuous curves in the soft coordinate system. In conclusion, we suggest several avenues for theoretical and practical research.