Abstract
This chapter introduces the basic concepts of Total Semantics based on transreal numbers, a new field of mathematics in which division of zero is allowed. In the logic created from the transreals, it is possible to evaluate non-propositional objects such as sensations or feelings as having truth value. By covering all semantical possibility, transreal semantics enables a “poetical discourse” based on images and an immediate intuition of reality, an intuition that is both sensory and mental. This describes, and defines, a new type of epistemic operation—a sentient one—and leads to the possibility of developing a formal epistemic logic that assesses how we obtain or acquire knowledge from the perspective of sentient agents.
TopIntroduction
What is a number? Answering such a question is a most difficult philosophical task. Therefore, while simplistic, the author will take the following thesis as an answer: a number is an entity, an object, which represents a certain measure of something—the extent of elements of a set, the size of a line segment, the intensity of a gravitational field, etc. Finally, a number is an object (or a property, if we understand it that way) that we relate to the notion of “quantity”, one of the basic categories of Aristotle's logical-metaphysical thought.
According to Aristotle (1991), a quantity may be continuous or discrete. Examples of continuous quantities are time and space, quantities that can be subdivided indefinitely without finding a minimal part that is the fundamental constitutive unity of these quantities. On the other hand, discrete quantities, as such finite numbers or the linguagem, cannot be subdivided indefinitely. At any given moment, a discrete quantity, through successive subdivisions, is reduced to its fundamental constitutive unity.
Therefore, according to the Western mathematical tradition that has been developing since Classical Greece, what are the sets or systems of numbers related to the dichotomous division of quantities between continuous and discrete ones? For continuous quantities, the answer is the associated system of numbers (the system of numbers that can measure a continuous quantity) which is that of real numbers, usually represented as 
. For discrete quantities, the related system of numbers is that of natural numbers, represented usually as 
.
But what if we want to consider quantities that, in addition to being continuous or discrete, are also infinite and, to some extent, indeterminate? In this case, human creativity is put to work side by side with logical and analytical thinking, and then alternative number systems, such as surreal or hyperreal numbers, appear to account for such “bizarrely” infinite or indeterminate quantities. In this chapter, then, we will present one of these number systems, which will be the basis for the development of a “Sentient Logic”: the transreal numbers.
Key Terms in this Chapter
Transreal Numbers: An extension of real numbers created by James A.D.W. Anderson, an English computer scientist, around 1997 (Anderson & Dos Reis, 2015a). This extension allows the division over zero, which is forbidden in the realm of real numbers.
Nullity: A number not comparable regarding to its magnitude with any other number. It has no defined size and, for this reason, represents the Indeterminate translated into mathematics.
Epistemic Logic: As a subfield of epistemology (knowledge), the term refers to logical approaches to knowledge, including beliefs and other related aspects of knowledge. This would include identifying epistemic principles and determining the logical relationships among diverse concepts.
Sentience: An “intelligence” inclusive of feeling and thinking among the infinite scales of fractal universal electromagnetic awareness (Schafer, 2019).
Infinity: In mathematics, this is the representation of a number beyond a specific assignable quantity; a number that cannot be counted. The symbol is ¥.
Total Semantics: A semantics in which all logical possibilities are modelled (Anderson et al., 2015). A Total Semantics of transreal numbers provides a new way of approaching epistemic logic.