There exists a grave misconception at present that the Aristotelian conception of logic (which we can call binary logic) is the only type of logic. Binary logic rests on an objective conception of reality and explicitly excludes the mind. For instance, when a mind produces thought, the thought is both a part of the mind and separate from it. After producing the though, the mind doesn't become ignorant. Nor is the mind ignorant when the idea isn't being thought. If we extend this problem to a conception of the body consistent with the existence of the mind, then we have to revise binary logic into non-binary logics in which two things can be both distinct and yet inseparable. Alternative logics have been previously developed by Buddhists (who created a four-valued logic) and Jains (who created a seven-valued logic). The universalism of binary logic is not just a myth but also problematic in conceptualizing non-object realities.
All over Vedic philosophy, the term “non-difference” is employed to describe reality. In Vedānta, this idea is called abheda (non-different). In Smriti, the same idea is called advaya (non-dual). In Sāñkhya, the same concept is described by the term avyatireka (non-exclusive). And there are other similar terms for it.
This conception of reality is quite distinct from the manner in which modern Western logic conceives reality as either identical or different. In this logic, two things are either separate or united; they cannot be non-separated. Finally, you must either choose between one thing or the other; you cannot be non-exclusive.
These principles of modern logic govern everything else, including mathematics, numbers, equations, conceptual thinking, and all theory formation. We also know that this logical system of reasoning leads to mathematical incompleteness, which then leads to scientific incompleteness, as Gödel’s theorems demonstrate.
The fundamental intuitions underlying modern logic are based on physical things. For example, either there are two separate things, or there is just one thing. But this separateness undergoes a dramatic change when we introduce concepts. For example, if you see two cars, they are different as individual things, but they are identical as conceptual entities. Then, if you are thinking only about concepts, then ‘cows’ and ‘horses’, are conceptually different, but also not different as ‘mammals’. Then, if you think of contexts where the “same thing is said using different words”, two contexts are different in words but have the same meaning.
Thus, everything that is held sacrosanct in modern logic, mathematics, rational thinking, and modern science, comes crumbling down, if we broaden our material ontology to include concepts and contexts in our systems of reasoning.
The alternative system of reasoning requires modalities such as individual, contextual, and conceptual, such that two cars can be different as individuals, but the same as the concept. Likewise, the same words in a different context can mean differently, whereas different words in another context can mean the same thing. The use of modalities changes our notion of logic, reasoning, and science.
This system of reasoning is not known today, but its ideas are presented pervasively in Vedic texts. How can an alternative system of reasoning and logic be formalized in the same rigorous manner that modern logic is? How will these systems of reasoning be different from the current systems? How does this new way of non-exclusive thinking lead to a new conception of numbers as ideas? And once we have developed an alternative notion of numbers, leading to new mathematics, how will the reality be reconceived using this rationality and logic?