In this series of articles, Aniruddha Singhal explores the difference between the two systems of logic, one prevalent in India and other dharmic societies, and other prevalent in the West, with its roots in the Aristotelian logic. The Aristotelian logic is bivalued, while the Indian system of logic is not. In the first part of the series, the roots of Aristotelian logic are explored.
Aristotelian logic, also known as western logic, is a bivalued system with excluded middle. In 1930 the limitations of western logic were shown by Kurt Gödel in his landmark Incompleteness Theorem. Some of its implications will be discussed here. The alternative logic systems such as Vedantic and Jain Saptbhangi logic system are India’s offering to logic and will be discussed here. The Boolean Vector Matrix Formulation of these system as developed by Dr. G. N. Ramachandran (1922-2001) will also be discussed in the article.
We will also talk about the fundamental change that such an eastern logic system can bring by offering a fresh outlook to science and society.
First time in the history of human knowledge axiomatic logic system was formally presented by Euclid in ‘Elements’ around 300 B.C. Since then axiomatic system and deductive logic are twin foundations of mathematics, science and philosophy. It is the basis on which all scientific studies are conducted, every research is done and conclusions are drawn.
The axiomatic system is built on a set of self-evident truths called axioms. Axioms require no proof and are laid as foundational premises on which the whole system of thought is built by deductive logic. That is why axioms are of great importance. If axioms are incorrect, all further deductions derived from it will be incorrect. Particularly in mathematics it is difficult to carry on an incorrect axiom for long because the system is so rigorous that a contradiction will soon emerge from an incorrect axiom. Such a contradiction indicates fault either in axiomatic foundation or in method of deduction.
Unlike mathematics, a society can carry on an incorrect axiom for long. Unstated subconscious assumptions and ambiguities of language makes detection of faults in their arguments a difficult task. Such incorrect axioms are not found and pointed out until it is very late. For example, the ‘axiom of identity’ which separate eastern and western societies in their philosophical outlook is how they see the human body. Western thought emphasizes the axiom “I am this body, therefore…” while eastern thought is based on an alternative axiom i.e. “I am not this body, therefore…” It is a matter of debate which of these axioms is correct.
Rene Descartes (1596-1650) said “I think, therefore I am”. In this axiomatic statement he pointed out that thought is primary and the sense of being comes after it. His axiom has dominated western philosophy and society for a long time, and still influence their thought and behavior. On the contrary in the east, the opposite axiom “I am, therefore I think” is considered to be true and places the sense of being above anything else.
Next step which is as important as a set of axioms is the system through which we make deductions, i.e. system of logic. In this essay we are primarily concerned with the systems of logic. Logic systems can be divided into eastern and western categories.
Western logic was founded by Aristotle (384–322 BC). He was trained by Plato for 20 years in Athens Academy. Later Aristotle taught princes and wrote many books. Most volumes have been lost but enough has survived to shape western history. Aristotle wrote several volumes on logic and reasoning, put together as “The Organon”. In it he discussed the method of deduction and interpretation. The Organon presented bivalued system of true or false, in which nothing exists in between these two values hence middle in between them is excluded. His basic idea can be summarized in his own words as “Everything must either be or not be, whether in the present or in the future”.
In thirteenth century, Christian Theologian Thomas Aquinas polished, debated and extended the works of Aristotle to create subject of Christian Theology. The Catholic Church owes a great deal to Aristotle who himself was a Pagan not a Christian. To a large degree Aristotle still defines what is philosophically correct in logic and reasoning.
Aristotle’s work had deep influence on the way of thinking and due to which bivalued logic system became foundation of all Christian, theological, philosophical and scientific works of the west. Thus, dualistic way of seeing and thinking was firmly established in western world.
The Aristotelian states of logic can be represented with a vector of one element taking value -1 and +1 along with a trivial state of initial ignorance as shown in table (I).
Table (I) – Vector representation of Aristotelian logic
We can imagine a one dimensional interpretation space with only three points on a line. Points (1), (-1) and (0) respectively representing “It is”, “It is not” and trivial state of initial ignorance. On this one-dimensional space the midpoint (0) represents zero-dimensional space of initial ignorance. By the tools of logical inquiry, knowledge proceeds in directions of either “It is” or “It is not” in one dimension.
Figure 1 – Geometric representation of trivial state
Figure 2 – Geometric representation of Aristotelian logic
Today we do not accept Aristotle’s philosophy in our physical theories. Why should we then accept his logic in our reasoning and computer design? What lies outside our Aristotelian boundary? What about multivalued or “fuzzy” logic?