"In scrubbing and cleansing your profound mirror Are you able to rid it of all imperfections? In loving the common people and breathing life into the state, Are you able to do it without recourse to wisdom? With nature’s gates swinging open and closed Are you able to remain the female? With your insight penetrating the four quarters Are you able to do it without recourse to wisdom?"
Ames, Roger; Hall, David (2010-05-12). Dao De Jing- Chapter 10: A Philosophical Translation (Kindle Locations 844-846) Chapter 10. Random House Publishing Group. Kindle Edition.
"The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function."
F. Scott Fitzgerald
Ever since the time of Aristotle, logic (and most philosophy) in the western world has been constrained to obey the principle of non-contradiction. This simply means that what is true, and what is false do not overlap. If something is said to be true it cannot also be false, and if something is said to be false it cannot also be true. Intuitively this seems to be a reasonable logical axiom. It is so reasonable on it's face that it has rarely been questioned. How could we after all, make any sense of our language if truth and falsity overlapped in logical space?
As it turns out the principle of non-contradiction may not be so logically necessary as it seems. Graham Priest is a mathematician/logician /philosopher who has been making arguments for a para-consistent logic that allows for the possibility of true contradictions for 25 years. This type of logic goes back more than 2,500 years to the original Buddha in what is known as the Catuskoti. The Catuskoti (sometimes termed as the tetralemma) is translated to mean the four corners (or the four possibilities). Much argument in Buddhist philosophy takes this form with Nagarjuna (known as the 2nd Buddha) the most well known practitioner. The four possibilities include not only standard conceptions of truth or falsehood, but also seemingly contradictory possibilities such as both (true and false) or neither (true or false).
Why Turn to a Para-Consistent Logic?
Is there a good reason however, to allow for the ambiguity and lack of consistency that arguments from the four possibilities entail when the classical logic of mutually exclusive truth and falsehood seems so intuitive? Graham Priest points to how unavoidable contradictions seem to be persistent in language, mathematics, and the physical world. Here is an example from language known as the liars paradox.
This sentence is false.
Due to it's self-reference one cannot determine the sentence to be purely true or false with out contradicting it's own contents.
Another example comes from the world of numbers. It is true that the sets of (a) natural numbers, (b) even natural numbers, (c) odd natural numbers, and (d) prime numbers are all infinite sets. It is then a contradiction that b,c, & d are all also subsets of a. How could there be an infinite number of odd numbers if the number 2 is not an odd number? This is a true contradiction.
There is also the great mathematician Kurt Godel who showed that any logical system:
- could not be both consistent and complete; and also
- could not prove itself consistent without also proving itself inconsistent.
Another example comes from the mathematics behind the phase changes of a fluid. The emergence of new properties due to phase change is often accompanied by mathematical singularities (infinities). The concept of emergence calls into question a reductionist approach to science. For example, even if fundamental physics were to find an ultimate 'Theory of Everything' that links the very small (quantum mechanics) to the very large ( theory of relativity) that 'Theory of Everything' would be of little use in describing the new properties that come into play as we move from physics to chemistry, biology and the problems facing modern social beings.
Finally of course there is the inconsistent logic in play in interpreting the results of the double slit experiments from quantum mechanics itself. See this article for a brilliant description of how the logic of the four possibilities might best model those results.
Given these examples, it seems that a logic which allows for more than just the standard two possibilities of truth and falsehood might better model the inevitable unknowns and inconsistencies we are presented with as we attempt to navigate and make sense of the world.
How About the Drawbacks to a Para-Consistent Logic?
The main argument against para-consistent logic ( including dialetheism which is the type of para-consistent logic that Graham Priest argues for ) is called the 'Argument from Explosion'. This is something like a 'slippery-slope' argument. Basically if something can be said to be both true and false then anything can be argued and argument itself loses all meaning. But Priest makes clear that only some, (not all) contradictions are true. Thus dialethism is not vulnerable to the 'Argument from Explosion'.
Another argument is the 'Argument from Exclusion '. This argument claims that a sentence is only meaningful if it rules something out. Again Priest argues that dialtheism ( & also the logic of the Catuskoti ) is also protected against this argument for similar reasons as to the 'Argument from Explosion'. This dialetheism link also describes how this form of para-consistent logic is also protected from another argument known as the 'Argument from Negation'.
So How Does All This Relate to the Catuskoti (the Four Possibilities)?
Interpretation of the Catuskoti is notoriously difficult, especially for those steeped in western cultural logic. In this paper (warning : requires some familiarity with logic to digest), Priest outlines his interpretation of Nagarjuna's writings on the Catuskoti. An important concept that Priest relies upon when understanding contradiction from Nagrarjuna's perspective is a Buddhist principle that distinguishes two types of reality, conventional reality and ultimate reality. Priest uses these two types of realities ( or truths ) along with the four possibilities to connect his interpretation to a many valued logic of relational semantics. In this logic 'conventional truths' are knowable but 'ultimate truths' are not. This relates to the Buddhist concepts of non-separability & emptiness whereby any thing or concept only exists in relation to some other thing or concept. Thus every thing or concept is empty of it's own pure essence and therefore must be known in context or relation to other things..
This also reminds me of the physicist Neils Bohr's conceptions of truth and contradiction as Bohr famously said:
"There are trivial truths and the great truths. The opposite of a trivial truth is plainly false. The opposite of a great truth is also true."I find this way of thinking about truth, contradiction and the limits of knowledge both very reasonable and of great practical importance. I believe it is easy to confuse our models or maps of the the world for the ontology of what ultimately exists. Instead of seeing these models as useful descriptions we often take them as ultimate truth. Our models can be based on subjective perceptions and beliefs or on objective science yet each of these methods tends to avoid contradiction rather then embrace it's inevitability.
On the other hand I think it is also dangerous to fall into the trap of using the inevitability of ultimate uncertainty and contradiction as a crutch to avoid the hard work of subjectively and objectively discovering that which we can. Whether from the viewpoint of an individual or a society, much of what currently is uncertain, and much of what is currently unknown is waiting to be discovered. Not all contradictions are 'true' contradictions, but it is our current uncertainties that can serve as the springboard to fill in some of those gaps.
