Gottfried Wilhelm Leibniz (1646 – 1716 CE) was a German philosopher and mathematician who invented calculus around the same time as Isaac Newton. Newton tried to confusingly name everything after himself, with many types of newtons, so science kept one kind, the newton as unit of force, and used Leibniz less self-serving system of notation for the rest of Calculus and physics instead. Leibniz’s first job was alchemist’s assistant, and then he became a lawyer’s assistant, perhaps with more worldly success. Leibniz published little during his lifetime, and to this day no definitive collection exists of his various and disparate writings. His most famous writings are his Monadology and his Discourse on Metaphysics.
Leibniz invented the binary system still used by computers today, which may or may not give way to something else like quantum computers in the future, which do not rely on two separate values such as 1 and 0. Leibniz was a sinophile, who loved Chinese culture, studied Chinese thought that was available to him, and invented his binary system inspired by the Yi Jing (I Ching), the ancient Chinese binary divination system that represents all possible situations with solid and broken lines just as Leibniz’s binary system represents all numbers with ones and zeros. Leibniz was communicating with Christian missionaries in China, and he, like some of the missionaries, believed that Europeans could learn much from Confucianism that was in line with Christianity. An admirer of the Chinese abacus, Leibniz was one of the most important innovators of the mechanical calculator, an early computer which employed his binary system.
Like Descartes and Spinoza, Leibniz believed that God created the world as a rational, mechanical apparatus. Because of this, Leibniz famously argued that this is the best of all possible worlds. As God is omniscient, God was aware of all possible worlds before creation, and chose this to be the created world, so it must therefore be the best. Of course, many who ponder the problem of evil, the theological problem debated for centuries about how suffering in a rational world is possible, would question this assertion. Leibniz came up with a pure deductive understanding of the world, which was quite unlike how we experience it. The infinite, the eternal, is for the mathematician Leibniz an infinite series of distinct points, the elementary particles of the universe, eternal and indivisible, like the atoms (“without cut”) of the ancient Indian and Greek atomists. This infinite plurality is entirely made of mind, and each is its entire universe, what Leibniz calls a pre-established harmony.
Leibniz is a strange, outlying philosopher, one I mention but don’t cover extensively for Modern European Philosophy, but he is very important to the history of logic, not only because he helped invent the binary computer, which is where many modern logics live, but he is also one of the first to articulate something Aristotle argues for but doesn’t draw into an explicit principle, what some logicians still call the Law of Non-Contradiction, but most others refer to as the Principle of Non-Contradiction, as it is more understood by secular scholars today as something psychological, but was, for Aristotle, Avicenna, Aquinas and others a law of the universe, one that Nagarjuna of India, Heraclitus of Greece and Hui Shi of China would certainly deny.
The Principle of Non-Contradiction, or PNC for short, can be stated as: If a statement is true, then its negation is false, and if a statement is false, then its negation is true. For example, if the statement Leibniz is a logician is true, then the statement Leibniz is not a logician is false, and vice-versa. Kant and Russell, advocates of logic and the Principle of Non-Contradiction, studied the work of Leibniz intensely, advocating this principle. Russell, who founded Logical Positivism, the basis of Analytic Philosophy, the dominant school of philosophy in the Anglophonic world, argued that we can base all logic, mathematics and certain, objective science on the single truth of the Principle of Non-Contradiction. In formal logic, Graham Priest at CUNY is one of the most famous critics of the principle, arguing, like Hui Shi and Heraclitus, that at least some contradictions are possible, such that Leibniz could be a logician but also not in different ways for different purposes which are both valid.
What is a contradiction? It is an argument, between sides, in its most visible and audible form. We can contradict ourselves, and argue with ourselves, which isn’t the same thing, but similar. A simple contradiction in speech, between two sides, whether or not there are one or two people, or more, is I turned the lights off and No, you didn’t, which is a difference of opinion. If we assume there isn’t a relative dimmer switch, and the lights could go more on or more off, then one side is right and the other side is wrong. Switches are designed as bifurcating devices, as a serious dilemma that could go one way or the other, like the contradiction over the switch. Checking to see if the lights are on or off can resolve the dilemma, debate and contradiction, unless there is a twist, and it turns out, given further evidence and experience, that the switch doesn’t work, or the lights were turned off by someone else later, or its the wrong switch, or the wrong set of lights that should never be turned off, unlike all the others. Even given the simple, bifurcating device of the switch there is a complex human situation that can easily involve contradiction between differently interested parties.
If this is what a contradiction is, then what is the principle of non-contradiction saying? That people can’t get into debates? Aristotle, like Kant long after him, and like Russell long after Kant, argued that not all questions can be solved correctly, but some can, and it is the task of the human mind to use reason to solve what can be solved completely. As for favorite flavor of ice-cream, there are some flavors that would turn heads in many cultures, but it is a matter of subjective taste and opinion. However, insofar as math and logic are supposed to ideally work, in these matters and questions there are true, objective and singular answers that are not relatively true, but absolutely true, much as many would say that two and three together making five is universally, objectively, absolutely and even ideally true, beyond practices or culture.
The principle of non-contradiction applies to these truths, such that if someone contradicts what is absolutely true, they are necessarily wrong. If two and three make five, in all possible ways, then anyone who says two and three make four or six is simply and completely wrong, regardless of their upbringing, practices, cultures, or success, as it is not about what pragmatically works, but what is positively true. Much as Kant argues about morality, it isn’t what keeps the ship sailing, but what is true that we should believe, even if it means disaster and we all go down with the ship, because right is what is overall important, not results. This is the major issue we will talk through again in many ways between positivism and pragmatism.
Aristotle, Farabi, Avicenna, Aquinas, Leibniz, Kant and Russell are some of the central thinkers who made the principle of non-contradiction what it is today. Aristotle said that skeptics like Heraclitus are no better than plants, understanding nothing. Avicenna said those who say fire and beatings are and are not good, only relative evils, should be burned and beaten until they stop saying such things, which is hopefully a joke at the expense of skeptics like al-Ghazali, the Sufi mystic, who criticized Avicenna and was criticized by Averroes. Russell argues that Mill, Dewey and all other instrumentalists, utilitarians and pragmatists, who argue truth is not ideal but relative cannot say anything with certainty, and have to investigate all possibilities continuously to absurdity, such as wondering if we truly did have coffee with breakfast.
I suggest Wittgenstein was right, as a pragmatic person myself, when he said the trick is to see we can stop and start doing philosophy when we want to, as we may never hit bedrock and have the final understandings or answers to anything. Like Priest, Wittgenstein wrote that contradictions need not be false. Logic which excludes all contradictions as nonsense is only a small part of the ways we use language. Lewis Carroll certainly thought so. Aristotelian logic is not the genuine basis of all reasoning any more than trigonometry is the genuine basis of all geometry. (LWPP1 525) The logic of language is more complicated than it looks. (LWPP2 44)
Wittgenstein wrote that some believe in the excluded middle, that a statement cannot be both true and false, but it is rather that true and false divide the field of possibilities, but not always into exclusive parts. Wittgenstein gives the sad example, “Have you stopped beating your wife?” which is not simply a yes or no question, as if someone has never beaten their wife, it is true in one sense, as I am not currently engaging in domestic violence, and false in another, as saying yes implies that I used to, in a way that resembles subalternation. (RPP1 274) The comedian Mitch Hedberg joked, I used to do a lot of drugs… I still do, but I used to, also. Wittgenstein said contradictions are not catastrophes to be feared and avoided, but problems that require engagement with contrary judgements. (Z 685-9) This is likely why Wittgenstein had a deep appreciation of the nonsensical problems found in the arguments of Wonderland.
I I tell you A is true and false, am I telling you nothing, or am I telling you a great deal? If I say nothing I say nothing, even if the silence implies something significant, but if I say something contradictory, that Steve is good and Steve is not good, that the girl with the curl in the middle of her forehead is sometimes very, very good, but when she’s bad she’s horrid, I’m saying at least two things, if not implying more. An argument tells us a lot about both sides, and often shows us things both sides can’t fully see about each other. Humor and nonsense are directly contradictory, just as fantasy contradicts reality by degree, and they teach us much about us.
Leibniz has two other principles which are important in the history of formal logic, and which Kant and Russell both support. The second is the Principle of Identity of Indiscernibles: If two things are without any discernible difference, then they must be not two things, but identical, the same single thing. Of course, if two things are in different locations or exist at different times, this is a discernible difference, one that would shoot any instance of the principle down. Many illustrate this today with the example of two types of minerals labeled equally as jade in ancient China, before humanity had the technology to tell the difference.
The third is the Principle of Sufficient Reason: If something exists, there must be a reason why it exists the way it does. Leibniz believes that the world was teleologically created by God, who controls all in this best and most rational of all possible worlds, and so he assumes that each thing can be rationally explained because each thing was rationally created by an intellect superior but similar to our own. Many secular modern people hold this principle, but without teleology it is difficult to argue that humanity can come up with intelligible reasons that explain apples entirely, or anything else. Are there sufficient reasons apples exist and behave as they do? What is sufficient enough for this? Is it to our satisfaction, or are there objective reasons, in number, that exist independently?
There is one last principle that should be mentioned and strangely isn’t as much, the Principle of Bivalence: A statement must be true or false, not neither true nor false. This is somewhat the inverse of the Principle of Non-Contradiction, that a statement must not be both true and false together, which is the third of Nagarjuna’s four things, and the Principle of Bivalence is the fourth. Leibniz, Kant, Russell and others are focused on non-contradiction, not bivalence. This could be because Aristotle himself wanders in his answer whether neither good nor bad is both or neither, and concludes it is more-so, relatively speaking, neither, which means Aristotle allows for Nagarjuna’s fourth but not third thing. It seems those who argue for non-contradiction think both sides can’t be right, but both sides could be wrong, as if correct is exclusive, but incorrect is inclusive, regardless of how much we all have common sense.