Buddhism, like Christianity, celebrates a day of Ascension (which is known as 'Himmelfart' in German). After achieving enlightenment under a bodhi tree, Lord Buddha ascended to the Tavatisma heaven and expounded his doctrine to the Gods. Since the Gods live in the realm of certitude, the Buddha's ascension certifies the truth of his teachings even if we humans are incapable of hearing and digesting the 'divyadhvani' of that divine sermon.
If all possible states of the world are known- as is the case for the gods in the Tavatisma Heaven- then all propositions are either true or false. The law of the excluded middle holds true once all possible states are disambiguated and rendered 'predicative' (i.e. there is a first order logic which is complete). Sadly, if there is 'Knightian Uncertainty'- i.e. if all possible states of the world are not known (indeed, it is not even known if two seemingly different states of the world genuinely are so- e.g wave/particle duality), then propositions are merely informative for some particular purpose. They have no truth value. Pragmatism rules. However, pragmatics won't seek to 'maximize' its own utility. It will be 'regret minimizing'. One corollary is that FOMO- fear of missing out- will obtain. In other words, bandwagons will be jumped on even if they are deeply silly. Also ontologically dysphoric goods and services will be embraced more or less hypocritically because they promote better correlated equilibria. Moreover love of Knowledge- i.e. Philosophy- will be mostly masturbatory and motivated by malice. The aim may not be to jizz so copiously in your own eye as to go blind, but, in these matters, the last thing you are concerned with is where you are aiming.
About a decade ago, Graham Priest (on whom I've posted elsewhere) wrote an article for Aeon titled
Beyond true and false
The Buddhist doctrine is that Truth is One and without a Second. Here Absolute Truth (paramārtha satya) is meant. It is not dependent, or indeed causally connected in any way with conventional or provisional truth (saṁvṛti satya).
Buddhist philosophy is full of contradictions.
No. Its Philosophy is stupid which is why it is consistent. It is the religion which isn't consistent. That's why it works perfectly well as a Faith.
Still, from the logical point of view- if there were no Boddhisattvas- and hence no monks and nuns- then everybody would eventually become a Pratyeka Buddha anyway. Indeed, Shramanism, as an alternative to Ritualistic Brahmanism- is more of a karmic snake than a ladder because, by adding states to the system, it prolongs entropy maximization or 'heat death'. Indeed, the Boddhisattva is an ouroboros. In other words, instead of facilitating, or having their own 'Liberation' facilitated, the Arhat gets trapped into seeking to bring all to Moksha thus creating a closed loop. This is why the Ajivikas specified a finite number of lives for all beings before they hit the jackpot. But this also meant that the Ajivikas died out. Religion is a service industry. Repeat business is important. Theism, with karma instead of bodily resurrection, is bound to go in a Vaishnav direction such that everybody wants to be reborn to do the Lord's will. In other words, Heaven or 'Kevalya' remain empty because everybody is obsessed with helping everybody else to... share that obsession.
Now modern logic is learning why that might be a good thing.
No. Modern logic has learnt that you can make a lot of money in Software and Expert Systems and AI and so forth. Contradiction doesn't matter. Productivity- i.e. drudgery- does.
Western philosophers have not, on the whole, regarded Buddhist thought with much enthusiasm.
Unless they have- e.g. Schopenhauer, Krause, Nietzsche, Heidegger etc. But there were ancient Greeks who were Buddhists.
As a colleague once said to me: ‘It’s all just mysticism.’
Brouwer could be called a Mystic. His intuitionistic logic, which does not have the excluded middle, turned out to be very useful.
This attitude is due, in part, to ignorance.
Stupidity. Everybody has had access to relevant texts for donkey's years. Back in the Sixties, you had to pretend to be into Zen or Che or both Che and Zen, just so as to get laid.
But it is also due to incomprehension. When Western philosophers look East, they find things they do not understand – not least the fact that the Asian traditions seem to accept, and even endorse, contradictions.
Dialectical logic, Intuitionistic logic, Fuzzy logic, dialethic logic etc. can do that.
Thus we find the great second-century Buddhist philosopher Nagarjuna saying:
about 'svabhava'.
The nature of things is to have no nature; it is their non-nature that is their nature. For they have only one nature: no-nature.
There is no contradiction here. It is obvious that the 'wicked' rock over which I tripped does not have a 'wicked' nature. It is made up of the same sort of atoms and molecules as we are.
An abhorrence of contradiction has been high orthodoxy in the West for more than 2,000 years.
Nonsense! Christendom- which is what the West is founded on- is itself founded on not just contradictions but ludicrous impossibilities- e.g. a Carpenter's son who is also 'Pantocrator'- the ruler of the Universe. But Krishna, who was born not in some fancy-shmancy manger but a fucking prison cell, is just like Christ in that respect. Indeed, there is no crazy shit in the West which can't be found in the East or the North or the South or Down fucking Under mate.
Statements such as Nagarjuna’s are therefore wont to produce looks of blank incomprehension, or worse.
Worse? Incontinence? Do Priest's colleagues shit themselves when he mentions Nagarjuna? Perhaps. If a very boring cunt sidles up to you and starts talking about Nagarjuna's Madhyamika what you need to say is 'Nagarjuna was great in ' Priya Priyathama. But he never made a movie called Madhyamika. You must be thinking of NTR.' On the other hand, if the boring cunt mentions Nietzsche, shit yourself immediately. You can always buy a new pair of trousers. You can never get back an hour of your life.
As Avicenna,
who was Muslim.
the father of Medieval Aristotelianism, declared:
Anyone who denies the law of non-contradiction should be beaten and burned
like I said, the dude was Muslim
until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.
If Avicenna had been sufficiently beaten and burned he would have renounced the law of non contradiction. Propositions are strategic. Sadly, they have no truth value for us because we evolved under Knightian Uncertainty.
One can hear similar sentiments, expressed with comparable ferocity, in many faculty common rooms today. Yet Western philosophers are slowly learning to outgrow their parochialism.
No. Philosophy became adversely selective of imbecility more than 50 years ago. Retardation not growth is what is displayed in that field.
And help is coming from a most unexpected direction: modern mathematical logic, not a field that is renowned for its tolerance of obscurity.
It turned out to be useful for the very boring and laborious business of programming computers and finding 'univalent foundations' so computer proof checking or even theorem proving could burgeon. But then, all sorts of arcane mathematical fields turned out to be very useful. But utility is not philosophy.
Let’s start by turning back the clock. It is India in the fifth century BCE, the age of the historical Buddha, and a rather peculiar principle of reasoning appears to be in general use.
The older Hindu schools of logic had an 'uncreated' Scripture. They needed a way to distinguish imperative from alethic statements in their inerrant scriptures. What this meant was finding ways to logically deduce that when Scripture says- 'butcher the cow'- the meaning is 'never harm a cow. Eat vegetables.' Buddhism was part of a movement which got rid of the older sacrificial religion which was too expensive for a sedentary population rather than a pastoral, warrior, elite.
This principle is called the catuskoti, meaning ‘four corners’. It insists that there are four possibilities regarding any statement: it might be true (and true only), false (and false only), both true and false, or neither true nor false.
The Jains went further and added more such combinations. However, all law courts or other fora of deliberation make such distinctions. A proposition may be a true statement of fact- e.g. I affirm my name is Vivek Iyer. It may be a false statement of fact- e.g. I affirm I am Beyonce. A proposition may contains an element of truth and an element of falsity- 'My name is Vivek Iyer though music lovers generally refer to me as Beyonce'. A proposition may not be statement of fact and may be neither true nor false- e.g. 'Beyonce would be a better name for me than Vivek'.
However, truth and falsity don't matter because deductions need to be verified. Moreover, verification always trumps deduction. Hercule Poirot deduces that the Duke was killed by the Butler. Sadly, the Duke gets up and removes the fake dagger stuck through his heart. He provides photographic evidence that he killed the Butler. Poirot then says 'I too fooled you. I'm actually Miss Marple. She is totally shit as a detective.'
We know that the catuskoti was in the air because of certain questions that people asked the Buddha, in exchanges that come down to us in the sutras. Questions such as: what happens to enlightened people after they die?
They stay dead. It is only if you fear death or wish to be reunited with your loved ones, that you are prepared to pay money for some priest or monk to lie to you.
It was commonly assumed that an unenlightened person would keep being reborn,
Guys who fucked with me in this life will get reborn as worms whom I will crush under my heel.
but the whole point of enlightenment was to get out of this vicious circle.
No. There was the notion that the enlightened person was happier and more successful- like the Butcher in the Vyadha Gita. So what if he dies in the same way that everybody else dies? Everyone farts, everyone shits, everyone dies.
And then what? Did you exist, not, both or neither? The Buddha’s disciples clearly expected him to endorse one and only one of these possibilities. This, it appears, was just how people thought.
No. They understood that the 'unenlightened' had certain fears and hopes which death would frustrate. Moreover, the moral economy of society needed 'ontologically dysphoric', rather than merely 'reputational', commodities so 'the books balanced'. All the various religious traditions were supplying more or less the same karmic 'vigyan' or 'praxis' in this respect. But there was 'product differentiation' at the level of 'matam' or doctrine. However, thanks to Umasvati, Nagarjuna & Sankara there was 'observational equivalence' between matam. In other words, dogmatic distinctions corresponded to no pragmatic differences. Still, if you wanted to take control of a particular Holy Place, you could say your sect had the correct dogma while the guys currently controlling that place had a corrupt version of it.
At around the same time, 5,000km to the west in Ancient Athens, Aristotle was laying the foundations of Western logic along very different lines.
They were foolish. Still, at a later point, his stupidity could be used by theologians discovering that rival sects were heretical and their adherents should be burnt at the stake.
Among his innovations were two singularly important rules. One of them was the Principle of Excluded Middle (PEM), which says that every claim must be either true or false with no other options (the Latin name for this rule, tertium non datur, means literally ‘a third is not given’). The other rule was the Principle of Non-Contradiction (PNC): nothing can be both true and false at the same time.
This was maintainable for 'admissible' propositions- i.e. ones which were probable and had a contingent truth maker.
Writing in his Metaphysics, Aristotle defended both of these principles against transgressors such as Heraklitus (nicknamed ‘the Obscure’). Unfortunately, Aristotle’s own arguments are somewhat tortured – to put it mildly – and modern scholars find it difficult even to say what they are supposed to be. Yet Aristotle succeeded in locking the PEM and the PNC into Western orthodoxy,
Not Western Christian orthodoxy or the orthopraxy of the Law Courts or of Poetry or Politics or anything else. For empirically testable propositions, however, these rules do apply- within certain limits and for certain practical purposes.
where they have remained ever since. Only a few intrepid spirits, most notably G W F Hegel in the 19th century, ever thought to challenge them. And now many of Aristotle’s intellectual descendants find it very difficult to imagine life without them.
It is in Islamic seminaries that you will find Aristotle's intellectual descendants though, perhaps, it might be wiser not to look for them there just in case you also stumble on an ISIS cell.
That is why Western thinkers – even those sympathetic to Buddhist thought – have struggled to grasp how something such as the catuskoti might be possible. Never mind a third not being given, here was a fourth – and that fourth was itself a contradiction. How to make sense of that?
By distinguishing imperative from alethic and then distinguishing 'intension' from 'extension'. Only if there is a well-defined extension do Liebniz's laws operate.
Well, contemporary developments in mathematical logic show exactly how to do it. In fact, it’s not hard at all.
It was never hard. However, it was only once computing power became cheap that logic began to pay for itself.
At the core of the explanation, one has to grasp a very basic mathematical distinction. I speak of the difference between a relatiom and a function.
No relations or functions exist if the underlying set is not well-defined. True, for some specific purpose, we may arbitrarily stipulate for this but this is a matter of utility not logic.
A relation is something that relates a certain kind of object to some number of others (zero, one, two, etc). A function, on the other hand, is a special kind of relation that links each such object to exactly one thing.
i.e. the 'output' is dependent on the 'input'.
Suppose we are talking about people. Mother of and father of are functions, because every person has exactly one (biological) mother and exactly one father. But son of and daughter of are relations, because parents might have any number of sons and daughters. Functions give a unique output; relations can give any number of outputs. Keep that distinction in mind; we’ll come back to it a lot.
The distinction to keep in mind is between predicates which are 'multiply realizable' (e.g. 'son of King Charles' is fulfilled by both William and Harry) and those which are unique (e.g. 'eldest legitimate son of King Charles')
Now, in logic, one is generally interested in whether a given claim is true or false. Logicians call true and false truth values. Normally, and following Aristotle, it is assumed that ‘value of’ is a function: the value of any given assertion is exactly one of true (or T), and false (or F). In this way, the principles of excluded middle (PEM) and non-contradiction (PNC) are built into the mathematics from the start. But they needn’t be.
You could have 'naive' or 'fuzzy' sets or 'non-standard' analysis of various types. But you could also apply mathematics to cooking and try to bake a cherry's π.
To get back to something that the Buddha might recognise, all we need to do is make value of into a relation instead of a function.
Buddha won't recognize this shit. His point is that the domain and codomain are empty. There are no functions or relations. Also, if you have found a function, why treat it as a relation? It's like finding your long lost Mummy and then treating her as an Aunty.
Thus T might be a value of a sentence, as can F, both, or neither. We now have four possibilities: {T}, {F}, {T,F} and { }.
Why stop there? Why not add { {T } { }}- as in the Buddha truly exists and there is no Buddha- and so forth?
The curly brackets, by the way, indicate that we are dealing with sets of truth values rather than individual ones, as befits a relation rather than a function.
The graph of a function has a similar representation.
The last pair of brackets denotes what mathematicians call the empty set: it is a collection with no members, like the set of humans with 17 legs. It would be conventional
but arbitrary
in mathematics to represent our four values using something called a Hasse diagram, like so:
{T}
↗ ↖
{T, F} { }
↖ ↗
{F}
Thus the four kotis (corners) of the catuskoti appear before us.
But the four corners of Nothing are themselves nothing.
In case this all sounds rather convenient for the purposes of Buddhist apologism, I should mention that the logic I have just described is called First Degree Entailment (FDE). It was originally constructed in the 1960s in an area called relevant logic. Exactly what this is need not concern us, but the US logician Nuel Belnap argued that FDE was a sensible system for databases that might have been fed inconsistent or incomplete information. All of which is to say, it had nothing to do with Buddhism whatsoever.
Then why mention it? The fact is, Buddhism wasn't about databases. It was a dogma (matam) associated with a spiritual praxis (vigyan) which was similar to that purveyed by rival sects. It didn't matter too much whether you worshipped Ganesha in a Buddhist or a Hindu or a Jain temple. Equally, a Brahmin purohit could be a Buddhist while performing Vedic rites for his patron- e.g. the Thai or Cambodian King's Purohita.
Even so, you might be wondering how on earth something could be both true and false,
The answer is because the 'intension' is not well defined. I say 'I am not feeling myself today'. There is some truth to this- I am not as I normally am- but it is false to suggest that if I am not feeling myself, it is because I am feeling up your wife.
or neither true nor false.
because they are imperative, impredicative, speculative, or nonsensical.
In fact, the idea that some claims are neither true nor false is a very old one in Western philosophy.
It is obvious that most statements aren't propositions or theorems. Furthermore, a claim may not be associated with any particular argument. I claim your wallet because I want the money in it.
None other than Aristotle himself argued for one kind of example. In the somewhat infamous Chapter 9 of De Interpretatione, he claims that contingent statements about the future, such as ‘the first pope in the 22nd century will be African’, are neither true nor false. The future is, as yet, indeterminate. So much for his arguments in the Metaphysics.
Aristotle, like Plato, was making a living by running an Academy (though his was called the Lyceum). Buddha was running a religious sect. His subject was not Physics and what is beyond Physics. It was soteriology- i.e. why you should give money to monks and hope to be reborn as a monk.
The notion that some things might be both true and false is much more unorthodox. But here, too, we can find some plausible examples. Take the notorious ‘paradoxes of self-reference’, the oldest of which, reputedly discovered by Eubulides
who discovered the 'masked man' or 'intensional' fallacy.
in the fourth century BCE, is called the Liar Paradox. Here’s its commonest expression:
This statement is false.
Where’s the paradox? If the statement is true, then it is indeed false. But if it is false, well, then it is true. So it seems to be both true and false.
That's one interpretation. Another is that all statements are false. Some, however, are useful or, at the least, informative or stimulative to the imagination.
Many similar puzzles turned up at the end of the 19th century, to the dismay of the scholars who were then trying to place mathematics as a whole on solid foundations. It was the leader of these efforts, Bertrand Russell, who in 1901 discovered the most famous such paradox (hence its name, Russell’s Paradox). And it goes like this:
Some sets are members of themselves; the set of all sets, for example, is a set,
No it isn't because it isn't well defined.
so it belongs to itself.
No. It isn't a set.
But some sets are not members of themselves.
No. This is ruled out in ZFC by the Axiom of Foundation or Axiom of Regularity
The set of cats, for example, is not a cat, so it’s not a member of the set of cats. But what about the set of all the sets that are not members of themselves?
There is no such set.
If it is a member of itself, then it isn’t. But if it isn’t, then it is. It seems that it both is and isn’t. So, goodbye Principle of Non-Contradiction. The catuskoti beckons.
No it doesn't. What beckons is a theory of types. Naive set theory generated paradoxes because of 'unrestricted comprehension'. Restrict this in a sensible manner- which is what we all do all the time- and the paradoxes disappear. Sadly, philosophers doubled down on availability cascades arising out of intensional fallacies (i.e. pretending a thing which can't be a set or a function, is actually a set or a function) of a puerile type.
Here you might wish to pause for a brief sanity check. Do scenarios such as these really break the chains of Aristotelian logic?
His logic, like his Physics, was shit. So what? He died long ago. Back then everything was shitty including Buddhism. Emperor Ashoka kept killing Jain monks because he was a Buddhist.
Well, an increasing number of logicians are coming to think so – though matters remain highly contentious. Still, if nothing else, examples of this kind might help to remove the blinkers imposed by what Wittgenstein called ‘a one-sided diet’ of examples.
Godel, Tarski, Gentzen, & Church made progress in the thirties- as had Brouwer & Weyl. Witlesstein and Russell had lost the plot by then.
We’ll need to keep those blinkers off as we return to those tricky questions that the Buddha’s disciples asked him. After all, what does happen to an enlightened person after death?
His body starts to smell bad. Worms eat him unless he is cremated.
Things are going to get only more disconcerting from here on in.
Disconcerting to Buddhists- sure.
The Buddha, in fact, refused to answer such queries.
More particularly when he was taking a dump.
In some sutras, he just says that they are a waste of time: you don’t need to bother with them to achieve enlightenment. But in other texts there is a suggestion that something more is going on. Though the idea is never really elaborated, there are hints that none of the four possibilities in the catuskoti ‘fits the case’.
There are hints that the Buddha could gain all sorts of super-powers if he really wanted them. I too often tell my disciples that I could easily use my 'siddhi' of gaining disciples so as to have disciples whom I could exploit financially. Sadly, my disciples- thanks to their devotion to me- have gained the siddhi of being invisible and immaterial which is why I am currently locked up in a lunatic asylum.
For a long time, this riddle lay dormant in Buddhist philosophy.
No. There were plenty of boring Buddhist philosophical sects which took pot-shots at each other. The preceptors of these sects had super-duper siddhis- you betcha.
It was only around the second century CE that it was taken up by Nagarjuna, probably the most important and influential Buddhist philosopher after the Buddha himself. Nagarjuna’s writings defined the new version of Buddhism that was emerging at the time: Mahayana. Central to his teachings is the view that things are ‘empty’ (sunya). This does not mean that they are non-existent; only that they are what they are because of how they relate to other things. As the quotation at the beginning of this essay explains, their nature is to have no intrinsic nature (and the task of making precise logical sense of this claim I leave for the reader to ponder; suffice it to say, it can be done).
Buddha's real innovation is 'kshanikavada'- momentariness. There is no past, no future, there is only the vanishing present illumined by the lightning flash of intentionality. That's why there's no point being a Buddhist. Just have an intention of getting Nirvana and that's what you've got. This is Zen 'satori' rather than boring Indian 'samadhi'.
The most important of Nagarjuna’s writings is the Mulamadhyamakakarika, the ‘Fundamental Verses of the Middle Way’. This is a profound and cryptic book, whose principle theme is precisely that everything is empty. In the course of making his arguments, Nagarjuna often runs through the four cases of the catuskoti.
Meanwhile, the Jains were using a seven case logic to justify pluralism. Incidentally, their view was more compatible with Purva Mimamsa (i.e. the older Vedantic philosophy) and thus Umasvati was praised by Madhva who revived Theistic Dualism. Still, the 'vigyan' was pretty much of a muchness, so people were Jain/Vaishnav or Shakta/Buddhist etc.
In some places, moreover, he clearly states that there are situations in which none of the four applies. They don’t cover the status of an enlightened person after death, for example.
The big problem Organized Religion faces is that guys you helped get to Heaven or Nirvana or whatever can't pay you for helping them do so. That's why a doctrine of momentariness and the essential emptiness of money (which is worthless because the future does not exist) is useful. Sadly, this can be bad for National Defence and building up 'the sinews of war'. That's why the Japs, to 'escape Asia' and 'join Europe', spent a little time burning down Buddhist temples and beating monks till they grew out their hair, got married, and got proper jobs. At a later point, Zen monks became ultra-Nationalist assassins. That's why the Japs had a soft spot- in the head- for Heidegger.
Why might that be? Nagarjuna’s reasoning is somewhat opaque,
It is 'ipse dixit'. He was preaching to the converted. His system- like Sankara's or Umasvati's, was simpler and expressed in Sanskrit- i.e. it was 'anything goes' hermeneutically speaking.
but essentially it seems to go something like this. The language we use frames our conventional reality (our Lebenswelt, as it is called in the German phenomenological tradition). Beneath that there is an ultimate reality, such as the condition of the enlightened dead person.
This is like the Aristotelian Hypokeimenon which undergirds the Endekhomenon of what is possible but not necessary- i.e. this contingent world we live in. For Nagarjuna and Sankara (whom Vaishnavs consider a crypto-Buddhists) Sansara is Nirvana (or the other way around). That's perfectly reasonable- more particularly if you are drinking Prosecco on a Sunny day beside the river.
One can experience this directly in certain meditative states, but one cannot describe it. To say anything about it would merely succeed in making it part of our conventional reality; it is, therefore, ineffable.
Sadly, no. People won't stop gassing on about it.
In particular, one cannot describe it by using any of the four possibilities furnished by the catuskoti.
Which is why the thing does not matter in the slightest.
We now have a fifth possibility. Let us write the four original possibilities, {T}, {F}, {T, F} and {}, as t, f, b and n, respectively. The way we set things up earlier, value of was a relation and the sets were the possibilities that each statement might relate to. But we could have taken value of as a function and allowed t, f, b and n to be the values that the function can take. And now there is a fifth possible value – none of the above, ineffable, that which lies beyond language. Call it i. (Strictly speaking, it is states of affairs that are ineffable, not claims, so our values have to be thought of as the values of states of affairs; but let us slide over this subtlety.)
We can say that what Truth and Beauty and Justice and Duty refer to is beyond language. These are 'intensions' whose extensions are beyond our ken.
If something is ineffable, i, it is certainly neither true nor false.
not if truth is the same ineffable thing. That's a tautology.
But then how does i differ from n, neither true nor false?
In the way I mentioned. Mention of ineffability added a new possibility. There has been informational enrichment.
If we are looking at individual propositions, it is indeed tricky to discern any difference. However, the contrast comes out quite clearly when we try to join two sentences together.
Look at the sentence ‘Crows can fly and pigs can fly.’ You’ll notice that it is made up of two distinct claims, fused together by the word ‘and’. Expressions that are formed in this way are called conjunctions, and the individual claims that make them up are known as conjuncts. A conjunction is true only if both conjuncts are true. That means it is false if even one conjunct is false. ‘Crows can fly and pigs can fly’, for example, is false as a whole because of the falsity of the second conjunct alone. Similarly, if p is any sentence that is neither true nor false, that means ‘p and pigs can fly’ is false. By contrast, if p is ineffable, then ‘p and pigs can fly’ is ineffable too.
Only by arbitrary stipulation. The more natural approach would be to say it is false. Pigs can't fly even if Beauty or Duty is ineffable.
After all, if we could express the conjunction, we could express p as well – which we can’t. So i and n behave differently in conjunctions: f trumps n and i trumps f.
Nope. Pigs can't fly. The thing is false.
What I have just described is an example of a many-valued logic, though not a common one.
What you have described is stupid.
Such logics were invented by the Polish logician Jan Łukasiewicz in the 1920s. He was motivated, as it happens, by Aristotle’s arguments that contingent statements about the future are neither true nor false.
No. He was motivated by scientific and technological advances. Western Europe developed things like logarithm tables precisely because applications of mathematics- including applications to mechanical calculators or the 'Analytical Engine' of Babbage & Lovelace- contributed so much to raising general purpose and total factor productivity which in turn meant more resources could be developed to 'pure' research. The growth of Statistics and Probability theory was closely connected with the expansion of State Capacity. But the Law itself was being mathematicised. If you look at the career trajectory of 'Senior Wranglers' (i.e. top Mathematicians) from Cambridge, we find many ended up in the legal profession. Indeed, that was the reason Anglo-Saxon jurisprudence resisted a crude type of Utilitarianism- only succumbing to that virus in my own life-time.
In order to make sense of such claims, Łukasiewicz came up with a third truth value. It is indeed striking how useful his invention proves in the context of Buddhist metaphysics, though once again, Buddhism played no part in inspiring it. His innovation is entirely the product of the Western philosophical tradition.
It was entirely the product of the industrial revolution and a new type of total war which created the need for better and better algorithms and the cryptography which could crack such algorithms.
On the other hand, if Łukasiewicz really wanted to get to grips with Buddhist thought,
He was a patriot. He wanted his country to do well- which it did after it was released from Communist thraldom. I suppose Buddhism- as an 'Aryan' Religion- would have smacked of the Sarmatism which was the ruin of his great country. Anyway, Poles have a good reason to dislike the Swastika- which, back then, was associated with Buddhism more than Hinduism.
he shouldn’t have stopped with his many-valued logics. Perhaps you have already seen what’s coming next…
If Buddhist or Jain logic (the latter could be 'fuzzy') mattered, they would be contributing to great advances in computer software. There may be great wisdom in the belief systems of the ancient Mayans or Yoruba or Kshatriyas of India. But, what there isn't is cool tech from which we can all benefit.
Philosophers in the Mahayana traditions hold some things to be ineffable;
But, if Time does not exist, language and speech don't matter because their ineluctable modality is time. For Buddhism, there is only 'cetana'- consciousness or, more precisely, intentionality. The Jains have rather a neat notion of 'divyadhvani' which is sidesteps the issue. The point is that language is merely a tool.
but they also explain why they are ineffable, in much the way that I did. Now, you can’t explain why something is ineffable without talking about it. That’s a plain contradiction: talking of the ineffable.
Nope. We can talk about why we aren't talking anymore. Indeed, that tends to be an interminable conversation.
Embarrassing as this predicament might appear, Nagarjuna is far from being the only one stuck in it.
He had no predicament. Like Sankara and Umaswati he put an end to scholastic disputes of a futile type. Later, theistic, sects could adopt their arguments so as not to have to invest in that tedious shite.
The great lodestar of the German Enlightenment, Immanuel Kant, said that there are things one cannot experience (noumena), and that we cannot talk about such things.
We can talk about anything we like. True, what we say won't correspond very well with what we experience or want to experience or can't experience. But language is merely a tool. So is logic. It obeys the law of increasing functional information if there is selective pressure- which currently obtains for economic/military reasons. My guess is that there was a discrete math tradition at the time when Indic systems of logic were developing but then the selection pressure stopped because India's warrior class (from which Buddhism and Jainism arose) was shit and thus the country came under foreign domination.
He also explained why this is so: our concepts apply only to things we can experience. Clearly, he is in the same fix as Nagarjuna. So are two of the greatest 20th-century Western philosophers. Ludwig Wittgenstein claimed that many things can be shown but not said, and wrote a whole book (the Tractatus), explaining what and why.
To his credit, he understood his mistake. Had Frank Ramsey lived, Anal-tickle philosophy would have been smothered in the crib.
Martin Heidegger made himself famous by asking what Being is, and then spent much of the rest of his life explaining why you can’t even ask this question.
You can ask any question you like- not just what is is but also what what what is what what. You are also welcome to eat your own shit.
Call it mysticism if you want; the label has little enough meaning. But whatever you call it, it is rife in great philosophy – Eastern and Western.
Heidi had shit for brains. Spoiled Catholics often do.
Anyway, what did Nagarjuna make of this problem? Nothing much. He didn’t even comment on it. Perhaps that’s not so surprising: after all, he thought that certain things might be simultaneously true and false.
No. He thought somethings are conventionally true but not from the ultimate point of view.
But later Buddhist philosophers did try to wriggle out of it, not least the influential 15th-century Tibetan philosopher, Gorampa.
Who negates not just phenomena but existence itself.
Gorampa was troubled enough by the situation that he attempted to distinguish between two ultimate realities: a real ultimate reality, which is ineffable, and a ‘nominal’ ultimate reality, which is what we end up talking about when we try to talk about the real ultimate.
This is also true when we talk about our beloved puppy dog. Anything you can say about it is nothing compared to what it is to us.
But wait a minute – the nominal ultimate is obviously effable: by definition, it is the reality that we can talk about.
It is a name. It may not correspond to anything real. My puppy dog is imaginary because Daddy wouldn't buy me a bow wow.
In that case, if we say that ultimate reality is ineffable and we are actually talking about the nominal ultimate, what we are saying is false.
Everything we say is false from some perspective.
Thus Gorampa’s proposal refutes itself.
Only in the sense that his fart refutes itself because it insists on trying to refute Einstein's General Theory of Relativity. At any rate, this is what happens with my farts.
Interestingly,
i.e. very very boringly
Kant made a similar move. He distinguished between two notions of noumenon, the realm beyond the senses: a positive one and a negative one. According to him, only the negative one is legitimate. We cannot talk about things of this kind; we just need to be aware of them to mark the limit of what we can talk about. Pardon? In explaining what they do, are we not talking about them? Well, yes, of course we are.
We may be, we may not. Arguably, we are talking about the interactions of a thing rather than the thing itself. All the doggy traits of our puppy dog aren't 'intrinsic'. The creature is actually a wolf.
The Gorampa/Kant predicament is, in fact, inevitable. If one wishes to explain why something is ineffable, one must refer to it and say something about it.
The solution is to say there are degrees of ineffability.
To refer to something else is just to change the subject.
The subject is changed by predication. 'Good dog!' is different from 'bad puppy!'
So we have now hit a new problem: the contradiction involved in talking of the ineffable.
But that contradiction is either ineffable or else self-contradictory.
In a sense, the possibility of a true contradiction is already accommodated by that both option of the catuskoti. (Our Western thinkers could not even say this much.)
We have been saying that Lord Jesus Christ is both fully God and fully Man for two thousand years. Whatever we see that is good in the West (and the reason Hindus or Muslims or Buddhists are so happy to settle in the West) grew out of that wholly contradictory or ineffable hypostatic union. But we could say the same thing about the fundamental credo of the religions of other countries which have achieved much for their citizens. I may like Capitalism and hate Communism, but I have to admit that the Chinese Communist Party lifted hundreds of millions of its people out of poverty by discovering that Marx said 'to each according to his contribution'.
Alas, our contradiction is of a rather special kind. It requires something to take both the values true and ineffable, which, on the understanding at hand, is impossible.
The solution is a pragmatic. We say 'this is true enough for this purpose but not for some other purpose'. Moreover, what we say about a thing always falls short for some purpose or from some perspective. The law, as Edward Coke pointed out to James I, is 'artificial reason'. Naturality- i.e. non arbitrariness- is far to seek.
Yet the resources of mathematical logic are not so easily exhausted.
For any given purpose, we can always find good enough 'univalent foundations'.
In fact, we have met something like this before. We started with two possible values, T and F. In order to allow things to have both of these values, we simply took value of to be a relation, not a function. Now we have five possible values, t, f, b, n and i, and we assumed that value of was a function that took exactly one of these values. Why not make it a relation instead? That would allow it to relate something to any number of those five values (giving us 32 possibilities, if you count). In this construction, something can relate to both t and i: and so one can say something true about something ineffable after all.
You can say something useful or informative. But truth is far to seek.
The technique we are using here is called plurivalent logic,
but univalent foundations seems more useful for stuff like computerised proof checking or, indeed, proof generation.
and it was invented in the 1980s in connection with the aforementioned paradoxes of self-reference. In fact, one of those paradoxes is not a million miles away from our ineffability predicament. It is called König’s paradox,
and throws light on the Jain distinction between countable and uncountable.
after the Hungarian mathematician Julius König who wrote it up in 1905, and it concerns ordinals.The paradox is as follows- 'If you can define the "least indefinable ordinal," then it's defined, contradicting the assumption that it's indefinable.;
Ordinals are numbers that extend the familiar counting numbers, 0, 1, 2, etc, beyond the finite. After we have been through all the finite numbers (of which there is, of course, an infinity), there is a next number, ω, and then a next, ω+1, and so on, forever. These ordinals share an interesting property with the counting numbers: for any set of them, if there are any members at all, there must be a least one.
In other words, ordinals are totally ordered like natural numbers.
How far, exactly, the ordinals go is a vexed question both mathematically and philosophically. Nevertheless, one fact is beyond dispute: there are many more ordinals than can be referred to using a noun phrase in a language with a finite vocabulary, such as English. This can be shown by a perfectly rigorous mathematical proof.
This is also true of real numbers. We can name a number- e.g. the number of farts I will fart in my life- without having any idea what that number might be.
Now, if there are ordinals that cannot be referred to in this way, it follows that one of them must be less than all the others, for that is true of any collection of ordinals. Consider the phrase ‘the least ordinal that cannot be referred to’. It obviously refers to the number in question. This number, then, both can and cannot be referred to. That’s our paradox.
It is an intensional paradox arising out of the intensional or 'masked man' fallacy and has been known since the 4th century BC.
And since it cannot be referred to, one cannot say anything about it.
One can say anything one likes. It's just that what one says won't be admissible for any useful purpose.
So the facts about it are ineffable; but we can say things about it, such as that it is the least ordinal that can’t be referred to. We have said ineffable things.
In 1904, G.K Chesterton wrote- 'Every time one man says to another, “Tell us plainly what you mean?” he is assuming the infallibility of language: that is to say, he is assuming that there is a perfect scheme of verbal expression for all the internal moods and meanings of men. Whenever a man says to another, “Prove your case; defend your faith,” he is assuming the infallibility of language: that is to say, he is assuming that a man has a word for every reality in earth, or heaven, or hell. He knows that there are in the soul tints more bewildering, more numberless and more nameless than the colours of an autumn forest; he knows that there are abroad in the world and doing strange and terrible service in it crimes that have never been condemned and virtues that have never been christened. Yet he seriously believes that these things can every one of them, in all their tones and semi-tones, in all their blends and unions, be accurately represented by an arbitrary system of grunts and squeals.
The similarities between this and our Buddhist paradox of ineffability are, you must admit, pretty unnerving.
Not really. Faith is founded on a mystery. The point about Nagarjuna or Sankara or Umasvati is that they give free range to Faith. You don't have to study abstruse systems of logic or master Sanskrit grammar to gain salvation.
But those who developed plurivalent logic were entirely unaware of any Buddhist connections. (I say this with authority, since I was one of them.)
Jain logic would be more to the point but it does not seem to have been studied by non-Jains. In particular, Jainism has a dynamic conception of substance.
Once again, the strange claims of our Buddhist philosophers fall into precise mathematical place.But is plurivalent logic useful in the same way as Voevodsky's univalent foundations? Perhaps it will be implemented on quantum computers or stuff yet more arcane.
There is, of course, much more to be said about all these matters. But we have now seen something of the lie of the land. So let me end by stepping back and asking what lessons are to be drawn from all this.
The lesson seems to be that Buddhist logic could have been useful but wasn't. That's why China rules Tibet rather than the other way around. On the other hand, Western work in mathematical logic has enabled computers to alter all our lives for the better.
One is a familiar one. Mathematical techniques often find unexpected applications. Group theory was developed in the 19th century to chart the commonality of various mathematical structures. It found an application in physics in the 20th century, notably in connection with the Special Theory of Relativity. Similarly, those who developed the logical techniques described above had no idea of the Buddhist applications, and would, I am sure, have been very surprised by them.
LEJ Brouwer, though a committed Christian, showed great interest in Hindu and Buddhist thought. Indeed, the pre-War Zeitgeist featured the revival of different strains of mysticism- e.g. Gnosticism, Russian onomatodoxy, Celtic tuirgen, and the Theosophical Society's attempt to fuse all the great spiritual traditions of the world together.
The second lesson is quite different and more striking. Buddhist thought, and Asian thought in general, has often been written off by Western philosophers.
But Western philosophy has been written off by the West. At one time it appeared useful. Perhaps it could catalyse creative collaboration between different Sciences. Sadly, it turned out that a mathematician or a scientist could have any belief whatsoever. If they did good work, they were useful. If not, they didn't matter. Godel did good work. Because he was a devout Christian, he produced a mathematical proof of God. Sadly, a computer detected the flaw in his proof.
How can contradictions be true? What’s all this talk of ineffability? This is all nonsense. The constructions I have described show how to make precise mathematical sense of the Buddhist views.
It was possible to make precise mathematical sense of Ptolemy's belief that the Sun goes round the Earth. But that belief was wrong.
This does not, of course, show that they are true. That’s a different matter. But it does show that these ideas can be made as logically rigorous and coherent as ideas can be.
Sadly, any crazy shit can be made logically rigorous and coherent.
As the Buddha may or may not have said (or both, or neither): ‘There are only two mistakes one can make along the road to truth: not going all the way, and not starting.’
Following Nagarjuna, we say 'Nirvana is Samsara'. The Grail is the Quest for the Grail. As for mathematical logic, it turns out that it was a very useful type of drudgery. But, it now appears, there will always be more mathematical logics than there are mathematicians. Meanwhile the law of increasing functional information applies. Where there is selection pressure on a specific function (e.g. making better generative A.I) then there will be evolution to greater complexity and diversity. I suppose there was a time when the Gangetic plain was relatively rich and wanted to get richer yet. This meant there was rapid development in all sorts of disciplines- linguistics, logic, mathematics, medicine etc. But the martial spirit of the people tended to flag. The Buddha came to be seen as a delusive avatar of Vishnu whose doctrine was intended to get invading barbarians to give up their war-like ways. The problem was that, if the rulers lose their martial qualities, they will be replaced by more ferocious rulers who are immune to the charms of Buddhism.
Meanwhile, Jainism was able to keep itself going amongst a mercantile elite and thus survived in India whereas Buddhism disappeared under the Islamic onslaught.
Under the British Raj, there was a revival of interest in Buddhism amongst Hindus and after Dr. Ambedkar- the leader of the 'untouchables'- converted to Buddhism, the number of Indian Buddhists has grown steadily. But there is no interest in Buddhist logic. Why? I think the answer is that it had already been accommodated by the Navya-Nyaya tradition which, au fond, is utilitarian and pragmatic. Dogma (matam) must be learned by priests so that they are able to explain how their sect differs from another sect, but orthopraxy (vigyan) is pretty much of a muchness.
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