Is it possible that some contradictions can be true? If so, how would that affect Bayesian Rationality as well as Theoretical Physics and Quantum Computing? This idea is called "Dialetheism", and through paraconsistent logics like Graham Priest's "LP", suggests a trivalent value system where some statements can be either True, False, or both True AND False simultaneously.
It might sound like a laughable claim to say that some contradictions can be "BOTH true and false" (this doesn't apply to ALL contradictions, just to paradoxes), but it could be extremely useful for things like quantum mechanics research, AI research, and effective altruist ethics (not to mention it's essential to understanding eastern religions). Let's go over the historical context of Dialethiesm to see what this idea of supposedly "valid contradictions" is all about.
INTRO TO WESTERN LOGIC (For those unfamiliar)
Aristotle was the first to try and categorize all core operations of the mind, deriving 3 laws of logic that we can take absolutely for granted, laws that self-evidently apply without question. Aristotle called them “the 3 Laws of Thought”... (well... technically he stole them from Plato):
"First, that nothing can become greater or less, either in number or magnitude, while remaining equal to itself ... Secondly, that without addition or subtraction there is no increase or diminution of anything, but only equality ... Thirdly, that what was not before cannot be afterwards, without becoming and having become”
1) The law of identity : P=P, also called the law of self-evidence, the idea that a thing is a thing. For example the sentence “the Universe is the Universe” is a self-evidently valid statement.
2) The law of excluded middle : P∨~P=T, the all encompassing idea that an option is an option. For example the sentence “Either the Universe exists OR the universe doesn’t exist” is a self-evident statement (show shakespeare’s to be or not to be).
3) The law of non-contradiction : Aristotle made a critical decision in the history of western civilization, he decided to add a 3rd law, P∧~P=F, the notion that we can’t have both a thing and not a thing. It’s the idea that contradictions, not just some, but ALL of them, are outright false. For example, if you were trying to answer the question "why is there something rather than nothing?", then the sentence “The universe exists AND the universe doesn’t exist” is a contradiction that should supposedly disqualify your argument.
Naturally, you might think there’s nothing wrong with calling a sentence "the universe does and doesn't exist" FALSE and Aristotle believed that too. Little did he know that not all contradictions are structured equally, because SOME contradictions have truth values that refer to themselves. Take the statement “Existence doesn’t exist”, or the statement “Non-existence exists”. Are these statements true or false? Let's find out.
The first to point out paraconsistency was the philosopher Epimenides, a man from Crete who realized he could say “all people from Crete are liars”, creating a self-referencing contradiction. These special contradictions are what the greeks came to call “Paradoxes”, because not only can they be true or false, but they can also be BOTH true and false or NEITHER true or false, a puzzle Pyrrho called “the Tetralemma”. This tetralemma was actually a big problem for Aristotle, because the 2nd law, the law of exclusion says all statements must be either true or false and yet we also can’t label paradoxes as solely true or solely false because they are self referential. In order to account for paradoxes, we’d first have to rewrite Aristotle’s 2nd law from this P∨~P = T to this P∨~P∨(P∨~P) = T and then rewrite the 3rd law from this P∧~P = F to either this; "P∧~P = F∨(T∧F)" OR this; "P∧~P = F∨(~T∧~F)".
What all this means is that there are now only two ways possible ways to answer what a paradox is.
OPTION A: The first way is to deem all contradictions false but a paradox as BOTH true AND false simultaneously, written like this; P∧~P = F∨(T∧F).
OPTION B: The second way is we interpret all contradictions as false but a Paradox as NEITHER true nor false, written like this; P∧~P = F∨(~T∧~F).
By pure circumstance, the Greeks decided to interpret paradoxes using method 2, Neither True NOR False, which disqualifies paradoxes from ever being used in arguments again, permanently banishing them from our civilization, and thus, banishing them from your very language. Most people in the world grew up in a civilization grounded in Aristotelian philosophy, thinking there’s absolutely nothing wrong with ignoring paradoxes, but what if we only think that way due to a lifetime of social conditioning? If you ask anyone on the street if paradoxes counted as sensible statements they’d probably say no, and not surprisingly, so did Aristotle.
INTRO TO EASTERN LOGIC (Also for those unfamiliar)
While Aristotle was laying the groundwork for modern language, science, and law, there was a philosopher on the other side of the world who couldn’t disagree more, Siddartha Gautama, the Buddha. If a contradiction is NEITHER True or False, then we can rewrite it as NOT TRUE and NOT FALSE, which we can then rewrite as contradictions are FALSE and TRUE, but that’s actually the same thing as all contradictions are TRUE and FALSE, so could Aristotle have been wrong?
1) The Chatuskoti : In the Sutras, one of the Buddha’s students curious about the afterlife asks him:
"Master Gotama, does master hold the view that after death, a Tathagata exists, where only one thing is true and anything else is false?".
The Buddha responds:
"Vaccha, I do not hold the view that a Tathagata exists after death. Nor do I hold the view that a Tathagata does exist. I also do not hold the view that a Tathagatha neither exists nor does not exist. Nor do I hold the view that a Tathagata exists and does not exist".
In Buddhist logic this is called the “Catuṣkoṭi“ or "four corners", written like this P∨~P = F∨T∨(T∧F)∨(~T∧~F), which means it’s the paradox of all paradoxes from which all other paradoxes arise. A superposition of “Oness” from which we derive all other possible systems of logic. It holds that the first foundation of all reality is a divine contradiction called “Sunyata” or "emptiness", a kind of superposition-like state that is neither true, nor false, nor true and false, nor neither true OR false.
2) The Law of Contradiction : 600 years later, the philosopher Nagarjuna, founder of Mahayana Buddhism, worked out that you can divide that one divine superposition into further superpositions. Rather than interpreting a Paradox as neither true or false like Aristotle did P∧~P = F∨(~T∧~F), Nagarjuna actually choses the first option, to interpret paradoxes as BOTH true AND false P∧~P = F∨(T∧F), what we call “a Dialetheia”. Because if the ground of reality is itself a paradox as the Buddha says, then it should subdivide into further paradoxes rather than further negations. Najarjuna’s version of Aristotle’s 3rd law was called the “2 truths doctrine”, mandating that all metaphysical systems must account for the possibility of Dialethias, what the Zen Buddhists came to call “Koans”.
The idea that a thing could both exist AND not exist sounds absolutely absurd to us today, but for the Buddhists it was just life as usual. The universe is both something and nothing, and for Nagarjuna, the idea that the universe caused itself is a perfectly valid statement, since he never posited any ironclad ban on contradictions like Aristotle did. Right now, you might be really tempted to ignore the 2 truths doctrine as mystical new age rambling, but when it comes to computers, the laws of physics, and the very language we speak, the validity of self-referential paradoxes couldn’t be a more serious matter. Unfortunately, after the Empire of the Buddhist King Ashoka fell, Buddhist libraries, abhidarma schools, and temples in India were burned and monks were slaughtered.
Nagarjuna faded into obscurity while Aristotle’s 3 laws spread around the world through European empires, forming the next 2000 years of global civilization. Zen Buddhism and Daoism were the only major philosophies in human history that ever permitted contradictions, but it was Aristotle who shaping the world’s universities, legal customs, and social institutions, all dictating what kinds of thoughts our minds can and can’t think.
Even in the west, there were only ever 5 major western thinkers to base their entire philosophy on dialetheic logic, the continental philosophers Georg Hegel, Friedrich Nietzsche, Martin Heideggar, Gilles Deleuze, and the ancient presocratic, Heraclitus. All of them were also largely ignored by mainstream science and mainstream religion, never never stopping to question whether or not our reality could secretly be a wondrous world of paraconsistent simultaneity. A higher plane of contemplation where our ethics, metaphysics, and overall understanding of reality could all be different. As Ludwig Wittgenstein once said, “the limits of our language mean the limits of our world”, and unfortunately, our logic dictates our language.
DIALETHEISM (The Real Argument Starts Here)
Up until the 21st century, Aristotle’s 3rd law went on ignored until one man, decided to bring the question of the Paradox back from the dead. Dr. Graham Priest is a distinguished professor of analytic philosophy at the city university of New York and he has spent his entire career working on one phrase, a pesky statement called “this sentence is false”. This is the well known “Liar’s Paradox”, the statement that everything being said is a lie. So if the liar is indeed lying, then the liar is telling the truth, which means the liar just lied, which means they’re also telling the truth (repeat ad infinitum).
The liar's paradox used to be nothing more than a party trick, until the 20th century, where we needed to take it seriously for us to ground mathematics and construct quantum computers and more advanced machine learning systems.
1) THE PROBLEM OF BIVALENCE
The Principle of Bivalence is the idea that a thing can't have 2 truth values, but is it legit?
a) The Paradox: We can write out the paradox as a syllogism. "This sentence is false is true", "This sentence is false is false", "therefore this sentence is false is BOTH True and False" (rather than NEITHER true or false), thereby violating aristotle’s 3rd law:
b) The Rebuttal: Interestingly enough, the Liar’s paradox supports Nagajuna’s interpretation, meaning Aristotle’s law of contradiction could be changed. However, this usually hand waved away with common rebuttal to the Liar’s paradox. If the liar’s paradox is both true and false then it’s not true. If the liar’s paradox is both true and false then it is not false, therefore therefore the liar’s paradox is actually NEITHER true nor false like Aristotle said:
c) The Rebuttal to the Rebuttal: At first glance, the rebuttal seems to have debunked the Liar’s paradox, but if we write out the logic we will see that all this rebuttal did was try to distract us from the actual problem. If we assume the conclusion of that rebuttal, where the liar’s paradox is neither not true or not false, then as the second premise we can point out that the phrase “Not true AND Not false” is just the same thing as “False AND True”, meaning we have proven that Aristotle was wrong and there’s no such thing as a statement that’s neither true nor false, leaving the only remaining interpretation of the Liar’s Paradox as true AND false:
2. ~F∧~T→T∧F ,
2) THE PROBLEM OF TRUTH VALUE GAPS
We could always just classify the liar's paradox as a so-called "truth value gap", meaning not only is it neither true or false, paradoxes aren't even deserving of a truth value.
Let’s give an example of a sentance without a truth value. “What’s your favorite color?” That’s a question, so it's an example of a TVG (truth value gap). But what about a sentence like “Existence doesn't exist”?
a) the paradox : One might say a paradox like that has zero truth value because we don’t respond to it with "that's true" or "that’s false". But let's try a special statement, “the present king of france is bald”.
b) the rebuttal : It makes an assumption that there is a present king of france, so it’s neither true nor false, but it MUST have a truth value because it is certainly in the category of "statement". I suppose we can infer that a Dialetheia works only in-so-far as it’s talking about things that actually exist (there is no such thing as a "king of france" today).
c) the rebuttal to the rebuttal : The statement “unicorns are white and not white” is not a Dialethia, it's quite fairly a TVG. But for something "present" that IS legitimately being talked about, a paradox is not a TVG, it's perfectly grammatical, doesn't commit category mistakes, and it doesn't suffer from failure of reference. Meaning things present at hand (like problems of quantum mechanics) do demand a truth value.
3) THE PROBLEM OF EXPLOSION
The issue of bivalence is solved, but if so, would a Dialethia mean that all of reality is subjective and a matter of opinion?
Well, no, it’s just saying SOME parts of objective reality are structured through paradoxes.
This kind of attack on Dialetheism is called "The Principle of Inference", but Logicians just call it "Explosion”, because what it suggests is that if we break the law of contradiction then people can just make any argument they want.
The principle of explosion has been around since the middle ages and it’s the very reason that Aristotle’s law of contradiction never gets questioned (because if we violate the law of contradiction, then people can pretty much say anything whatever, thereby making all of logic pointless). This is why it's strongly recommended that we must ABSOLUTELY NEVER break Aristotle's law.
The principle of explosion can be written as p and not p imply q ( ~P∧P→Q ) where p and not p is any given contradiction and q is whatever conclusion you feel like proving.
a) The Paradox : For example, take this ridiculous argument where we treat a contradiction as true: assume a contradiction like the universe does exist and doesn’t exist, an idea some eastern philosophers actually accept, premise two, either the universe doesn’t exist or unicorns exist, seems fair so far. But then we see the conclusion, if the universe does exists, which it does, then unicorns exist too.
b) The Rebuttal : In fact, not just unicorns, replace Q with whatever you want and it will be true. Considering contradictions true is a nightmare because you make literally any argument and have it be valid. The principle of explosion concludes that only way to avoid ridiculous arguments like that is to declare the first premise of the argument, the contradiction, as false, thus making unicorns and all other fantasies an unsound argument:
c) The Rebuttal to the Rebuttal : It seems like a rock solid rebuttal, however explosion misses one key detail, Dialetheism never made the assumption that EVERY contradiction is necessarily true. Considering a contradiction to be BOTH true and false is not the same thing as considering it true. Dialetheism says that if we reject most contradictions BUT accept the existence of self-referencing contradictions, AKA paradoxes, then we can violate Aristotle’s 3rd law without permitting ridiculous arguments that can claim whatever they want. Let’s look at that unicorn argument again, but instead of just having true and false, this time let’s allow 3 possible truth values, True, False, or Dialethia:
Premise one, "the universe does exist and doesn’t exist", instead of making this true or false we’ll make it a dialethia, as some eastern thinkers have posited. Premise two, "either the universe doesn’t exist or unicorns exist", the same premise as last time. Surprisingly, we see that using the Dialetheia still makes the unicorn argument false. Conclusion, if the universe exists then we still can’t infer that unicorns exist since that conclusion no longer follows from premise 2:
Overall, we’ve shown that you can still break the law of contradiction without being being allowed to say just anything, thereby deconstructing the principle of Explosion and challenging Aristotle’s 3rd law. In the 21st century, any system of logic that rejects explosion and considers paradoxes valid is what we call a "Paraconsistent logic”, while any logic that keeps the principle is called a “Classical Logic”.
Now note one key detail, I’m not saying Classical Logics should be done away with. Over 99% of scientists still use classical logic, and you know what, that’s perfectly okay, because they never have to deal with paradoxes. However, that last 1% of of scientists, like quantum physicists, have to deal with paradoxes all the time, so we can’t just force them to use classical logic too, they need a more accurate set of axioms. Today, there’s a desperate need to create computers, perhaps quantum computers, that can violate the law of non-contradiction so physicists can finally solve their problems. If we put Paraconsistent logics into computers it means they’re going to start collecting a lot more information and drawing a lot more conclusions, since there's now more than two possible truth values: "true", "false".. and "dialetheia".
In CLASSICAL LOGICS we only have "1" and "0"
In PARACONSISTENT LOGICS we now have "1" and "0" and "#". I imagine this is something that could come in handy for quantum computers.
Important Note: I’m not saying Aristotle was stupid for inventing the Law of Non-Contradiction, I’m just pointing out that paradoxes are exceptions to the rule. It’s the same principle behind Newton’s classical dynamics being not entirely accurate and getting replaced with Einstein’s General Relativity, it depends on what you’re using it for. 99% of scientists would get by just fine using Newton’s laws of mechanics, but physicists doing Black Hole research would need to use General Relativity to get a more accurate answer. This is the same reasoning for why we should expand Aristotle’s classical law of contradiction P∧~P=F into the paraconsistent law of contradiction P∧~P=F∨(T∧F), allowing us to discover the deeper philosophical truths of the universe.
4) THE PROBLEM OF GROUNDEDNESS
Now allow me to stop attacking Pseudo-strawmen and get to the REAL rebuttals posed by actual philosophers. The ultimate attack on the Liar's Paradox comes from the logician Saul Kripke.
a) The Paradox : Kripke says the Liar's Paradox isn't grounded and is just a viscious cycle of adding fake truth-predicates, thus making it a valueless statement.
b) The Rebuttal : Kripke's clarification of the paradox is called "groundedness", where we remove all "falsity predicates" from the statement, we're left with the root of the statement “this sentence”. In other words, if we use the symbol P to represent "this sentence" and the words "is false" represented as the falsity predicate on P (~), then we’ll see that just as "questions" lack truth value, the statement “this sentence” ALSO has no truth value.
c) The Rebuttal to the Rebuttal : This brings us to the “Proof of Revenge”, which Dr. Priest evokes as a rebuttal to Kripke using what’s called “the Strengthened Liar's Paradox”, where we change the syntax of “this sentence is false” into “this sentence is not true” or even better yet “this sentence is either not true or valueless”.
Now when you try Kripke’s rebuttal it no longer works. If Kripke, like before, points out that the statement is valueless, then it's not true, and if we admit it's not true then the statement "this sentence is either not true or valueless" is true, again giving us a Dialethia where both trueness and falseness are simultaneously valid.
Kripke’s method only works on phrases like “this sentence is true” because it has a redundancy of it’s own truth value. However, Kripke's groundedness DOES NOT work on “this sentence is false” which refers to the existence of itself then overturns it’s own truth value.
“This sentence is true” undetermines it’s own truth value while "this sentence is false" overdetermines it, which is to say it creates a truth value within a truth value.
5) THE PROBLEM OF REDUNDANCY
Philosopher Arthur Prior had his own response to Paradoxes, which was essentially, "so what?". SO WHAT if the phrase “this sentence” refers to it’s own existence? If that’s the case, then don’t ALL sentences refer to their own existence? If all sentences implicitly refer to their own truth, then "the proof of Revenge" is redundant.
a) The paradox : If I had a sentence that simply contained the word “False”, then should we say the very existence of the word false is ALSO a paradox? Obviously not, beacause the sentences “the sky is blue” and “it’s true that the sky is blue” are actually the exact same sentence in terms of their truth value.
b) The rebuttal : In the same sense, the statements “this sentence is false” and “it’s true that this sentences false” are ALSO the same sentence, so the liars paradox is tricking us into seeing a predicate that isn’t really there.
c) The Rebuttal to the Rebuttal : But there’s one intersting thing about Prior's rebuttal. Yes the statement “this sentence is false” can translate to “it’s true that this sentence is false”, however using Dr. Prior’s exact same logic, THAT sentence itself is also identical to the sentence “it’s true this sentence is true and this sentence is false”, which is, you guessed it, a Dialetheia.
However, Prior took this into account and pointed out one more interesting thing, we would be using a Dialethia to prove the existence of a Dialetheia, which in it’s own metalinguistic way, is a circular argument. We haven’t derived a dialethia from the argument, we’ve just asserted it, which makes it a contradiction.
Unfortunately for him, Prior is also using a circular argument, using the law of contradiction to prove that the law of contradiction is true. At first it appears we have a clash between 2 circular arguments with both sides begging the question, but there is a way out, Occam’s razor. What’s interesting is that we make fewer assumptions about logic to get Dialetheia than we do to get Dr.Prior’s answer with regard to what we call “Contingent Facts”.
Dr.Prior’s argument uses the context of a sentence to make his argument, requiring him to multiply beyond necessity. Meanwhile Dialetheia is self-evidently derived from the sentence itself and needs no comparison with other sentences in order to make its point. Lastly, Dialetheia is actually NOT a circular argument because it only makes 2 assumptions, the law of identity and the law of exclusion, while Prior has made 3 assumptions, the previous 2 plus the law of contradiction.
6) THE PROBLEM OF LANGUAGE
Mathematician and Philosopher Alfred Tarski launched one final attack on the liar’s paradox, claiming that it's just a problem of language. It’s a similar response to what we’d get from the Postmodernists, because perhaps this whole idea of the Liar’s Paradox isn’t a self-ecvident truth and it might just be mental masterbation for one simple reason. If we can find a language where the paradox doesn't exist, then Dialetheism is not self-evident, it’s just a social construct. To do this Tarski draws a distinction between quote "semantically closed languages" like english verses what are called "semantically open languages". While every human language on earth is a semantically closed language where you can use the language to talk about the language, the liar's paradox CAN'T be expressed in a semantically closed languages. A semantically closed language has 2 elements, one, that can refer to it's own expressions and two, that it contains the predicates true or false for semantic closure.
However, Tarski says we can create synthetic machine languages where self-referential sentences are blocked or artificial languages that don't use true or false as predicates. These are called "Semantically Open languages", and they tend to be useless to humans but nevertheless ARE possible. We can create what's called "an object language" that is structure such that it can't possibly talk about itself in terms of truthood or falsity. Without a language that can talk about itself, the liars paradox is just a social construct, a problem of human language and has no basis in logic. In fact, Tarski has even suggested getting rid of all human languages and creating a new artificial semantically-open language that all humans on earth could speak and this language would not have the same problems of confusion we do. In fact, we'd be able to do philosophy and science without misinterpreting each other or be able to purposely mislead each other. A language where it's impossible to lie. But Tarski’s Solution has one small problem.
While we can build a Semantically open language that can’t talk about itself, it’s not possible to construct one that can’t talk about language in general. For example, what’s to stop it from evolving a way to talk about other languages? Linguistic philosophers call the idea of a language that can talk about other languages “a Metalanguage” while the language being talked ABOUT is called “An Object Language”. The problem here is that there’s nothing stopping an object language from making predicate statements about the metalanguage, for example “this statement’s metalanguage is false”.
Essentially, Tarski can’t give us a bulletproof way to keep Dialeitheism out of our discourse and has failed to banish it to the realm of linguistic constructs.
No matter what we do, the dreaded liars paradox will keep returning to logic no matter what we do.
So clearly there’s only one thing left to do, instead of desperately trying to ignore the existence of Dialetheia, why don’t we just embrace them?
Why don’t we find a way our languages, mathematics, and laws of physics can work WITH IT rather than Against it?
CONCLUSION (What does Buddhist Dialetheism mean for AI research?)
If we can accept Nagarjuna's notion that "SOME contradictions are true" then this changes everything... metaphysics, epistemics, ethics, politics, and rationality itself. If so, we might have to rewrite some ingrained social, economic, and scientific laws to adjust for dialetheia, but for now I just want to see what it means for Friendly AI.
We're all familiar with how a lack of paraconsistent thinking can lead to misinterpretations of commands, leading to Eliezer's infamous Paperclip Maximizer scenario. Perhaps this is also why philosophers of mind (John Sealre, Ned Block, and David Chalmers to name a few) perceive machine minds as subordinate to human minds because the human brain can operate on paraconsistent logic (Roger Penrose's "Quantum Coherence Theory" of consciousness also seems to suggest this). So am I saying we should build systems capable of yeilding something other than true-or-false outputs... well.... yes and no (see what I did there :^) ), while I do believe most contradictions are false, I also think dialethic logics could be the key to building a more "friendly" AI. If an artificial intelligence can think in dialethic terms, it might finally be able to learn our values (thus we could reduce the existential risk of it turning on us).
Consider the ending to a video game many of you might be familiar with, "Portal 2", where the rogue archailect GLaDOS is unexpectadly defeated by a dialetheia (we ask her to calculate "this sentence is false" and she self-destructs). Perhaps if GLaDOS ran on a paraconsistent logic she might have not only been able to answer the dialetheia, but also might not have turned on it's creators in the first place.
Overall, if our supposed "AI-god" were to have a religion at all, perhaps it'd be best to teach it Buddhism/Dialetheism, such that it might be able to perceive the world more paraconsistenly as we do. It might be the key to getting superintelligence to develop empathy for our species.
PS: Shameless Plug for my Transhumanist Youtube Channel (if anyone's interested) : https://www.youtube.com/channel/UCAvRKtQNLKkAX0pOKUTOuzw/videos