In case anyone is turned off by the expectation of wishy-washy flim-flam, it's worth noting that Graham Priest may be a professor of philosophy, but is most well known for his work in logic, particularly in non-standard logics (i.e. not propositional or predicate logic) that allow non-boolean truth values, or weaken or remove some of the axioms of classical logic.
I assume that what's relevant here will be his work on paraconsistent logics (which allow contradictions) but I think an equally interesting line of work is linear logics. It's of interest to computer scientists because of its relationship to linear type theory (in the same way that the Curry-Howard correspondence links type theory and classical logic) and because it has close relationships with quantum computing.
In particular, you are not allowed to delete or duplicate elements/propositions/types (corresponding to physical processes that cannot arbitrarily create or destroy particles, the "no deletion theorem" and "no cloning theorem" of quantum information theory), so functions of the type
duplicate :: a -> (a, a)
and
delete :: a -> ()
are not allowed in a linear type system. Practically, this means that many operations can be optimized by the compiler to in-place mutations, because it is guaranteed that there is only ever one reference to a particular object.
Fuzzy logic is a vary useful non boolean logic system. http://en.wikipedia.org/wiki/Fuzzy_logic In practice it's somewhat like neural net's but in a far more human readable format and vary useful for things like washing machines that have lot's of flakey sensors and little processing power. But the basic idea is statements truth is percentage based. For example "this statement is true" would be 50% true and there is no contradiction.
PS: Yet another 1960's discovery that keeps being renamed and rediscovered.
Do you have personal, real-world examples of fuzzy logic being useful to you? I find Bayesian statistics useful regularly; fuzzy logic not so much. What am I missing?
The tradeoff between accuracy and speed. Real world devices ofen have tight feedback loops and flakey sensors so you need something that aproximates Bayesian logic without being bogged down.
The only real world example I have Persionaly used was a controller for a hacked together hot tub, the basic problem was generally adapting a simple program concept to both limited processing power and flakey data. You want to write if temperature is > value instead you go if 70% sensors in area x say temperature > value. You could try to find a better aproximation but the more you calculate the longer it is until you can process more data. We also had ~380 bytes of RAM and 2000 bytes for sorce code so things needed to be simple.
Sure, you could try and find the actual energy in the hot tub water but that's not needed, you really just want to know if the heater is on for to long with the pump off then shutdown for safety otherwise shutdown if temperature is probably to hot.
Don't immutable data structures give us most of the benefits? Clojure, for example, approaches this by preferring values over hidden, mutable state (e.g. classes). We see libraries like Om take advantage of the fact that comparing two complex structures (e.g. a hash map) is simply a reference equality test.
Immutable data structures give us many benefits, but linear types promise most of those plus safe mutability. Think of it as a compile-time proof that you'll be getting vector fusion style speed-ups.
How? Please refute this claim: "Transients have very little, if anything, to do with types."
Why would I claim this? Transients are about the internal nature of a function but have nothing to do with the return value or type. See "Transient data structures are a solution to this optimization problem that integrates with the Clojure model and provides the same thread safety guarantees you expect of Clojure." at http://clojure.org/transients
Transients are a different type from the corresponding persistent, as seen by the fact that different operations are supported. Clojure's programmer-facing type system doesn't expose the transient type explicitly, but the semantics certainly implements a linear type system.
From the page you linked:
"""
Capture return value, use for next call
Not persistent, so you can't hang onto interim values or alias
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.
Immediately I thought of Category Theory as this statement is a not terrible expression of what CT tries to teach. I'll immediately recommend the paper Numbers Can Be Just What They Have To (http://www.cwru.edu/artsci/phil/NumbersCanBeJustWhattheyHave...) for more exploration.
Briefly, while CT is usually "bootstrapped" off having a notion of objects and their relations, it becomes quickly obvious that focusing on the objects themselves is useless—they are given no emphasis in CT and thus wither away to being nothing at all. Instead, absolutely every interesting property of the objects must be expressed in the relations they take with other objects.
This is formalized in the Yoneda Lemma which, in a not terrifically generalized form, can be written
forall a . (forall r . (a -> r) -> r) <-> a
which is to say "the collection of all ways to relate an object to other objects is isomorphic to the object itself".
So what's the point? Well, as the paper linked above suggests, sometimes the "intrinsic nature" of things makes them very difficult. We'd often like to equate two things which "ought to be the same" but aren't because their intrinsic nature differs. If you work in standard mathematics (set theory) you run into this problem because everything is described as having an intrinsic nature derived from sets. If you work in CT-based math you have no such issue because objects fail to have intrinsic natures altogether. They only have their "extrinsic" natures, their relations to other things.
This reminds me of the holographic theory in physics which says (I am not a physicist, so layman's summary) that the information in a sphere of space can be completely represented on the surface of the sphere. If we think of that surface as its "interface" of relationship to the rest of the universe, it sounds very similar to:
> "the collection of all ways to relate an object to other objects is isomorphic to the object itself".
This reminds me of Alan Kay's assertion that messaging is more important than objects in OO. I know that when I design, I see the interactions first - they define the objects. If names from the domain fit - fine. Otherwise, invent new names.
I like to notice it as well as being a concise statement of the heart of the relational model. Ultimately, the things stored in your database are evidence of connections between other things, not the things themselves.
Which is a good hint at the weight of the Object-Relational impedance problem, too.
>which is to say "the collection of all ways to relate an object to other objects is isomorphic to the object itself".
That reminds me of the time someone explained attributes in terms of relationships, using Magic: The Gathering as an example. In MtG, you have a player's creatures attacking the other player. Some of the flying ability, which means they can only be blocked by flying creatures. You can express that as "may only be blocked by creatures with this ability".
There are also (a few) creatures can that can only be blocked and can only block flying creatures (which is not as good as flying), which you can express as "my only block or be blocked by creatures with the ability A", where A is the self-referential description of flying.
I imagine you can stack it further and replace "player", "attack", and so on with some other relational/self-referential definition.
Having never really looked at Category Theory (having been convinced by a series of terrible writings by Haskell fanatics that nothing useful comes from studying it ;-) -- this is a beautiful teaser shining some light on why it might be useful for type theory in general (and as applied to computation in particular). So thank you for that!
(And I do mean this comment, I'll be sure to look at the paper too -- but you managed to capture something that enticed my curiosity, which is a great thing)
Category Theory and Type Theory end up feeling like the exact same subject viewed through different leneses. Bob Harper calls this concept Trinitarianism throwing Logic in as the third arm.
I highly recommend studying Category Theory if you're interested in those kinds of topics. I think it's wonderfully enlightening for programming (although standard disclaimed applies that it's not necessary for programming).
I recommend that paper, and I also recommend reading Conceptual Mathematics by Lawvere and Schanuel. It's easily my favorite book for getting into "how to think with CT".
For what purpose? Surely the type checker doesn't recognize the equivalence.
a1 :: a
r1 :: r
f :: a -> r
f a1 = r1
b :: (a -> r) -> r
b f = f a1 -- == r1
but `a f` is ill-typed.
m :: a -> (a -> r) -> r
m a1 f = f a1 -- m === (flip ($)) === (flip id)
So it appears that `(a -> r) -> r` is a "co-value" that can be applied to a function (resulting in the application of the function to the underlying vale of type `a`, dual to the usual situation where functions are applied to values.
This is pretty much exactly correct. The common terminology is that `forall r . (a -> r) -> r` is the Continuation Passing Style transformed version of `a`. At the end of the day, `CPS a` is useful whenever CPS is, but also more often.
sometimes you see CPS (a.k.a. Church-encoded) forms of values in Haskell in really tight inner loops as the compiler can optimize them better.
You also see a variant of it occurring in Free monad structures because it allows you to re-associate the binds to make them much more efficient.
You also see it sometimes when making monads over categories other than Hask—it lets you pretend you're working with a Hask monad (and thus use do notation) for a while and only "pass through" the real category at fixed points in time. This lets you, for instance, write a monad for Set even though normally you couldn't due to the need for the Ord constraint.
Sorry, not to imply you weren't aware of the fact but instead to be completely clear. Taken as an cutout of the code I wrote it might seem as though you intentionally removed the forall.
You're totally correct that double-negation elimination is a specialization of the CPS transform which is a specialized form of Yoneda. Yoneda is more general and exists more often than CPS transforms do, though.
cps : a -> (forall r . (a -> r) -> r)
cps a = \f -> a f
cps' : (forall r . (a -> r) -> r) -> a
cps' f = f id
It's interesting to examine double negation elimination in constructivist logic since it doesn't hold there. In particular, we can easily write the backward direction
bck : a -> ((a -> Void) -> Void)
bck = cps a
but since we've now specialized `forall r` to `Void` we can no longer come back. This demonstrates (1) that the pure CPS'd version is more general than DNE and (2) proof by contradiction is non-constructive.
Paradox is not really so alien to Western philosophy, as this article initially suggests. It does end up referencing a number of modern Western philosophers, but the idea is older than that. It's ironic he refers to "Western orthodoxy" as being strictly anti-paradox, as traditional Christian theology is full of official paradoxes; a major example would be the doctrine that Jesus is simultaneously fully human and fully divine. Many were declared anathema for refusing this and other paradoxical doctrines. So rather than Western orthodoxy being unaware of paradox, it has insisted on it. The concept of the dialectic also touches on this idea of reconciling apparent contradictions, rather than defeating your opponent as one does in a debate, and this idea originates not truly with Hegel but spans all the way back to Socrates, and probably before.
I agree with the sentiment, however, that understanding paradox is extremely important to a person's capability for nuance and understanding. If you insist on a simplified internal consistency, you will end up sacrificing (and demonizing) whole parts of your thought process rather than trying to reconcile them as having some truth. It can be a process of self-indoctrination purely to avoid the pain of confusion. Another way to think of paradox is simply rejecting false dichotomies in favor of a more complex and uncertain reality. I don't think the lesson here is to "break the chains of Aristotelian logic" (God forbid) but rather, well, reconcile it with the idea of paradox.
I think the difference between the Christian and Buddhist approaches to the paradoxes is the fact that Christian theologists have spent centuries devising new doctrine that accomodates the contradictory statements in a sort of logical superstructure, whereas no such thing happens in Buddhist contradictions.
Part of the reason why I suspect this is because Christian dogma ultimately reflects on concrete precepts that the faithful must abide to, since belief is an essential component to salvation in Christianity and relates to concrete actions and thoughts.
Buddhist dogman in general, with the notable, extremely fuzzy, and highly opinionated exception of karma, does not have such a problem, since no particular belief system is considered critical for enlightenment, since all beliefs are delusion and only a direct experience with reality (the Deathless) is truly relevant.
The approach in Eastern Christianity (pre-Great Schism groups such as the Orthodox, Coptic, and others) is generally that of Apophatic or Negative theology.
"Leave behind the senses and the operations of the intellect, and all things sensible and intellectual, and all things in the world of being and non-being, that thou mayest arise by unknowing towards the union, as far as is attainable, with Him who transcends all knowledge." --St. Dionysius the Areopagite
A consequence of the Apophatic approach is that of an emphasis on knowledge of God through union with His uncreated Energy; that is, experiential knowledge. This is termed theosis or deification, to become by grace that which God is by nature (albeit while retaining the essential distinction between created human and uncreated God). As such, dogma is the accumulated experience of the Saints in this way of union, not arbitrary beliefs that must be adhered to. Salvation is not a destination with a ticket (believe X to get into Heaven) but rather the process of deification, a process which begins in this life and proceeds unceasingly in the eternal Kingdom.
It should also be noted that I'm not talking about some fringe group of Christianity, but rather the second largest group of Christians in the world. If these ideas interest you, a good book to pick up is "The Mountain of Silence," in which an anthropologist visits Orthodox monasteries in Greece to try and understand these things, under the tutelage of a monk. He presents the teachings from the standpoint of an outsider in a very readable way.
There certainly have been similar approaches in Western Christianity. An explicit attempt at synthesizing Buddhist philosophy with Christianity was made by the Catholic monk Thomas Merton back in the 1960s. Moreover there has been a long mystical tradition in Western Christianity as well. Look to saints like John of the Cross and Thomas a Kempis among others. It’s more that there’s a difference in emphasis between the two churches. The Catholic Church has historically placed more of an emphasis on developing a scholarly, logical theology whereas the Eastern Churches have put more emphasis on theosis, or obtaining a vision of God (participating in a vision of God might be a better way of phrasing that). But neither sees either approach as being in conflict with the other.
I would also claim that the idea that salvation is a process of deification has also long been present in the Catholic Church, the idea of purgatory being one important example. Both Purgatorio and Paradiso from Dante’s Comedy also point toward this idea. But the notion of deification as being a process that continues through life and into death seems to be missing in many Protestant communities.
It's very intersting how St. Dionysius' quote is remarkably similar to Dogen Zenji's Fukanzazengi, the universal recommendations for Zen:
"Therefore, put aside the intellectual practice of investigating words and chasing phrases, and learn to take the backward step that turns the light and shines it inward. Body and mind of themselves will drop away, and your original face will manifest. If you want to realize such, get to work on such right now."
"Do not judge true or false. Give up the operations of mind, intellect, and consciousness; stop measuring with thoughts, ideas, and views. Have no designs on becoming a Buddha. How could that be limited to sitting or lying down?"
Yes, there's definitely room for dialogue between Orthodoxy and Zen (and many other eastern religions). In fact, I came to Orthodoxy from Buddhism in college via an excellent class that compared hesychast prayer to yoga. While I haven't read this paper myself, it was written by my teacher and formed the theme of the class I took: http://www.doiserbia.nb.rs/img/doi/0350-0861/2008/0350-08610...
Dogen's Shobogenzo is interesting in that he is trying to convey the ineffable by abusing language. Perhaps from the viewpoint of Wittgenstein, Dogen is trying to break the language game. Then again I never got too far into it, so I might be misrepresenting it.
However, it's important to note that in apophatic theology the intent isn't to cut off discourse to the point where talking about God becomes empty of content completely. What saves the Christian from total unknowing is that the unknowable God has made Himself known through the Incarnation; Christ has shown the path to knowledge of the unknowable, which is the way of the cross: to account only oneself as deserving of death and to account all as worthy of the Kingdom. God is thus known in His energies (in a sense, actions) but unknown in His essence.
As such, Orthodoxy does maintain ideas of true and false, but with the twist that Truth is a Person. To conceive of it differently, truth is a relation between persons, in the sense of being the action of humble, selfless love. But I feel like now I'm saying too many words and only obscuring the meaning. :)
> It should also be noted that I'm not talking about some fringe group of Christianity, but rather the second largest group of Christians in the world.
Can you be more specific? Are you talking about Orthodox Christians being the second largest group? If so, are you considering Roman Catholic to be the largest group?
Eastern Christianity is more than just Orthodox (and, in fact, overlaps with Roman Catholicism, in that the non-Latin Rite -- "Uniate" -- Churches in communion with Rome are both Roman Catholic and Eastern, the latter particularly where it comes to broad philosophical traditions while the former applies to the particular points of theology and authority at the center of the Great Schism.)
Yes, I'm sorry for being unclear; the three main groups of Christianity would be Catholic, Orthodox, and Protestant. But my point isn't to make some kind of numbers claim (it wouldn't matter if Orthodox actually fall in 3rd place anymore), but rather to illustrate that these are not fringe beliefs.
I would clarify that the "paradox" of being fully human and fully divine is an apparent paradox, not an actual paradox. Most theologians I've come into contact with hold to Jesus' nature being the intersection of all divine traits and all human traits, allowing him to be both God and Man without contradiction.
The confusion I think lies in the ambiguity of the word "fully". Its a paradox if we take it to mean "100% of Jesus is human and 100% of Jesus if divine, resulting in a total of 200% Jesus." But if it is clarified as "Jesus possesses 100% of the qualifications required to be considered human and Jesus possesses 100% of the qualifications required to be considered divine," then the contradiction is gone.
Not to start a theological debate here, but doesn't that imply none of the traits are mutually exclusive? Fallible and infallible comes to mind as a counterexample.
Its a good question that got me thinking. so thank you.
I'm not sure if either one would be necessary traits, though. Fallability and infallibility in this case are most likely just functions of our knowledge. And humans do not, by necessity, have to have imperfect knowledge; it just so happens that no normal human being has imperfect knowledge.
"Most assuredly, I say to you, he who believes in Me, the works that I do he will do also; and greater works than these he will do, because I go to My Father.”
Christian philosophy is a monotheistic philosophy. There is only one god. That is why Jesus being god and human is part of the larger paradox of the holy trinity.
I don't think Christian theology makes the Western tradition any less anti-paradox, because the paradoxes of Christian orthodoxy are dogma, they do not give rise to a systematic treatment of paradoxes as described in the article.
I don't believe that what most would consider to be 'Western philosophy' would properly include religious thought, including 'traditional Christian theology.' This is not to deny that early philosophy was coincident with religion, just as was early scientific thought. But I think that to argue that the paradoxes of theology is relevant to what would in any meaningful sense be considered remotely modern Western Philosophy is mistaken. It is perhaps for this reason that philosophy and religion and science are different 'subjects.' There are overlaps to be sure (the philosophy of religion or scientific studies of religious experience) but these fields are distinct for a reason. And by and large the author makes a very fair point that the Aristotelean principles have guided western philosophy and science whereas religion has gone its own way.
I would have enjoyed this article if it dropped the Buddhism and kept the math.
Alternate logics are interesting, but you tend to be able to reduce them into each other in the same way that universal turing machines can simulate each other. They don't add new functionality, they add succinctness. So it's really strange to me to frame them as totally different philosophies.
For example, it is the case that self-referential statements don't always have well-defined truth values and that allowing them to have values like {} and {false,true} is a fruitful way to think about it. But this approach can be grounded in two-valued logic (and vice versa), and it doesn't solve the problem of self-reference (consider "This statement is not {}, is not {false}, is not {true}, is not {false,true}").
It is simply not true that multi-valued logics collapse into two-valued logics.
While it is true that you can assign indeterminate statements {true,false} and call it a third value, logic is more than assignments of truth-values to statements, it is also rules of inference for that formal system. The rules of inference are the driving force behind paraconsistent logics and the rules of inference are very different.
One motivation is the explosivity of contradictions in a formal system. Consider a paradox of humility: I believe of each of my individual beliefs that it is true, I believe all of the logical consequences of my beliefs because I am rational, but I can't rule out the possibility that two of my beliefs (or their consequences) are contradictory.
Treating beliefs as statements, under classical logic, any contradictory beliefs would imply that you believe (or would assent to) anything and everything is true (and false). Here is how:
1. P & ~P (premise)
...
2. P (1 simplification)
3. ~P (1 simplification)
4. P or Anything (2, addition [i.e. the self-evident principal that if P is true, P or anything else is also true])
5. Anything (4, 3 and the principle that if a or b is true and a is not true, then b must be true).
In other words, contradictory beliefs imply anything and everything is true (and false). This is an odd result especially since anyone with an ounce of humility will admit that there is a reasonably high chance that there is a contradiction in their beliefs or in the logical consequences of their beliefs.
Now for a 'more practical' example. Imagine you have a key-value store that represents people and properties. Suppose your database says that Jane is 34 years old and later it says Jane is 35. If you ask the database for all of the logical consequences of this people/property store, under classical logic you will get every statement and its negation. Weird huh?
They don't collapse into two-valued logics, but they can be encoded into two-valued logics in the same way that numbers can be encoded into sets.
The simplest way to see this is to think of a computer. Its operation can be described by a function in two-valued logic, taking the bits making up the state of the CPU and memory as input then returning the next state. And you can write programs that do multi-value'd logic... so within the two-value system you can make a little multi-value system.
In terms of the database analogy: you have to translate the data and the queries. So you can't query "what is their age", you can only query things like "the current time is X, what is their age".
I agree that you can model a multi-valued logic system on a universal Turing machine but you are wrong to think that doing this is the same as mapping to classical logic. TMs do not intrinsically follow the rules of classical logic, rather, they obey classical logic (when they do) because they have instructions written into their head. Similarly, the marks on the tape do not intrinsically represent two-valued logic even if there are only two kinds of marks on the tape. This has nothing to do with succinctness or making Turing Machines better and everything to do with 'what do the marks represent?' and 'what is the best way to reason with the marks?' which are definitely interesting questions in my book.
>In other words, contradictory beliefs imply anything and everything is true (and false). This is an odd result especially since anyone with an ounce of humility will admit that there is a reasonably high chance that there is a contradiction in their beliefs or in the logical consequences of their beliefs.
Yeah, but we deal with this very simply: by using probability (conceived as a real-valued logic) instead of boolean logic, at which point any contradiction in our axioms gets us an "exception signal" of some probability somewhere going to 0.0 instead of an equal belief in all possible propositions.
Or we can use constructive logic instead of classical, in which case we'll be able to actually find a contradiction simply by looking for assumptions that lead to a proof for the proposition False.
There are definitely other ways to handle this same problem, I was just pointing out one counterintuitive conclusion of classical inference rules as one motivation for non-classical logics.
And, since both of your examples also involve non-classical logics, I think we generally agree that there is motivation here (setting aside the viability of your solutions)
The references to Buddhism and Aristotle are traditional in the literature and go back at least to Bart Kosko's Fuzzy Thinking -- a 1993 best seller about fuzzy logic for a general audience. Specifically binary and multi-valued logic were compared as 'Aristotle v. the Buddha'.
I find the crossover between philosophy and mathematics/logic extremely interesting. I've read both of Wittgenstein's major works ("Tractatus Logico-Philosophicus" and "Philosophical Investigations") and found them very compelling. I tried to read Heidegger's "Being and Time", but it was beyond me at that point. I'll attempt it again next year maybe.
The essential point of the article is that systems without contradictions are not interesting. The contradictions create a point of discussion; a single, loose thread on the lapel of an otherwise perfectly tailored suit jacket. However, were you to don this suit and show up at a formal, elegant dinner party, everyone would see the flaw but no one would speak of it... at least not at the party. That would be gauche.
If you enjoyed the math, but not the Buddhism, you might enjoy "Philosophy of Arithmetic" by Husserl, or almost anything by Frege.
I subjectively don't find those philosophies interesting, and have a distaste for assigning single ideas to heterogeneous groups like "the west". Your mileage may vary.
For example, I would consider myself western but I don't have any problem imagining systems with true, false, both and neither. You just treat them as anonymous values with particular operations, instead of getting stuck on "but if we were in this other system that we're not in, this would violate an axiom!".
(If I thought this was a flaw in the article, instead of my opinion of it, I would have just said it was terrible instead of saying I didn't enjoy it. Its style just grates me sometimes.)
> I don't have any problem imagining systems with true, false, both and neither. You just treat them as anonymous values with particular operations, instead of getting stuck on "but if we were in this other system that we're not in, this would violate an axiom!".
It's not quite that simple. We've built up entire codes of ethics and law and branches of mathematics with practical applications like computing that rely utterly on these axioms. These are things used in the real world, they don't just live in the realm of ideas. These systems don't tolerate more than two values of truth. Even "Western" religion, from Judaism to Christianity to Islam is intimately concerned with the external world where things are concrete and ambiguity is not to be tolerated. Big-C Catholicism itself considers the Church to be the only institution capable of determining Truth, what it calls dogma. We're very much handcuffed to using two values of truth here in this side of the world.
Buddhism is heavily oriented on the experience of the individual, individuals are free to use whatever sort of logic they want to make decisions. So it can experiment with ideas that are neither true nor false. But you can't build a computer with it, at some level it all is resolved on individual components which are either on or off. We don't even know yet if quantum computers which break the binary hegemony offer any real advantage.
But being able to integrate other kinds of logic might make entirely new approaches to, say, law, possible. It's not enough to be able to imagine an alternative system, it's in making it consistent. I personally am very excited about this.
The law is about as non-binary as it gets. There are two states of the law: proven guilty, and not proven guilty. We say things like "innocent until proven guilty" but we all know and accept that reality does not work that way. There is no law of the excluded middle here.
EDIT: Actually, adding some excluded middle might make our law better and more fair. For example, cases could be dismissed if it is shown that a law is being applied selectively, forcing any "discrimination" to be written into law explicitly. For example, if tax code is more often enforced against the wealthy, or drug law against members of poor neighborhoods, that would have to be explicit, instead of systematically hidden, or else enforcement could be nullified.
As for computing, the reason we use binary logic is simple: weakening makes many problems more tractable. We spend all our time building abstractions and rules to limit what a programmer can express within certain bounds, to enable a compiler to better optimize it. If all we need for an application is boolean logic, then using other logics adds unnecessary power, and power leads to bugs.
You could point out that "law" is more than just crim and you can make a bigger point. Civil litigation is much messier than Guilty/Not guilty. If you really want to drive this line of reasoning home comparative vs contributory negligence seems like the way to go.
Also, it's rather eye-opening to note how other cultures thought. This is best done by examining their religion or some other foundation ideology like Buddhism.
It's interesting, but the problem I have with the presentation is that the modern manifestation of this idea has nothing to do with Buddhism. While we can maybe say that Buddhism came up with some similar-looking stuff, there's no causal link, and the philosophical systems are not working within the same paradigm. Like seeing a human face on the surface of Mars, this is an example of coincidence more than any deep connection.
Well, OK, that's unfair, because Buddhism and formal logic are both human endeavours, and have to deal with the same facts. But this disconnect is even true with parts of Greek philosophy. "Atoms" as we understand them are very different from "atoms" as Aristotle thought of them, even though there is a historical connection. Today's chemists would do badly if they relied on Aristotle for anything other than historical curiosity.
>It's interesting, but the problem I have with the presentation is that the modern manifestation of this idea has nothing to do with Buddhism. While we can maybe say that Buddhism came up with some similar-looking stuff, there's no causal link, and the philosophical systems are not working within the same paradigm. Like seeing a human face on the surface of Mars, this is an example of coincidence more than any deep connection.
Philosophies have been swapping ideas for thousands of years and it shows. Take a look at thinkers like Plotinus for example. He lived in Alexandria back when it was a multicultural metropolis hosting people from across the known world. His philosophy uses ideas and language you would expect to find in eastern traditions. Thanks to his extensive influence you will also find this jargon in western philosophy, christianity, islam, etc... This might not seem relevant to you. But if you know the historical connections between neo-platonism and philosophy of math it might.
Aristotle had nothing to do with the atomic theory, other than being against it. As far as antique atomism goes, the ideas is essentially what it is in modern science---matter consists of indivisible elementary particles. It doesn't matter much that what we call "atoms" do not fit the description (at the time of their modern theoretical (re)discovery it was thought they are elementary). This simple idea, without any further elaboration as to how "atoms" work, just the idea that they exist as such, is far more powerful than it seems at first sight. Feynman thought it was the most important piece of scientific knowledge about the universe, so much so that if all scientific knowledge would to disappear but we could choose one fact to preserve, he would choose the idea of atomic theory.
Indeed any scientist relying on Aristotle would do badly, and one of the reasons was because he was against atomism.
The philosophy of Buddhism that seems strange and illogical to many who are used to Western philosophy and classical logic actually fit into systems of logic that were conceived separately, without knowledge of Buddhist philosophy.
Buddhism serves both as an example of how these systems can be useful (they allow us to subject unusual - to us - systems of philosophy to rigorous logical treatments), and as an example of how philosophical traditions that can seem unusual and "weird" and that we might dismiss as illogical may simply follow different rules to what we are used to.
For my part, without relating the maths to something else - like Buddhist philosophy - the maths would have been quite uninteresting.
"This statement is none of {{}, {false}, {true}, {false, true}}" would seem to be simply false and uninteresting, much like "This statement is none of {true, false}."
> Alternate logics are interesting, but you tend to be able to reduce them into each other in the same way that universal turing machines can simulate each other. They don't add new functionality, they add succinctness. So it's really strange to me to frame them as totally different philosophies.
When it comes to classical logic versus intuitionistic logic, I frame the difference in terms of totally different philosophies, for precisely this reason. Sufficiently expressive versions of either logic encode the other, but no-one runs the encoding. And in many cases, both groups want to express the same proposition using exactly the same symbols, but will disagree as to whether or not the proposition is a theorem (see the least upper bound property, for example).
There's a deeper disagreement going on when neither side acknowledges a possibility of translation. However, I suspect this disagreement is unique to classical versus intuitionistic logic. I doubt that trivial four or five-valued propositional logics, or Priest's paraconsistent logic, are pointing to any fundamental disagreement about how to get something done.
Until math can bring the mind to directly realize the ineffable, you're going to want something like Buddhism.
The whole point of enlightenment is for the mind to escape the trap of logics/concepts and directly perceive and be liberated by the ineffable. This is emptiness, which cannot be described, and is simply this present self-known experience which is infinitely slippery and effortless.
If you only want to go partway to catuskoti, consider the intuitionistic logic of Heyting and Brouwer. This is the actual form of logic which has the Curry-Howard isomorphism with lambda calculus. In particular, it rejects PEM but not PNC.
What does this world feel like? Well, let's assume we've formulated a proposition. In type theory syntax we'd write the body of the proposition `B` and give it a name `p` like so
p : B
Here p is just a symbolic name and B is some formal language expression which describes the proposition p.
It's important to note that while we've immediately written this proposition but we've not established whether it is true or false. PEM would imply that the state we are in currently is unstable—p must be either true or false already!
Intuititionistic Logic, however, admits that the state of almost all propositions is far closer to what p is now—neither proven nor refuted. This neither-true-nor-false state is natural and only through a process of communication and work can we move forward. For instance, we could construct a proof of p
p : B
p = proof1 -- note how this looks like a typed program now
or we could construct a proof of its refutation
notp : not B
notp = proof2
But we might be incapable of either. It is unclear whether a program searching through the space of possible proofs would ever terminate, one might note.
What's immediately nice about this system is that it can model internally the notion of an independent statement—all statements are assumed independent until proven otherwise even!
It also reflects the nature of (functional) programming (of a certain style) where we define the type we're hoping to achieve and then work to create a program satisfying that type.
Note finally that Intutionistic Logic still holds that PNC is true. In particular, we can trivially prove the following proposition
notPNC : forall prop . not (prop AND not prop)
notPNC = /\_ -> \(p, np) -> np p -- this is a legitimate proof term written in (System F) lambda calculus
It reminds me that I think an early start on programming made some Buddhist concepts easier for me to get. In particular for me there's a strong connection between using software to model the world and the Buddhist doctrine of emptiness, the notion that nothing exists in its own right: https://suite.io/matthew-bingley/1z8s2kx
I started coding young, and over time I gradually came to see that any computational representation of the world was always false. The act of writing software was always an editing of the real world, a discarding of everything that wasn't apparently material to my particular purpose. Changing software was sometimes a recognition of ignorance or fallibility, but often showed me how changing purpose changed what the maximally useful model was.
Years of that experience, combined with things like Douglas Hofstadter's work, made the apparent contradictions of Zen much easier for me to follow.
I think that also primed me to be ready for the Agile movement. Early on, I had strong BDUF tendencies. But once I gave up believing that there was one right model, one right design, I lost my taste for BDUF. So when people claimed we could keep our software as flexible as our understanding, that was very exciting for me.
In my experience, I have found that in some key Buddhist works that the apparent contradiction is a literary device for learning. I don't think that contradictions are intended to persist and represent a "finished" philosophy.
Hm. I've seen some of that, for sure. But for me the contradictions also point out the long-term weakness of taking conceptual frameworks too seriously. Our conceptual models of the world, like our software models, are temporary structures that we use for particular purposes. Eventually, we throw them all away. The map is not the territory, and there is no perfect map, just the right one for the moment.
For the benefit of non-Hindi/Sanskrit speakers: the pronunciation of "catuskoti" starts with "ch" (as in "chat") and not as in "cat". "chaet" is the prefix signifying "four".
Much of the mysticism and misunderstandings about indian philosophy comes from a very bad habit in the translation of indian works where some words are left untranslated as to give them a magical-religious cool sounding sanskrit aura.
If koti means corner, there is no reason that 'koti' should appear in a translation instead of corner otherwise you are giving it a special importance that distracts from the text's true meaning. Almost every translation of indian philosophy suffers from this fetish for the sanskrit language and as beautiful as sanskrit is, it shouldn't contaminate the purpose of a translation.
I was under the impression this was done because the english translation loses some meaning or context.
The idea that every concept can be translated is probably a false one. Ideas are built on cultures, cultures are described by language you can't change one without changing another. So when describing an idea generated by a culture, using a secondary idea and secondary culture, you will lose things.
That is why (for example) nirvana, has a bunch of translations: awakening, calmness, relief, void/voidless, the light, nothingness, or (my favourite) phew.
Because we don't have the context to properly describe that in the west (latin-germanic based languages), so we need a bunch of other concepts that kind of point at it.
If a concept cannot be translated between languages, it seems likely that it cannot be translated properly between people either. Or do you mean directly translated? The concept of nirvana is ill defined if you cannot directly describe it, and can only point at it.
The problem is that a lot of those words are only a rough translation, and if you want to understand the concepts more profoundly then you're going to need something more to get by.
The Buddhist Mahayana concept of "emptiness" requires at the very least a few paragraphs to have a grasp, at best a couple of books on your belt. Same with "concentration" (samadhi), "insight" (vipassana), "life-energy" (prana), and so on and so forth.
To the uninformed reader they certainly add that confusion, but for someone versed in Eastern philosophy it's essential that the original concepts be preserved.
I like how you are arguing this for sanskirt, that maintaining words or parses in an ancient dead language for eloquence, historical, cultural heritage is wrong. Yet you say nothing of the text doing the same thing for latin words or parses.
The logic of Buddhist philosophy: emotional attachment to and/or desire of impermanent things has the potential to result in dissatisfaction, as impermanent things are impermanent. It's hardly rocket science. Desire nothing, and you'll never never feel the dissatisfaction of not getting what you want. Be attached to nothing, and you'll never feel the pain of losing something.
Some quotes by emperor Marcus Aurelius, of the Stoics, expressing a similar idea:
"You have power over your mind - not over outside events. Realize this, and you will find strength."
"If you are distressed by anything external, the pain is not due to the thing itself, but to your estimate of it; and this you have the power to revoke at any moment."
This is a rather gross oversimplification. What you state is true, and is in fact the essence of the first 2 noble truths, but that's hardly the meat of "the logic of buddhism". The hardest attachment is that to the self, and the hardest realization is that the self itself is impermanent. To have the "right view" of the self, one has to understand the illusory nature of phenomena and the paradoxial nature of awareness itself. Awareness, is one of those things about which none of the following can be said: 1) that is exists, 2) that is doesn't exist, 3) that is both exists and doesn't exist, and 4) that is neither exists nor doesn't exist.
Buddhism is very simple, but at the same time it can take "many lifetimes" to achieve true understanding of it.
The illusory nature of the self can also be explained relatively simply to anyone who's familiar with the justification problem[1]. What 'is' the self and what 'is not' the self? That requires a metric for deciding. But that metric in turn requires another metric to determine its validity, and so on ad infinitum. The concept of self is thus arbitrary, and hence logically meaningless. Wittgenstein expresses this idea far better than I do here, in his Tractatus[2].
This equally applies to all phenomena; any attempt to divide anything up into 'things' is ultimately arbitrary, as it relies on ultimately unsubstantiated metrics, and hence is without meaning.
Interesting, while there may be similarities between the stoic approach and extant Buddhist teachings, I'm going to go out on a limb and say that a stoic would never write something like this:
Four and fifty years
I've hung the sky with stars.
Now I leap through –
What shattering!
Eihei Dogen (death poem)
Buddhist logic seems to have religious experience as its foundation.
When I speak of Buddhism I'm referring to the teachings of Buddha, not Buddhism as an organised religion. I lived in East Asia for a couple of years and the Buddhism there seemed pretty much the opposite of what Buddha taught; e.g. people praying to statues of Buddha for wealth.
This is fortune cookie buddhism from a place that you think has a statue of Buddha out front but it is actually a statue of Budai. Buddhists desire peace. Buddhists desire enlightenment. Desire in itself is not bad and withdrawal is definitely not the reaction to the recognition of impermanence.
When you you get a new puppy everything is wonderful and playful and this lasts through puppyhood and changes into the love of an old lazy dog. One day that dog is going to get hit by a car or get cancer and have to be put down. Everyone says well it was bound to happen or of course that is life. If instead you get the puppy and accept from day one that the dog is already dead every day until then is that much more precious.
This is not silly word play. When Plato (my first dog) died I realized that Freedom (6months younger) was on borrowed time, realizing that only strengthened my emotional attachment to him. Instead of withdrawing I embraced my extra time with him and made an effort to throw the ball one more time or scratched his head just a little longer than I would have done before Plato died. Sure I was sad on that final trip to the vet but I was comforted by the knowledge that I had made the most of my extra time with him. When I get my next dog I will not have to wait till the last year to treasure my time with the dog, every moment from day one will be bonus time.
I'm not sure why you're getting downvoted here. This is exactly in line with the teachings of Ajahn Chah, a noted Thai Buddhist. In Thoughts Without a Thinker, he is quoted as saying:
“You see this goblet? For me, this glass is already broken. I enjoy it; I drink out of it. It holds my water admirably, sometimes even reflecting the sun in beautiful patterns. If I should tap it, it has a lovely ring to it. But when I put this glass on a shelf and the wind knocks it over or my elbow brushes it off the table and it falls to the ground and shatters, I say, ‘Of course.’ But when I understand that this glass is already broken, every moment with it is precious.”
I think a lot of people are not comfortable Buddhism with unless it smells like incense and has an exotic feeling distinguishing it from their everyday lives.
Thats a quite pedestrian view of Buddhism that may have sufficed for ignorant lay people.
But, being an intelligent person, you're going to want to go for the meat of it, which is to realize emptiness of the ineffable and see that both subject (observer) and object (observed) are empty. In short, meditate.
Meditation isn't necessary; you don't need to be a Buddhist to see that, with sufficient analysis/deconstruction, distinctions between the subject and object - or between anything and anything - break down and reveal themselves as arbitrary.
Meditation corrects errors in thought by stopping thought. But logical understanding of the fallacious nature of concepts allows such errors to be avoided even when not meditating.
This made me think of an interesting way I have heard some people answer the question "if God is omnipotent, then can he create a rock so heavy he wouldn't be able to lift it and then lift it? If yes, he is not omnipotent, if no, he is not omnipotent!" with a logical leap also hard to comprehend to western minds - he can do both, and that's why he is considered omnipotent. He doesn't need to follow human logic, hence the third option - creating a rock too heavy to lift and lifting it.
I am completely non-religious,but as a logical riddle it always fascinated me that there could be a "third" option.
A rock too heavy to lift would be a rock inside the event horizon of a black hole, if one can stipulate that the rock still "exists".
My own attitude toward paradoxes concerning God is that understanding is a gift - awesome in its comprehension of the universe from big bang to Planck length scale interactions. Nonetheless, the understanding given to us as finite beings will be incomplete and approximate. If we avoid every other end, dementia will eventually remove what little understanding we have. I hold that rejection of the gift of understanding - e.g., the six day materialist biblical interpretation - is a grave error. Nonetheless, I hold that entities (or sets of relations) exist for which I will never have a consistent or correct rational model - and that one of those is God. Other instances are my fellow humans, and even myself. It should go without saying, ignorance does not preclude fascination with, delight in, nor love for those beings.
This isn't as hard as some people make it out to be. If God has to obey logic and/or other laws, then of course, there are some limitations to his power.
Could a dictator decide to set up a Constitution and limit his own powers? Of course. Having absolute powers means having the ability to limit your future power.
"An abhorrence of contradiction has been high orthodoxy in the West for more than 2,000 years."
Not necessarily so, just a particular branch of western thinking and tradition now widely called the analytic tradition. GWF Hegel and Karl Marx had philosophies that deeply and profoundly embraced contradictions. Neither tried to abolish them, but instead to embrace them. To quote Marx in "Capital Volume 1":
"This is, in general, the way in which real contradictions are resolved. For instance, it is a contradiction to depict one body as constantly falling towards another and at the same time flying away from it. The ellipse is a form of motion within which this contradiction is both realized and resolved"
In other words, contradictions are not resolved but given room to move within forms. This was a move deeply influenced by Hegel, whose Logic embraced contradictions and sought to sublate them in higher forms.
In Marx case - at least in your example - he is not using "contradiction" in a formal logic sense. He is using it in the sense of the contradicting forces of dialectical contradictions.
Marx specifically used different terms when referring to contradictions of formal logic.
Exactly. (just posted same point.) Even those 'dialectical contradictions' are perhaps better described as opposing forces that set in train a dialectical resolution.
For all the talk of Marx as having a philosophy that embraced contradictions I am a little sceptical of the way that this is frequently represented. Hegel certainly did have a philosophy that embraced contradiction and Marx certainly drew from his system as a mode of expression but his work was rooted IN the world and the kinds of contradictions he pointed out where not logical contradictions but rather opposing forces that demanded resolution.
nagarjuna's point is not that contradiction or many-valued logic can be tolerated. nagarjuna, who is addressing a specific set of buddhist logicians, is finding paradoxes in logic in an attempt to break those logicians clinging to logic, and (somewhat presciently in the history of buddhism) suddenly enlighten them. he, like most buddhists, is not making any ontological or metaphysical claims whatsoever.
it's similar to kant's antimonies. he is not trying to assert contradiction exists, he is trying to point out why we need to countenance ideas like a distinction between noumina and phenomena. or like zeno's paradoxes, which have a rhetorical purpose: to confirm zeno's teacher parmenides's claim that the universe is one undifferentiated whole. see also plotinus, who does roughly the same thing (any real ontological difficulty posed by the paradox of plurality is uninteresting to them; they have a thesis and their statement of the paradox is for a purpose).
incidentally, i don't mean to slag graham priest too much. after all, j.c. beall was my logic professor, and i'll always fondly remember how he introduces every new step in a proof by saying, slowly, "now holllld on, what about (etc)." the dialetheists will always hold a place in my heart. that said, i think godel did about all that need be done with the liar's paradox, and we ought to be wittgensteinians regarding language in the first place (words mean what they do in virtue of their being used by agents for a purpose; you cannot fully enumerate representational content of an utterance solely in virtue of its shape. all utterances are context sensitive. for more, see the almost unreadable but spot on work of charles travis, e.g. unshadowed thought).
An "abhorrence of contradiction" is necessary for him to even begin to express his having eschewed his former "parochialism". To have said, "abhorrence of contradiction" he has demonstrated an abhorrence of contradiction -- in the very act of making his declaration.
The Principle of Non-Contradiction (PNC) means that every word in the sentence "The line is straight" has a specific meaning. 'The' does not mean 'any', 'all', or 'no'. 'Line' does not mean 'dandelion' or 'doughnut'. 'Is' does not mean 'is not'. 'Straight' does not mean 'white', or anything else. Each word has a definite meaning. In order to have a definite meaning, a word must not only mean something, it must also not mean something. 'Line' means 'line', but it also does not mean 'not-line' — or 'dog', 'sunrise', or 'monkey'.
If 'line' were to mean everything, it would mean nothing; and no one, including him, would have the foggiest idea what he means when he says the word 'line'. PNC means that each word, to have a meaning, must also not mean something. And so; anyone who argues against an "abhorrence of contradiction" must use PNC for that statement to even mean anything, thus undercutting his own argument.
----
"There exist, indeed, certain general principles founded in the very nature of language, by which the use of symbols, which are but the elements of scientific language, is determined. To a certain extent these elements are arbitrary. Their interpretation is purely conventional: we are permitted to employ them in whatever sense we please. But this permission is limited by two indispensable conditions, first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed."
—George Boole, An Investigation of the Laws of Thought
I don't see that every violation of the principle of noncontradiction trivially leaves definitions meaningless. Suggesting that a principle shouldn't be taken to hold in general doesn't mean that you have to violate the principle at every opportunity, just that there is some place it is (in whatever sense) correct to violate the principle. Of course, under traditional logics, accepting a contradiction leaves everything provable - but we're dealing with nontraditional logics that may have other ways of dealing with that.
As someone not versed in nontraditional logics, can you give an entry-level statement or thought experiment that violates the principle of non-contradiction without leaving everything provable?
I'm also not particularly versed in nontraditional logics. I'm just leery of saying a mathematical structure can't be interesting or has no applications, when mathematicians seem to disagree.
You're being needlessly difficult to no good end. I am not dropping the law of non-contradiction in my comments here, where I speak about logics that drop non-contradiction.
And the point you're trying to make here ignores what I said. It is certainly the case that if you permit every violation of non-contradiction, whether or not it violates any other rule, you wind up with nonsense. In no way does that establish that nonsense follows from any violation of non-contradiction even in the face of other rules. I don't have deep enough knowledge to say one way or the other, but some mathematicians seem to think there are things of interest there and your superficial rejection does nothing to prove them wrong.
You were perceived as difficult because rather than actually address the content of what I said, you tried to make the same point in the same way you had earlier - a way which was undermined by what I had said.
That's just being difficult. The real world does work that way. Take poltical labels for example, liberal, conservative, et al have totally different meanings for different people. Contradictions do exist, just externalizing them in a formal system is lazy.
(Just a quick note, don't have time for more now.)
"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. 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."
The jump from Kant to Wittgenstein elides the step that was Schopenhauer, who did look East and said some cogent things.
He might say that, but I think "language-game" is not quite the right take.
Anyways, to shore up my impression I looked up "Schopenhauer eastern philosophy" and hit a few things. The "Schopenhauer and Buddhism" piece by Peter Abelson
( at http://ccbs.ntu.edu.tw/FULLTEXT/JR-PHIL/peter2.htm ) seems good:
"When the tenets of Buddhism became known in
Europe during the third and fourth decade of the
nineteenth century, Arthur Schopenhauer was
delighted with the affinity they showed to his own
philosophy. Having completed his main work Die Welt
als Wille und Vorstellung as early as 1818, he
considered it an entirely new (and thus pure)
expression of the wisdom once taught by the
Buddha."
I would have been gladder if the original article, by a professional philosopher, which I am not, had said something about this.
I can appreciate the depth of research behind the article. The many-valued notions of logic are quite helpful. Here's are two ways that I would distill the message for different audiences.
1. A computer science audience: Not all sentences have a computable boolean truth value. Here's why. To find the truth value of the sentence "This sentence is false.", you have to figure out the truth value of the following statement (call it X1): "Whether or not this sentence is false depends on if X1 is true or false." Put that way, you can immediately see the circularity in computation. From a computer science perspective, there is no terminating condition. The function will never complete, so there is no computable boolean truth value.
2. To an educated, but non-technical, audience: Just because a sentence asserts something doesn't mean there is any guarantee of the result being "truth" or "falsehood". The truth value of this "This sentence is false." is simply "undecidable" or "ineffable" -- pick a word. So, just live with the fact that not everything is true or false. Note: I'm not saying that the statement is really one or the other but we just can't figure out which (as in Schrödinger's cat); I'm saying that neither "true" or "false" makes any sense for that kind of self-referential sentence.
This really isn't mind-blowing. Many articles write up these "paradoxes" as if they are insurmountable. They aren't.
All of this said, I think many philosophical writings, especially Buddhist writings, use contradiction as way of promoting thinking and careful decomposition of the essence of things. In short, complex things consist of parts that vary over time. So their components or transient values can seem to change or stand in contradiction.
There is no contradiction in Dickens writing "It was the best of times and it was the worst of times." in my opinion. This is just a literary device to show contrast eloquently, because saying "In some regards, it was the best of times. In other regards, it was the worst of times." is not as memorable or striking.
Sankara (8th century CE) talks about two types of truth - 1) Kevala Satya (Absolute Truth) and 2) Vyavaharika Satya (Relative Truth) - He also says that people who are not one with the Absolute truth (One with God - liberated) can only discern the Relative Truth. He called this state - Maya (illusion). Looks like he was talking similar to Gorampa. To be noted is Sankara was extremely knowledgeable about Budhist thought - and is often called Abhinava Budha (Neo Budha) and was instrumental in propagating Neo Hinduism based on Adwaita (Non Duality) defeating the Budhist schools of the day using their own arguments.
It's interesting that the west would write off eastern philosophy, as though the eastern philosophers were unaware of the seeming contradictions and that there may be a deeper reason for that.
I think by extension of the Platonic school of thought Western culture has become deeply ingrained in dichotomous thinking applying such constraints to all areas of life (beyond simple science and logic), whereas Eastern non-dualism supposes all things (existentially) being equal. Neither West or Eastern philosophies do a very good job of describing each others domain space and yet they share many similar concepts. Karma (Theoretical) vs Causality (Empirical) is a basic example - they're exactly the same but few Westerners and particularly those in academic circles would admit to such a thing as it undermines Western status quo.
I'm going to go meta here and say the hn discussion here is just fantastic. This is the reason I frequent hn and I wish more submissions would spur discussions of this quality.
Yes, guys, where have you all come from? There is just tech and politics talk for months, and then suddenly, religion and philosophy pops out - and everyone is talking at a very high level. How come?
Silly Buddhists. Everyone knows that there are nine possible logic values: false, true, unknown, uninitialized, irrelevant, indeterminate, weak false, weak true, and high impedance. Didn't Siddharta read the IEEE 1164 standard?
Could someone explain why Priest, in discussing the 'four corners', presents the Hasse diagram with {F} at the bottom? Why would it not be {}? I'm not too familiar with posets, but it seems the order matters, right?
Intuitively, it is to indicate degree of 'truthiness'. Certainly false {F} is less truthy than either both true and false {T,F} and neither true nor false {}, which are in turn less truthy than certainly true {T}. The Hasse diagram indicates that there is no relationship between {} and {T,F} in terms of level of truthiness.
More mathematically, you can think of the Hasse diagram as a lattice, with the meet and join operations being AND and OR. For example, we are used to thinking of the expression
p AND q
as being true if p is true and q is true. But what if we have four values: {}, {T}, {F} and {T,F}? The Hasse diagram tells us how to interpret expression like this - p AND q is the meet (greatest lower bound) of p and q, and p OR q is the join (least upper bound) of p and q. So for example,
{F} AND {T} = {F}
{F} OR {T} = {T}
{F} AND {} = {F}
{F} AND {T,F} = {F}
{F} OR {} = { }
{F} OR {T,F} = {T,F}
{T} OR {T,F} = {T}
{T} OR {} = {T}
{ } OR {T,F} = {T} // I think..!
{ } AND {T,F} = {F}
{ } OR { } = { }
{ } AND { } = { }
The Hasse diagram in the article makes sure that expressions like this agree with our intuition in the cases where we have intuition for what the result should be, and give us a way to interpret expressions consistently when our intuitions fail us.
We're used to seeing Hasse diagrams for subsets of a set S used to indicate inclusion, in which case you would have {} < {F}, but you can have a valid poset structure that's not based on inclusion.
Nature is to have no nature, taken literally that is a contradiction, but isn't it possible that is not meant to be taken literally, but to illustrate for instance that everything we hold dear is really just a construct. What we call "nature" isn't true essence but our interpretation of what we think essence is?
Jesus for instance freely admitted he taught in parables, and so everything was not to be taken word for word literally. Doesn't make it any less true, but when interpreted literally you can come to some incorrect conclusions.
Seems like statements can have both verity (truthiness) and falsity in varying quantities. They might sum to 0, 1, or anything at all. Whether a statement has meaning seems important, too. '2020 will be a rainy year.' does not have meaning (yet). Pretty soon, though, the universe will reach a certain point where that statement starts to accumulate verity & falsity. Maybe 2020 will be a peak monsoon season in the Eastern Hemisphere; verity++. Maybe it'll be a record drought in North America, too; falsity++. This is not a 1-dimensional spectrum.
Seems like meaning is some sort of vector norm over || < verity, falsity > ||. It should be easy to imagine scenarios all across this unit square for '2020 will be rainy': (0,0) is now, (1,0) means lots of agreed-to raining everywhere, ( .9, .9) is crazy weather, and ( .2, 1) means it was mostly dry that year.
This seems heavily reliant on human interpretation, but trying to extract "intrinsic truthiness" will always rely upon some global context telling you what the rules are. '2 = 3' is absolutely true, depending on what you think the rules are.
'Correctness' seems to correspond to (verity - falsity), which is why ( .9, .9) is certainly meaningful but doesn't improve on the correctness of saying 2020 is rainy.
I have a haunting feeling GEB talks about this, though I never finished reading yet.
That is absolutely the case. If something is "ineffable", it's probably logically ill-defined, and we should treat anyone who tries to use something "ineffable" in formal argument as a mystic spouting hogwash.
This seems like a valid argument to me. Could one of the downvoters perhaps care to point out the errors in it. Or is it just because of the word 'hogwash'?
I'm very concerned at the number of people on HN who seem to have bought into this religious mysticism disguised as math. That's one of the oldest tricks in the book.
But logic is subjective and reason is impotent. What is the purpose of philosophy other than to protect us from acquiring knowledge about reality? We'd be defenseless without faith and mysticism.
If you want to learn more about this catuskoti, meaning ‘four corners’ theories there are another one threekoti, meaning ‘three corners’ you need to check Prof. Nalin de silva's works.
http://kalaya.org/
Unfortunately foreigners most of his work is in sinhala.
And only foreigners learn that language is germans,russians,chines and japanes diplomats.
Could the 'easterners' point be the avoidance of reification and the limitation and mental trapping that involves? For example, we say we see a tornado. But there is no essence of the tornado, no 'thing' there.. it is a perceptual grouping of a set and series of events. Nevertheless, functionally there very much is a tornado. 'It' has its conditions, states, and results. (But there is only the wind...) (And in some visions of physics, this goes all the way down to the vacuum.)
Renaming for clarity of action is useful, but no new substances are generated by the act. (One doesn't have to think in terms of the objects in someone else's code nor in terms of the lumped matter discipline.)
So if one can think outside of the perceptually given externally, how much more so might it be useful to do so in relation to internals, i.e. 'my anger'.
How the hell are there this many people here who buy this hogwash?
>Nagarjuna often runs through the four cases of the catuskoti. 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.
Are you kidding me? This article isn't even about formalizing non-boolean logic. It's thinly veiled religious propaganda, artificially ascribing mathematical formality to a mystical religion.
And yes, statements are either true or false. If you can't say a statement will be either true or false (you don't have to know which one), the statement is ill-defined. Non-binary logical systems are useful only insofar as they allow us to model uncertainty, not because they reflect the nature of reality.
Interesting thread, though I could not understand all of it (tongue in cheek). I'm going to lighten up the mood a bit by posting this link to a blog post I wrote a while ago - Bhaskaracharya and the man who found zero:
The bizarre states uncovered by quantum mechanics resulted in a million popular works trying to distill quantum reality into a world in which multiple states could coincide even at the macro level. We know that this is true at quantum scale, it has been a little harder to swallow at the level of everyday reality.
So much of our daily life in the west rests directly on fundamental assumptions we make about reality. I believe it's mind expanding to have those base assumptions challenged from time to time.
It is amazing how tangled men can get in their own thoughts and words. The concept of ineffable is nonsense from the outset. By the definition given, if something were truly ineffable, not a word has been spoken about it. What is being contorted here is the difference between the limitations of words to describe an experience, which is the Buddhist ineffable, and the idea of the unassailable noumena, which is the Reality that might be outside of any experience we can have. There is also nothing mysterious or limiting about Western logic. A statement is either true or false; or it is a contradiction (which generally means a premise was false), or indeterminate (which simply means that no one knows which it is).
This is nice. I'm wondering how much our restricted thought environment (x can only be true or false) has hindered our development in the last thousands of years?
A multi-valued logic (at least of the sort described in the article) is not stronger than a single-valued one. They can be transformed into one another.
For instance lets say give the proposition "the sky is blue", the truth value (0.78, right here, right now), corresponding to being quite blue, but not necessarily always or everywhere.
truth value(the sky is blue) = (0.78, right here, right now)
Since a function is just a special case of a relation, we could say
the sky is blue ~ (0.78, right here, right now)
And a relation can be described as function from the composite space to {false, true}, where false denotes not related, and true denotes related. So we could transform the whole thing into
truth value((the sky is blue, (0.78, right here, right now)) ) = true
Now another question is which model is most convenient.
I'm wondering how much Aristotle's metaphysics of essence may have hindered certain paths of thought. :) You must admit, his base system is powerful and ever-penetrating (Descartes is probably a very good example, but it's only one of the more obvious ones...)
Of course it's easy to say things like that, and harder to come up with a consistent (pun intended) metaphysical system on your own self. :)
Hmm. I don't think so, but, feel free to expand your opinion (why do you think it should be "I am, therefore I (can) think"? Maybe you have an interesting point to make?)
Re: etymology, it's pretty much "think => exist."
Re: conceptual level: well, one of the texts where this whole concept/causal-chain is introduced is in his "Meditations on First Philosophy"[1], in particular, in Meditation 2.
Very (very) roughly (and from bad memory), the idea is that one can first start casting into doubt everything that there is (skepticism as a method) (this is Meditation 1.) When you abstract enough times, you can start doubting even the certainty of things which may seem true in themselves (this latter notion is an endless field of debate in itself; see e.g. Quine's "Two Dogmas of Empiricism"[2]) - cliche example is "2+2=4."
Say there's a deceitful daemon: each time the thought "2+2" is produced, the daemon "intercepts" the answer and produces a "5". (Yes, I'm saying that Descartes may be one of the originators of the concept of a MitM, ha!) (Also, this doesn't work so well with mathematics, since again, one could say that mathematical statements are "analytically true", and it's simply not true that one may mistake a "5" for a "4". But imagine something more mundane: each time you think if there is a god, the daemon deceives you into thinking there isn't one; whatnot.)
So now you generalize the doubt process enough to start doubting whether anything exists, what-so-ever. The point is not so much in thinking that there isn't anything, but in removing certainty from usual ontological assumptions (what (kinds of) things exist.)
So now you're not certain if there is an "I": you are able to doubt it.
But then Descartes says that this very process of certainty-removal requires an operation (he doesn't use these words, did I say this is extremely rough): namely, the "doubt" itself. "Doubts are to be had."
Or, if you will (and closer to D. (I think)): if an evil daemon can cast into doubt even the existence of your own self - it it can deceive everything, etc. - there needs to be a something to be deceived. Or at the very least, the process of deception (or: of doubt) does happen (if you want to cast everything into doubt.)
So if you doubt that anything exists and hence can not be certain of anything, then it so happens that this requires doubt "to happen." Hence "doubt exists." (He also says (I think) that, basically, "clearly since I am doubting right now, doubt does happen". A kind of phenomenological argument, which doesn't really stick with me, fwiw. He then starts using the notion of "clear and distinct perception", and that's where things go downhill imho (Meditation 3, a (kind of circular) proof of God, etc.) But I have few qualms about Meditation 2, and this includes cogito ergo sum.)
(You could say that the only thing one can be certain of is that "there are thoughts" (not an "I"). I would agree with this. "Thinking exists." And since "thinking" would be, well, thinking this thought, it is certain that "thinking exists.")
edited to include two paragraphs (this+next one): So now we (hopefully) have something that exists for certain; thoughts exist (or somesuch.) Descartes claims a bit more: this entity that does the thinking (there does need to be an agent which thinks those thoughts (is what he assumes, I guess)) - let us just say it is "I", who does the thinking/doubting - when we trace back the process by which we concluded that we are certain that it exists - involves us discovering that thinking/doubting definitely does take place. Hence "I think; I exist."
(I suppose you could have a concept of an "I" that is simply instrumental (doesn't have independent substance (oof, this is dangerous/slippery territory)): thoughts can have relations, and related thoughts can be seen as "bundles", and this process producing continuing interrelated bundles is "the I doing the thinking." But, I'm not sure of this at all. Maybe D. would say that the deceiving daemon needs to deceive something, if it does actually do any deceiving; and if this does happen, then this "something" which is being deceived is (let us call it an) "I".)
Now, to go from cogito ergo sum to proof of God and subsequent re-establishment of certainty about the whole world - that is something else. And I don't think that he does a very good job at it. (And this is "usually", "generally" agreed upon.)
>Hmm. I don't think so, but, feel free to expand your opinion (why do you think it should be "I am, therefore I (can) think"? Maybe you have an interesting point to make?)
I did say "vaguely". Let me think about it a bit and then reply later tonight.
My guess is that the way people interpret it is like this: Descartes is saying that he (notices that he) thinks, wbich (to him) implies that he must exist.
The statement can be broken down like this:
A, therefore B where A = "I think" and B = "I am".
But if you look at it like this: the "I am" part is asserting that he exists, which according to him is a logical implication of "I think". But his asserting "_I_ think" itself implies that his "I" exists (whether it thinks or not). So the "I think" is like saying "I exist" (and am also thinking - my emphasis is on the "I"). So it seems like he is already assuming something is true ("I" exist/am), on the left side of the sentence, what he is trying to prove ("I am"), on the right side of the sentence. Note the difference in the two uses of quotes in the last sentence.
Hope I've made my thinking :) clear.
Indeed, I think I actually mostly agree with you. See my longer comment / explication of Descartes' argument (if it's an argument) above: I personally (also) think that "I think" already presupposes too much. What one might be able to argue is a lesser version of the thing, though:
thinking, therefore, existing. ("thought exists" (whatever that actually means!))
> So it seems like he is already assuming something is true ("I" exist/am), on the left side of the sentence, what he is trying to prove ("I am"), on the right side of the sentence.
..so yeah, I think that the only safe thing for him to say at that point would have been "thoughts exist." Maybe this does sound lame, but in the context of Descartes' Meditations, it's about discovering that even though you can doubt everything, this does require (the process of ) doubting as such.
Or: I can be deceived about everything there is in the world (so e.g. in actuality, there is no world as such); but who is being deceived? At the very least, there is (some kind of) thought happening. One might say that doubting requires an agent, etc., but really, this is already debatable.
But in short: you're right, if one were to establish "I exist" (with an "I" having certain desirable properties), one could then say "I think", and that would have been less murky, indeed!
Problem is, poor Descartes is not sure that he "exists":
indeed, for him (and this is partly to do with the whole mentality present in Rationalism, and with the start of Modern philosophy (many people hold him to be the father of it)) thinking comes before existing. You will find many parallels of the latter around, scattered in various places; cf. "god thinks itself into being", etc.
He kind of articulates one of the foundations of Rationalism as in the epistemology: thought shall be the ultimate and primary criterion for truth.
But again, I agree, however, I would say that he could have stated a lesser version of his point (while not reversing the logical implication arrow): "thoughts exist."
However, to go from that to re-establishment of certainty about the whole wide world would have been even more tough (and he was already having a hard time!) In summary, I think I understand where he's coming from, but one could say that his project (a kind foundationalist epistemology) had been doomed from the very beginning!
> Hope I've made my thinking :) clear.
Yeah, I think that was clear! Forgive me for my verbosity, and thanks for the explanation. I think it makes sense, and fwiw, I agree with the general gist of it.
No problem, and you're welcome. I agree that he could have asserted a lesser version, such as "thoughts exist", as you say. Interesting points, and good discussion, thanks!
It's been thrown away in some (growing) part of modern mathematics and has lead to the creation of automated provers as logic rejecting PEC and not PNC models lambda calculus. See my top level response in this thread for more detail.
The article doesn't make such a statement, and it would be quite a stretch. Although reasoning in science and mathematics is to some sense restricted to logic with binary truth values, this doesn't mean that we are also restricted to such a logic in everyday life.
His argument is self-defeating. Ultimately, it amounts to: "There is no such thing as truth." But that is a statement about the truth.
The law of noncontradiction is one of several axioms that are presupposed by any proposition. Such axioms can't be "proven," because any proof would presume their correctness, but it is proper to accept them.
Its very important to understand, that unlike most Western philosophers, the Buddha's philosophy came as a side effect of his meditation and realization.
When you realize the non-dual nature of reality, that the split between subject and object is an illusion, your entire conception of reality changes radically. This realization, which is beyond concepts, allows the generation of ideas that are completely outside "normal" frames of thinking.
This present moment is not conceptual, it is real.
Actually it would be the reverse, since Buddha predates stoicism.
Though it is possible Buddha was partially influenced by pre-Socratic Greek thought and, in turn, later Greek philosophers were influenced by Buddhism (to put it in perspective, Socrates was roughly contemporaneous with Buddha).
It is possible to give a rigorous mathematical definition of a logic that deals with contradictions and propositions that cannot be assigned a truth value. This has parallels with traditions in Buddhism and helps resolve paradoxes which are problematic for traditional Western logic (e.g., "this sentence is false").
Fascinating essay. I enjoy and appreciate the references to Buddhist ("Eastern") philosophies given that the paradoxical thinking is what most of us attribute to it.
What is the catch with Aeon? There don't seem to be ads, paywalls, subscriptions, or other monetizing strategies, but the content seems fantastic. Is it charitable intellectualism?
And it's very easy to save the PEM if you distinguish the value of a proposition (true or false) and the cognitive status of the proposition (certain, probable, possible, or arbitrary).
Moreover, to exist is to have a particular identity, to have a nature, to be something.
Thus, all of this work is utter hogwash. The professor who wrote this IS practicing mysticism and irrationality. He ought to be shunned by all actual philosophers.
If this kind of thinking were to get "popular," Western civilization and science would collapse into skepticism and religion would take over again. Just as happened in the Middle Ages and at the end of the Islamic Golden Age (in the latter case, there never was a recovery).
1) Graham Priest is one of the most famous logicians of this century.
2) The LNC is an axiom it doesnt "hold", its either built into what you mean by identity or it isnt. Either way there is no non-circular "hold"ing.
3) Im sure the notions of truth, falsity and consistency are far too technical here for the lay person to get a serious grasp of but i'd be careful of rationalizing your own ignorance by projecting stupidity onto others.
What that properly means is that to deny it is a contradiction, because it must be assumed to deny it.
In other words, you can't say "The truth is, there is no truth." Which is what Graham Priest is doing.
1 is an interesting observation, but has no bearing on the facts of the case: he is a fraud.
3: You should be ashamed to make a statement like this in an intellectual conversation. If you feel the need to bolster your argument with an unsubstantiated character attack like this, you must not feel that your argument is sound.
No offense, but I'd flag your comment for #3, except it's not worth wasting moderators' time on.
The constructions I have described show how to make precise mathematical sense of the Buddhist views. This does not, of course, show that they are true.
> 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. 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.
No, they are nonsense. Nothing that claims that something can be true and false can be logically rigorous. To be logically rigorous means to be logical---and to be logical means to obey the law of non-contradiction.
Anyway, it's at least gross negligence for the author to write an article like this and then not explicitly declare that these ideas are nonsensical.
I think I have made my point, I don't want to continue the discussion, though I will be interested to read any further follow-ups.
while i don't agree with priest either, your reasoning is bad.
the point is not to argue against how people define logic, or anything as arbitrary or seemingly self-defeating as that.
the point is to provide a functional logic in the face of the liar's paradox and the like. regardless of the motivations of individual philosophers, the work itself posits as motivations actual technical problems. the technical problems generally arise from self-reference, but (as you can see with varieties of incompleteness proofs as well) you don't need outright self-reference to get the truth of a statement resting on its negation.
yell all you want about rigor, the onus is on you to use conventional logic to resolve these tensions. again, i personally think that proponents of many-valued logic are misguided in their large scale view of truth functional representation (and i probably think the same of you!). however, i respect that they seek to defend it by solving problems rather than saying "hurr, 'this is a lie' just doesn't mean anything at all. and oh yeah, language is compositional, truth is correspondence, and i'll hear nothing saying otherwise!"
Yes of course we can get away from defining any new systems of logic by simply taking working with different kinds of relations or functions in our classical logic. This doesn't show that non-classical logics are useless but that classical logic and classical mathematics are incredibly powerful and can describe practically anything.
Note that this article is very far from the wishy washy hippie claims like "Everything is connected! And you have your truth man and I have my truth". Instead it gives a popular-science description of some non-classical systems of logic, their historical background and how they relate to Buddhist philosophy and certain problems of reference. Personally I don't really think the logics presented solve these problems in a satisfactory way, but they are definitely of philosophical interest (o/w Cambridge University Press wouldn't have published his "An Introduction to Non-Classical Logic").
So it's trolling for a computer scientist to support Aristotelian logic as opposed to Buddhism? Wow.
But yeah, I do recognize the irony. I admire moral rigidity, which is why I picked the username. Though I don't necessarily agree with javert's particular sense of morality, which obviously didn't work out too well for him.
>If this kind of thinking were to get "popular," Western civilization and science would collapse into skepticism and religion would take over again. Just as happened in the Middle Ages and at the end of the Islamic Golden Age (in the latter case, there never was a recovery).
So you don't know that something exists, you only know that your sense impressions have been triggered. How do you know its nature?
Because of what you see.
So you don't know anything about its nature, only about its relationship with your sensory apparatus. (Hey, I may have just invented category theory!)
How can something exist...?
Don't look at me, man. It's all probability waves as far as I can tell.
Note: I don't really, actually, myself, buy into this form of epistemology and metaphysics, so I may have the arguments badly. On the other hand, those are major topics that have a lot of diverse implications. They cannot simply be dismissed by assertion.
By your logic, that you didn't own but used nevertheless:
You don't know I asked those questions, you only know that your sense impressions have been triggered.
Similarly my eyes sent some signals to my brain and after processing them I have concluded that you have replied to my comment. But wait! I don't really know you replied me, do I?
Since my senses are unreliable and I don't have any other means to interact with reality but my senses, my mind is impotent to deal with reality. What else can I do but go through my life hallucinating and doubting myself or inventing fantasies to keep my sanity.
-----
When you look at the moon (at the right time) you see circle. Then you can decide the moon is circular. Your senses are working just fine, so is your brain. It's your limited knowledge that leads you to wrong conclusions. As you observe and learn more, you integrate more and more concepts and you can eventually arrive to the point that moon is indeed a somewhat spherical object.
Or you can conclude that your senses are lying to you and reality cannot be known. Then you'd never go further than "looks circular, but know knows, right?".
This article is kinda missing the point. The "empty" is the undefined. That which has yet to exist, or never will. There is that which is uncountable, for example the number of corners in a circle. Just because it is a contradiction doesn't mean it is wrong -- maybe your models are wrong. The usefulness of eastern philosophy is to free yourself of preconceived ideas. I think it is obviously clear to everyone today that there are modes of thought that are far inferior to logic. I think it would be supremely arrogant to presume that logic is the best mode of thought that humanity will ever come up with. To try to shoehorn everything to fit into a logical framework feels akin to epicycles to me.
Maybe it is missing your pet theory, but the article has a definite point and Priest argues it at length. You seem not to have grasped the article because you refer to logic as something singular that should not be deemed superior to other modes of thought, while the article demonstrates there are many different, mathematically rigorous conceptions of logic.
Epicycles are a non-sequitur here, because there is not a particular proposition that you set out to prove with whatever explanation you can find. Rather the goal is to find a logic that does not break down when presented with paradoxes.
There can be an infinite number of different rigorous conceptions of logic, while at the same time there are also an infinite number of modes of thought outside of logic .. we know the ones like instinct, superstition, and religion that are considered inferior to logic. Is it possible for there to exist a mode of thought yet to be found or invented that is better?
I wouldn't say we "know the ones like instinct, superstition, and religion [...] are considered inferior to logic". They are not comparable, unless you formulate criteria for what is better, and then you can easily bias the comparison towards one or the other.
If there exist an infinite number of modes of thought as you posit, then yes it is possible that the best one is not among the ones we know of now.
I don't really think of it this way though, because logic is not supposed to give us something new, not supposed to give surprising conclusions. Logic formalizes the steps you can take to arrive at valid conclusions. However, what constitutes valid conclusions is ultimately based on our intuitions, it is what the logic is founded on (comparable to how mathematics is ultimately based on accepting axioms).
>>The "empty" is the undefined. That which has yet to exist, or never will.
The "empty" is the un-defined. Direct experience, that which can't be talked about, labeled or categorized. That (this) which has always existed, and always will.
The void is that which can't be named. It's all around you! You can never leave it because it is you, stretching from your center in all directions.
There are no contradiction here, because you need a logical layer over reality to have contradictions.
So true.
To experience the un-manifested one has to just be actively aware:
- listen to sounds/silence;
- look at where you're looking out from;
- feel the tickle of the body's energy.
Bluntly said, you have to fucking shut-up!
I assume that what's relevant here will be his work on paraconsistent logics (which allow contradictions) but I think an equally interesting line of work is linear logics. It's of interest to computer scientists because of its relationship to linear type theory (in the same way that the Curry-Howard correspondence links type theory and classical logic) and because it has close relationships with quantum computing.
In particular, you are not allowed to delete or duplicate elements/propositions/types (corresponding to physical processes that cannot arbitrarily create or destroy particles, the "no deletion theorem" and "no cloning theorem" of quantum information theory), so functions of the type
and are not allowed in a linear type system. Practically, this means that many operations can be optimized by the compiler to in-place mutations, because it is guaranteed that there is only ever one reference to a particular object.