Hacker News Reader

Comment by Strilanc | Hacker News Reader

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}").

View on HN

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'.

http://en.wikipedia.org/wiki/Bart_Kosko

http://www.amazon.com/Fuzzy-Thinking-The-Science-Logic/dp/15...

oh heck yes! There are a number of great references in this area going back to the early 1980s!! [0] [1]

[0] http://ccbs.ntu.edu.tw/FULLTEXT/JR-PHIL/kasulis1.htm [1] http://enlight.lib.ntu.edu.tw/FULLTEXT/JR-PHIL/thomas.htm

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 would have enjoyed this article if it dropped the Buddhism and kept the math.

May I ask you why? Personally I enjoy reading about the origin of some particular idea/philosophy, even when it's purely anecdotal.

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.

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.

> law

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.

I believe in law it is perfectly possible to settle with no admission of guilt. Are you then innocent or guilty?

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.

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.

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.

>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.

http://en.wikipedia.org/wiki/Plotinus

But this is part of the point of the article:

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.

The takeaway from the east vs west thought for me was that paradoxes were actually embraced by both societies and are an inherent aspect of logic.

> I would have enjoyed this article if it dropped the Buddhism and kept the math.

People are different that way, aren't we. Something about not being able to please all demographics.

Reply on news.ycombinator.com