Which resolution of the liar's paradox are you using?
First, let's take a special case of paradoxes. I'll describe it as "negation selfreffering" statements. One exemple might be Russel's parodox.
I will take the most simple example, "The liar's paradox", wich states:
P: "This statement is false." ("This" reffers exactly to the statement it belongs to.)
Now let's take a false statement:
F: "The sky is green."
Ok. I'm interested in the character of the conjunction and disjunction of the two statements. (I know it's more philosophy than math, but I think I found a way to mathematicaly model it; I wouldn't post the model yet, to not influence your opinion).
My opinion is:
(P and F) is paradoxical,
(P or F) is false.
What's your opinion?
Which resolution of the liar's paradox are you using?
Thank you for your interest.
I do not use a resolution. It is more philosophy.
But if a resolution is necessary, I intend the resolution that Godel used in his two incolpeteness theorems. Probably the Saul Kripke resolution would fit the best.
I do not want to get a mathematical model. That will be my task to do. I just have an opinion and I am also interested in other people (maybe more smarter than me) opinions and possible explanantions.
Thx
What are you referring to by "the resolution that Godel used in his two incolpeteness theorems"?
Anyway, the problem with your question is that you've not indicated a method for evaluating truth, falsehood, or paradoxicalness of conjunctions and disjunctions that may have paradoxical conjucts or disjuncts.
Ok. To make it simple, I state again. I do not ask for something to be evaluated (boolean, Heyting, MV-, etc... algebras). Ackbeet asked for a "resolution". I wanted something philosophical. Know, if a resolution is necessary, I gave an example about what I consider it reflects the best way I interpret the notion of a "paradox".
So, in my interpretation:
P is paradoxical, it's clear.
F is false, it's clear.
I am interested if (P and F) "have the same truth characteristics" as P, and if (P or F) "have the same truth characteristics" as F.
You have the freedom to chose for yourself what "have the same truth characteristics" means. I mean something abstract through the notion of "paradox", this is the reason I labeled my problem "philosophical".
So, I need an opinion (the opinion that makes the most sense to its author) and an explanation attached to it.
Thx for your interest.
I think what OP meant by not asking for an evaluation is the fact that there are no precise rules of evaluation yet. This is not a problem where there is a precise definition for the trith value of a conjunction and where one is asked to apply this definition to evaluate some concrete conjunction. Instead, the OP asks to evaluate a specific example of a conjunction with the intent of finding out people's intuition: does it seems that according to a reasonable definition, this particular conjunction should be true, false, paradoxical or something else?
There are obviously various options. Say, we have three truth values: T, F and P. Two reasonable options are to put (x1 or x2) = max(x1, x2) and (x1 and x2) = min(x1, x2) where x1, x2 are in {T, F, P} and either F < P < T or P < F, P < T. The second ordering gives (P and F) = P, (P or F) = F, just like in the OP. It corresponds to degrees of knowledge where definite information (true or false) is considered superior to lack of information. The first ordering seems natural to me because one can think of P as "diverging". In this case, one does not have to wait for P to evaluate (P and F): since the second conjunct is false, the whole conjunction has to be false.
Maybe it does not matter much what seems natural to most people. If the OP thinks that there is some interesting and original mathematics for the definition he/she has in mind, I would encourage him/her to pursue this.
That makes a lot of sense. I hadn't thought to interpret that way.
Another framework that might work: Three truth values, T, F, and P.
P and F = F
P and T = P
P or F = P
P or T = T
P and P = P
P or P = P
Essentially P is the same as "indeterminate".
Incidentally, the decision to let P and F = F and to let P or T = T reminds me of short-circuit evaluation.
Edit: I called this "another framework" but on closer inspection it is exactly the same as emakarov's first ordering. I read too quickly and thought emakarov's first ordering only differed from the OP in evaluating (P and F) because that was the only one specifically mentioned, but it also differs in evaluating (P or F).
Looks like emakarov's use of the word "diverging" is the same as my use of the word "indeterminate". And it also seems this idea is encapsulated in the notation as used in this context.
Edit 2: Reading more on strictness analysis, I see the use of the word "diverge" is standard for this concept.
Edit 3: Poking around more on the internet, the truth table for this setup is here, where the third truth value is simply named "Unknown".
Thanks.
Both last posts helped me a lot. Moreover, emarkov helped me in an other thread too. The truth is I am studying for my PhD degree in informathics and I am interested a lot in the problems of "Foundational chrisis of mathematics", "Hilbert Program", "Godel's incompleteness theorems", etc. The study isn't finished yet so I want not to post incomplete information.
P.S. I'm pleasently surprised that people on this forum refer to wikipedia.
P.P.S: I'm using a four valued logic.
Thx
For what it's worth, Terence Tao gave a talk on mathematical research and the internet and wrote on a slide that "On the largest and most well known wiki –Wikipedia –the quality of the maths articles is steadily increasing" (slide 21 of updated version) and during the speech (as seen on video) says (didn't transcribe the stuttering) "But of course Wikipedia itself has got plenty of great mathematical articles now. If you want to learn any basic concept in mathematics, for example, that of a group.. (I couldn't make a few words out, he mumbled I think) the articles there are really quite good now. It didn't used to be that way, but they continuously improve. This is the thing. I mean, a textbook is static; you write a textbook, and it's just there, right, it doesn't change. Or you can make new editions, but it changes very slowly. But the articles here, I mean, they change, and quite generally for the better. This article ['Group (mathematics)'] in particular is quite good, actually" (starting from time 22:08).
Mathematical research and the internet « What’s new
I think it's okay to refer to Wikipedia once in a while to get a feel for some topic or another or to scout out a particular item. It's just that I take it with a grain of salt, and I don't rely on it.
One of the main problems, among others, is that the articles are not INTEGRATED with one another. So, it's not possible to get a good systematic treatment of an entire subject such as mathematical logic, set theory, analysis, etc.
It's good that the articles are improving. But that it they are improving while textbooks are static is not to much endorsement. My friend's weight and physique have been static for a few decades - he's in great shape - while my weight has steadily been improving (dropping) and my physique improving since I've been on a diet. Which is better? The one that's real good and has always been real good, or the one that is merely changing toward being good?
(emphasis added)
Seems your language is a little misleading?
Agreed that it is hard to learn from Wikipedia as you would from a textbook. I don't think Wikipedia was designed for this. In the same way, it is much better to consult a mathematics textbook on group theory if you wish to learn group theory, rather than consult Encyclopedia Britannica, although the latter could still be useful for broad overview.
But for example my last reference to Wikipedia above was to show a truth table. This is a small tidbit of information that does not require an integrated treatment of a broad subject. I think Wikipedia is well designed for such things.
Agreed. My main reason for quoting though was that Wikipedia can have articles of sufficient quality to justify a reference, at the discretion of the one who is referencing.
Wikipedia does frequently disappoint me for specialized subjects. When the quality is not good, I try not to use it as a reference. However, sometimes it is hard for me to find another suitable reference that is freely available online, so I end up saying "This reference is a bit hard to follow and/or lacking, but you might try it."
I understand that you see a trend of over-reliance on Wikipedia, which bothers you. I think the point is well taken, but because of the language you use, you end up sounding a little prejudiced against a useful resource, in my opinion.
The statements are consistent. I said that I dislike that people refer to Wikipedia RATHER than good books (or other more carefully written and edited sites). That is consistent with saying Wikipedia is okay for occasional lookups as long it is not relied upon to the exclusion of good books(or other more carefully written and edited sites).
Whatever it was designed for, my point is that I think it is a bad idea to use it that way. Also, it's not even a good source for pinpointed definitive definitions and formulations.
Yes, I granted that.
And better would be to quote a well written book, published article, or even Internet site that is more reliably written and edited.Wikipedia can have articles of sufficient quality to justify a reference, at the discretion of the one who is referencing.
What I said is that I dislike people relying on it rather than using much better sources; and that is based on both my experiences with it and on its own makeup. That's not prejudice.I understand that you see a trend of over-reliance on Wikipedia, which bothers you. I think the point is well taken, but because of the language you use, you end up sounding a little prejudiced against a useful resource, in my opinion.
P.S. What is especially annoying to me is that more and more there are people (I'm not at all saying you are one of them) who habitually cite Wikipedia (for mathematics) while not also getting a more integrated perspective. What happens is that you have people with these dangling, out of context, notions about this and that and they end up basically just throwing Wikipedia quotes at one another rather. And since the articles are not integrated or even uniformly edited in any way, quotes (especially definitions) from one article don't always line up with those from other articles. And worst are people who demand things like "You're not right unless you can show me in Wikipedia." So, in general, as I said at first, (regarding mathematics) I do dislike that people more and more turn to Wikipedia RATHER THAN better, more integrated, more careully edited, more carefully conceived books and articles.
Even more fundamentally, it's interesting to me that I do very much appreciate hyperlinking, the "network" nature of our knowledge (refer even to Quine's notion of Neurath's ship built at sea), but that it doesn't always suit mathematics very well. I mean, I feel that it is better to learn certain mathematical subjects from the BEGINNING, adding definition and theorem along the way, in the traditional way, as opposed to working BACKWARDS to learn definitions and theorems and then learn the definitions and theorems that were used in regress. And Wikipedia is used too much by people in that backwards way. A made up example: What's the definition of "cardinal"?, so then have to look up the definition of "ordinal", so then have to look up the definition of "well ordered", so then have to look up the definition of "leastness", etc., rather than starting with the primitives and working FORWARD, which is much more EFFICIENT and intellectually organized.