Curry's paradox

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xris
 
Reply Sat 22 Nov, 2008 05:00 am
@jknilinux,
I bow most reverently to your knowledge and i will enquire further but my logic which i pride myself on tells me that it is false concept...a play on words and formula...i judged it on its results not its reasoning, so how is that no logical?
 
jknilinux
 
Reply Sat 22 Nov, 2008 05:56 am
@jknilinux,
Why thank you Very Happy

Anyway, you're not using logic. You're using hunches.
 
xris
 
Reply Sat 22 Nov, 2008 06:10 am
@jknilinux,
jknilinux wrote:
Why thank you Very Happy

Anyway, you're not using logic. You're using hunches.
no hunches here... you gave me the conclusions and i concluded it was nonsense..sorry..
 
xris
 
Reply Sat 22 Nov, 2008 06:17 am
@xris,
I observed a debate once with two advocates of this method of logic..they threw formula at each other for weeks in between abuse and weeping and they never did agree..not once..
 
jknilinux
 
Reply Sat 22 Nov, 2008 06:54 am
@jknilinux,
What do you mean by "this method of logic"?
 
xris
 
Reply Sat 22 Nov, 2008 07:24 am
@jknilinux,
jknilinux wrote:
What do you mean by "this method of logic"? Do you mean, by any chance, the method logic uses? There weren't any other methods here...
I hope your logic can deduce my answer..
 
jknilinux
 
Reply Sat 22 Nov, 2008 07:34 am
@jknilinux,
Has anything I said offended you? If so, I'm sorry.

Curry's paradox was meant as a sort of disproof of logic- from it, everything follows, so logic's gotta be wrong. It's somewhat metaphysical in nature, since it may show that logic is man-made - that it is not an aspect of the "true" reality. But, without logic, can you conclude anything at all?

So, that's the gist of it. You may be able to find out more on google. good luck!
 
xris
 
Reply Sat 22 Nov, 2008 08:06 am
@jknilinux,
jknilinux wrote:
Has anything I said offended you? If so, I'm sorry.

Curry's paradox was meant as a sort of disproof of logic- from it, everything follows, so logic's gotta be wrong. It's somewhat metaphysical in nature, since it may show that logic is man-made - that it is not an aspect of the "true" reality. But, without logic, can you conclude anything at all?

So, that's the gist of it. You may be able to find out more on google. good luck!
You have not offended me .just stirred by grey matter..yes i will google it thanks.
 
validity
 
Reply Sat 22 Nov, 2008 02:05 pm
@jknilinux,
jknilinux wrote:
Validity-

Thank you for wanting to know more. Your lack of ignorance is the only reason why I'm bothering to write this.


Thankyou for your time.

jknilinux wrote:
In my post, I actually said "If it is raining and it is not raining, then the moon is made of gouda cheese".


I did notice that, however since we were discussing Curry's Paradox I thought it more direct to deal with "if it is raining then the moon is made of gouda cheese", since this is more like "if x, then y". Either way your imparting of knowledge to this forum is appreciated by me.

jknilinux wrote:
So, here's the proof:

Well, first, let's symbolize this...
p = "it's raining"
q = "the moon is made of gouda cheese"

These are the logic symbols required:
* = "and"
v = "or"
~ = "not"

So, this is symbolized as:
If (p*~p) , then q

-or-

It's raining and it's not raining, then this implies the moon is made of gouda cheese.

So, here are the rules of logic we'll use:

1: Addition, aka: "from x, we get x v y"
This is like saying "if it's raining, then it's raining or it's not raining." It works for whatever x and y stand for. So long as x was true, it's true.

2: Simplification, aka: "from x * y, we get x, and we get y"
This is like saying "If there's cake and cheese, then there's cake"

3: Disjunctive syllogism, aka: "from x v y, and ~x, we get y"
This is like saying "If you have 1 or 2, and you don't have one, then you have 2"

So, let's prove this:
We have p*~p, and we'll prove q.

From p*~p, we get p, by simplification.
From p, we get p v q, by addition.
From p*~p, we get ~p, by simplification.
From ~p, and p v q, we get q, by disjunctive syllogism.
So, if p*~p, then q.

So, if it's true that it's raining and it's not raining, then the moon is made of gouda cheese.

Of course, it's never raining and not raining at the same time, so we don't need to worry about this. This isn't the paradox I was talking about, by the way.


Since it is never raining and not raining at the same time, and as I mentioned before, there is something wrong with the Simplification Rule and its application to all things. In the example of "If there's cake and cheese, then there's cake" is true since you can have cake and cheese at the same time, and as you mentioned you can not have it's raining and it's not raining at the same time, the rule has an error in it? Maybe there needs to be a new rule such as if it's raining or it's not raining, but that only gives raining or not raining, which is equally useless. Is not it?

jknilinux wrote:
Anyway, proving curry's paradox will take too long here, as it goes into metalogic and metalinguistics, so I'll give up on this.


That saddens me.

jknilinux wrote:
Really, though, we should have a thread here explaining basic logic, at least. I wonder what people talk about in the logic forum if they don't even know modus ponens. It's gotta be the most basic part of deductive logic there is.
Sorry for going on about this, but it's a bit disappointing is all. Anyway, I'm glad one person was interested in logic, in the logic forum.


I look forward to your thread on basic logic.

So, where do you think the paradox (if there is one :rolleyes: , ha ha) in Curry's Paradox stems from?
 
jknilinux
 
Reply Sat 22 Nov, 2008 06:25 pm
@jknilinux,
There's nothing wrong with simplification. Although you can prove anything if a contradiction is true, a contradiction is never true, so everything's still alright. You can never have "it's raining and it's not raining" be true. If there's virga, then it is simply false that if it's raining, the street is wet. This doesn't mean that it's both raining and not raining.

Also, the statement "I'm human or I'm an alien" is true. However, you can't really get much out of this alone, aside from adding random stuff like "I'm human or I'm alien or I'm a cow". You can't break it apart into "I'm human" and "I'm an alien" because Only one of them is true; hence the "or" in the middle.



Anyway, curry's paradox is not of the form "if x then y". It is a self-referential statement, x, defined as "if x then y".
There's a "propositional logic symposia" thread on truth tables, in case you're not familiar with them. The truth table for if... then is as follows:

F = False
T = True
a,b = variables

| a | b | if a then b |
---------------------
| F | F | TRUE |
| F | T | TRUE |
| T | F | FALSE |
| T | T | TRUE |

So, no matter what a and b are, "if a then b" will be false if and only if a is true and b is false. Think about it, substituting true and false statements for a & b.

So, with the statement x defined as "if x then y", let's see if it's true or not.

If it's false, then that's because "if x then y" is false. So, x is true and y is false. So, if it's false, then it's true. Thus, it can't be false.

So, it's true. If it's true, then what it means will be true as well, so, given that x is true and "if x then y" is true, what is y's truth value? Looking at the table above, we see that the only place where x is true and "if x then y" is true is where y is also true. So, y must be true. So, everything is true.


You have a point, though- I'll work on making that logic thread.
 
validity
 
Reply Sun 23 Nov, 2008 11:04 pm
@jknilinux,
jknilinux wrote:


F = False
T = True
a,b = variables

| a | b | if a then b |
---------------------
| F | F | TRUE |
| F | T | TRUE |
| T | F | FALSE |
| T | T | TRUE |

So, no matter what a and b are, "if a then b" will be false if and only if a is true and b is false. Think about it, substituting true and false statements for a & b.

So, with the statement x defined as "if x then y", let's see if it's true or not.

If it's false, then that's because "if x then y" is false. So, x is true and y is false. So, if it's false, then it's true. Thus, it can't be false.

So, it's true. If it's true, then what it means will be true as well, so, given that x is true and "if x then y" is true, what is y's truth value? Looking at the table above, we see that the only place where x is true and "if x then y" is true is where y is also true. So, y must be true. So, everything is true.


I do not understand how "if it is not raining, then the street is wet"

ie | F | T | TRUE |

is assigned a TRUE. Could you please clarify.
 
validity
 
Reply Mon 24 Nov, 2008 08:42 pm
@jknilinux,
Okay jknilinux and xris, try this.

With the [quote]I do not understand how "if it is not raining, then the street is wet"[/quote] aside, I now see that

"If this sentence is true, then the moon is made of cheese"

or any variation ie

"If this sentence is true, then <insert whatever you like here>"

is a perfectly valid sentence and if the sentence were true, then the moon would indeed be made of cheese or anything would be true, but not becuause of the truth of the sentence. As the sentence is not saying the moon is made of cheese, it is saying if the moon is made of cheese then the sentence is true. The truth of the sentence comes the moon being made of cheese. Embarrassingly simple.

However it would seem that in going from this to

"If it is raining or not raining, then the moon is made of cheese"

it must be a different case. The paradox is not seen in the "If this sentence is true, then the moon is made of cheese" and yet I can not do anything else but claim paradox in "If it is raining or not raining, then the moon is made of cheese". Because this statement is placing the truth outside of a relationship between the sentence and the moon.

With this in mind I can only see the paradox stemming from substituting "it's raining or not raining" with "if this were true". I know it sounds absurd, but I guess what I am saying is that "if it's raining, then it's raining or it's not raining." is not a truth ie from "it's raining or it's not raining" it can be shown/demonstrated/observed that it is raining and it is not necessarily true that it is raining. (Oh dear the men in the white coats are coming)

Thoughts? (Apart from the men in the white coats that is)

PS Is this what you meant by
jknilinux wrote:
Curry's paradox was meant as a sort of disproof of logic- from it, everything follows, so logic's gotta be wrong. It's somewhat metaphysical in nature, since it may show that logic is man-made - that it is not an aspect of the "true" reality. But, without logic, can you conclude anything at all?

So, that's the gist of it. You may be able to find out more on google. good luck!
 
xris
 
Reply Tue 25 Nov, 2008 05:12 am
@validity,
Well i need more than luck...if my head explodes its not wet...
 
validity
 
Reply Tue 25 Nov, 2008 01:53 pm
@xris,
xris wrote:
Well i need more than luck...if my head explodes its not wet...


Laughing

I do have a degree of confidence in my resolution though. I am hoping to get an analysis from jknilinux as I can see how to resolve Curry's Paradox.
 
Zetetic11235
 
Reply Sat 6 Dec, 2008 08:05 pm
@validity,
If x then y is false iff(if and only if) x is true and y is false, but this means that if (if x then y) is false, then x is false. So we have P(x,y) is false => X is false. Godel did a lot of work on self-referential statements. In fact these type of statements tie in directly to the incompleteness theorem.

This paradox touches on some fairly deep ideas delineated here: Curry's Paradox > Notes (Stanford Encyclopedia of Philosophy)
 
jknilinux
 
Reply Sat 6 Dec, 2008 09:09 pm
@Zetetic11235,
"if (if x then y) is false, then x is false."

and x is true...
 
Zetetic11235
 
Reply Sun 7 Dec, 2008 12:31 pm
@jknilinux,
jknilinux wrote:
"if (if x then y) is false, then x is false."

and x is true...


Sorry, I actually gave the wrong address for the curry's paradox page: Curry's Paradox (Stanford Encyclopedia of Philosophy)

Anyway. In response, I put up the wrong sentence form. It should be something like If (if x then y) is true, then y is true. Now, that is not really the corresponding sentence form. There is an imbedded infinite loop, so the sentence looks more like: If(If(If(If................(if x then y)is true)is true........), then y is true. This cannot be evaluated by basic predicate logic.

Also, P^~P is always false, the reason that P^~P can imply anything is that it is always false. There is really no need to take ~P by it self as P^~P is patently invalid and thus implies anything. This is due to the fact that x=>y is false iff True=>False.
 
jknilinux
 
Reply Sun 7 Dec, 2008 07:12 pm
@jknilinux,
You seem to be saying that self-reference should be disallowed. That works for curry's paradox, but not for the strengthened liar and many other paradoxes. Besides, by disallowing self-reference, you disallow an infinity of OK statements. You even disallow consciousness in a sense, since it is by definition self referential, since it derives from Tarski's semantically closed thought. That's the big point, and the one that makes me think self-reference should be kept.

And yes, I know P*~P is always false, which is why it's so bad when we have something that says it's true, like a paradox.

And thanks for the link! If anyone wants an overview of the paradox, be sure to check it out!


validity-

Ugh. I'm reaallly sorry, I didn't see your post.

Anyway, F => T (False thing implying true thing) is totally different from ~R => W (if it's not raining, then the street is wet.)

All false things are contradictions. A better example, then, is "If it is raining AND NOT raining, then the street is wet." Since I just showed that from any from any contradiction we can derive anything we want, then it is true that "if it is raining and not raining, then the street is wet".

Anyway, with regards to your solution:

The sentence isn't equivalent to "if the moon is made of cheese, then this sentence is true."

I'm afraid I'm not understanding your point, but I'd love to help.
You're right in a sense: the truth of the statement comes from the self-reference. The paradox is seen in the "If this sentence is true, then the moon is made of cheese" bit, because I proved a contradiction. However, the paradox is totally different from "If it's raining and not raining, then the moon is made of cheese"- that isn't a paradox at all, since it just shows the problem we have if there's a true false thing. Since there aren't any true false things, we don't have a paradox... except for curry's case...


Once again, I'm really sorry I didn't see your reply... you know how it is.

btw: zenetic: awesome- the Fibonacci sequence... and You thought I wouldn't notice
 
Zetetic11235
 
Reply Sun 7 Dec, 2008 09:02 pm
@jknilinux,
You seem to be saying that self-reference should be disallowed. That works for curry's paradox, but not for the strengthened liar and many other paradoxes.

Not at all! I'm simply stating that predicate logic is wholly inadequate for this sort of problem. Nothing can be disallowed from consideration, but I think that you are using a spoon for a jackhammer.

The suggested proofs on the stanford page are unfortunately over my head. I am not well versed in truth theory or any any of the other logics aside from standard propositional/predicate logic. I just haven't had the time to delve into them. Hopefully I'll be taking some courses on mathematical logic at some point, and hopefully that will give me some insight. After all, as I said, Godel's theorems tie in heavily to self reference.

And yes, I know P*~P is always false, which is why it's so bad when we have something that says it's true, like a paradox.

The paradox doesn't say that anything known to be false is true, but rather shows that the system itself is flawed, or at least inappropriate for the question we are asking. That aside, I don't see how the paradox shows that p^~p is true. I also wanted to point out that the fact that a false statement implies any other statement doesn't mean that it implies that said statement is true, just that the implication is true. e.g.

p^~p=> the moon is made of gouda cheese is tautological, but the implication doesn't say anything about the nature of the moon. Tautology doesn't necessitate anything but the truth value of the sentence.

That is not to say that you were not aware of that, but I think that you did not make it sufficiently clear.


About the fibbonacci sequence: I kinda just stuck it on there, and yes, most people don't seem to notice. Its always good to see another who recognizes it. Its an interesting sequence.
 
jknilinux
 
Reply Sun 7 Dec, 2008 10:55 pm
@Zetetic11235,
zeteticphi:

you said:

"There is an embedded infinite loop... This cannot be evaluated by basic predicate logic."

Since infinite loops are a result only of self-reference, it seemed to me that you were saying something about self reference. Perhaps you could explain your viewpoint in greater detail.

Also, a paradox is defined as an apparently sound statement that proves P*~P.
Curry's paradox is a paradox because it can be defined as x : "x => P*~P", from which we get P*~P.

And sorry I didn't make myself clear on that, but yes, I know.


btw, Did anyone else here notice it?
 
 

 
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