Curry's paradox

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Reply Tue 18 Nov, 2008 03:18 am
Hi everyone,

I was wondering if anyone here had any idea on how to solve Curry's paradox. It's basically the statement x defined as "x implies y". If it's false, then x is true, so x is true. If x is true, then y is true, so everything is true.

GAAA!!! IS MODUS PONENS WRONG?
 
validity
 
Reply Wed 19 Nov, 2008 01:26 am
@jknilinux,
I have never heard of Curry's Paradox. Honestly, I am a little unsure of the paradox.

Using an example from Wikipedia

If this sentence is true, then Santa Claus exists.

Is the paradox that you can not argue against the existence of Santa Claus because the first part of the sentence is true? But I do not see any truth in the sentence to show it is.
 
jknilinux
 
Reply Wed 19 Nov, 2008 02:10 am
@validity,
Sorry, here's an explanation...

Take the statement x:"x => y", (aka x, defined as "x implies y", aka the statement "if me, then y")
The only way implication, aka if...then, can be false, is if the "if" part is true and the "then" part is false. For example:

If it's raining, then the street is wet (AKA R => W). This can only be false if it's raining and the street is not wet. So, it can only be false if R is true and W is false.

So, back to x. If x is false, then what it means must be false, so it is false that "x => y". This can only be false if x is true and y is false. So x is true. However, we just assumed x is false! So, if x is false, then x is true and false, so x cannot be false. So, x must be true...

If x is true, then what x means must be true. So, x is true, if x then y is true, so y must be true.

Uh, one problem...

What was y? Here's the awful truth: y was anything. :devilish: You can put anything in y: "I exist", "I don't exist", "I both exist AND don't exist"... I can prove anything with this!
 
xris
 
Reply Wed 19 Nov, 2008 06:23 am
@jknilinux,
It only looks like a game of words rather than a philosophical statement and even less a mathmatical equation that makes sense...I suppose its a good bit of impressionable nonsense that philosophers use to encourage the lay thinker his as thick as a plank..
 
jknilinux
 
Reply Wed 19 Nov, 2008 01:12 pm
@jknilinux,
xris-

Do you mean you think it can't be represented in formal logic? Watch me:
x:"x => y"
 
xris
 
Reply Wed 19 Nov, 2008 01:21 pm
@jknilinux,
sorry but with my little knowledge all it tells me the unknown x is equal to something greater than the unknown y...yes?
 
jknilinux
 
Reply Wed 19 Nov, 2008 01:41 pm
@jknilinux,
Oh, sorry... I guess my ascii art isn't too good. The => is an implication.
Rewritten:

The statement x is defined as "if x then y".

-or-

If this statement, then y.
 
validity
 
Reply Wed 19 Nov, 2008 02:31 pm
@jknilinux,
To me, it appears that there is something wrong with allowing x to imply all y. I do not think you can ignore some kind of correlation or causality relationship between x and y.

In the example: If it's raining, then the street is wet, is not necessarily true in all cases eg Virga - Wikipedia, the free encyclopedia nor is it necessarily possible that one drop of rain reaching the street means the entire street as wet.

I ponder more...
 
xris
 
Reply Wed 19 Nov, 2008 02:42 pm
@validity,
I think i was right the first time...
 
jknilinux
 
Reply Wed 19 Nov, 2008 03:50 pm
@jknilinux,
validity-

If virga occurs, then "if it's raining, the street is wet" is no longer true. This means it must be raining, and the street is not wet, aka virga.
 
validity
 
Reply Wed 19 Nov, 2008 05:26 pm
@jknilinux,
xris wrote:
I think i was right the first time...


xris, please accept my apology for not acknowledging your post prior to making mine, as looking back it may seem as though I was ignoring what clearly is a similar arguement to mine. Smile

jknilinux wrote:
validity- if virga occurs, then "if it's raining, the street is wet" is no longer true.


Then I still do not see a paradox, if the "if x then y" is not robust to all situations and is situation dependant ie sometimes true and sometimes not true, then what value is it to say if "x then y" if it is only true some of the time.
 
jknilinux
 
Reply Wed 19 Nov, 2008 06:07 pm
@validity,
Saying "if x then y" is simply contingent- you're right. But the self-referential statement x defined as "if x then y" must be tautologous, because the only way x can be false is if x is true and y is false- contradiction. So, x must be true.

xris- sorry, I may have misunderstood you. Could you explain?
 
jknilinux
 
Reply Fri 21 Nov, 2008 02:18 pm
@jknilinux,
So, are you guys just giving up?

validity- If you still disagree, look up Modus Ponens.
xris- Like I said, no matter what, it can be represented in Predicate logic.
 
xris
 
Reply Fri 21 Nov, 2008 02:46 pm
@jknilinux,
jknilinux wrote:
So, are you guys just giving up?

validity- If you still disagree, look up Modus Ponens.
xris- Like I said, no matter what, it can be represented in Predicate logic.
For what ive seen its not very logical, just using unknowns to create a false concept of reality...
 
jknilinux
 
Reply Fri 21 Nov, 2008 03:09 pm
@xris,
There's no unknown here.

It does require an understanding of modus ponens, though.
 
validity
 
Reply Fri 21 Nov, 2008 03:21 pm
@jknilinux,
jknilinux wrote:
So, are you guys just giving up?

validity- If you still disagree, look up Modus Ponens.


I'm not giving up, more trying to understand why it is a paradox. I still do not see the paradox. I will read up on Modus Ponens.

jknilinux wrote:
Saying "if x then y" is simply contingent- you're right. But the self-referential statement x defined as "if x then y" must be tautologous, because the only way x can be false is if x is true and y is false- contradiction. So, x must be true.


x being defined as "if x then y" I understand, I make use of my correlation or causality reference. But there needs to be an establishment that x and y are related for the "then" bit to work. It seems to me to be ridiculous to suggest that if x is unrelated to y, then x implies y.

jknilinux wrote:
If virga occurs, then "if it's raining, the street is wet" is no longer true. This means it must be raining, and the street is not wet, aka virga.


Correct. But that is because virga is defined as rain that does not reach the ground. The logic only seems to be justified if x is defined as or causes y, ie "if x then y" is true. There is no justification for "if x then y" if x and y are un-related.
 
jknilinux
 
Reply Fri 21 Nov, 2008 03:33 pm
@jknilinux,
There doesn't need to be a relation at all. If something's false, then it implies everything.
Like "it's raining and it's not raining, therefore the moon is made of gouda cheese". This is actually true.

I'm honestly a bit surprised that people in the logic forum don't know what implication or modus ponens means. Please read up on it before you respond.

I'm really not trying to be harsh, just making a recommendation.

Curry's paradox, specifically, requires a bit of knowledge of metalogic and Tarski's work in metalinguistics, plus a good grasp of predicate logic in order to understand it.
 
xris
 
Reply Fri 21 Nov, 2008 03:40 pm
@jknilinux,
jknilinux wrote:
There doesn't need to be a relation at all. If something's false, then it implies everything.
Like "it's raining and it's not raining, therefore the moon is made of gouda cheese". This is actually true.

I'm honestly a bit surprised that people in the logic forum don't know what implication or modus ponens means. Please read up on it before you respond.

I'm really not trying to be harsh, just making a recommendation.

Curry's paradox, specifically, requires a bit of knowledge of metalogic and Tarski's work in metalinguistics, plus a good grasp of predicate logic in order to understand it.
To be honest i look at the results of a supposed revelation in philosophical formula and if it results in wet roads when it aint raining and the moon is made of blue cheese im not really interested..
 
validity
 
Reply Fri 21 Nov, 2008 10:35 pm
@jknilinux,
jknilinux wrote:
There doesn't need to be a relation at all. If something's false, then it implies everything.
Like "it's raining and it's not raining, therefore the moon is made of gouda cheese". This is actually true.

I'm honestly a bit surprised that people in the logic forum don't know what implication or modus ponens means. Please read up on it before you respond.

I'm really not trying to be harsh, just making a recommendation.

Curry's paradox, specifically, requires a bit of knowledge of metalogic and Tarski's work in metalinguistics, plus a good grasp of predicate logic in order to understand it.


Thank you for your recommendation. However, I do not know how much you recommend I should read before I respond, so I do hope 30 minutes is enough.

Could you please show me, in a step wise fashion, with consideration to my lack of knowledge in logic symbols (may I recommend words to begin with) how the statement "if it is raining then the moon is made of gouda cheese" is determined to be in fact true. I ask as I still see the paradox as being caused by an error in assigning truth to y by the truth of x ie if x is true then y is true, when there has been no establishment of the relation of y to x.
 
jknilinux
 
Reply Sat 22 Nov, 2008 01:20 am
@jknilinux,
xris-

If you don't understand something, be quiet.

Validity-

Thank you for wanting to know more. Your lack of ignorance is the only reason why I'm bothering to write this.
In my post, I actually said "If it is raining and it is not raining, then the moon is made of gouda cheese". 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.

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

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.
 
 

 
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