Curry's paradox

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validity
 
Reply Tue 9 Dec, 2008 02:15 am
@jknilinux,
jknilinux wrote:


validity-

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


Thats okay Smile

I am really confused with what is going on here... in this thread that is.

I think I need to stop and go back to square one and rethink this through. Give it some full attention. I thought I had something but it would seem I am confusing myself.

I need a lonely mountain to sit with.

Thank you jknilinux for your time. I will be back, hopefully with more sense.
 
jknilinux
 
Reply Tue 9 Dec, 2008 09:57 am
@jknilinux,
That link zetetic provided is a good resource, although it starts to get a bit over my head as well toward the end. The top might be a good resource for beginners, though.

And don't be afraid to take a stab at it, or explain your idea in greater detail so I can understand it. Sometimes all a problem needs is an outsider to question the basic assumptions that the old-timers made in the first place!
 
ACB
 
Reply Mon 29 Dec, 2008 05:47 pm
@jknilinux,
I propose the following additional rule of logic:

For any 'x' that can never be true, "if x then....." has no truth-value.

In other words, if it is logically impossible for x to be true, any statement beginning "if x then....." is incoherent and meaningless, so it is neither true nor false, so it is not true. That should solve the whole problem!
 
xris
 
Reply Tue 30 Dec, 2008 01:14 pm
@ACB,
Is this a vindaloo curry or a korma...if its x= vindaloo then its true you need a lager..if x = korma its not true you need a lager...but have one just in case..so x could always be true if y= lager...
 
noumenon
 
Reply Tue 30 Dec, 2008 08:41 pm
@jknilinux,
ACB-

All false things imply everything, via the principle of explosion. So it does have a truth-value- it's true.

However, jk recently told me the answer, and no, there's nothing wrong with logic...
 
ACB
 
Reply Wed 31 Dec, 2008 10:42 pm
@jknilinux,
I understand how, according to the accepted rules of logic, false things imply everything. What I do not understand is why the rules of logic are formulated in such a way as to permit this.

I am particularly puzzled as to why necessarily false statements (i.e. contradictions) are allowed to imply anything. Is not a contradiction completely meaningless, and therefore tantamount to gibberish? How, therefore, can it have any logical function?

Consider the following sentences:
1. If it is raining and not raining, then the moon is made of cheese.
2. If rain thinks hot memory squares, then the moon is made of cheese.
The first sentence is said to be true. Is the second sentence true also? If not, why not? 'It is raining and not raining' is no more meaningful than 'rain thinks hot memory squares'; both phrases are nonsense. (And if the second sentence is true, will any random sequence of words be OK?)

Why not just deny contradictions a truth-value? If they are regarded not as false but simply nonsensical, they cannot form part of any meaningful implication.
 
hdeutsch
 
Reply Sat 31 Jan, 2009 02:08 pm
@jknilinux,
I think Curry's Paradox might be formulated in a slightly different way than usual. Tell me if I'm wrong!

Let Impl(v, w) be a binary predicate added to Q (Robinson's arithmetic) with the associated axiom:

(I ) Impl([A], ) < - > A - > B.

Here [A] etc. is the Godel number of the sentence A.

Let Impl(v, [p]) be the formula that results from Impl(v, w) by replacing the variable w by the term [p] denoting the Godel number of p--a specific sentence letter of the object language. Then it would appear that the diagonal lemma will apply to yield a sentence C such that

(1) C < - > Impl([C], [p]).

Then one can use just the transitivity of implication, the rule of modus ponens, I, and 1 to prove p. Is this right?
 
Zetetic11235
 
Reply Sat 31 Jan, 2009 04:43 pm
@ACB,
ACB wrote:
I understand how, according to the accepted rules of logic, false things imply everything. What I do not understand is why the rules of logic are formulated in such a way as to permit this.

I am particularly puzzled as to why necessarily false statements (i.e. contradictions) are allowed to imply anything. Is not a contradiction completely meaningless, and therefore tantamount to gibberish? How, therefore, can it have any logical function?

Consider the following sentences:
1. If it is raining and not raining, then the moon is made of cheese.
2. If rain thinks hot memory squares, then the moon is made of cheese.
The first sentence is said to be true. Is the second sentence true also? If not, why not? 'It is raining and not raining' is no more meaningful than 'rain thinks hot memory squares'; both phrases are nonsense. (And if the second sentence is true, will any random sequence of words be OK?)

Why not just deny contradictions a truth-value? If they are regarded not as false but simply nonsensical, they cannot form part of any meaningful implication.


The sentence is true, the logical structure of the sentence has absolutely nothing to do with the meaning of it. Logic is a system similar to mathematics, in fact; it's more similar to mathematics than to natural language.

To better understand the If, then statement, understand that it can be built from or(v) and negation (-) thusly: p->q <=> -p v q since an 'if, then' is false only when the antecedent (p) is true, and the consequent (q) is false, so it is thought of from the standpoint of a promise which is kept in all other circumstances and not kept if you do what is asked of you and you don't get a reward.

So here denote A=go get me a chair; B=I'll give you a dollar. To set up the implication we have:
A -> B which in its English translation says If you get me a chair, then I will give you a dollar. Clearly this promise is not broken if you don't give him a chair, but he still gives you a dollar,which is the False implies True situation. The promise is ONLY broken if you give him a chair and he does not give you a dollar, so A ^ - B (A and Not B).

Remember that all of the logical laws (at least at the basic level we are talking about here) can be derived from disjunction (OR) and Negation (NOT). In actuality the reason that the english language has so little bearing on the Formal Language of logic, is that they have as much to do with each other as Mathematics and Shakespear. There is a whole separate branch of the study of Natural Language which is entirely different. The main applications of logic are in Computer Science and mathematics. Lingustics is not in the domain of logic, though there is some intersection.
 
ACB
 
Reply Sun 1 Feb, 2009 02:46 pm
@Zetetic11235,
Zetetic11235 wrote:
The sentence is true, the logical structure of the sentence has absolutely nothing to do with the meaning of it. Logic is a system similar to mathematics, in fact; it's more similar to mathematics than to natural language.


But logical statements depend on the meanings of the words that express them. In your example 'If you get me a chair, then I will give you a dollar', the words mean what they do by linguistic convention. The words or letters are not mathematical symbols like 1+1=2. The phrase 'get me a chair' has a certain meaning. But 'if you get me a chair and don't get me a chair' would be nonsensical, i.e. it would have no meaning.

What I am suggesting is that a sentence containing a meaningless conditional should be considered meaningless, period. It should be regarded as an improperly formed sentence, which could not be either true or false. I think it is generally agreed that an 'if A then B' sentence should be grammatical, e.g. both the A and B elements should contain (or at least imply) a verb, otherwise the sentence cannot form a logical statement. So why not extend this caveat so as to outlaw internal contradictions?

Surely the 'if' part of a promise must, at the very least, mean something. Otherwise, nothing has been promised.
 
Zetetic11235
 
Reply Sun 1 Feb, 2009 03:02 pm
@ACB,
Then you should go learn the difference between the study of syntax and the study of semantics and read this: Relevance Logic (Stanford Encyclopedia of Philosophy)
 
odenskrigare
 
Reply Thu 26 Feb, 2009 02:38 pm
@jknilinux,
jknilinux wrote:
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?


Hm

So...

If 'x -> y' is false then x = true and y = false, necessarily. Those two values are the only two for which an implication can be false.

Which means the implication x = x -> y itself is true Laughing

Well what happens if you expand x?

x -> y
(x -> y) -> y
((x -> y) -> y) -> y

etc.

Doesn't the infinite recursion get in the way somehow? I'm not a graduate math or philosophy student, but I can see how that might present an issue.
 
 

 
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