If this sentence is true....

[Qfwfq]

Well, I must say I disagree with all three of you. Curry's paradox is not logically equivalent to the liar's paradox, for one thing Craig it's negation-free.

 

If this sentence is true, then sausages are good.

 

Whatever assert one places after 'then' can be argued consequentially true by applying the inference rules of formal logic, mainly using modus ponens. There is contradiction only when the assert is one known to be false, but of course this is formally an extra axiom and not something consequential. Without an axiom "sausages are not good", there is no logical contradiction at all.

 

Rather than the liar's paradox, it is more like the fact that admitting one contradiction being possible (supposing both A and not-A as axioms), it can be argued that any assert is true. The difference is that Curry's argument doesn't need to suppose any axiom at all; the trouble lies in that kind of structure itself, formally implying:

 

A <=> (A => B)

 

This gives the bootstrap effect from which B can be deduced without any further assumption. It isn't necessary to suppose A being true as an axiom, or even the implication; the co-implication gives the bootstrap effect.

 

The main question is: Is it sufficient (and necessary) to eliminate self-reference (as by Tarski's prescriptions) in order to avoid the problem?

[Kriminal99]

Lol @ fake paradoxes... not that any paradoxes are real...

 

If Earth didn't blow up, then the statement is false.

 

The original sentence allows that even "if this statement is false" the earth still could have blown up. This is irrelevant.

 

But it says that "if this statement is true" it blew up for certain. Therefore if the earth didn't blow up the statement is simply false...

 

As for self reference regarding truth... Even the real paradox isn't really a paradox so much as it is trying to hold two different definitions of a word at the same time.

 

The minimalist definition of truth (context in which an idea of truth arises) does not in any way allow a statement to dictate it's own truth. Rather truth is a parameter that is determined through external means .

 

Consider these other so called paradoxes.

 

An apple is true. True or false? (Apples are not true or false)

A banana is false. True or false? (Bannanas are not true or false)

An apple is red. True or false? (Which apple? Not enough info for anyone to ever determine if it is true or false therefore it is neither)

This statement is true. (What statement? True or False cannot take this as input because it has no claims about external things that can be verified or cause contradictions.)

[jedaisoul]

Lol @ fake paradoxes... not that any paradoxes are real...

See below...

If Earth didn't blow up, then the statement is false.

I agree.

As for self reference regarding truth... Even the real paradox isn't really a paradox so much as it is trying to hold two different definitions of a word at the same time.

I agree that apparent contradictions often arise due to the use of different definitions. But once the definition is agreed, those contradictions are resolved. That is not a paradox. However, self referencing statements are paradoxical when they contain a self negation: A = not A.

Consider these other so called paradoxes.

An apple is true. True or false?

A banana is false. True or false?

An apple is red. True or false?

This statement is true.

None of these are paradoxes:

a) The first two are nonsensical.

:phones: The third is sometimes true and sometimes false, depending on the apple in question. That is not a paradox, it is a difference of fact.

c) The fourth is a truism. I.e. It is, by definition, true.

 

However, if you amend the last one to "This statement is false" then you have a paradox. This is because:

a) If the statement is true, then it is false.

B) If the statement is false, then it is true.

 

You do not need to be able to externally verify whether it is true or false. The contradiction is internal. You cannot resolve it as either true or false, as in the process it negates the option you chose.

[Qfwfq]

Ahem, the problem Krim is in the foundations of logic itself, I'm seeking a discussion on how the trouble could be avoided. Note that the sentence does not assert its own truth, it only refers to it, while determining an assert's truth through external means is not the concern of logicians such as Curry or Tarski. Tarski's approach is also by means of avoiding self-reference, I ask whether it is necessary and sufficient. Unfortunately the argument is rather subtle and folks here seem to have difficulty appreciating it.

 

I started this thread in a jocular manner but let's get down to business; although the Stanford article spells the argument out in detail it does it in a less simple way, especially with the list it uses, let me do it more plainly. First we must show the implication is true, which can be done by supposing:

 

Hyp: The sentence is true.

 

and showing that the implied clause follows as consequence. As the sentence is the implication, this is true by the hypothesis itself, therefore by modus ponens it does follow:

 

Th: Planet Earth exploded.

 

Now showing that the thesis follows from the hypothesis means the implication is true, not just under the hypothesis. This requires no assumption, it is sufficient that a sentence may talk about its own truth. Using again the fact that the sentence is the implication (the other way around) and modus ponens trivially completes the argument.

 

Logically, this is no self-contradiction. It only makes logic useless and of course implies contradiction indirectly, by the fact that the argument works for every implied clause (and hence for its negation as well).

 

If this sentence is true, then God exists.

 

If this sentence is true, then God does not exist.

 

Absolutely catastrophic, if the problem isn't somehow avoided.

 

An apple is true. True or false? (Apples are not true or false)

A banana is false. True or false? (Bannanas are not true or false)

I'd rather say meaningless because 'an apple' is not a well-formed assert (subject without predicate).

 

An apple is red. True or false? (Which apple? Not enough info for anyone to ever determine if it is true or false therefore it is neither)

Not every apple is red; not true and therefore false by tertium non datur. "A zebra has stripes" is true (according to experience) but the same would hold if at least one zebra had no stripes. From the logician's point of view, these are not determined without some axiom about apples or zebras.

 

This statement is true. (What statement? True or False cannot take this as input because it has no claims about external things that can be verified or cause contradictions.)

Free choice of axiom, without which it is not determined. Neither choice is inconsistent, a formal system may include either.

[Qfwfq]

I agree that apparent contradictions often arise due to the use of different definitions. But once the definition is agreed, those contradictions are resolved. That is not a paradox. However, self referencing statements are paradoxical when they contain a self negation: A = not A.

Er, nearly missed that, and I had forgotten to disagree about the use of different definitions. This is not a problem in Curry's paradox and note also that is contains no self-negation.

[jedaisoul]

I agree that apparent contradictions often arise due to the use of different definitions. But once the definition is agreed, those contradictions are resolved. That is not a paradox. However, self referencing statements are paradoxical when they contain a self negation: A = not A.

Er, nearly missed that, and I had forgotten to disagree about the use of different definitions. This is not a problem in Curry's paradox and note also that is contains no self-negation.

I'm not sure what your meaning is:

a) Are you saying that different definitions can give rise to paradoxes?

:phones: Are you disagreeing with my statement that A = not A gives rise to a paradox?

 

I agree that Curry's paradox is not self negating. I did not say that paradox only arises when statements are self negating. That is a different issue. However, I've read Curry's paradox, and so far I have been unable to determine in what sense it is paradoxical. So perhaps we are using different meanings for the word "paradox"?

[jedaisoul]

Consider the following list of sentences, named ‘The List’:

• Tasmanian devils have strong jaws.
  
• The second sentence on The List is circular.
  
• If the third sentence on The List is true, then every sentence is true.
  
• The List comprises exactly four sentences.

Although The List itself is not paradoxical, the third sentence (a conditional) is.

I cannot see why it is asserted that the third statement is paradoxical. The third statement is self-referencing, but that does not automatically make it paradoxical. It seems to me that:

a) The thirds statement is true if the other three are true.

:phones: It is false if any of the other statements are false.

 

The first statement may be interpreted as a generalisation, so it may be true even though an individual Tasmanian devil may not have strong jaws. Similarly, it may be assumed that it is referring to live Tasmanian devils. In that context, it is true. If, on the other hand it is taken to be axiomatic, then it is false. Dead Tasmanian devils do not have strong jaws.

 

The fourth statement is true.

 

The only problem I see is whether the second statement is circular? Merely stating that it is does not make it so. Therefore I deduce that the second statement is not circular, i.e. it is false. Therefore the third statement is also false, irrespective of whether the first statement is true or false. Where is the paradox????

 

I'd be interested to hear your explanation Q, as I find the Stanford article less than illuminating. I acknowledge that it may be caused by a failure of comprehension on my part, that's ok. I'd just like a pointer as to where I am going wrong???

[Qfwfq]

I only meant that those things aren't pertinent to Curry.

 

I'd be interested to hear your explanation Q, as I find the Stanford article less than illuminating. I acknowledge that it may be caused by a failure of comprehension on my part, that's ok. I'd just like a pointer as to where I am going wrong???

Did you read my post #21? I put it more simply than the Stanford article.

[ughaibu]

Qfwfq: Have you heard of Yablo's paradox? It uses neither conditionals nor self reference, you might find it interesting: http://www.mit.edu/~yablo/pwsr.pdf

[jedaisoul]

...First we must show the implication is true, which can be done by supposing:

 

Hyp: The sentence is true.

 

and showing that the implied clause follows as consequence. As the sentence is the implication, this is true by the hypothesis itself, therefore by modus ponens it does follow:

 

Th: Planet Earth exploded.

 

Now showing that the thesis follows from the hypothesis means the implication is true, not just under the hypothesis. This requires no assumption, it is sufficient that a sentence may talk about its own truth. Using again the fact that the sentence is the implication (the other way around) and modus ponens trivially completes the argument.

Ok Q I have read this, and I had not commented on it, as I did not wish to seem stupid. But here goes... Basically, I just do not get what you are saying here. Are you saying that "The Planet Earth exploded" must be true simply because it is preceded by a hypothesis "The sentence is true"?

 

You set out to "show... that the implied clause follows as consequence". That is something that, IMHO, you have not done. You have not shown any consequential relationship between the statements.

 

Why should the assertion that something is true (which is implicit anyway) mean that it is? "If this sentence is true, the Planet Earth exploded", is identical to the assertion "The planet Earth exploded". Saying "If this sentence is true" adds nothing. It is trivial.

 

Also, why do you assert that "showing that the thesis follows from the hypothesis means the implication is true, not just under the hypothesis"? Even if you had shown that the thesis follows from the hypothesis (which you have not done), I cannot see why it should then be true not just under the hypothesis? Why???

 

Logically, this is no self-contradiction. It only makes logic useless and of course implies contradiction indirectly, by the fact that the argument works for every implied clause (and hence for its negation as well).

 

If this sentence is true, then God exists.

 

If this sentence is true, then God does not exist.

 

Absolutely catastrophic, if the problem isn't somehow avoided.

Why is this catastrophic? Why does it make logic useless? I hope my comments do not offend, but I think there is something wrong if a trivial (but true) hypothesis means that any asertion following it must also be true. It is simply not meaningful to base an assertion about God, or anything else, on the hypothesis "this sentence is true". The reason is that, if you substitute the phrase "God exists" for the reference to it "this sentence", you arrive at "If God exists is true, then God exists", which is true, but trivial. It is a truism, as you have merely stated the same thing twice.

 

If your logic does not reflect this, then, IMHO, it is faulty. However, I expect that what you said was meaningful to you, which means that I just do not understand what you are saying. Can you express these thoughts in another form?

[ughaibu]

"If this sentence is true, the Planet Earth exploded", is identical to the assertion "The planet Earth exploded". Saying "If this sentence is true" adds nothing

Consider it in the form:

1) if I'm correct, the Earth has exploded

2) sentence 1 is true, therefore I'm correct

3) as I'm correct, the Earth has exploded

Why is this catastrophic? Why does it make logic useless?

It's catastrophic because one can prove anything, including, as Qfwfq demonstrated, direct contradictions.

[Qfwfq]

Thanks for the link Ughaibu, I hadn't come across that one! :hihi: It supports my suspicion of self referentiality not being the dragon to kill. I'll need more time to discuss further...

 

Consider it in the form:

1) if I'm correct, the Earth has exploded

2) sentence 1 is true, therefore I'm correct

The subtle point is in showing that the implication (1) is true, that's where Jedai is stuck:

Also, why do you assert that "showing that the thesis follows from the hypothesis means the implication is true, not just under the hypothesis"? Even if you had shown that the thesis follows from the hypothesis (which you have not done), I cannot see why it should then be true not just under the hypothesis? Why???

I think your trouble is from not distinguishing between the implication (A=> being true and the hypothethis (A) being true.

 

What I'm dicussing is a failure of formal logic, which most people attribute to self-reference. The aim of formal logic is to model what you call "showing a consequential relationship between statements" i. e. determining whether a statement is true on the basis of other facts; it models the process as a calculus for determining the value (true or false) of statements. It is clearly troubling if there is a way to apply formal logic which "calculates" any statement as true. Sorry if I can't make a better effort at the moment.

[freeztar]

Using ternary logic, couldn't we say the statement is unknown.

 

In other words:

A=unknown

B=false

 

Hence "not A" is unknown, even if we assign b a value of "true".

 

Yes/no?

[jedaisoul]

Consider it in the form:

1) if I'm correct, the Earth has exploded

2) sentence 1 is true, therefore I'm correct

3) as I'm correct, the Earth has explodedIt's catastrophic because one can prove anything, including, as Qfwfq demonstrated, direct contradictions.

Sentence 1) is only true if the Earth has exploded. As the Earth, demonstrably has not exploded, 2) is untrue. Therefore 3) does not follow.

 

You seem to be confusing the correctness of a conditional sentence "if I'm correct, the Earth has exploded" (which contains two statements) with the correctness of both statements. That is not necessarily so. Two falsehoods have the same effect as two truths on the validity of the overall sentence.

 

To explain, I'll reverse the statement, to be:

"The Earth has exploded, if I am correct"

 

So:

a) If the Earth has exploded then the statement "The Earth has exploded" is true. Therefore the statement "if I am correct" is also true. Thus the sentence "The Earth has exploded, if I am correct" is true. But...

 

:( If the Earth has not exploded, the statement "The Earth has exploded" is false. Therefore the statement "if I am correct" is also false. Thus the sentence "The Earth has exploded, if I am correct" is true because both are false, which is equivalent to stating:

"The Earth has not exploded, if I am incorrect (in stating that it has)".

 

Which is true.

 

So rule #1 is, the correctness of a compound sentence does not necesarily require its component parts to be correct. That must be established by other means (usually external fact).

[Qfwfq]

Anyway, :doh: I realized that Yablo's construct runs into difficulty in Tarski's framework. Each [imath]S_n[/imath] would have to be in the meta-language of the language of [imath]S_{n+1}[/imath], there could be no language L to start with but instead there would be the "meta-est" language. I'd say Tarski avoids Yablo's construct too.

 

Yes/no?

The argument on Curry's construct does not rely on tertium non datur, it would remain valid in terniary logic. It does not suppose anything, not even the sentence, it determines the value of this (as true) and hence of the implicant and hence of the implied clause.

 

Basically the bootstrap effect is due to identification of implicant and implication. Jedai appears to hold that, because of this, it is a truism and not a problem; at the same time he holds that the argument can't determine the implied clause being true, which is actually a rather sane opinion but still in conflict with the previous position. I still can't figure what your actual opinion is Jedaisoul...:confused: In any case the problem is to be solved by finding not a fault in the argument but instead what's wrong with the formalism.

 

When I first heard the matter from an old friend, and realized the bootstrap is due to identification of implication and implicant, my first thought was of perhaps tweaking modus ponens to exclude such cases and my friend agreed. It was only later I thought of the fact that this identification is caused by self-reference, which Tarski avoids as being the cause of the liar's paradox. In the light of this it may seem neat to go by Tarski and kill both birds with one stone, but I'm still not so sure. The self-reference in Curry's construct:

 

1) A = (A => B)

 

implies the co-implication:

 

2) A <=> (A => B)

 

which itself is enough to determine the vaues of (A => B), A and B as being all three true. There is a subtle semantic difference between the = and the <=> in the above two constructs, the second of which is not such a catastrofic problem because it has no consequence unless the <=> is supposed true (axiom), or demonstrated (which might be possible in some manner that depends on what A and B are). It does not follow from the mere existence of the construct.

 

While self-reference implies the existence of construct 1) which in turn implies 2), and this for any B, how sure can we be that no other mechanism exists that gives a construct [imath]{\cal S}(B)\equiv X\Leftrightarrow(X\Rightarrow B)[/imath] where X is some magic beast unrelated to self-reference, in which the co-implaction is true for any B? I don't know how this could work, but can it be proven there is no such thing? This was the point of the word 'sufficient' in my question last week:

The main question is: Is it sufficient (and necessary) to eliminate self-reference (as by Tarski's prescriptions) in order to avoid the problem?

while the word 'necessary' could perhaps be addressed by tweaking modus ponens as I hinted above. Something like:

 

If the implication A => B and the implicant A are both true,

independently of each other

, we conclude B is true.

 

Whaddya think? Any more logicians around?

[jedaisoul]

The subtle point is in showing that the implication (1) is true, that's where Jedai is stuck:I think your trouble is from not distinguishing between the implication (A=>:confused: being true and the hypothethis (A) being true.

 

What I'm dicussing is a failure of formal logic, which most people attribute to self-reference. The aim of formal logic is to model what you call "showing a consequential relationship between statements" i. e. determining whether a statement is true on the basis of other facts; it models the process as a calculus for determining the value (true or false) of statements. It is clearly troubling if there is a way to apply formal logic which "calculates" any statement as true. Sorry if I can't make a better effort at the moment. :doh:

I'd agree. See my comments to ughaibu above. If formal logic cannot distinguish between the validity of a compound sentence and the validity of the component statements then it is useless.

[Qfwfq]

Gosh Jedaisoul, we seem to have concurrency problems all the time, we should live on different sides of the world!:confused:

 

If formal logic cannot distinguish between the validity of a compound sentence and the validity of the component statements then it is useless.

No, sorry, the problem in logic isn't this, I meant that by not understanding the argument you apparently weren't making that distinction. The argument shows A => B is true by calculating B = true in the case A = true.

 

You seem to be confusing the correctness of a conditional sentence "if I'm correct, the Earth has exploded" (which contains two statements) with the correctness of both statements. That is not necessarily so. Two falsehoods have the same effect as two truths on the validity of the overall sentence.

No, Ughaibu is not confusing these, he simply didn't spell out the argument better.

 

Consider construct 2) in my post above, as an axiom rather than as a consequence of the self-referential construct 1) and regardless of what the devil A and B are. B can be calculated without further assumptions and is true; you need to suppose A = true only in the sub-calculation for (A => :doh: = true. This would not be disturbing because it follows from 2) as an axiom. What is disturbing is that it is consequential to the existence of statements of the form: "If this statement is true, then B."