Paradoxes

[MJ]

There is no paradox here. It is just a statement expressed with a lack of discipline characteristic to Domino’ random train of thought…

 

Here is the statement, again, for the sake of accuracy: “I am an Italian. All Italians are liars” (though traditionally this “paradox” is conveyed as “One Corsican said all Corsicans always lie.”)

 

There are two possibilities for the above referenced narration – at least in a logical framework where a statement is either false or true:

 

1. it is false

2. it is true.

 

If we assume that it is false (case #1), this implies that either the person is not Italian or there are Italians who are not liars.

 

If we assume that the statement is true (case #2), then the person is indeed an Italian and all Italians are liars, but then he must be liar too, so he must be either non Italian or at least one Italian must be a non-liar. Since this assumption (#2) leads to contradiction, that means the assumption is false.

 

Therefore we are left with a conclusion that the statement is false and therefore, again, either the person is not Italian or there is at least one Italian who is not a liar.

 

P.S. I just noticed that there may have already been a similar explanation given to the "paradox."

Edited December 6, 2003 by MJ

[Sip]

I agree with MJ although the fuzzy version may be a cooler explanation.

 

"Those who speak, know nothing." Is that a paradox?

[DominO123]

There is no paradox here. It is just a statement expressed with a lack of discipline characteristic to Domino’ random train of thought…

 

Here is the statement, again, for the sake of accuracy: “I am an Italian. All Italians are liars” (though traditionally this “paradox” is conveyed as “One Corsican said all Corsicans always lie.”)

 

There are two possibilities for the above referenced narration – at least in a logical framework where a statement is either false or true:

 

1. it is false

2. it is true.

 

If we assume that it is false (case #1), this implies that either the person is not Italian or there are Italians who are not liars.

 

If we assume that the statement is true (case #2), then the person is indeed an Italian and all Italians are liars, but then he must be liar too, so he must be either non Italian or at least one Italian must be a non-liar. Since this assumption (#2) leads to contradiction, that means the assumption is false.

 

Therefore we are left with a conclusion that the statement is false and therefore, again, either the person is not Italian or there is at least one Italian who is not a liar.

 

P.S. I just noticed that there may have already been a similar explanation given to the "paradox."

MJ, it is somehow pleasent that you say that the paradox is only a product of my random train of thought, when the answer you give is exactly the same as I gave previously when I did not understood really the question.

 

As I said above in one of my posts, Armat question was unclear(was not exactly the same as the paradox). I clarified this by providing the source(the work and a brain braking quotation). Before answering by rejecting an old paradox(recognised as a paradox for centuries), you must understand what is the question(which in your answer you show that you do not understand the implication, the same way I did not understand exactly Armat in his polls).

 

Here again, you can ignore me by considering that what I am going to write as a random thought.

 

 

 

I will post something here, and I hope you will realise by yourself that my "random thought" has nothing to do with this paradox.

 

 

 

Consider Statement A.

 

Statement A: "Statement A is not true."

 

Is Statement A true?

 

Statement A is not true. Argument 1 explains why.

 

Argument 1:

SUPPOSE Statement A is true.

Then the proposition that Statement A states is true.

But Statement A states that Statement A is not true.

So, Statement A is not true, contrary to our initial supposition.

So, IN FACT, Statement A is not true.

 

Unfortunately, Statement A can't be not true either.

 

Argument 2:

2.1 SUPPOSE Statement A is not true.

2.2. Then the proposition that Statement A states is not true.

2.3 But Statement A states that Statement A is not true.

2.4 So Statement A is not not true, contrary to our initial supposition.

2.5 So, IN FACT, Statement A is not not true.

 

Your mind should be blown. You should not be saying, "That's puzzling." You should be saying, "My mind is exploding."

 

Argument 1 proves that Statement A is not true. Argument 2 proves that Statement A is not not true. We have proved a contradiction, which is impossible.

 

A fundamental principle of logic is the Law of the Excluded Middle, which states that every statement is either true, or not true, and never both. It is just a straightforward consequence of the definition of the word "not." For any well-defined characteristic (say, "onchyness"), and for any thing (say, "Ralph") , if Ralph is not onchy, then...Ralph is not onchy!

 

However important you think this paradox is, it is more important. If contradictions are possible, then anything's possible.

 

So how can we resolve the paradox?

 

Inadmissible Statements

One approach is never to utter or think Statement A or statements like it. Then we never have to worry about the paradox.

It's easy to avoid Statement A, but how do we know what statements are similar enough to Statement A to cause a problem? Strategy Alpha will keep us safe.

 

Strategy Alpha: Never make any statement that refers to a statement.

 

That rules out Statement Alpha and other statements liable to cause the same problem.

 

Unfortunately, Strategy Alpha rules out many useful and harmless statements. For example:

Statement B: Two plus two is five.

Statement C: Statement B is not true.

According to Strategy Alpha, Statement C is inadmissible--we can't say Statement B is false. So Strategy Alpha is too stringent.

 

Statement A not only refers to a statement, it refers to itself, Statement A. Perhaps the root problem is this self reference.

 

Strategy Beta: Never make a statement that refers to itself.

 

Strategy Beta does rule out Statement A, but it suffers from the opposite problem to Strategy Alpha: it's too weak to prevent the paradox.

 

Statement D: Statement E is not true.

Statement E: Statement D is true.

 

Statement D can't be true.

Argument 3:

SUPPOSE that Statement D is true.

Then Statement E is not true.

Then Statement D is not true--which contradicts our original supposition.

So, IN FACT, Statement D is not true.

 

But Statement D can't be not true.

Argument 4: SUPPOSE that Statement D is not true.

Then Statement E is true.

Then Statement D is true--which contradicts our original supposition.

So, IN FACT, Statement D is not not true.

 

Again we face paradox. Hofstadter illustrates this version of the paradox with Escher's drawing of two hands drawing each other.

 

Preventing a statement from referring to any other statement is too strong. Preventing a statement from referring to itself is too weak--if we only prevent it from directly referring to itself. We need to rule out indirect self reference as well.

 

Strategy Gamma: Don't make any statement that refers to any statement that...that refers to any statement that refers to the original statement.

 

That'll do it.

 

But there are still two problems with Strategy Gamma.

 

To implement Strategy Gamma we have to keep careful track of all our statements to make sure that there are no chains of self reference. That vigilance could be irksome for any lengthy argument; but, if we ignore Strategy Gamma, we can't be sure we won't be led into contradiction.

Strategy Gamma only avoids the Epimenides Paradox. It is an emergency stopgap. What we ultimately want is to resolve the paradox. The Epimenides Paradox appears to prove that contradictions are possible, or that the Law of the Excluded Middle is false. But if that's true, then we can never be sure that avoiding the Paradox is enough to avoid paradox; all our arguments are suspect.

How can we resolve the paradox?

 

 

Meaninglessness,

or, "It depends what your definition of is is."

Statement F: Statement F is true.

Statement F doesn't lead us into any paradoxes. Supposing it's true only confirms that it's true. Supposing that it's not true only confirms that it's not true. But we still have the problem of decidng which it is--is Statement F true or not true?

 

It seems that there's no way to tell. How can that be? What extra information could possibly turn up that would help us to decide?! Since Statement F is only about Statement F, and we know Statement F, it seems we should be able to decide, right now and for sure, whether or not it's true. F is for Fishy.

 

Perhaps Statement F is meaningless: it doesn't express any proposition at all. Perhaps Statement A is meaningless. Perhaps that's the resolution of the paradox--Statement A is neither true nor not true, just meaningless.

 

To accomodate this possibility, from now on let's consider not "Statements" but "Sentences".

 

Sentence A: Sentence A is not true.

 

Unfortunately, it is not proper to say that Sentence A is neither true nor not true, but meaningless.

 

Sentence G: Zebras are white.

 

You might think Sentence G is both true and not true.

Argument 5:

Zebras are partially white; therefore Sentence G is true.

Zebras are only partially white; therefore Sentence G is not true.

Therefore, Sentence G is both true and not true.

 

We seem to have violated the Law of the Excluded Middle again.

 

Similarly, you might think that Sentence G is neither true nor not true. It's not true, because zebras are only partially white; it's not not true, because zebras are partially white.

 

Nevertheless, Sentence G does not pose a contradiction. The Law of the Excluded Middle says that Ralph is either onchy or not onchy; but it applies only if onchyness is well defined.

 

Sentence H: That suit is smart.

 

Argument 6:

The suit looks good; therefore Sentence H is true.

The suit does not have a high IQ; therefore Sentence H is not true.

Therefore, Sentence H is both true and not true.

 

Argument 6 doesn't overturn the Law of the Excluded Middle, since it uses two different meanings for the word "smart". If we specify that "smart" means "good-looking", then Sentence H is just true. If we specify that "smart" means "intelligent", then Sentence H is just not true.

 

Similarly, but more subtly, the seeming contradiction in Argument 5 stems from a shift in the definition of "white". If we define "white" to mean "completely white" (ignore the circularity, you know what I mean) then Sentence G is just not true. If we define "white" to mean "partially white" then Sentence G is just true. As long as we stick to a single definition, we don't face any contradiction.

 

Similarly, but more subtly, when we deal with Sentence A we must be careful about our definition of the word "true".

 

Sentence I: Green virtue swims under fastness.

 

Is Sentence I true or not true?

 

Definition i: "True" means "expressing a correct proposition".

 

According to Definition i, Sentence I is not true, since, being meaningless, it doesn't express any proposition at all.

 

Definition ii: "True" means "not expressing an incorrect proposition".

 

According to Definition ii, Sentence I is true, since it doesn't express an incorrect proposition, since it doesn't express any proposition at all.

 

Even though Sentence I is meaningless, we can still determine whether or not it's true, so long as we define carefully what we mean by "true".

 

Let us adopt Definition i, which seems more natural.

 

Recall that, in our attempt to resolve the Epimenides Paradox, we supposed that Sentence A is neither true nor not true, because it is meaningless. With Definition i in hand, however, we can see that "meaninglessness" is not a distinct third possibility. Rather, meaninglessness is a form of untruth.

 

If Sentence A is meaningless, then, using Definition i, Sentence A is not true. But we have seen that supposing that Sentence A is not true implies, via Argument 2, that Sentence A is not not true. Thus, our attempt to use meaninglessness to resolve our dilemma fails. The paradox lives.

 

Or does it?

 

 

 

Meaninglessness, Mark II

Let us from now on adopt Definition i of "true".

And suppose that Sentence A is meaningless.

 

Then Sentence A is not true.

 

Earlier, we saw in Argument 2 that supposing that Sentence A is not true implies that Sentence A is not not true, enveloping us in contradiction.

 

Argument 2 seems ironclad, but in fact it is flawed, as we can see now that we recognize the possibility of meaninglessness.

 

The flaw is step 2:

2.2 Then the proposition that Sentence A states is not true.

 

If Sentence A is meaningless, then Sentence A doesn't state any proposition at all--it's just a string of words. Therefore 2.2 doesn't hold, therefore Argument 2 doesn't hold, and therefore no contradiction arises.

 

To summarize.

 

Sentence A is meaningless.

Therefore, Sentence A is not true.

And Argument 2, which attempts to prove that Sentence A is also not not true, fails: Argument 2 refers to the proposition which is expressed by Sentence A, but Sentence A doesn't express any propositions.

Indeed, we can now prove that Sentence A is

not true, and

meaningless.

Argument 7:

7.1 Sentence A is either not true or not not true, but not both. (Law of the Excluded Middle)

7.2 Sentence A is not true. (By Argument 1. Do you see why Argument 1 still holds even though Argument 2 does not?)

7.3 If Sentence A is meaningful, then Sentence A is not not true (by Argument 2)--contradicting 7.2.

7.4 Therefore, Sentence A is meaningless.

Thus, we have resolved the Epimenides Paradox.

 

Or have we? In fact, meaninglessness embroils us in even worse confusion than before! See why?

 

 

Meaninglessness: Ugh

There are two problems with the meaninglessness approach.

First: Sentence A is weird, but it doesn't seem meaningless. It has a clear subject, a sentence. And it has a clear predicate adjective, untruth. And we know that truthfulness is just the kind of thing that we ordinarily talk about when we talk about sentences. So we have an independent argument that Sentence A is perfectly meaningful.

 

I merely mention this first problem, because the second problem is decisive anyway.

 

Namely: We have decided that Sentence A is meaingless and not true. I.e., we have concluded

i. Sentence A is a meaningless sentence.

ii. Sentence A is not true.

We think that i and ii are both true.

But expand i:

i'. "Sentence A is not true" is a meaningless sentence.

 

Return to Argument 7, which we had hoped had settled this whole matter; reexamine 7.2. Behold! It's Sentence A! But how can an argument employ a sentence that that same argument proves is meaningless!? It can't.

 

So what do we do now?

 

 

http://www.ocf.berkeley.edu/~sjblatt/notes...es/nottrue.html

[MJ]

MJ, it is somehow pleasent that you say that the paradox is only a product of my random train of thought, when the answer you give is exactly the same as I gave previously when I did not understood really the question.

 

As I said above in one of my posts, Armat question was unclear(was not exactly the same as the paradox). I clarified this by providing the source(the work and a brain braking quotation). Before answering by rejecting an old paradox(recognised as a paradox for centuries), you must understand what is the question(which in your answer you show that you do not understand the implication, the same way I did not understand exactly Armat in his polls).

 

Here again, you can ignore me by considering that what I am going to write as a random thought.

 

 

 

I will post something here, and I hope you will realise by yourself that my "random thought" has nothing to do with this paradox.

 

Huh?

Edited December 6, 2003 by MJ

[Sip]

Domino, here's the breakdown:

 

Your post:

 

2060 words

9961 characters

 

MJ's post:

 

221 words

998 characters

 

Do you see the factor of 10 there? I've got 2 words for you: Ockham’s razor.

[DominO123]

Huh?

 

MJ, read my first answers to the paradox, I answered like you did, but this was before I understood that Armat was presenting the Epimenides paradox(which the way he did it was not really).

 

Read what I posted(the quotation), you will see what is the problem here.

Edited December 6, 2003 by Fadix

[MJ]

Err...

[DominO123]

Err...

OK MJ, lets put that clearer for you and Sip, contrary to what Armat question imply, the guy is really Italian.

 

I wonder if this is why you guys claim it is not a paradox. If he is really Italian, could his statment that every Italians are liars be right?

 

This is what you should try to answer.

[MJ]

OK MJ, lets put that clearer for you and Sip, contrary to what Armat question imply, the guy is really Italian.

 

I wonder if this is why you guys claim it is not a paradox. If he is really Italian, could his statment that every Italians are liars be right?

 

This is what you should try to answer.

I guess this is another example of your earlier confession that your thinking is random in its charatcter - perhaps you called it statistical.

[DominO123]

Do I have to replace the statment A as the Italian, and the not true as a liar for you to unsertand? Or can you just not do that by yourself?

 

Consider Statement A.

 

Statement A: "Statement A is not true."

 

Is Statement A true?

 

Statement A is not true. Argument 1 explains why.

 

Argument 1:

SUPPOSE Statement A is true.

Then the proposition that Statement A states is true.

But Statement A states that Statement A is not true.

So, Statement A is not true, contrary to our initial supposition.

So, IN FACT, Statement A is not true.

 

Unfortunately, Statement A can't be not true either.

 

Argument 2:

2.1 SUPPOSE Statement A is not true.

2.2. Then the proposition that Statement A states is not true.

2.3 But Statement A states that Statement A is not true.

2.4 So Statement A is not not true, contrary to our initial supposition.

2.5 So, IN FACT, Statement A is not not true.

 

 

 

I may have a random thought, but you on the other hand don't have the humility to accept that you are not understanding what is the essence of the paradox presented, so you must claim it isen't one, like if people in all those centuries that considered it as a paradox were all dumb idiots.

Edited December 6, 2003 by Fadix

[Armat]

Sorry guys for possible intensity here. My initial statement was correct and straight from the book by Hofstadter.

Epimenides said, “I am a Cretan” “ all Cretan are liars”

So I just replased the Cretan for Italian. No confusion. Domino is right we should not debate whether he is Cretan, but he is Cretan! He says so. No speculations. The paradox is that this statement does not fall into typical false or true category. It is a loop!

[MJ]

I may have a random thought, but you on the other hand don't have the humility to accept that you are not understanding what is the essence of the paradox presented, so you must claim it isen't one, like if people in all those centuries that considered it as a paradox were all dumb idiots.

I cannot read what you write - alomost ever - it is just impossible to read. Therefore I cannot reply to your "arguments."

 

As far as the other people being dumb or smart is concerned, it is always a bad tone to claim credibility at someone else's expense.

Edited December 6, 2003 by MJ

[DominO123]

Sorry guys for possible intensity here. My initial statement was correct and straight from the book by Hofstadter.

Epimenides said, “I am a Cretan” “ all Cretan are liars”

So I just replased the Cretan for Italian. No confusion. Domino is right we should not debate whether he is Cretan, but he is Cretan! He says so. No speculations. The paradox is that this statement does not fall into typical false or true category. It is a loop!

Armat, the reason I said that you did not exactly refered to the paradox, is because in your sentence you forgot to mention that the first statment is an absolute truth, and Hofstadter demonstration is that the "word" is the "word" this kind of logic, where the ellement is the ellement, or the thing is the thing or whatever... (in the work it is clear the first statment is absolutly true).

 

 

MJ, I am not trying to claim credibility on someones else expense, obviously you do not understand, or better don't want to understand what the paradox is about. OK true, Armat first reference could be misunderstood, because we could question if he is an Italian, but after Armat clarified that his point was about the Epimenides paradox, I posted a quotation to refer to the problem of the paradox in question. A quotation that it is possible that you did not read, because in this quotation it refer clearly that the first proposition as an absolute truth. Like a word is a word(is it really, because from this paradox I can deny it being a word, and THIS IS what it implies, from it you can reject pratically everything.).

 

If my posts could not be understood, I quoted from someone else just after I knew what was exactly Armat allusion...

Edited December 6, 2003 by Fadix

[Armat]

Therefore we are left with a conclusion that the statement is false and therefore, again, either the person is not Italian or there is at least one Italian who is not a liar.

He is Italian! Hence he is also lying but at the same time he is telling the truth about his lying hence the loop. Since the statement cannot be logically correct meaning all Italians are liars then it is false but he is telling the truth about his lying. Loop again.

[DominO123]

(in the work it is clear the first statment is absolutly true).

OK! Armat, it isen't really, since from the paradox I could reject it after I consider it as absolute truth...

[MJ]

"Those who speak, know nothing."  Is that a paradox?

“Those who speak, know nothing.”

 

Let’s assume that it is true. Then the above speaker knows nothing, therefore he would not know what those who speak may know or not. Therefore, him/her being a speaker, that statement cannot be true. Thus we arrived to this contradiction because we assumed that the statement was true. Therefore, our assumption was false. Therefore, those who speak may know something. This is the only valid implication of the quoted statement with absolutely no built-in paradoxical structure.

Edited December 6, 2003 by MJ

[MJ]

Sorry guys for possible intensity here. My initial statement was correct and straight from the book by Hofstadter.

Epimenides said, “I am a Cretan” “ all Cretan are liars”

So I just replased the Cretan for Italian. No confusion. Domino is right we should not debate whether he is Cretan, but he is Cretan! He says so. No speculations. The paradox is that this statement does not fall into typical false or true category. It is a loop!

Armat, the reason I said that you did not exactly refered to the paradox, is because in your sentence you forgot to mention that the first statment is an absolute truth, and Hofstadter demonstration is that the "word" is the "word" this kind of logic, where the ellement is the ellement, or the thing is the thing or whatever... (in the work it is clear the first statment is absolutly true).

 

 

MJ, I am not trying to claim credibility on someones else expense, obviously you do not understand, or better don't want to understand what the paradox is about. OK true, Armat first reference could be misunderstood, because we could question if he is an Italian, but after Armat clarified that his point was about the Epimenides paradox, I posted a quotation to refer to the problem of the paradox in question. A quotation that it is possible that you did not read, because in this quotation it refer clearly that the first proposition as an absolute truth. Like a word is a word(is it really, because from this paradox I can deny it being a word, and THIS IS what it implies, from it you can reject pratically everything.).

 

If my posts could not be understood, I quoted from someone else just after I knew what was exactly Armat allusion...

Cut the BS, Domino.

[DominO123]

I give up.

[DominO123]

Sip, this for you, now don't tell me I am not clear, I have writen as short and clear as possible.

 

He is an Italian(FACT), this sentence is the truth, the thing is that the real story is about an Italian that declare that all Italians are liars(always lie). Him being an Italian is a given fact.

 

Now, he claim that all Italians are liars.

 

 

First possibility. Suppose that all Italians are liars.

 

This means that any claim that this Italian makes are lies.

Therefore his claim is not true.

His claim was: Every Italians are liars.

 

ANSWER: FALSE

 

------------------------------

 

Second possibility. Suppose that all Italians are not liars.

 

This means that him being a liar, would make a false claim.

Therefore his claim will not be true.

His claim was: Every Italians are liars.

But the problem here is that he admit that Italians are liars, so the claim not being truth(liars) shows that he is telling the truth by saying all Italians are liars(so admitting he is a liar).

 

ANSWER: TRUE

Edited December 7, 2003 by Fadix

[THOTH]

Bottom line...you can't believe everything you read....

[Sip]

Domino, MJ's assumption is "liar in strong sense". I also make that assumption ... which means liar always lies.

 

If he is telling truth (i.e. statement is TRUE), then he can't be Italian.

If he is telling lie (i.e. statement is FALSE), then it works.

 

So the conclusion is, he is lying and the statement is FALSE.

 

 

The question is not whether or not his claims can happen or not. The question is whether the statement is TRUE of FALSE in the logic sense. The only logical conclusion here is "FALSE". "I am itallian AND all italians lie" cannot be TRUE. But as TB pointed out, 1/2 of it can be true.

[Sip]

Second possibility. Suppose that all Italians are not liars

If all italians are not liars then by direct implication the statement "All italians are liars" is logically FALSE. ( "not p" implies p is FALSE)

[Armat]

"He is itallian AND all italians lie" cannot be TRUE.

Sip slow down reread the simple argument.

you got one right but he is telling the truth about his lieng hence true statement.It is a loop going in circle.

[DominO123]

OK Sip! Lets give another clearer picture this time... suppose that you Sip(YOU) SAY:

 

Every Sips are liars...

 

This is what you are not taking in account... the person is saying... you Sip is saying that all Sips are liars... you are Sip and you are a Sip.

 

Is your claim true?

 

 

First possibility. All Sips are liars.

 

Any claim that Sip makes are lies.

Therefore Sips claim is not true.

Sips claim was: Every Sips are liars.

Conclusion: The statment: Every Sips are liars is not true.

 

ANSWER: FALSE(not true)

 

------------------------------

 

Second possibility. All Sips are not liars.

 

Sips being a liar, makes false claims.

Therefore Sips claim will not be true.

Sips claim was: Every Sips are liars.

But the problem here is that Sip admit that Sips are liars, so the claim not being truth shows that Sip is telling the truth by saying all Sips are liars.

Conclusion: Sip makes a false claim, but he do admit that he makes false claims, because he admit Sips are liars.

 

ANSWER: TRUE(not not true)

Edited December 7, 2003 by Fadix

[Sip]

I should have been more precise in my assumptions:

 

1) I assume liar in the strong sense.

2) I assume a logical "AND" conjunction between the two sentences.

 

Either half can be true on it's own. We don't have enough information. But both together cannot be true. Thus, with an "AND", the entire statement has to be FALSE.