If this sentence is true....

[Qfwfq]

If this sentence is true, then Planet Earth exploded on December 31st, 1999.

 

:confused:

 

So, must we conclude that it exploded?

[Boerseun]

Er... uhm, like... you know, ah...

 

What sentence??? :confused::confused:

[Jet2]

sentece? or sentence?

[InfiniteNow]

If this sentece is true, then Planet Earth exploded on December 31st, 1999.

 

:)

 

So, must we conclude that it exploded?

 

Wouldn't it depend on which calendar you used? I'd guess... No... we cannot logically conclude that without additional information.

[Pyrotex]

Qfwfq is referring to the sentence, "If this sentence

("If this sentence

("If this sentence

("If this sentence

("If this sentence

("If this sentence

( ... )

is true, then Planet Earth exploded on December 31st, 1999")

is true, then Planet Earth exploded on December 31st, 1999")

is true, then Planet Earth exploded on December 31st, 1999")

is true, then Planet Earth exploded on December 31st, 1999")

is true, then Planet Earth exploded on December 31st, 1999")

is true, then Planet Earth exploded on December 31st, 1999.

[Qfwfq]

What sentence???

See Pyro's clever answer...:confused:

 

Wouldn't it depend on which calendar you used?

No, it doesn't. If you want, let's make it:

 

"If this sentence is true, then Planet Earth exploded when Neil Armstrong set foot on the Moon."

 

:) I just knew some Clever Dick would nitpick on the arbitrarity of dates! :doh:

 

Anyway it's easy to argue that the implication is true. The sentence is the implication, and therefore is true.

This makes the implicant true and, so... therefore......:phones:

[Pyrotex]

See Pyro's clever answer...:hihi:...Anyway it's easy to argue that the implication is true. The sentence is the implication...

Okay, mister smart guy.

 

The sentence is "true" only in the sense that IF the sentence is true, then Earth exploded. The sentence contains two (2) conditional paths, and if either of them is true, then the sentence is true. Only the False path is True and therefore the Earth did not explode. And therefore, the sentence has one True path and the sentence is "true".

 

The sentence is "true" only in the sense that IF the sentence is true, then Earth exploded. The sentence contains two (2) conditional paths, and if either of them is true, then the sentence is true. Only the False path is True and therefore the Earth did not explode. And therefore, the sentence has one True path and the sentence is "true".

 

The sentence is "true" only in the sense that IF the sentence is true, then Earth exploded. The sentence contains two (2) conditional paths, and if either of them is true, then the sentence is true. Only the False path is True and therefore the Earth did not explode. And therefore, the sentence has one True path and the sentence is "true".

 

The sentence is "true" only in the sense that ....

 

Uhhh... where was I...??? :shrug:

[CraigD]

If this sentence is true, then Planet Earth exploded on December 31st, 1999.

This is two logical steps from the canonic form of the famous “liar paradox”, so I’d call it a variation.

 

Formally (“=” means “is”, “->” “implies”, and “'” “not”) , we have given

A = A -> B (“If this sentence is true, then …”)

'B (“Planet Earth did not explode on 12/31/1999”)

 

Algebraically, taking the contrapositive, and the given ‘B, it follows

A = 'B -> 'A (“If Earth didn’t explode on 12/31/1999, this sentence is false”)

A = ‘A (“This sentence is false”)

 

As the first wikipedia link describes, this is an old and consequential paradox, and a large reason that many generations of mathematicians have felt something between profound unease and vehement hatred for self reference.

[Buffy]

My name is Yon Yonson I work in Wisconsin, I work in a lumber mill there. The people I meet as I walk down the street, they say "What's your name?" And I say,

"My name is Yon Yonson I work in Wisconsin, I work in a lumber mill there. The people I meet as I walk down the street, they say 'What's your name?' And I say,

'My name is Yon Yonson I work in Wisconsin, I work in a lumber mill there. The people I meet as I walk down the street, they say "What's your name?" And I say, ......

 

Ad nauseum, :)

Buffy

[Qfwfq]

Well, Craig is coming the closest, but not quite and I still haven't gotten over my flu.:naughty: It isn't really like the liar's paradox, although there's self reference in both cases. The gross difference is that, in this case, it isn't impossible to assign true or false; the argument I outlined concludes TRUE.

 

"If this sentence is true, then

Claudia Schiffer

is mine, all mine!"

 

Don't worry about Pyro's infinite regress Buffy, it isn't really necessary! Just identify all the copies because they are one and the same.

[LaurieAG]

"If this sentence is true, then Planet Earth exploded when Neil Armstrong set foot on the Moon."

 

Hi Qfwfq,

 

If Planet Earth exploded when Neil Armstrong set foot on the Moon then the sentence is true otherwise the sentence is false.

[jedaisoul]

If this sentence is true, then Planet Earth exploded on December 31st, 1999. So, must we conclude that it exploded?

No. The statement is simply false. Perhaps I'm thick, but to me, this is trivial.

[CraigD]

"If this sentence is true, then Planet Earth exploded when Neil Armstrong set foot on the Moon."

If Planet Earth exploded when Neil Armstrong set foot on the Moon then the sentence is true otherwise the sentence is false.

In standard logical terms, Laurie’s statement does not follow from Qfwfq’s.

 

What Laurie appears to be suggesting is that if we have given as true

A -> B (“A implies B”)

Then

B -> A (“B implies A”)

must also be true

 

These statements are known as converses of one another, and are not both guaranteed to be true under the usual rules of logic.

 

They aren’t true for common-sense examples, either. For example

If Alice removed the book from the shelf, the book is no longer on the shelf.

does not guarantee

If the book is no longer on the shelf, Alice removed the book from the shelf.

(Perhaps Bob removed it.)

 

What does follow in all cases from

A -> B

is

‘B -> ‘A (“not B implies not A”)

 

These statements are known as contrapositives of one another.

 

Taking the original sentence as true

If this sentence is true, then Planet Earth exploded when Neil Armstrong set foot on the Moon.

The contrapositive, which must also be true, is

If Planet Earth

did not

explode when Neil Armstrong set foot on the Moon, then this sentence

is not

true.

 

Since “Planet Earth did not explode when Neil Armstrong set foot on the Moon” can be assumed to be true (please, moon landing hoax conspiracy theorists, don’t respond :shrug:), The sentence can be evaluated (simplified) to

This sentence is not true.

which is the liar paradox in its canonic form. Therefore the original statement is logically equivalent to the liar paradox.

 

I maintain that both this and the original statement

If this sentence is true, then Planet Earth exploded on December 31st, 1999.

are logically equivalent to the liar paradox.

[jedaisoul]

I maintain that both this and the original statement

If this sentence is true, then Planet Earth exploded on December 31st, 1999.

are logically equivalent to the liar paradox.

Well thanks for that. However, I disagree. The liar's paradox "this statement is a lie", is not conditional. The two statements you refer to are conditional, and are simply untrue. There is no paradox in them, as there is in the liar's paradox.

[Pyrotex]

...I maintain that both this and the original statement

If this sentence is true, then Planet Earth exploded on December 31st, 1999.

are logically equivalent to the liar paradox.

Excellent proof, Craig. Cool beans and smooth cheese! :)

I like how the contrapositive allows you to "drop the conditional", as it is, in fact, consistent with reality. If the sky is obviously blue, then starting a sentence with "if the sky is blue" is rather redundant.

 

Are you really from Crete? :) :hihi:

[CraigD]

The liar's paradox "this statement is a lie", is not conditional. The two statements you refer to [eg: If (1:this sentence is true), then (2:Planet Earth exploded on December 31st, 1999)] are conditional, and are simply untrue. There is no paradox in them, as there is in the liar's paradox.

If the truth value of statement #2 were unknown, or know to be true, the paradox would be possibly or certainly avoided. However, statement #3 (“Planet Earth exploded on December 31st, 1999”) is known to be false. “Earth did not explode on 12/31/1999” is thus true. This allows the equivalent statement “If Earth did not explode on 12/31/199, this sentence is false” to be simplified to “This sentence is false”, which is the Liar Paradox.

 

In short and in general, a conditional (“if A then B”) can be reduced to a constant truth value if A is know to be constant and true (in which case it reduces to “B”), or B is know to be constant and false (in which case it reduces to “not A”).

 

Here’s one of my previous logical algebraic manipulations, with words, not punctuation

1. If A then B (if this sentence is true, then Earth exploded on 12/31/1999)
   
2. If not B then not A (if Earth did not explode on 12/31/1999 then this sentence is false)
   
3. Not B (Earth did not explode on 12/31/1999)
   
4. Not A (this sentence if false)
   

# 1 and 3 are givens. # 2 and 4 follow from the usual rules of formal logic.

 

Put tersely:

If ((If A then :) and not :hihi: then not A

 

There’s no logical inconsistency in this statement (it’s true for all TRUE/FALSE values of A and :) until we introduce the self-referential nature of A:

If A then (If A then :)

 

We than have a paradox, because:

• If A is true and B is false, then A is false, contradicting the given, A is true.
  
• If A is false and B is false, then A is true (“if FALSE then FALSE” is true), contradicting the given, A is false.
  

The only resolution of the paradox is for B to be true, in which case A can be either true of false, and “if A then B” remains true in either case.

[jedaisoul]

If the truth value of statement #2 were unknown, or know to be true, the paradox would be possibly or certainly avoided. However, statement #3 (“Planet Earth exploded on December 31st, 1999”) is known to be false. “Earth did not explode on 12/31/1999” is thus true. This allows the equivalent statement “If Earth did not explode on 12/31/199, this sentence is false” to be simplified to “This sentence is false”, which is the Liar Paradox.

Perhaps I can explain why I disagree...

 

The liar's paradox "This statement is false" can be expanded to:

"If this statement is true then this statement is false".

 

The paradox arises because the satement cannot be both true and false. I.e. the statement is of the form "If A then not A".

 

Whereas the statement:

If this sentence is true, then Planet Earth exploded on December 31st, 1999.

is of the form "If A then A". It allows:

a) If A is true then A is true.

B) If A is false then A is false.

So that statement is not contradictory. It may be true, or false, but not paradoxical.

 

However, if Q had said:

"If this sentence is false, then Planet Earth exploded on December 31st, 1999."

Then there would be a conflict, because it is in the form "If not A then A".

 

Interestingly we can modify the the liar's paradox as follows:

"If I am a liar then I would say that I am not a liar".

In this case there is no conflict because it is quite possible that a liar would say that he was not a liar, because he was lying.

 

On the other hand, if he said:

"If I were an honest man, I would say that I am a liar".

Then, again, there is a conflict. If he were an honest man, then he would not be a liar in the first place. Therefore, the statement "I would say that I am a liar" would be false. The statement would only be true if he is a liar, in which case he is not an honest man.

 

I'm not sure how you analyse these algebraically to reveal the difference, but there clearly is one.