Paradoxes

[Alliteration]

How would you interpret the following statements? Are they true? False? Neither/both?




"This sentence is false."

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"The following sentence is true."
"The previous sentence is false."

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"Everything I say is a lie."

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"Albany is the capital of New York, but I do not believe that."

[tBone]

Meh, they are logical paradoxes

Except for the last one, which is basically saying "I acknowledge that the dictionary says A, but I believe B anyways."

[Bandit224]

Wheatley (Portal 2 robot) says the answer to all of them is "True", so I side with Wheatley

[DemonXia]

How would you interpret the following statements? Are they true? False? Neither/both?

"This sentence is false."

This makes this sentence "False", which means that this sentence is "True". The sentence then tells me that it is "False", which means it is "True". This creates an infinite loop of which you'll never get out.

"The following sentence is true."

This will make the following sentence "True". Moving on.

"The previous sentence is false."

This will make the previous sentence "False", which means that this sentence is also "False", so that makes the previous sentence "True", which will then tell me that this sentence is "True" making the previous sentence "False" again. Another infinite loop.

"Everything I say is a lie."

This tells me that everything you say is "False". So that makes this sentence "False", which then tells me that everything you say is "True". This sentence then tells me that everything you say is "False" and that again tells me that everything you say is "True". And another infinite loop.

"Albany is the capital of New York, but I do not believe that."

As a result it means that this sentence could be "True" or "False", because there is no sentence that will tell me if it is "True" or "False". It can not be both or neither as that is impossible.

I hope it's not to hard to follow.

[Evals]

How would you interpret the following statements? Are they true? False? Neither/both?

I think anybody with a decent sense of logic can argue the obvious case for each. Most of these, at least to me, are very basic paradoxes. There are far more interesting ones, in my opinion. This is especially true for higher-level math, but I've got the feeling most aren't qualified for that.

You might enjoy it: "If God can do anything, can he create a boulder which He cannot lift?"

[shell]

I always learn things around you, dude! *Smile*
I wasn't familiar with word paradox's, only the reference to them with time travel.

So, naturally.....Google and I became familiar once again. I think I shall marry Google next month...we after all, have intimate for some time now. *giggle*

Okay, so...here comes my two cents worth on this....

***

How would you interpret the following statements? Are they true? False? Neither/both?

"This sentence is false."

I will go with true on this one. My mind won't go into a loop with it, just goes one time. If the sentence is false, then that makes it true.

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"The following sentence is true."
"The previous sentence is false."

I will go with both on this. This one my mind does see the infinite loop, therefore making it both false and true.
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"Everything I say is a lie."

First I would say, stop lying, gosh darn it!! *lol*
I will go with true on this. Again, my mind stops and does not loop on it. If everything that is said is a lie, then what you just said was a lie, making it true.

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"Albany is the capital of New York, but I do not believe that."

First, because I didn't know, I checked to see if indeed Albany is the capital. *blush*
I would say that this is true. This is a statement of fact, but the writer of the fact doesn't believe the information to be true.

How'd I do teach....do I get an A? *giggle* What if I offer you an apple???



*wicked smile*

[Alliteration]

How'd I do teach....do I get an A? *giggle* What if I offer you an apple???

Everyone get an A, as the question I was asking was how YOU interpret them. I'll take the apple anyway, though...I don't particularly like apples, but I would like a better view of what's behind it!

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Anyway, here's my view. I think that the feature of these sentences that's most important is that they point out the difficulty in translating between natural languages and formal languages. For example:

"This sentence is false." might be translated into formal logic as something like this:

Code: Select all

Theorem S: ∀x(Cxy → ¬x)

Leading to this proof:

Code: Select all

1) ∀x(Cxy → ¬x)2) ∃!x(Cxy)C) ∃!x(¬x)

Which, when translated back into English, would look like this:

Theorem S: For all x, if x is a sentence with content y, then x is false.

1) For all x, if x is a sentence with content y, then x is false.
2) There exists exactly one x such that x is a sentence with content y.
C) Therefore, There exists exactly one x such that x is false.

Now, there may be other ways to translate this first sentence (and the others as well). But clearly, there's a "babelfish effect" - what happens here is what happens when you put an English sentence into an online translator, turn it into German, then into Japanese, then into Hungarian, then back into English. You don't get the same sentence you started with. Similar things apply for the other paradoxes I listed, however....the last one is a bit different.

"Albany is the capital of New York, but I do not believe that."

This one is more of an epistemic paradox. When someone says something like "pizza is delicious", or "Mexico is south of the U.S."; there's an implicit "I believe that..." tacked on to the front of the statement - they wouldn't say it if they didn't believe it. But in this case we have:

(I believe that) Albany is the capital of New York, but I do not believe that.

So, maybe it's the case that no one can sincerely utter this sentence?

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You might enjoy it: "If God can do anything, can he create a boulder which He cannot lift?"

I have a lot to say about this, but it will have to wait. Check back later :P

[whacker44]

Well the earlier examples seen are variants of situations that arise because of auto-referencing btw Godel's theorem says that are non-trivial logic systems will contain them ..Its interesting that
similar visal (see Esher's drawings, Penrose triangles..) and auditory sysrems (Bach's hmmm i am sick can't think of the title lol). They are undecidable !!

[Alliteration]

Well the earlier examples seen are variants of situations that arise because of auto-referencing btw Godel's theorem says that are non-trivial logic systems will contain them ..Its interesting that
similar visal (see Esher's drawings, Penrose triangles..) and auditory sysrems (Bach's hmmm i am sick can't think of the title lol). They are undecidable !!

If you're going to use Douglas Hofstadter's ideas, at least give him credit...

[whacker44]

actually i was teaching these ideas before..Godel< bach and Esher appeared..they were pretty common in AI circles....particulary amongst those involved with recursive sustems..howere the reference is good..i am sick with a terrible flu nd not writing an academic papae for this venue !!

[whacker44]

Btw my favorite paradox was always Russell's Paradox it seem to cause the moost head aches.Russell's Paradox (Stanford Encyclopedia of Philosophy)

[whacker44]

ooops not sure if its the flu or the use of inprivate browsing when i browse porn sites...but messed up th link
http://plato.stanford.edu/entries/russell-paradox/

[Alliteration]

You might enjoy it: "If God can do anything, can he create a boulder which He cannot lift?"

Alright...I'm going to dissect this apparent paradox several different ways; and I'm going to try to avoid most of the philosophical jargon, so everyone can understand.

First let's look at God. Can God do, literally, anything? Most people these days say no. For example, God can't make 2+2=5, he can't cause a flood that was not caused by him, and he can't create a square circle. I don't like this answer, though, as it says basically "God cannot create true contradictions". The problem is that *I* can create true contradictions, using paraconsistent logic. A better thing to say would be something like "God cannot create true contradictions in a system which does not allow them".

That brings us to a second point...is there anything else which might limit what God can do? The answer is yes...his own nature. Yes, God is omnipotent, but there's also other things about him. He's also perfectly good - and this means he can't do something immoral (such as torture us all for shits and giggles). The point here is that yes, there's something he can't do, but it's not because of a lack of power. Which brings us to a third point...

Let's examine this rock for a second. What's it like? Is it just massively heavy? If that's all, then there's two things to point out here...one, that God isn't a physical being, so he's probably not doing any lifting at all; and two, that there's an upper limit to how heavy the rock can be, based on the physical properties of the universe. And of course, a rock that heavy would collapse under it's own mass. Basically, "God cannot do anything inconsistent with his own nature".

Now, maybe the rock isn't just really, really heavy. Maybe it's really slippery as well. If so, then again there's no lack of power if God cannot lift the rock.

A bit more complicated of an answer is this...God never actually does anything in the sense that we do things; he merely causes things to be, by willing them.

A final way to view this question goes back to what I said at the beginning...such a rock would be a logical contradiction, as it's a rock that an all-powerful being cannot lift. People like to think that this question points to some fault in the idea of God, but maybe it points to a fault in the idea of a God-defeating rock.

[Alliteration]

Oh, and for the more philosophically minded...

Omnipotence (stipulative definition) = x is omnipotent iff it is not the case that there is some state of affairs, p, such that x is unable to bring about p at least partially because of a lack of power in x.

( Wielenberg, Erik (2000). Omnipotence Again. Faith and Philosophy 17 (1):26-47.
http://philpapers.org/archive/WIEOA)

[whacker44]

and there is always Universal acid..not relevant i know but..i felt like mentionning it