The Liar Paradox

[Orphia Nay]

The first thing said in the first post in this thread is not true.



(Discuss.

I'm hoping I've formulated the paradox in a 'correct' way. I read about the liar paradox yesterday and keep thinking about it.)

[TragicMonkey]

I always phrase it thusly:

True or false?: this statement is a lie.

(Although actually "the first thing said in the first post in this thread" is the title "The Liar Paradox", which is a name and not an assertion, and thus probably isn't true or false.)

[Orphia Nay]

I always phrase it thusly:

True or false?: this statement is a lie.



Hi, TragicMonkey. Yes, that's a good version.


(Although actually "the first thing said in the first post in this thread" is the title "The Liar Paradox", which is a name and not an assertion, and thus probably isn't true or false.)

Ah, but "The Liar Paradox" is the title, not the actual post. I already thought of that.

Cheers.

[TragicMonkey]

Ah, but "The Liar Paradox" is the title, not the actual post. I already thought of that.

In that case, I fall back to the position that nothing is said in that post: it's written, not said.

Ha!

:mdance:

[Paul C. Anagnostopoulos]

The liar is the same man who shaves all the men who don't shave themselves. What a guy!

~~ Paul

[TragicMonkey]

The liar is the same man who shaves all the men who don't shave themselves. What a guy!

~~ Paul

What?

[Meffy]

Is the liar a vampire? And what color is his or her hat?

[CFLarsen]

"It's true!"

[Roboramma]

True or false?: this statement is a lie.
Neither.

Where's the paradox? I don't see why every statement must be either true or false.

[Meffy]

I don't see why every statement must be either true or false.
All azure threnodies have been trained in cloud maintenance.

[edit] Okay, that was a little on the cryptic side. What I meant was that besides true and false, nonsense is a third case. My example was one variety of nonsense: gibberish, dadaistic style. Statements that lead only to self-inconsistent conclusions are another class of nonsense.

[HighRiser]

This paradox?

The following statement is true: The preceding statement was false.

[Meffy]

That's an example of my second class of nonsense -- statements that lead only to self-inconsistent conclusions. It's neither true nor false, it's a sequence of words that doesn't reflect a possible state of things.

[edit] Note that I've not taken philosophosophy classes. There might well be proper names for these concepts. If so, let me know.

[Terry]

What?

http://en.wikipedia.org/wiki/Barber_paradox

[c4ts]

Is the paradox in the statement, or the dichotomy?

[sphenisc]

All azure threnodies have been trained in cloud maintenance.



I think this statement is true.
It can be regarded as of the form

p -> q

Given that no azure threnodies exist, (since azureness is not a possible property of a threnody), then we have

(~p & ( p -> q))

which is tautologically true.

[fuelair]

"It's twue! It's twue!!" M.Kahn Blazing Saddles

[Meffy]

I'm doubtful that a statement having no meaning can have truth. Or falsity. Madewine Kahn notwitstanding, she makes her own rules.

[TragicMonkey]

Neither.

Where's the paradox? I don't see why every statement must be either true or false.

If you buy into the notion that the universe has an underlying order which can be rationally understood, then everything conceivable either is or isn't. We call "true" the state of things that are. "False" is those that aren't.

[Wheezebucket]

This thread is blowing my mind.

[DreadNiK]

All azure threnodies have been trained in cloud maintenance.



Argh this one is making me ponder. If there are no such things as azure threnodies, then all of them (i.e none) have been trained in a non-existent ability. So all "non-existing plural" have been trained in "non-existent skill" seems reasonable. And if it was in fiction literature it could be referring to existing things. So it is certainly more coherent than "this statement is a lie". It is logically coherent but "has no reality?"

[Meffy]

Near enough as I can make out. Trained philostomophers may differ (I hope at least some do) but that's more or less the idea as I understand it. It's an utterance made of words, but it doesn't really say anything.

[c4ts]

While there are many statements that don't fit into the dichotomy of "true or false," that does not mean these statements are meaningless or devoid of information. Are you familiar with Wittgenstein's language games?

[CplFerro]

I always phrase it thusly:

True or false?: this statement is a lie.


Dear TM,

A lie about what? I don't think this statement gives enough information to tell us what it's about, which makes it a textual fragment.

Cpl Ferro

[Roboramma]

If you buy into the notion that the universe has an underlying order which can be rationally understood, then everything conceivable either is or isn't. I'm not sure I agree with that, but anyway it would only have to be true if everything in the universe had an underlying order and could be rationally understood. That's not necessary for most of the universe to be rationally understood.

For instance, if I said, "this electron is right here and is moving this fast, exactly, true or false?" (well, strickly speaking, it might be possible to say "false", but certainly not "true") an answer isn't even possible. That doesn't mean the universe can't be rationally understood.

But that's all a digression.

We call "true" the state of things that are. "False" is those that aren't.

What do we call things that don't actually refer to anything? The statement "this is a lie" doesn't refer to anything about our universe, until some content is added about what it's lying about.

[Raphael]

Those that enjoy this sort of thing should try Raymond Smullyan's "What is The Name of This Book?" Hours of Logic fun.

[TragicMonkey]

Dear TM,

A lie about what? I don't think this statement gives enough information to tell us what it's about, which makes it a textual fragment.

Cpl Ferro

The truth (or lie) the statement "this is a lie" makes about itself. It's not a fragment. It's a complete sentence that makes a claim. That it's making a claim about itself is, of course, what makes it a paradox.

[TragicMonkey]

I'm not sure I agree with that, but anyway it would only have to be true if everything in the universe had an underlying order and could be rationally understood. That's not necessary for most of the universe to be rationally understood.

For instance, if I said, "this electron is right here and is moving this fast, exactly, true or false?" (well, strickly speaking, it might be possible to say "false", but certainly not "true") an answer isn't even possible. That doesn't mean the universe can't be rationally understood.


The assumption underlying the view that the universe is rational is that the truth is out there, and that things either are or aren't a certain way, whether we know now, or can ever know, which they are. Your question about electrons might be flawed, but only because we may be ignorant of the actual nature of the thing. What it is, it is, regardless of human opinion, understanding, or interest.

It would probably be better put as "the universe can be rationally understood, if we had enough knowledge of it." As it happens, we will probably never acquire enough knowledge to understand even a fraction of everything. But we either take it on faith that if we could see it all, it would all be ordered in a manner capable of being understood by reason, given good enough reasoning....or we chuck it all in and join the mystics.


But that's all a digression.

What do we call things that don't actually refer to anything? The statement "this is a lie" doesn't refer to anything about our universe, until some content is added about what it's lying about.

The statement refers to itself. It's a thought resident in the universe, and expressed therein. The content is there as well: it's self-referential. Hence the paradox, because it's self-referential in a manner that contradicts itself no matter which way you go. It's actually an example of what things would be like in a irrational universe. If the universe weren't rational, it wouldn't be a paradox in the first place.

[Orphia Nay]

In that case, I fall back to the position that nothing is said in that post: it's written, not said.

Ha!

:mdance:

Pretty good try, but I refuse to be swayed by your cute monkey dance. Besides:

say
v. said, (sd) say·ing, says (sz)
v. tr.

1. To utter aloud; pronounce: The children said, “Good morning.”
2. To express in words: Say what's on your mind.
3.
1. To state as one's opinion or judgment; declare: I say let's eat out.
2. To state as a determination of fact: It's hard to say who is right in this matter.
4. To repeat or recite: said grace.
5. To report or maintain; allege.
6.
1. To indicate; show: The clock says half past two.
2. To give nonverbal expression to; signify or embody: It was an act that said “devotion.”
7. To suppose; assume: Let's say that you're right.

http://dictionary.reference.com/search?q=said

I must say that I am enjoying your contribution to this thread.

I've got more to add later, time permitting.

[Meffy]

While there are many statements that don't fit into the dichotomy of "true or false," that does not mean these statements are meaningless or devoid of information. Are you familiar with Wittgenstein's language games?
No, and I doubt I'd understand them. Trained philosters can doubtless run rings 'round my pitiful attempts at making sense of the world. But I've seen philosophical arguments that disprove the arguer's own existence and so forth, and been unable to spot the errors in them. So I put no faith in my ability to judge philosophical ideas. Those disagreeing (or agreeing, for that matter) are welcome to dismiss anything I say on the subject as so many uneducated gropings, and I wouldn't deny it. I don't expect to convince anyone.

[Roboramma]

The assumption underlying the view that the universe is rational is that the truth is out there, and that things either are or aren't a certain way, whether we know now, or can ever know, which they are. Your question about electrons might be flawed, but only because we may be ignorant of the actual nature of the thing. What it is, it is, regardless of human opinion, understanding, or interest. Maybe, or maybe not.
As to the electron - maybe the electron really does exist in a certain place, moving at a certain velocity. Maybe it is spread out in some way and only has a definite position when that's measured. Both of those are interpretations of quantum physics, and based on what we know we can't distinguish between them.
Yet, no matter which one is true about this universe, the universe is still understandable - quantum physics still holds, I can still say that the sun will rise tomorrow, or that a rock will fall when I drop it, even if I can't say that this electron is either here or it isn't.

I'm not saying that it definitely doesn't have a definite position and momentum that we just can't measure at the same time, I'm only saying that if it doesn't, that doesn't mean that the universe isn't understandable. It just means that there are limits to our understanding of it.
But that's not anything new - there are limits - we never will learn that electron's position and momentum at the same time, even if it does have them.

It would probably be better put as "the universe can be rationally understood, if we had enough knowledge of it." How is that different from saying "the universe cannot be rationally understood" considering that we cannot get that knowledge?

As it happens, we will probably never acquire enough knowledge to understand even a fraction of everything. But we either take it on faith that if we could see it all, it would all be ordered in a manner capable of being understood by reason, given good enough reasoning....or we chuck it all in and join the mystics. I disagree.
I'm pretty much a reductionist - I think the way things work on one level is just the sum of the interactions of it's component parts. But that doesn't mean that this has to be true, or that everything has to be related through cause and effect, or whatever, for us to understand the things that we do understand.
There might be things about the universe that have no truth value. That doesn't change the truth value of all the things that we do know about.

Newtonian physics explains a lot about how the universe works. When Einstein came along, he didn't change any of those explanations, he just expanded them. Similarly, if there are places where physics and even rational inquiry break down, that doesn't change the fact that they work perfectly well in the places that we have been using them so far.

I don't know that there are such places, but I don't think that being open to the possibility is the same as joining the mystics.

Geez, I didn't mean to get so serious in this thread.

[c4ts]

No, and I doubt I'd understand them. Trained philosters can doubtless run rings 'round my pitiful attempts at making sense of the world. But I've seen philosophical arguments that disprove the arguer's own existence and so forth, and been unable to spot the errors in them. So I put no faith in my ability to judge philosophical ideas. Those disagreeing (or agreeing, for that matter) are welcome to dismiss anything I say on the subject as so many uneducated gropings, and I wouldn't deny it. I don't expect to convince anyone.

Wittgenstein's idea of the word game was that words don't just contain information, they can contain instructions. One example is builder's English in a fictional example, such as "insert tab A into slot B." If you approach the true/false paradox as instructions, you are instructed to place the object within a set, and the paramaters for this object are defined by one of the sets. Like if the builder told you to "insert tab A into tab A," and the definition of "tab A" relied on its relationship to "tab A." You can't construct truth or falsehood when the language is circuitously defined.

Of course, then there is Saul Kripke's theory of groundlessness, where even false statements can obtain a truth value while circular statements cannot. I haven't read Kripke, though, so I'm not entirely sure how this works.

[Elind]

The first thing said in the first post in this thread is not true.



(Discuss.

I'm hoping I've formulated the paradox in a 'correct' way. I read about the liar paradox yesterday and keep thinking about it.)

Maybe it's here, but I haven't read all the posts yet; but it brings to mind the one about the directory of all books that don't have their title mentioned within the book.

[DickK]

I think this statement is true.
It can be regarded as of the form

p -> q

Given that no azure threnodies exist, (since azureness is not a possible property of a threnody), then we have

(~p & ( p -> q))

which is tautologically true.

Not sure about your initial reasoning but your final term can be trivially disproved for cases where p is true, can't it?

[TragicMonkey]

Maybe, or maybe not.
As to the electron - maybe the electron really does exist in a certain place, moving at a certain velocity. Maybe it is spread out in some way and only has a definite position when that's measured. Both of those are interpretations of quantum physics, and based on what we know we can't distinguish between them.

Which is a limitation of ours. The electron is how it is, regardless of whether we understand it or not. We may argue our ideas of it, but that won't change the thing itself.

I'm not saying that it definitely doesn't have a definite position and momentum that we just can't measure at the same time, I'm only saying that if it doesn't, that doesn't mean that the universe isn't understandable. It just means that there are limits to our understanding of it.

I agree.

But that's not anything new - there are limits - we never will learn that electron's position and momentum at the same time, even if it does have them.

We can't be certain that we'll never be able to. We might make more discoveries, or get better technology, or understand it in a more accurate way later on.

How is that different from saying "the universe cannot be rationally understood" considering that we cannot get that knowledge?

It's the difference between "I cannot bench press 400 lbs" and "400 lbs cannot be bench pressed".

I disagree.

I'm pretty much a reductionist - I think the way things work on one level is just the sum of the interactions of it's component parts. But that doesn't mean that this has to be true, or that everything has to be related through cause and effect, or whatever, for us to understand the things that we do understand.
There might be things about the universe that have no truth value. That doesn't change the truth value of all the things that we do know about.

Newtonian physics explains a lot about how the universe works. When Einstein came along, he didn't change any of those explanations, he just expanded them. Similarly, if there are places where physics and even rational inquiry break down, that doesn't change the fact that they work perfectly well in the places that we have been using them so far.

I don't know that there are such places, but I don't think that being open to the possibility is the same as joining the mystics.

I'm not wedded to physics. Boil it down to logic. A is A. Absolutely, always, and everywhere. If it weren't, then the universe wouldn't be a rational one. To quote Chesterton, the universe is "only infinite physically, not infinite in the sense of escaping from the laws of truth." Reason presumes its own monopoly on reality, because if it weren't the only game in town, then it wouldn't work at all. It's either everything or nothing. If A is A but only sometimes, then we have nothing. Permit the universe to be irrational, and we not only know nothing, but we can never know anything.

[Meffy]

Wittgenstein [...] Kripke
Sounds like the kind of thing I'm interested in but have trouble wrapping my head around. There's not much head to wrap with. Now, if Hofstadter would write a nice gentle introduction...

[c4ts]

Sounds like the kind of thing I'm interested in but have trouble wrapping my head around. There's not much head to wrap with. Now, if Hofstadter would write a nice gentle introduction...

The point is, you were right when you said the liar paradox was neither true nor false, but you were wrong when you said it wasn't self-inconsistent. It's not total nonsense because it carries instructions.

[Meffy]

Ah, okay. I have half a clew, not enough to make fast the boat, but mine own. :-)

[Roboramma]

I'm not wedded to physics. Boil it down to logic. A is A. Absolutely, always, and everywhere. If it weren't, then the universe wouldn't be a rational one. To quote Chesterton, the universe is "only infinite physically, not infinite in the sense of escaping from the laws of truth." Reason presumes its own monopoly on reality, because if it weren't the only game in town, then it wouldn't work at all. It's either everything or nothing. If A is A but only sometimes, then we have nothing. Permit the universe to be irrational, and we not only know nothing, but we can never know anything.

Maybe, but that's just the way it is. We really can't know anything for certain.
A has always been A when we've looked. But it might not be tomorrow. It seems to me a pretty good bet that it will go on being A, but we can't know that for sure.
A is usually A. A has been A.

As far as we know the speed of light is 3 * 108 m/s. It might be different tomorrow, but it's unlikely. We can't assume that it can't change though. We hope that if it does change there is some pattern that we can figure out, some underlying cause of that change that we can understand.
I don't see how this is necessarily true, though. Nor do I see that it being false undermines all of science.

There are patterns in the things we observe in the universe. These patterns seem to imply underlying causes. But those underlying causes being the way things actually are isn't necessary for us to notice and take advantage of the patterns.

Nor would one thing being completely random mean that other things don't follow patterns.

[TragicMonkey]

Maybe, but that's just the way it is. We really can't know anything for certain.
A has always been A when we've looked. But it might not be tomorrow. It seems to me a pretty good bet that it will go on being A, but we can't know that for sure.
A is usually A. A has been A.

No. A is A, absolutely, always, and forever. If you think otherwise, then you are rejecting reason itself. Which might not be wrong, but it is irrational. Reason is like a ship: if you leave the tiniest crack open for irrationality, the whole thing sinks. There can be no disclaimers on reason "only sometimes" because that goes against the whole nature of reason, which does presuppose a rational universe.



As far as we know the speed of light is 3 * 108 m/s. It might be different tomorrow, but it's unlikely. We can't assume that it can't change though. We hope that if it does change there is some pattern that we can figure out, some underlying cause of that change that we can understand.

But what we can assume is that the speed of light is the speed of light, and that if we measure it differently tomorrow there will be a rational explanation, even if we don't know what it is. We can't say there's not a reason for it, even if the previous reasons we thought out are wrong.


I don't see how this is necessarily true, though. Nor do I see that it being false undermines all of science.

There are patterns in the things we observe in the universe. These patterns seem to imply underlying causes. But those underlying causes being the way things actually are isn't necessary for us to notice and take advantage of the patterns.

Nor would one thing being completely random mean that other things don't follow patterns.

It's back to the Rationalists and the Empiricists. The Empiricists seem more, well, reasonable to us, because we are the children of their ideas. At first glance, science seems to be entirely empirical. But it's not. The Rationalists are harder to fathom these days, because they went back to first causes and started trying to use reason to explain the universe; rather than observe what is, then seek an explanation, they thought of what could be (and what could only be) and worked forwards. "I think, therefore I am" is the real start of all human reasoning. Reason is a structure, simple at the beginning but growing in complexity as it grows upward and supports its spinoffs like Logic and Science. But the foundation remains the same, and that's the assumption of a rational universe. A is A, and if it's not, then everything falls down, because Reason won't be reasonable any more, logic won't be logical, and science will be mysticism because you will have to take on faith that the universe is rational at the moment and in the specifics any given theory is attempting to describe. Science is an attempt to rationally explain a rational universe; it's simply not equipped to handle an irrational one.

[a_unique_person]

How about the AmyWilson Paradox.

Everything I say is true. :)

[sphenisc]

Not sure about your initial reasoning but your final term can be trivially disproved for cases where p is true, can't it?

You're right, I think?

I was trying to express the idea that

'If something is an azure threnody then it has been trained in cloud maintenance.'

as p->q

and

'There are no azure threnodies.'

as ~p

it seems logical that p-> q must be true, but apparently I fouled up my propositional logic, perhaps someone could educate me? Cheers

[LW]

it seems logical that p-> q must be true, but apparently I fouled up my propositional logic, perhaps someone could educate me? Cheers

What you are after is the notation:

$\{ \neg p \} \vDash p \rightarrow q


This should be read as: The sentence p -> q is the logical consequence of the premise not p.

The semantics is that whenever we have a valuation that satisfies all formulas in the premise side, then the formula on the consequence side is true.

[drkitten]

it seems logical that p-> q must be true, but apparently I fouled up my propositional logic, perhaps someone could educate me? Cheers

Alternatively, you could express what you want as pure propositional logic with

~p -> (p -> q)

which is tautologically true, as can be validated by a truth table. In English, this would translate
as "if p is false, then the statement "if p, then q" is true no matter what q is."

But there's actually a huge debate -- or was, more accurately -- about how the semantics of a statement that "all A are B" should apply when there are no A to begin with.

Classical (Aristotelian) logic, for example, would hold that the following syllogism is valid (it's technically the Darapti syllogism):


All M are L
All M are S
Therefore, some S are L


This is obviously false if we allow true universal sentences without referent


All dragons are fire-breathing animals
All dragons are flying animals
Therefore, some flying animals breathe fire



But to an Aristotelian, a sentence of the form "all M are L" implies that not just that "no M are not L," but also that "some M are L," and therefore that "some M exist."

Modern first-order logic assumes the contrary; a statement of the form "for all x, P(x)" specifically does not imply the existence of such an x.

[Belz...]

The first thing said in the first post in this thread is not true.



(Discuss.

I'm hoping I've formulated the paradox in a 'correct' way. I read about the liar paradox yesterday and keep thinking about it.)

"The Liar Paradox" ? Sounds like an episode of the old Star Trek.

[Meffy]

Could be the legendary lost third Harry Mudd episode. (Speak not to me of animated series; of such I know not, nor do I care to acknowledge its existence.)

[gnome]

I think this statement is true.
It can be regarded as of the form

p -> q

Given that no azure threnodies exist, (since azureness is not a possible property of a threnody), then we have

(~p & ( p -> q))

which is tautologically true.

I disagree... I know an azure Threnody; and she is not trained in cloud maintenance. Disproof by counterexample. In fact, I also disproved your claim that there are no azure Threnodies.

(her internet nic is Threnody, and she wore a lovely blue dress when I first met her)

[Meffy]

Hmph. Evidence that she is still clad in cerulean?

[gnome]

Could be the legendary lost third Harry Mudd episode. (Speak not to me of animated series; of such I know not, nor do I care to acknowledge its existence.)

Bleah! I speak to you of it anyway--a pity you are not versed, as it contains quite interesting material. Including an excellent tale by Niven that has been unfortunately excised by lawyers.

[Meffy]

To tell the truth, I sometimes have a hard time watching the original series (the one true series). The cheepnis can be palpable. Visible even. But I'm loyal, cheesiness and all.

[Dr Adequate]

Is "What is a question which is its own answer?" a question which is its own answer?

[Meffy]

Is "What is a question which is its own answer?" a question which is its own answer?
"Now you've did it!" -- Kayo Mullins

If this were an original Star Trek episode, the thread would now have to ignite the Starfleet Standard pyrotechnic charge under each panel.

[gnome]

Hmph. Evidence that she is still clad in cerulean?

Irrelevant... at one time there was an azure threnody that lacked this claimed universal property--it has not been qualified to presume only recent threnodies.

[gnome]

"Yields falsehood when preceded by its own quotation" yields falsehood when preceded by its own quotation.

[Dr Adequate]

Okay, try this:

"Either this sentence is false, or this sentence is a logical paradox which cannot be evaluated as true or false."

Now, suppose it's false. Then neither the first or the second half of the disjunction is true, so in particular the first alternative ("this sentence is false") can't be true, and we have arrived at a contradiction.

On the other hand, suppose it's true. Then either "this sentence is false" is true, which gives us a contradiction, or "this sentence is a logical paradox which cannot be evaluated as true or false." But this statement is also inconsistent with the hypothesis that the sentence is true.

So finally, supppose that the sentence is a paradox. But if it was, then since this is what the second clause of the disjunction says, if it was a paradox it would be true, in which case ... it wouldn't be a paradox.

BWAHAHAHAHA.

Ergo, it is neither true, nor false, nor paradoxical.

[Meffy]

Maybe it's metadoxical.

[SuperCoolGuy]

The first thing said in the first post in this thread is not true.

Evidence?!?!?!

[GreedyAlgorithm]

The way I've always thought about these things is as pointers. Language constructs are pointers to some concept... and the fact that I can type

int *p = 0.5;

does not keep me awake at night. If our brains had decent compilers we wouldn't have this problem. Scratch that. If our brains had *perfect* compilers we wouldn't have this problem. I'm not about to disparage the brain, it's too amazing for me. See what I mean about a better compiler?

[Tirdun]

Either this sentence is false, or this sentence is a logical paradox which cannot be evaluated as true or false.

I'll approach this as a boolean, since I program.

if (($S==FALSE)||((!$S == TRUE)&&(! $S == FALSE))):
return $S;
endif;

If the sentence is FALSE it ignores the OR and resolves, returning TRUE (it is TRUE that $S is FALSE)

if $S is TRUE, then the first option fails and is ignored. The second boolean ($S is neither true nor false) resolves as FALSE since theses are mutually exclusive and the && cannot be resolved as TRUE. Thus then entire boolean resolves as FALSE.

so if the sentence is inherently true, it resolves as FALSE. If it is false, it resolves as TRUE. So the code can be rewritten as


return !$S;


and the sentence is "If this sentence is true, it is false and vice versa" :D

[gnome]

A retraction:

I have spoken to the azure Threnody--and it turns out that she knows quite a bit about cloud maintenance. She's created clouds with dry ice, and knows a technique for seeding clouds in the atmosphere... so there may be something to that proposition after all.

[drkitten]

I'll approach this as a boolean, since I program.

if (($S==FALSE)||((!$S == TRUE)&&(! $S == FALSE))):
return $S;
endif;

If the sentence is FALSE it ignores the OR and resolves, returning TRUE (it is TRUE that $S is FALSE)



I don't think you quite understand the nature of the paradox.

The problem is not that you can't write such a boolean expression.

However, you're separating the return value from the value itself -- i.e. "IT" is true that "$S$ is false. The problem is that this is not possible : "IT" and "$S" are the same exact object, and therefore it's not possible for "IT" to be true and "$S" false.

If you approach it as a "real" programming language problem (none of the PHP stuff), then the basic problem is that propositional logic has fixed-point semantics, and there's no fixed point in the code fragments you wrote.

[Meffy]

I have a message for Threnody: "Cheer up, goth!" =^_^=

*ducks and runs*

[Orphia Nay]

"It's true!"

Hahaha!
":)"

This paradox?

The following statement is true: The preceding statement was false.

Another good one.

This thread is blowing my mind.

I'm glad to see others seem to be enjoying it. I was reading about it in a book called 'The Philosophy Gym' by Stephen Law. It's a fun read, but slow going, because I keep stopping to do 'work-outs'.

"Yields falsehood when preceded by its own quotation" yields falsehood when preceded by its own quotation.

A tree just fell in the forest of wood in my head.


Okay, try this:

"Either this sentence is false, or this sentence is a logical paradox which cannot be evaluated as true or false."

Now, suppose it's false. Then neither the first or the second half of the disjunction is true, so in particular the first alternative ("this sentence is false") can't be true, and we have arrived at a contradiction.

On the other hand, suppose it's true. Then either "this sentence is false" is true, which gives us a contradiction, or "this sentence is a logical paradox which cannot be evaluated as true or false." But this statement is also inconsistent with the hypothesis that the sentence is true.

So finally, supppose that the sentence is a paradox. But if it was, then since this is what the second clause of the disjunction says, if it was a paradox it would be true, in which case ... it wouldn't be a paradox.

BWAHAHAHAHA.

Ergo, it is neither true, nor false, nor paradoxical.

Oooh, this one's fun. Let me try.

The statement is a logical fallacy, specifically a false dichotomy - the options are either 'it is false' OR 'it cannot be evaluated as true nor false' - but another option is that it could be true.
OK, so it's not true, because it's a false dichotomy, but it is a true 'false dichotomy'. Damn, that's a logical fallacy too - equivocal use of the word 'true'! Still, if the statement is a logical fallacy, therefore it's false (more equivocation?), therefore...
it's true, in which case the statement made is false, then it must be true, etc etc... so I see it as true and false and thus paradoxical, and thus... true, in which case etc etc...
and thus it is circular reasoning, and therefore false, and oh, carp.

Backing up a bit. The statement is a logical fallacy, therefore it is wrong, and neither false nor a paradox. :p


Evidence?!?!?!

Damn you. :) Fine. What evidence do you require to prove that it's not?

[drkitten]

Oooh, this one's fun. Let me try.

The statement is a logical fallacy, specifically a false dichotomy - the options are either 'it is false' OR 'it cannot be evaluated as true nor false' - but another option is that it could be true.
OK, so it's not true, because it's a false dichotomy, but it is a true 'false dichotomy'. Damn, that's a logical fallacy too - equivocal use of the word 'true'! Still, if the statement is a logical fallacy, therefore it's false (more equivocation?), therefore...
it's true, in which case the statement made is false, then it must be true, etc etc... so I see it as true and false and thus paradoxical, and thus... true, in which case etc etc...
and thus it is circular reasoning, and therefore false, and oh, carp.

Backing up a bit. The statement is a logical fallacy, therefore it is wrong, and neither false nor a paradox.


I'm not sure I'm following. Are you asserting that a fallacy cannot be a paradox, and vice versa?

Because I think that's a category error. A statement cannot be a fallacy; a fallacy is a type of argument. An argument is cannot be true or false, but valid or invalid (and of course, a fallacy is by definition a type of invalid argument). And a logically fallacious argument can still have a true conclusion; you cannot conclude that "if the statement is a logical fallacy, therefore it's false," both because of the category error and because you're committing the Fallacist's Fallacy.

If what we're looking at is the simple propositional semantics of a statement, then there really are only three choices -- a statement is either true, false, or neither. And, of course, all three choices lead to contradiction ("left as an exercise to the student" -- proof by intiimidation), therefore the universe will shut down and reboot in approximately 10... 9.... 8....

[sphenisc]

A statement cannot be a fallacy

] you cannot conclude that "if the statement is a logical fallacy, therefore it's false,"

Surely you can conclude that, on the basis that ~p->(p->q) as you explained in #43?

Or maybe the universe's rules of logic changed during the reboot?

Is everyone keeping up? :confused:

[drkitten]

Surely you can conclude that, on the basis that ~p->(p->q) as you explained in #43?

No.

Some elementary definitions.

A proposition is a statement with propositional content, i.e. either true or false.

An argument is a connected series of propositions, linked in a purportedly logical fashion, such that the truth-value of a given proposition is assured by the truth-value of the previous statements in the chain.

A valid argument is an argument where the truth of the conclusion is a necessary consequence of the truth of the premises.

An invalid argument, also known as a fallacy is an argument where the truth of the conclusion is not a necessary consequence of the truth of the premises. In other words, in a valid argument, it is not possible for the premises to be true but the conclusion false. In a fallacy, it is possible (but not necessary).

A well-known fallacy is that of the undistributed middle. All A are B, All A are C, therefore all B are C. But it's fairly easy to construct an argument of this form in which the conclusion happens to be true.


All cats are mammals
All cats nurse their young
Therefore, all mammals nurse their young.


This argument is therefore fallacious, but the conclusion is still true. It just doesn't follow from the argument given.

To assert otherwise -- that the conclusion of a fallacious argument is automatically false -- is to commit the well-known "Fallacist's Fallacy (http://www.skepticwiki.org/wiki/index.php/Fallacist%27s_Fallacy),"

[sphenisc]

I'm quite happy to accept these definitions, and the possibility that an invalid argument may have a true conclusion.

However, you don't explain how that applies to the the two statements of yours that I quote.


A statement cannot be a fallacy.



"if the statement is a logical fallacy, therefore it's false,"


The first is ~p, the second is p->q.

~p->(p->q) is a tautology

in LW syntax (p->q) is a logical consequence of {~p}, so we can conclude (p->q), given ~p.

....

Having re-read your post, I think you meant "if the argument is a logical fallacy, therefore it's false,".

This will teach me to analyse posts semantically as well as syntactically before rushing to print... or maybe not :(

[drkitten]

I'm quite happy to accept these definitions, and the possibility that an invalid argument may have a true conclusion.

However, you don't explain how that applies to the the two statements of yours that I quote.[/QUIOTE]

The first statement is a direct quotation, i.e. it's my words, and I stand by them.

The second statement is not my writing (you'll notice that it's a quotation, with attribution to orphia ney). In context, I had hoped that it was apparent that I was trying to illustrate the problem with the reasoning underneath the inappropriate phrasing -- evidently I failed, for which I apologize.

[QUOTE]
Having re-read your post, I think you meant "if the argument is a logical fallacy, therefore it's false,".

This is closer. Technically speaking, of course, an argument is neither true nor false. A full and formal statement of the fallacist's would be something like "if the argument is a logical fallacy, therefore its conclusion is false."

So it looks like you understood what I had intended on the second reading, at least.

[c4ts]

To assert otherwise -- that the conclusion of a fallacious argument is automatically false -- is to commit the well-known "Fallacist's Fallacy (http://www.skepticwiki.org/wiki/index.php/Fallacist%27s_Fallacy),"

If no logical arguments for C can be made without fallacy, is it still a fallacy to find C false?