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Science Forum Index » Logic Forum » Liar's Paradox
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| raydpratt |
 Posted: Wed Apr 02, 2008 4:58 pm |
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When we call a statement true or false, we are saying that we have
compared that statement to a given fact or principle and either found
corraboration or contradiction. For example, if we say "All apples
are cubes that glow in the dark," we can only judge that statement as
being true or false by comparing the statement to what we know about
apples.
In the Liar's Paradox, we say "This very statement is false." The
alleged paradox is that if the statement is true, then it is false as
it claims, but if it is false as it claims, then it has stated the
truth and cannot be false, ad infinitum. Properly understood,
however, there is no paradox.
To judge the truth or falsity of "This very statement is false," we
must compare the statement not only to itself as explicitly required,
but also to what we know about finding the falsity of any general
statement. This is implied by the use of the term "false," much like
the term "apple" would require us to compare a statement to what we
know about apples. With the Liar's Paradox, the very fact of the
supposed paradox proves that the statement cannot be definitively
proven false. And as such, the statement is ultimately true because
it admits that it falsely asserts that it is provably false.
Similarly, in the Truth Teller, we say "This very statement is true."
Although there is no alleged paradox, the statement's self-reference
to its own veracity is not sufficient evidence for its truth. The
informal fallacy called petitio principii, or begging the question,
occurs where a questioned fact is called in as proof of that fact, and
such proofs are always illegitimate. However, we can go further here
and say that the statement is definately false because it falsely
claimed that it was provably true.
Very Respectfully,
Ray Donald Pratt |
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| Marshall |
| Posted: Wed Apr 02, 2008 5:53 pm |
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On Apr 2, 6:58 pm, raydpratt <raydpr...@gmail.com> wrote:
Quote:
And as such, the statement is ultimately true because
it admits that it falsely asserts that it is provably false.
That doesn't help. If the statement is true, then that
means "this statement is false" is true, which means
that it's false. Simply declaring it to be one way or
the other doesn't resolve the problem. In fact, it's
clear enough on the face of it that even simply
arbitrarily picking either true or false on the assumption
that there's some justification to do so doesn't help.
Some statements just don't possess a truth value.
Quote: Similarly, in the Truth Teller, we say "This very statement is true."
Although there is no alleged paradox, the statement's self-reference
to its own veracity is not sufficient evidence for its truth.
I take the view that the paradoxical aspect isn't the real issue
with the Liar's Paradox; rather it's the infinite regress, which
is every bit as present in the Truth Teller as in Liar. Neither
statement can be assigned a truth-value because both
statements require that they be assigned a truth value
before they can be assigned a truth value.
Marshall |
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| William Elliot |
| Posted: Thu Apr 03, 2008 12:00 am |
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On Wed, 2 Apr 2008, raydpratt wrote:
Quote: In the Liar's Paradox, we say "This very statement is false."
You're the victim of propaganda that's intended to
mislead you from the real liars' paradox which is:
A lie told by the privileged, is not a lie.
This is akin to other liars' paradoxes such as
"The emperor has no cloths."
and the invisible white elephant. |
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| Marshall |
| Posted: Thu Apr 03, 2008 5:53 am |
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On Apr 3, 5:58 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
Quote: Marshall says...
I take the view that the paradoxical aspect isn't the real issue
with the Liar's Paradox; rather it's the infinite regress, which
is every bit as present in the Truth Teller as in Liar. Neither
statement can be assigned a truth-value because both
statements require that they be assigned a truth value
before they can be assigned a truth value.
Then we have three types of sentences, true, false, and
sentences involving infinite regress.
I hope that list is not meant to be exhaustive.
Quote: But then what about
the sentence
This sentence cannot be assigned a truth value, since it
involves infinite regress.
Meh. It requires orders of magnitude more machinery to
formalize than the previous ones, and what would be the
lesson learned in doing so? That trying to model natural
language formally is a bad idea? I already know that.
Marshall |
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| Marshall |
| Posted: Thu Apr 03, 2008 7:11 am |
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On Apr 3, 8:38 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
Quote: Marshall says...
On Apr 3, 5:58 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
But then what about the sentence
This sentence cannot be assigned a truth value, since it
involves infinite regress.
Meh. It requires orders of magnitude more machinery to
formalize than the previous ones, and what would be the
lesson learned in doing so? That trying to model natural
language formally is a bad idea?
It doesn't have anything to do with natural language,
specifically. It has to do with the meaning of the
predicate "true".
I have no idea what you're driving at.
My default conception of "true" is that it is one of
two values in the boolean domain, and my default
conception of evaluating expressions is within a
computational model. I apologize if this is not the
default conception for the more usual denizens of
sci.logic.
I can translate "this sentence is false" into a very
simple computational model involving only naming,
booleans, and equality. The translated expression
is malformed under strict evaluation and does not
terminate under lazy evaluation, in both cases because
of infinite regress.
You say your question doesn't have anything to do with
natural language specifically; does that mean you have
a particular computational model in mind in which to
examine the sentence? If so, I'd be interested to hear it;
if not, then it seems to me we're back to the question of
trying to use natural language as a formal language,
which is a waste of time in my humble and perhaps
unimaginative opinion.
Marshall |
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| Marshall |
| Posted: Thu Apr 03, 2008 7:20 am |
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On Apr 3, 8:12 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
Quote:
As usual, "This statement is not true" yields a paradox even with
Marshall's third possibility.
I suppose I see what you mean, but I have to point out that
under a wide range of evaluation models, "not true" and
"false" are exactly the same thing, so your above sentence
would be the same as the liar's paradox.
I may just be overly pragmatic in my thinking, but I really
don't see the value to the logician of riddles. Natural language
has ambiguous, nondeterministic, context-sensitive semantics
by design and intent. It's great for poetry, lyrics, literature,
and even prose when we don't have a formal model yet.
But given the wobbliness of it, and the lack of an exact
semantics, we shouldn't be at all surprised by the discovery
of weirdly behaving corner cases.
Marshall |
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| Marshall |
| Posted: Thu Apr 03, 2008 8:48 am |
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On Apr 3, 9:29 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
Quote: Marshall says...
On Apr 3, 8:38 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
It doesn't have anything to do with natural language,
specifically. It has to do with the meaning of the
predicate "true".
I have no idea what you're driving at.
That the Liar Paradox has nothing to do with natural language,
specifically.
Okay, perhaps I wasn't explicit enough with my original reply.
My reaction to your statement, "The Liar Paradox has nothing to
do with natural language" is yes, the same issue shows up in
other contexts besides natural language ones, which can be
seen by translating Liar's from its natural language form into a
formal language. However I don't understand *why* you're saying
this particular sentence at this point. I hypothesize it is to make
some particular further point, but I don't see what point it is you
are trying to make. So I would like to ask you to explicitly state
the relevance of this sentence to the rest of the discussion, or
else state that you intend it as a standalone point which you now
accept that I acknowledge.
Was that clear?
Quote: My default conception of "true" is that it is one of
two values in the boolean domain,
Well, a *sentence* is neither equal to the boolean
value true nor is it equal to the boolean value false.
So how does it make any sense to call a statement
true or false?
Um, I'm guessing you're asking leading questions as
part of some sort of pedagogical technique. In what way
does the end of my sentence, which you snipped, not
answer your question?
Some formal languages have a boolean type and evaluation
rules which will assign a boolean value to an expression
of that language, in which case we may say as informal
shorthand that an expression equals a value. "1+1" does
not equal 2 but 1+1 does in fact equal 2, within an
appropriate computational model. The expression "false or true"
does not equal true, but false or true = true, under an
appropriate computational model. (Under a different model,
"false" might be a variable name and not a boolean literal
value.)
Is that clear?
Marshall |
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| Marshall |
| Posted: Thu Apr 03, 2008 8:57 am |
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On Apr 3, 9:41 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
Quote: Marshall says...
On Apr 3, 8:12 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
As usual, "This statement is not true" yields a paradox even with
Marshall's third possibility.
I suppose I see what you mean, but I have to point out that
under a wide range of evaluation models, "not true" and
"false" are exactly the same thing, so your above sentence
would be the same as the liar's paradox.
No, "false" and "not true" are not the same thing. A false
statement is one whose *negation* is true.
In boolean algebra, false and not-true are identical.
In other contexts this does not hold.
Marshall |
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| Daryl McCullough |
| Posted: Thu Apr 03, 2008 8:58 am |
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Marshall says...
Quote: I take the view that the paradoxical aspect isn't the real issue
with the Liar's Paradox; rather it's the infinite regress, which
is every bit as present in the Truth Teller as in Liar. Neither
statement can be assigned a truth-value because both
statements require that they be assigned a truth value
before they can be assigned a truth value.
Then we have three types of sentences, true, false, and
sentences involving infinite regress. But then what about
the sentence
This sentence cannot be assigned a truth value, since it
involves infinite regress.
--
Daryl McCullough
Ithaca, NY |
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| Marshall |
| Posted: Thu Apr 03, 2008 8:58 am |
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On Apr 3, 9:43 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
Quote: Marshall says...
I may just be overly pragmatic in my thinking, but I really
don't see the value to the logician of riddles.
Natural language has ambiguous, nondeterministic,
context-sensitive semantics by design and intent.
The Liar Paradox doesn't have anything specifically to do with
natural language.
Do you see your reply as having some relevance to what
I wrote? I cannot see any. If you did not intend any, no
problem, but I thought I'd check.
Marshall |
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| MoeBlee |
| Posted: Thu Apr 03, 2008 9:17 am |
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On Apr 3, 11:57 am, Marshall <marshall.spi...@gmail.com> wrote:
Quote: In boolean algebra, false and not-true are identical.
In other contexts this does not hold.
In a Boolean algebra with exactly TWO elements in the carrier set,
yes.
MoeBlee |
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| Marshall |
| Posted: Thu Apr 03, 2008 9:49 am |
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On Apr 3, 11:16 am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Quote: Marshall says...
On Apr 3, 9:43 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
Marshall says...
I may just be overly pragmatic in my thinking, but I really
don't see the value to the logician of riddles.
Natural language has ambiguous, nondeterministic,
context-sensitive semantics by design and intent.
The Liar Paradox doesn't have anything specifically to do with
natural language.
Do you see your reply as having some relevance to what
I wrote?
Yes. You said: "Natural language has ambiguous, nondeterministic,
context-sensitive semantics by design and intent". I responded
by saying that that has nothing to do with the Liar Paradox
(the subject of this thread). Maybe you didn't intend for your
statement to have any relevance to this thread?
Have you found success in the past with the technique of
repeating the same phrase over and over again in response
to requests for clarification? Is your expectation that I'm
going to understand the relevance on the third or fifth
or seventh try?
Your proposed sentence:
This sentence cannot be assigned a truth value, since it
involves infinite regress.
is written in natural language, is it not? The properties of
natural language have to have at least *some* relevance,
since you're asking about a sentence in natural language,
and this is true even if the specific properties in question
are not properties that are exclusive to natural language.
Yes, I understand that there are manifestations of things
very much like the Liar's Paradox in other contexts, some
of which are formal languages. Russell's paradox, for example.
This does not negate my claim that natural language can be
ambiguous. If you have some further point to make about
it, it's clear that I'm not going to get what it is simply by
repeated reading of the same phrase you used initially;
you'll have to be more specific if you intend to communicate
whatever it is you have in mind. If that is not your goal, that
is also fine.
Marshall |
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| Marshall |
| Posted: Thu Apr 03, 2008 10:04 am |
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On Apr 3, 11:06 am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Quote: Marshall says...
Um, I'm guessing you're asking leading questions as
part of some sort of pedagogical technique.
Look, you made a post about the Liar Paradox. I don't
think that your post made very much sense. But if you're
not interested in defending it, I'll let it drop.
If you think something I wrote didn't make much sense,
I'd invite you to point out what you object to specifically.
You could perhaps also explain what reasoning you used,
and ideally even supply an alternative view that you
feel is superior. I could respond with counter-arguments,
or I might abandon my view if I feel yours is better. Or
I might point out some specific context-dependence, or
we might find that our different backgrounds have caused
us to employ different definitions for terms we are both
using.
Did the rest of what I wrote, which you snipped, sufficiently
answer your questions about my earlier meaning, or is
it still unclear?
Marshall |
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| Marshall |
| Posted: Thu Apr 03, 2008 10:51 am |
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On Apr 3, 12:13 pm, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Quote: Marshall says...
On Apr 3, 11:06 am, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Marshall says...
Um, I'm guessing you're asking leading questions as
part of some sort of pedagogical technique.
Look, you made a post about the Liar Paradox. I don't
think that your post made very much sense. But if you're
not interested in defending it, I'll let it drop.
If you think something I wrote didn't make much sense,
I'd invite you to point out what you object to specifically.
I don't think that would be very fruitful. Let's drop it.
I am disappointed, but I shall respect your wishes.
I hope I may be forgiven for making an intuitive leap,
but please allow me to express the wish that you feel
better as soon as possible.
Marshall |
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| Jesse F. Hughes |
| Posted: Thu Apr 03, 2008 11:12 am |
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stevendaryl3016@yahoo.com (Daryl McCullough) writes:
Quote: Marshall says...
I take the view that the paradoxical aspect isn't the real issue
with the Liar's Paradox; rather it's the infinite regress, which
is every bit as present in the Truth Teller as in Liar. Neither
statement can be assigned a truth-value because both
statements require that they be assigned a truth value
before they can be assigned a truth value.
Then we have three types of sentences, true, false, and
sentences involving infinite regress. But then what about
the sentence
This sentence cannot be assigned a truth value, since it
involves infinite regress.
We can say that this sentence isn't true, but we can't say whether it
is false or involves infinite regress. That doesn't seem particularly
paradoxical, though it is analogous to the Truth Teller situation that
Marshall mentions.
As usual, "This statement is not true" yields a paradox even with
Marshall's third possibility.
--
"The papers are currently at journals. [When published,] make no
mistake, there will be no place on this planet where you can hide.
Remember, I'm not talking about something vague here. I'm talking
about publication in journals." James S. Harris. Wow. Journals. |
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