The fallacy of strengthened liar's paradox.
Newberry
false? It should not be difficult to answer. Tractatus Logico-
Philosophicus says: "In order to tell whether a picture is true or
false we compare it with reality." [2.223] When we attempt to compare
"This sentence is meaningless" with reality we find that it is not
comparable with anything. It is not a picture of a fact; it is
meaningless.
We can analyze the situation further:
Case A: P is true.
If P is true. Then it is the case that it is meaningless. But then it
cannot be true. This is a contradiction. Therefore P is not true.
Case B: P is false.
If P is false then it is not the case that it is meaningless. It is
the opposite of what it claims. This is a contradiction. Therefore P
is not false.
Case C: P is meaningless.
If P is meaningless then nothing is the case. There is no
contradiction. In a three valued logic (T, F, M) we conclude that P is
meaningless.
It is not correct to say that if the sentence is meaningless than what
it SAYS is true. This argument assumes that it becomes TRUE half way
through the argument and then it IS THE CASE that it is meaningless.
Thus the sentence confirms our initial assumption that it was
meaningless. But if it stays meaningless all the time it confirms
nothing.
Clearly, if the sentence does not have any meaning then it does not
have the meaning that it is meaningless.
This gives us the basic insight that all self-referential, paradoxical
sentences, including possibly Goedel's sentence, are probably
meaningless.
William Elliot
> Let P be the sentence "This sentence is meaningless."
> Is it true or false?
It's meaningless and a good thing that it's meaningless else
you'd get migraine headache flip flopping, is it true, is it
false. We recommend instead you consider, she loves me, she
loves me not.
Let Q be the sentence "This sentence is nonsense."
> We can analyze the situation further:
Mind your P's and Q's:
Of that which is meaningless
one speaks not but nonsense.
Riddle of the day.
Does this notion need to be notarized?
----
David C. Ullrich
<newbe...@gmail.com> wrote:
>Let P be the sentence "This sentence is meaningless." Is it true or
>false? [...]
>
>This gives us the basic insight that all self-referential, paradoxical
>sentences, including possibly Goedel's sentence, are probably
>meaningless.
You're jumping a bit from one example to _all_ such sentences.
But much more important: There's nothing _literally_ self-referetial
about "Godel's sentence" - your lovely anlysis is irrelevant there.
The sentence in question is just an ordinary assertion about positive
integers, with no problem whatever regarding what it means,
any more than there's a problem with "If n and m are even positive
integers then n + m is even."
************************
David C. Ullrich
Newberry
> On Mon, 24 Dec 2007 15:11:50 -0800 (PST), Newberry
>
> >Let P be the sentence "This sentence is meaningless." Is it true or
> >false? [...]
>
> >This gives us the basic insight that all self-referential, paradoxical
> >sentences, including possibly Goedel's sentence, are probably
> >meaningless.
>
> You're jumping a bit from one example to _all_ such sentences.
It is a bit conjecture.
> But much more important: There's nothing _literally_ self-referetial
> about "Godel's sentence" - your lovely anlysis is irrelevant there.
> The sentence in question is just an ordinary assertion about positive
> integers, with no problem whatever regarding what it means,
> any more than there's a problem with "If n and m are even positive
> integers then n + m is even."
One of those number is the Goedel number of the sentence ITSELF.
herbzet
Under a different coding scheme the same sentence does not refer
to itself.
--
hz
John Jones
I can't start this car. You have not said what sort of object P a
sentence is, nor even specified what sentence you refer to by 'it'.
Newberry
>
> - Show quoted text -
"This" in the sentence "This sentence is meaningless" obviously refers
to the sentence in quotation marks.
Newberry
I was mainly disputing the idea that the strengthened liar defeats the
3-valued approach. The side note about Goedel's sentence is just an
afterthought. But anyway here is what Torkel Franzen says about self-
reference:
QUOTE:
But sentences constructed in the proof that every arithmetical
property P has a provable fixpoint are self-referential in a stronger
sense: they are sentences A of the form
There is an m such that m has the property P and property Q
where it is provable in PA that the only number that has the property
P is the Goedel number of the sentence itself. It is in this sense
that the sentence A "says of itself it has property Q."
END OF QUOTE [p. 45]
raydpratt
is a false proposition." As the argument pointing out the paradox
goes, if such a statement is taken as true, then it must be false as
it claims; but, if it is false as it claims, then it has made a
truthful statement about its falsity and must be taken as true, and
thus the argument begins anew, ad infinitum.
Normally, as a matter of logic, a statement declared to be true or
false is a statement that was compared to some referent fact or
concept and either found to match or not to match.
Thus, when we say that a statement is true or false, we imply or show
that such a comparative process can be and has been carried out.
With that process in mind for how to comparatively find a statement to
be true or false, let us look closer at the exact meaning of a
statement that gives rise to "the Liar's Paradox":
"This very statement is a false proposition."
This statement, "the Liar's Paradox," ascribes to itself the quality
of being false.
That statement must point to and be compared to a referent fact or
concept if we are to determine the truth or falsity of the statement.
The Liar's Paradox only points to and invites comparison to the very
concept of how to find the truth or falsity of a statement, but that
concept by itself does not prove a statement to be true or false. And
so, the Liar's Paradox, by default, has truthfully stated that its own
statement is false in declaring itself to be false. The final outside
truth value of the Liar's Paradox is that the Liar's Paradox is true.
Conversely, to clarify the point, let's consider the opposite
statement that "This very statement is a true proposition."
The above statement, "the Truth Teller," points to and ascribes to
itself the quality of being true by comparison to a referent fact or
concept, but the only concept implicitly pointed to is the very
process for finding truth or falsity, and that concept by itself does
not prove anything true or false. Thus, the final outside truth value
of the statement is false.
Someone will urge, of course, that we have not dispensed with "the
Liar's Paradox" or "the Truth Teller" because we must change our final
outside truth values because they are contradicted by the self-
assigned truth values of the statements themselves.
We should counter, of course, that the self-assigned truth values of
either the positive or negative versions of the statement were a part
of the very matter that we judged with a correct, comparative process
for determining the truth or falsity of the statements. Thus, we may
reject as unsound the argument that we must change our final outside
truth values based solely on the contradictory truth values alleged in
the statements themselves.
Thus, with "the Truth Teller," we have correctly disagreed that it
states a true proposition.
Likewise, with "the Liar's Paradox," we have come to correctly
understand the statement and have agreed with its assertion that it
makes a false proposition -- thus, there is either no contradiction in
giving the statement a final outside truth value of true or, at the
very least, we have correctly disagreed with the statement's self-
asserted final outside truth value of being false.
In short, we prefer a sound process for determining the truth or
falsity of propositions, not an unsound rule that changes our final
judgments with a simple decree of disagreement.
This should be the logical death of "the Liar's Paradox," and a very
welcome death for those of us who value logic as a practical matter.
Many of us have been doubtlessly tortured by the endless sing-song
chants of "if true, then false, and if false, then true," etc.,
printed endlessly onto the paper remains of ever more dead trees.
Let's work together to spread and even sharpen the logical necessity
of the long-awaited demise of "the Liar's Paradox." It's dead!
Very Respectfully,
Ray Donald Pratt
djr...@bath.ac.uk
"2++exp(+)=8". It is a concept that can be applied to mathematical
statements. If you are applying it to english-language statements,
that is different. The statement "this statement is meaningless" is
not meaningless in terms of being grammatically incorrect. It is just
an english-language mess.
Newberry
Could you please succintly state your thesis?
David C. Ullrich
A totally wrong afertthought.
> But anyway here is what Torkel Franzen says about self-
>reference:
>
>QUOTE:
>But sentences constructed in the proof that every arithmetical
>property P has a provable fixpoint are self-referential in a stronger
>sense: they are sentences A of the form
>
>There is an m such that m has the property P and property Q
>
>where it is provable in PA that the only number that has the property
>P is the Goedel number of the sentence itself. It is in this sense
>that the sentence A "says of itself it has property Q."
>END OF QUOTE [p. 45]
And in particular, it's not _literally_ self-referential,
so that any paradoxes or meaninglessnesses associated with
sentences like "this sentence is false" simply don't come up.
************************
David C. Ullrich
Gene Ledbetter
>
> Let P be the sentence "This sentence is meaningless." Is it true or
> false? It should not be difficult to answer. Tractatus Logico-
> Philosophicus says: "In order to tell whether a picture is true or
> false we compare it with reality." [2.223]
When someone says to you, "This sentence is ...", you should stop him
and ask, "What sentence are you referring to?"
No sentence exists. A sentence will not exist until one has been
completed. You cannot evaluate the meaningfulness of an unstated
statement.
Gene Ledbetter
Newberry
OK, so is "This sentence is meaingless" true or false?
Daryl McCullough
>When someone says to you, "This sentence is ...", you should stop him
>and ask, "What sentence are you referring to?"
But it is perfectly clear which sentence is being referred to.
--
Daryl McCullough
Ithaca, NY
Daryl McCullough
>
>Let P be the sentence "This sentence is meaningless." Is it true or
>false? It should not be difficult to answer. Tractatus Logico-
>Philosophicus says: "In order to tell whether a picture is true or
>false we compare it with reality." [2.223] When we attempt to compare
>"This sentence is meaningless" with reality we find that it is not
>comparable with anything. It is not a picture of a fact; it is
>meaningless.
>
>We can analyze the situation further:
>Case A: P is true.
>If P is true. Then it is the case that it is meaningless. But then it
>cannot be true. This is a contradiction. Therefore P is not true.
>
>Case B: P is false.
>If P is false then it is not the case that it is meaningless. It is
>the opposite of what it claims. This is a contradiction. Therefore P
>is not false.
>
>Case C: P is meaningless.
>If P is meaningless then nothing is the case. There is no
>contradiction.
That analysis is just silly. It doesn't clarify anything at all.
You came to the conclusion that P is meaningless. Was that
a meaningful conclusion, or not?
Perhaps you want to say that it is never meaningful to say that
something is not meaningful?
G. Frege
McCullough) wrote:
>
> Gene Ledbetter says...
>
>> When someone says to you, "This sentence is ...", you should stop him
>> and ask, "What sentence are you referring to?"
>>
> But it is perfectly clear which sentence is being referred to.
>
Especially if it is written down (i.e. an inscription).
F.
--
E-mail: info<at>simple-line<dot>de
Newberry
wrote:
> Newberry says...
>
>
>
>
>
>
>
> >Let P be the sentence "This sentence is meaningless." Is it true or
> >false? It should not be difficult to answer. Tractatus Logico-
> >Philosophicus says: "In order to tell whether a picture is true or
> >false we compare it with reality." [2.223] When we attempt to compare
> >"This sentence is meaningless" with reality we find that it is not
> >comparable with anything. It is not a picture of a fact; it is
> >meaningless.
>
> >We can analyze the situation further:
> >Case A: P is true.
> >If P is true. Then it is the case that it is meaningless. But then it
> >cannot be true. This is a contradiction. Therefore P is not true.
>
> >Case B: P is false.
> >If P is false then it is not the case that it is meaningless. It is
> >the opposite of what it claims. This is a contradiction. Therefore P
> >is not false.
>
> >Case C: P is meaningless.
> >If P is meaningless then nothing is the case. There is no
> >contradiction.
>
> That analysis is just silly. It doesn't clarify anything at all.
> You came to the conclusion that P is meaningless. Was that
> a meaningful conclusion, or not?
Absolutely.
> Perhaps you want to say that it is never meaningful to say that
> something is not meaningful?
P: Q is meaningless
Q: This sentence is meaningless
P is true, Q is meaningless. Since Q is not a picture of a possible
fact it is meaningless. That is what P says. Therefore P is true. Q
cannot say about itself that it is meaningless. Since it does not have
any meaning it cannot speak, so to speak.
Similarly "the sentence 'this sentence is meaningless' is meaningless"
is true.
Somebody on this board pointed out the above some time ago. I am
bringing this subject again because it is generally accepted that the
strengthened liar defeats the 3-valued approach. It seems to me
clearly false. Again, I am not the only one who figured that "this
sentence is meaningless" is meaningless. But a lot of folks still did
not get it.
Newberry
So why did Goedel say "the similarity with liar's paradox leaps in the
eye"?
Newberry
I did not say that it was grammatically incorrect. In fact the
sentence is grammatcally perfect. "Meaningless" means that it does not
have any meaning.
Meaningfulness id a semantic concept, grammatical correctness is a
syntactic concept.
David C. Ullrich
Because the proof is similar to the liar's paradox?
"Similar to" is not the same as "is the same as".
The liar's paradox is in fact somewhat paradoxical;
suggesting that there is hence something paradoxical
about the proof of Godel's theorem is just silly.
Because there's nothing literally self-referential
in the proof.
************************
David C. Ullrich
Daryl McCullough
>
>On Dec 26, 9:13=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> >Let P be the sentence "This sentence is meaningless." Is it true or
>> >false? It should not be difficult to answer. Tractatus Logico-
>> >Philosophicus says: "In order to tell whether a picture is true or
>> >false we compare it with reality." [2.223] When we attempt to compare
>> >"This sentence is meaningless" with reality we find that it is not
>> >comparable with anything. It is not a picture of a fact; it is
>> >meaningless.
>>
>> >We can analyze the situation further:
>> >Case A: P is true.
>> >If P is true. Then it is the case that it is meaningless. But then it
>> >cannot be true. This is a contradiction. Therefore P is not true.
>>
>> >Case B: P is false.
>> >If P is false then it is not the case that it is meaningless. It is
>> >the opposite of what it claims. This is a contradiction. Therefore =A0P
>> >is not false.
>>
>> >Case C: P is meaningless.
>> >If P is meaningless then nothing is the case. There is no
>> >contradiction.
>>
>> That analysis is just silly. It doesn't clarify anything at all.
>> You came to the conclusion that P is meaningless. Was that
>> a meaningful conclusion, or not?
>
>Absolutely.
>
>> Perhaps you want to say that it is never meaningful to say that
>> something is not meaningful?
>
>P: Q is meaningless
>Q: This sentence is meaningless
>
>P is true, Q is meaningless. Since Q is not a picture of a possible
>fact it is meaningless. That is what P says. Therefore P is true. Q
>cannot say about itself that it is meaningless. Since it does not have
>any meaning it cannot speak, so to speak.
Your resolution is not a resolution at all. It doesn't *resolve*
anything. *Why* is Q meaningless? Normally, a simple sentence can
be understood through (1) figuring out what the subject of the
sentence is, and (2) figuring out what is being said about that
subject. In the case of Q, the subject is Q itself. What is being
said about the subject is that it is meaningless. So what is
meaningless about Q? How *can* it be meaningless? Your analysis
doesn't actually explain anything.
Your resolution is actually inconsistent, as well. Rather than
talking about "This sentence", suppose that a guy named "Bob"
compiles a book that lists all the known paradoxes. It's called
"Bob's Book of Paradoxes". For example, it might go like this:
1. "All Cretans are liars" said the man from Crete.
2. This sentence is false.
3. Can God create a rock too heavy for him to lift?
4. Let R be the set of all sets that are not elements of themselves.
.
.
.
42. Sentence number 42 of "Bob's Book of Paradoxes" is meaningless.
.
.
.
Now, as shown above, sentence number 42 makes the claim that sentence
number 42 is meaningless. So it's just like "This sentence is meaningless".
So by your analysis, sentence number 42 is meaningless. So we conclude:
Sentence number 42 of "Bob's Book of Paradoxes" is meaningless.
But that *is* sentence number 42! Your "resolution" leads to a
meaningless conclusion in this case. So it's not a sound way
to reason. Your "cure" is as bad as the problem.
In my opinion, there is a simple, and very general resolution to
all semantic paradoxes, and that is to realize that semantic predicates
such as "true", "false", "meaningful", "meaningless" are all relative
to an *interpretation* of the words. You can't call a sentence "true"
unless you know what the nouns are supposed to refer to, and you know
the definitions of the predicates, and you know what collection of
objects the quantifiers range over. So there isn't just one truth
predicate or meaningfulness predicate, there is one for each possible
interpretation. So if we look at sentence P below:
This sentence is meaningless.
we can't say whether P is true, false, or meaningless until we
specify an interpretation of the terms. The usual interpretation
of "This sentence" in these kinds of games is that it refers
to the sentence it occurs in. But what about the predicate
"is meaningless"? As I have suggested, there is no absolute
notion of meaningfulness. So to know what "is meaningless" might
mean, we have to pick a standard for interpreting sentences.
So let I be some consistent, coherent way of interpreting sentences,
and let "is meaningless" be interpreted using I. Then sentence
P has the meaning:
"This sentence is meaningless" is meaningless under
interpretation I.
That's a perfectly meaningful sentence, but it *isn't* meaningful
under interpretation I. It's meaningful (and true) under an
interpretation I' that *extends* I.
Similarly,
This sentence is false.
can be interpreted to mean
"This sentence is false" is false under interpretation I.
That's a perfectly meaningful sentence, but it *isn't* meaningful
under interpretation I. So it isn't false under interpretation I.
So that sentence is meaningful (and false) under an
interpretation I' that *extends* I.
Finally,
This sentence is not true.
can be interpreted to mean
"This sentence is not true" is not true under interpretation I.
which is meaningless under interpretation I, but true under an extended
interpretation I'.
Newberry
Why is it significant that Goedel sentence is not literally self-
referential but just self-referential?
Newberry
wrote:
But it is not sentence 42. Obviously it is a different sentence..
> Ithaca, NY- Hide quoted text -
Newberry
wrote:
Again, this has already been discussed:
QUOTE:
> >This leads immediately to tokenism as
> > the thesis that different tokens
> >of a same non indexical sentence can
> > have different logical values
> >when used in different logical contexts.
DUH. That is basic. That is trivial. That has been known
since Frege if not before. Frege's personal example in
the relevant paper was "Russia and Canada quarreled today."
Obviously that has different truth-values depending on what
day it is.
END OF QUOTE
http://groups.google.com/group/sci.logic/browse_frm/thread/22e12a7a46f65939/4da50441d92aa73f?lnk=gst&q=tokenism#4da50441d92aa73f
Daryl McCullough
>Again, this has already been discussed:
Yes, I know. It's well known that the three-value "resolution"
is worthless.
Daryl McCullough
>
>On Dec 27, 5:54=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>>42. Sentence number 42 of "Bob's Book of Paradoxes" is meaningless.
>> So by your analysis, sentence number 42 is meaningless. So we conclude:
>>
>> Sentence number 42 of "Bob's Book of Paradoxes" is meaningless.
>>
>> But that *is* sentence number 42!
>
>But it is not sentence 42. Obviously it is a different sentence...
That's ridiculous. A "sentence" is a sequence of characters. Sentence
42 is the sentence whose first character is 'S', whose second character
is 'e', whose third character is 'n', etc. It's the *same* sentence.
You can certainly talk about *occurrences* of sentences in which
the same sequence appearing in two different books are different
occurrences. But Sentence 42 specifically talks about the sequence
of characters. If you like, we can change the example:
42. The sequence of characters appearing at position 42 of "Bob's Book
of Paradoxes" is a meaningless sequence of characters.
Your tokenism *doesn't* address the problem! It's *not* a resolution
at all.
Daryl McCullough
>Why is it significant that Goedel sentence is not literally self-
>referential but just self-referential?
Only because it is possible to ban literal self-reference, but it
is not possible to ban indirect self-reference (at least not without
severely handicapping the ability to use language).
Marshall
>
>
>
> > Let P be the sentence "This sentence is meaningless." Is it true or
> > false? It should not be difficult to answer. Tractatus Logico-
> > Philosophicus says: "In order to tell whether a picture is true or
> > false we compare it with reality." [2.223]
>
> When someone says to you, "This sentence is ...", you should stop him
> and ask, "What sentence are you referring to?"
Interrupting someone mid sentence is rude. I cannot endorse
your approach.
> No sentence exists. A sentence will not exist until one has been
> completed. You cannot evaluate the meaningfulness of an unstated
> statement.
If you'd just let the guy finish talking, you'd have a stated
sentence to evaluate. Your critique only applies for as long
as it takes the guy to reach the end of the sentence.
Marshall
G. Frege
<marshal...@gmail.com> wrote:
>>
>> When someone says to you, "This sentence is ...", you should stop him
>> and ask, "What sentence are you referring to?"
>>
> Interrupting someone mid sentence is rude. I cannot endorse
> your approach.
>
Moreover a sentence is a "semantical unit". You simply have to wait
until it has been finished (i.e. completely uttered).
>>
>> No sentence exists. A sentence will not exist until one has been
>> completed. You cannot evaluate the meaningfulness of an unstated
>> statement.
>>
> If you'd just let the guy finish talking, you'd have a stated
> sentence to evaluate.
>
Right.
Newberry
wrote:
Those are still two occurrences of sequences of characters. They
express two different propositions. The one tagged with 42 is Q and
the one about it is P.
P: Q is meaningless
Q: This sentence is meaningless
These two occurrence are homonyms so to speak. It is a well know fact
that natural language sentences cannot be parsed by syntax alone, but
must be evaluated according to the context. In your example one has
conveniently the tag "42" attached to it.
Newberry
wrote:
We get a contradiction anyway. All we have to do is to add
Ex (P_ultimate(x,#F)) -> F (1)
to the system.
Goedel's sentence states its own truth when interpreted. The new axiom
above is the formalization of the interpretation. It states the truth
if PA is consistent. It is so because all its axioms are manifestly
true and they cannot possibly be inconsistent with themselves. Then
since PA is consistent then (1) is equally manifestly true.
LauLuna
> Let P be the sentence "This sentence is meaningless." Is it true or
> false? It should not be difficult to answer. Tractatus Logico-
> Philosophicus says: "In order to tell whether a picture is true or
> "This sentence is meaningless" with reality we find that it is not
> comparable with anything. It is not a picture of a fact; it is
> meaningless.
>
> We can analyze the situation further:
> Case A: P is true.
> If P is true. Then it is the case that it is meaningless. But then it
> cannot be true. This is a contradiction. Therefore P is not true.
>
> Case B: P is false.
> If P is false then it is not the case that it is meaningless. It is
> is not false.
No. There's no contradiction here. It can have a meaning and be false.
Nevertheless, pronouncing it false seems unsatisfactory because it
relies on Bivalence for sentences, which we know not always holds.
> Case C: P is meaningless.
> If P is meaningless then nothing is the case. There is no
> contradiction. In a three valued logic (T, F, M) we conclude that P is
> meaningless.
>
> It is not correct to say that if the sentence is meaningless than what
> it SAYS is true. This argument assumes that it becomes TRUE half way
> through the argument and then it IS THE CASE that it is meaningless.
> Thus the sentence confirms our initial assumption that it was
> meaningless. But if it stays meaningless all the time it confirms
> nothing.
>
> Clearly, if the sentence does not have any meaning then it does not
> have the meaning that it is meaningless.
>
> This gives us the basic insight that all self-referential, paradoxical
> sentences, including possibly Goedel's sentence, are probably
> meaningless.
No. You have to distinguish propositions and sentences. A sentence is
a chain of symbols; a proposition is the information content an
interpreted proposition conveys. No proposition is about itself but a
proposition can be about the sentence that expresses it:
(1) this sentence has five words
Gödel's G is alike (1), not alike the Liar. G, meta-theoretically
intrepreted, refers to a string of symbols in a formal language,
whatever its possible interpretations may be.
Regards
LauLuna
I must say I don't understand your new axiom; it seems like a
reflection principle, but I'm not sure.
But there is no way you could make Gödel's sentence state its own
truth. Remember Tarski theorem: there is no arithmetical predicate
expressing the truth predicate for arithmetical sentences. Since
Gödel's contains only an arithmetical predicate, it cannot be
interpreted as stating or denying its own truth.
Regards
Daryl McCullough
>
>On Dec 27, 9:52=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> Newberry says...
>>
>> >Why is it significant that Goedel sentence is not literally self-
>> >referential but just self-referential?
>>
>> Only because it is possible to ban literal self-reference, but it
>> is not possible to ban indirect self-reference (at least not without
>> severely handicapping the ability to use language).
>
>We get a contradiction anyway. All we have to do is to add
> Ex (P_ultimate(x,#F)) -> F (1)
>to the system.
What I argued is that there *is* no P_ultimate.
>Goedel's sentence states its own truth when interpreted.
No, it doesn't. It states its own unprovability.
Daryl McCullough
>
>On Dec 27, 9:45=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> 42. The sequence of characters appearing at position 42 of "Bob's Book
>> of Paradoxes" is a meaningless sequence of characters.
>
>Those are still two occurrences of sequences of characters.
Yes, but it doesn't talk about *occurrences*, it talks about
*sequences*. The character sequences are the same.
>They express two different propositions.
No, they don't. In both cases, the subject is the sequence of
characters 'T' 'h' 'e' ' ' 's' 'e' 'q' 'u' 'e' 'n' 'c' 'e' ' '
'o' 'f', etc. In both cases, the predicate is "...is meaningless".
Your "resolution" doesn't work.
Newberry
wrote:
> Newberry says...
>
>
>
> >On Dec 27, 9:52=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
> >wrote:
> >> Newberry says...
>
> >> >Why is it significant that Goedel sentence is not literally self-
> >> >referential but just self-referential?
>
> >> Only because it is possible to ban literal self-reference, but it
> >> is not possible to ban indirect self-reference (at least not without
> >> severely handicapping the ability to use language).
>
> >We get a contradiction anyway. All we have to do is to add
> > Ex (P_ultimate(x,#F)) -> F (1)
> >to the system.
>
> What I argued is that there *is* no P_ultimate.
because it results in a contradiction
>
> >Goedel's sentence states its own truth when interpreted.
>
> No, it doesn't. It states its own unprovability.
I meant unprovability.
Newberry
> On 28 dic, 04:54, Newberry <newberr...@gmail.com> wrote:
>
>
>
>
>
> > On Dec 27, 9:52 am, stevendaryl3...@yahoo.com (Daryl McCullough)
> > wrote:
>
> > > Newberry says...
>
> > > >Why is it significant that Goedel sentence is not literally self-
> > > >referential but just self-referential?
>
> > > Only because it is possible to ban literal self-reference, but it
> > > is not possible to ban indirect self-reference (at least not without
> > > severely handicapping the ability to use language).
>
> > We get a contradiction anyway. All we have to do is to add
> > Ex (P_ultimate(x,#F)) -> F (1)
> > to the system.
>
> > Goedel's sentence states its own truth when interpreted. The new axiom
> > above is the formalization of the interpretation. It states the truth
> > if PA is consistent. It is so because all its axioms are manifestly
> > true and they cannot possibly be inconsistent with themselves. Then
> > since PA is consistent then (1) is equally manifestly true.
>
> I must say I don't understand your new axiom; it seems like a
> reflection principle, but I'm not sure.
>
> But there is no way you could make Gödel's sentence state its own
> truth.
I meant it states its own unprovability.
Remember Tarski theorem: there is no arithmetical predicate
> expressing the truth predicate for arithmetical sentences.
BTW, I have derived a contradiction here.
> Since
> Gödel's contains only an arithmetical predicate, it cannot be
> interpreted as stating or denying its own truth.
>
> Regards- Hide quoted text -
Newberry
wrote:
First of all, strictly speaking sequences of characters are not
sentences. They do not express any propositions and they are all
meaningless.
Secondly, the grammatical subject is actually "The sequence of
characters appearing at position 42" But in the sequence 42 it does
not refer to anything. In the other sequence it refers to the sequence
of characters at 42.
Daryl McCullough
>
>On Dec 28, 6:16=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> >> 42. The sequence of characters appearing at position 42 of "Bob's Book
>> >> of Paradoxes" is a meaningless sequence of characters.
>>
>> >Those are still two occurrences of sequences of characters.
>>
>> Yes, but it doesn't talk about *occurrences*, it talks about
>> *sequences*. The character sequences are the same.
>>
>> >They express two different propositions.
>>
>> No, they don't. In both cases, the subject is the sequence of
>> characters 'T' 'h' 'e' ' ' 's' 'e' 'q' 'u' 'e' 'n' 'c' 'e' ' '
>> 'o' 'f', etc. In both cases, the predicate is "...is meaningless".
>
>First of all, strictly speaking sequences of characters are not
>sentences. They do not express any propositions and they are all
>meaningless.
That's completely silly. A sentence is certainly a sequence of
characters (although not every sequence of characters is a sentence).
>Secondly, the grammatical subject is actually "The sequence of
>characters appearing at position 42" But in the sequence 42 it does
>not refer to anything. In the other sequence it refers to the sequence
>of characters at 42.
You're being ridiculous. It's the *same* sequence of characters!
The following sequence of characters
The sequence of characters appearing at position 42 of "Bob's Book
of Paradoxes" is a meaningless sequence of characters.
*is* the sequence of characters appearing at position 42 of "Bob's
Book of Paradoxes". That same sequence of characters *also* appears
in other places (for instance, in this post it appears several times).
Your resolution is clearly nonsensical. It is inconsistent.
If we let Q be the sequence of characters
Q is a meaningless sequence of characters.
Then, is it true that the above sequence is a meaningless
sequence of characters? If so, then the conclusion is
Q is a meaningless sequence of characters.
which is a sequence of characters, and the name of that
sequence of characters is Q. Your proposal has the unacceptable
property that a logical deduction from true premises can be
a meaningless sequence of characters.
Your "resolution" is not a resolution at all.
--
Daryl McCullough
Ithaca, NY
LauLuna
wrote:
>
> - Show quoted text -
I think yours is an attempt to avoid tokenism by explicitly typing all
sentences. As I see it, what your interpretations provide are simply
'different logical levels'.
The problems are:
1. English is already an interpreted language; adding a requirement of
interpretation to English sentences leads to an infinite regress.
That's why I think your way of typing is not the most fortunate one.
2. It's very dubious that sentences can be explicitly typed or
distributed into logical levels in such a way that all expressions get
syntactically disambiguated and tokenism is no longer needed. Assume
there is a collection C of syntactic marks or types (expressing
levels, interpretations or whatever is required for disambiguation) by
which natural language can be so extended that any expression gets
logically disambiguated. C should be a part of a grammar able to
disambiguate all expressions; as such C must be nameable. Now consider
LC 'this sentence is meaningless in all levels/interpretations in C'
Meta-LC 'LC is meaningless in all levels/interpretations in C'
It seems we need a new level not in C for LC, contrary to what was
assumed about C. So, C does not exist.
I mean there is no grammar able to incorporate our capability of
distinguishing different tokens of a same expression-type as
possessing different logical and semantical values.
Regards
LauLuna
>
> - Show quoted text -
Yes, I think you're right here.
Any language as expressive as natural language is doomed to logical/
semantical ambiguity that can only be solved by the logical context. I
don't think there is a syntactic device able to solve that; we are
condemned to tokenism while we still wish to keep the whole expressive
power of natural language.
Regards
LauLuna
> First of all, strictly speaking sequences of characters are not
> sentences. They do not express any propositions and they are all
> meaningless.
I think you'd rather say that propositions are not sequences of
characters. As Daryl McCullough says, sentences are commonly taken as
purely syntactic objects.
> Secondly, the grammatical subject is actually "The sequence of
> characters appearing at position 42" But in the sequence 42 it does
> not refer to anything. In the other sequence it refers to the sequence
> of characters at 42.
I think you're perfectly right on this.
LauLuna
>
> - Show quoted text -
But this is an indexical sentence. Tokenism is the claim that even non
indexical expression-tokens (or occurrences, perhaps) of the same
expression-type may be logically non equivalent.
I think you're on the right way. Please, let me suggest again a
distinction between sentences (as syntactic objects) and propositions
(the semantic objects sentences use to express according to some
linguistic codes). This can show, for instance, that there is no
strict self-reference in Gödel sentence.
The distinction between expression-types and expression-tokens is
necessary as well.
Let me add that, while this kind of solution seems appealing and
intuitive, it is by no means fully developed and clarified. E.g. why
and when is self-reference impossible?
Consider the three following sentences:
A) this sentence has five words
B) this sentence is false
C) all propositions are either true or false
All of them seem to be self-referential in some way but only B) seems
pathological.
Why?
Newberry
wrote:
> Newberry says...
>
>
>
>
>
>
>
> >On Dec 28, 6:16=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
> >wrote:
> >> >> 42. The sequence of characters appearing at position 42 of "Bob's Book
> >> >> of Paradoxes" is a meaningless sequence of characters.
>
> >> >Those are still two occurrences of sequences of characters.
>
> >> Yes, but it doesn't talk about *occurrences*, it talks about
> >> *sequences*. The character sequences are the same.
>
> >> >They express two different propositions.
>
> >> No, they don't. In both cases, the subject is the sequence of
> >> characters 'T' 'h' 'e' ' ' 's' 'e' 'q' 'u' 'e' 'n' 'c' 'e' ' '
> >> 'o' 'f', etc. In both cases, the predicate is "...is meaningless".
>
> >First of all, strictly speaking sequences of characters are not
> >sentences. They do not express any propositions and they are all
> >meaningless.
>
> That's completely silly. A sentence is certainly a sequence of
> characters (although not every sequence of characters is a sentence).
>
> >Secondly, the grammatical subject is actually "The sequence of
> >characters appearing at position 42" But in the sequence 42 it does
> >not refer to anything. In the other sequence it refers to the sequence
> >of characters at 42.
>
> You're being ridiculous.
That's OK. We want to have some fun too.
It's the *same* sequence of characters!
> The following sequence of characters
>
> The sequence of characters appearing at position 42 of "Bob's Book
> of Paradoxes" is a meaningless sequence of characters.
>
> *is* the sequence of characters appearing at position 42 of "Bob's
> Book of Paradoxes". That same sequence of characters *also* appears
> in other places (for instance, in this post it appears several times).
>
> Your resolution is clearly nonsensical. It is inconsistent.
>
> If we let Q be the sequence of characters
>
> Q is a meaningless sequence of characters.
>
> Then, is it true that the above sequence is a meaningless
> sequence of characters? If so, then the conclusion is
>
> Q is a meaningless sequence of characters.
>
> which is a sequence of characters, and the name of that
> sequence of characters is Q. Your proposal has the unacceptable
> property that a logical deduction from true premises can be
> a meaningless sequence of characters.
>
> Your "resolution" is not a resolution at all.
What about "it is raining"? If it rains today and I utter it today it
is true. If it will not rain toorrow and I will say it tomorroe it
will be false. And it will be exactly the same sequence of characters.
How about this:
xyz
This is immediately below "xyz."
This is immediately below "xyz."
Newberry
I meant to say that G says about itself that it is not provable. Since
it is indeed not provable it is true. We decided it by
metamathematical considerations. The methamatematical consideration
can be formalized using
Ex (P(x,#F)) -> F (a)
(if there is a proof of Godel number of F then F)
Daryl claims that adding (a) to T is equivalent to T + G. So now the
question is if we can construct P_ultimate, which would prove all of
them i.e.
G+G'+G" ... +Gomega+Gomega'+Gomega"+ ... +Gomega^omega + ... etc.
Daryl seemed to claim in a previous thread that such a P_ultimate can
be constructed. Now he is saying that it cannot.
Newberry
>
> > - Show quoted text -
>
> But this is an indexical sentence. Tokenism is the claim that even non
> indexical expression-tokens (or occurrences, perhaps) of the same
> expression-type may be logically non equivalent.
>
> I think you're on the right way. Please, let me suggest again a
> distinction between sentences (as syntactic objects) and propositions
> (the semantic objects sentences use to express according to some
> linguistic codes). This can show, for instance, that there is no
> strict self-reference in Gödel sentence.
>
> The distinction between expression-types and expression-tokens is
> necessary as well.
>
> Let me add that, while this kind of solution seems appealing and
> intuitive, it is by no means fully developed and clarified. E.g. why
> and when is self-reference impossible?
>
> Consider the three following sentences:
>
> A) this sentence has five words
>
> B) this sentence is false
>
> C) all propositions are either true or false
>
> All of them seem to be self-referential in some way but only B) seems
> pathological.
>
>
> - Show quoted text -
I will expand on this shortly. Did you write any paper about tokenism?
URL?
LauLuna
>
> - Show quoted text -
Thanks for your explanation.
Well, your (a) is certainly a kind of reflection principle. I'd say PA+
(a) is at least as strong as PA+G, but surely Daryl is right on this.
What is clear, from Gödel's theorem, is that no ultimate theory can be
reached by adding successive versions of G or (a). If we start with a
consistent arithmetical theory what we obtain is always again a
consistent arithmetical theory and there will always be an unprovable
true sentence.
BTW, assume you have that ultimate system US (no pun), i.e. a sound
and complete arithmetical system. Then the predicate BEW (i.e.
'provable') for US will represent the truth predicate for US language.
Then Gödel sentence G_US, in its meta-theoretical interpretation, will
become a version of the Liar. Then we will have a contradiction in
arithmetic itself, since G_US expresses equivalent propositions under
its meta-theoretical and its arithmetical interpretation.
This is the relationship between Gödel's procedure and the Liar. But
as you can see, Gödel falls into no paradox. In a rough and general
way, it can be said that limitation theorems (such as Gödel's,
Tarski's, Church's, Turing's) reveal some limits and those limits are
what paradoxes violate. This is why, as Gödel wrote, we can use almost
any epistemic paradox to produce a limitation theorem.
Best regards.
Daryl McCullough
>What about "it is raining"?
That isn't relevant to this discussion. The problem there
is that "it is raining" has an implicit time parameter,
namely, the time that it is uttered. So its meaning is
ambiguous if you don't know the time it was uttered.
But paradoxical sentences do *not* rely on ambiguity of
reference. So basing your resolution of the paradoxes
on ambiguity is just incorrect. Look again:
This sentence is not true.
There is no ambiguity about what "this sentence" refers to.
By convention, it refers to the sentence it appears in:
"This sentence is not true.". Pretending that there is
an ambiguity of reference is just incorrect. There is no
such ambiguity. A resolution of the semantic paradoxes
that relies on referential ambiguity is just an *incorrect*
resolution.
It's possible to restrict oneself to a restricted language
in which all noun phrases are unambiguous. That's the case
with mathematical theories. In a sufficiently expressive
(but still unambiguous) language, it is possible to construct
sentences equivalent to
"This sentence has property P"
for every expressible property P of (codes of) sentences. But
the liar paradox applied to such languages shows that there
is no expressible property P that holds of a (code for) a
sentence if and only if that sentence is true.
So it is just a mistake to believe that the problem with
"This sentence is not true" has anything to do with difficulties
of reference. The difficulty is with the predicate "is true"
(or "is meaningful" or "is false"). Your analysis is just not
correct.
Daryl McCullough
>
>On Dec 27, 2:54=A0pm, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>The problems are:
>
>1. English is already an interpreted language;
Not completely. English is for most practical purposes
an interpreted language, but that doesn't mean that *every*
expressible English sentence has a unique interpretation,
or that every word of English has a consistent, unambiguous
definition. The liar paradox in particular shows (in my
opinion) that there is no unique meaning to the semantic
predicates "is true", "is false", "is meaningless", etc.
We *usually* know what those words mean, but not always.
In particular, in the sentence
This sentence is not true.
we really don't know what "is not true" means.
>adding a requirement of interpretation to English sentences
>leads to an infinite regress.
I didn't *require* extra interpretation. I simply made the
claim that the semantic paradoxes can be understood by making
interpretations explicit.
>2. It's very dubious that sentences can be explicitly typed or
>distributed into logical levels in such a way that all expressions get
>syntactically disambiguated and tokenism is no longer needed.
As I understand it, tokenism is just a special case of what I'm
talking about. The meaning of a sentence depends on the interpretation
of its words, and which interpretation is appropriate may depend on
the context in which the sentence was uttered.
>Assume there is a collection C of syntactic marks or types (expressing
>levels, interpretations or whatever is required for disambiguation) by
>which natural language can be so extended that any expression gets
>logically disambiguated. C should be a part of a grammar able to
>disambiguate all expressions; as such C must be nameable. Now consider
>
>LC 'this sentence is meaningless in all levels/interpretations in C'
That shows that the collection C cannot be expressed in natural language.
It doesn't show that C does not exist.
Newberry
wrote:
The point is that two different instances of the same sentence can
have different meanings. There is simply no a priori requirement that
they cannot have different meaning.
Looking at your particular example;
42: The sentence 42 is not true.
56: The The sentence 42 is not true.
The second instance actually says:
56: "The The sentence 42 is not true" is not true.
Newberry
But we actually do have a contradiction. The axioms of PA are
manifestly true. truth cannot be inconsistent with itself, therefore
PA is consistent. This is just as manifestly true as the axioms
themselves. Now Goedel's sentence tells the truth if PA is consistent.
But it is manifestly true that PA is consistent. Therefore
Ex (P(x,#F)) -> F (a)
is just as manifestly true as the rest of the axioms. But if PA is
consistent then the entire hierarchy
T, T' = T + G, T" = T' + G' ... is consistent. Therefore if we can
construct
Ex (P_ultimate(x,#F)) -> F (1)
that can prove the entire hierarchy it is also true. But (1) results
in a contradiction.
Marshall
> On Thu, 27 Dec 2007 15:21:32 -0800 (PST), Marshall
>
> >> completed. You cannot evaluate the meaningfulness of an unstated
> >> statement.
>
> > If you'd just let the guy finish talking, you'd have a stated
> > sentence to evaluate.
>
> Right.
Thanks.
Say, all this talk about "this sentence is false" raises a
related question for me that I haven't seen discussed.
Would you (or whoever) care to comment on this sentence:
This sentence is true.
The liar's sentence has this xor sort of property in that it's
false if it's true and true if it's false. The sentence I gave
has a related property: it's true if it's true and it's false
if it's false. (Did I get that right?)
In both cases we have to decide the truth of the sentence
in order to decide the truth of the sentence. However
in the liar's sentence, any such decisions are inconsistent,
but in the truth teller's sentence, any such decisions are
consistent.
I am tempted to say that the liar's sentence is overspecified,
and the truth teller's sentence is underspecified, both in
an infinitely recursive way.
Or am I way off base here?
Marshall
ChrisX
"Marshall" <marshal...@gmail.com> wrote in message
news:09d73875-90ec-4350...@z26g2000pre.googlegroups.com...
I read the rest of the thread but I'd still say both sentences
are meaningless. They're meaningless because if
you interpret the individual words and word-order in the
usual fashion, the result doesn't actually specify anything.
So I don't think either sentence is an over-specification like
a contradiction or an under-specification like a tautology,
but a failure to specify, like division by zero. It's just one of
those facts about the concept of truth that any sentence
whose constituents seem to say no more than that it is
true or false actually says nothing. That's not even very
surprising really because you'd normally decide whether a
sentence is true or false based on what it says, and these
ask you to evaluate them based on nothing.
Love and respect
Chris
Newberry
Yes, the problem with both sentences is that when you try to compare
it with reality you get stuck in an infinite loop. None of them is a
picture of a possible fact. They are both meaningless.
Newberry
> "Marshall" <marshall.spi...@gmail.com> wrote in message
Absolutely correct. But some people claim that "This senetnce is false
or meaningless" is self-contradictory.
> Love and respect
> Chris- Hide quoted text -
djr...@bath.ac.uk
> On Dec 25, 5:42 pm, djr...@bath.ac.uk wrote:
>
> > Meaninglessness is when a statement is grammatically incorrect, like
> > "2++exp(+)=8". It is a concept that can be applied to mathematical
> > statements. If you are applying it to english-language statements,
> > that is different. The statement "this statement is meaningless" is
> > not meaningless in terms of being grammatically incorrect. It is just
> > an english-language mess.
>
> I did not say that it was grammatically incorrect. In fact the
> sentence is grammatcally perfect. "Meaningless" means that it does not
> have any meaning.
>
> Meaningfulness id a semantic concept, grammatical correctness is a
> syntactic concept.
I did not say that you said that it was grammatically incorrect. My
point was that whatever you think about the statement, it has no
relevance to mathematics and mathematical logic. You will never make
any Russell's paradoxes out of it. You could make a fuss about it, but
not a mathematical fuss.
Marshall
> Meaninglessness is when a statement is grammatically incorrect, like
> "2++exp(+)=8". It is a concept that can be applied to mathematical
> statements.
Hmmm. Do you mean to imply that every syntactically well-formed
mathematical expression is meaningful? What is the meaning of
1/0?
Marshall
Daryl McCullough
>Looking at your particular example;
>42: The sentence 42 is not true.
>56: The The sentence 42 is not true.
>
>The second instance actually says:
>56: "The The sentence 42 is not true" is not true.
Let's get rid of the extra "The".
42. Sentence number 42 is not true.
56. Sentence number 42 is not true.
Sentence 42 is the *same* sentence as sentence 56.
And there are *no* ambiguous referring expressions.
So if 42 is meaningless, then so is 56.
djr...@bath.ac.uk
Yes, or at least statements in elementary number theory, for example
(I don't wish to get into a debate about the continuum hypothesis).
The statement "1/0" is not syntactically correct.
Marshall
> On Dec 29, 10:33 pm, Marshall <marshall.spi...@gmail.com> wrote:
>
> > On Dec 25, 5:42 pm, djr...@bath.ac.uk wrote:
>
> > > Meaninglessness is when a statement is grammatically incorrect, like
> > > "2++exp(+)=8". It is a concept that can be applied to mathematical
> > > statements.
>
> > Hmmm. Do you mean to imply that every syntactically well-formed
> > mathematical expression is meaningful? What is the meaning of
> > 1/0?
>
> (I don't wish to get into a debate about the continuum hypothesis).
Okay.
> The statement "1/0" is not syntactically correct.
An intriguing statement! Can you supply further justification?
It appears syntactically correct to me, but perhaps I am not
clear what you mean by that.
Further thoughts:
What is the meaning of the expression
1/(x-1)
When x=2? When x=1? It is the same syntax in both cases, isn't it?
Marshall
Kenneth Doyle
I think that what he's saying is that because "sentence number 42" is the
name of a sentence then we can substitute the actual sentence for its
name, in sentence number 56, giving:
Sentence number 42 is not true, is not true.
I think that somehow strengthens the case that sentence number 42 is
indeed meaningless. Not sure exactly why.
djr...@bath.ac.uk
"1/(x-1)" is not a statement. Something like "1/(x-1)<1 when x>2" is a
syntactically correct statement, and a true statement.
Newberry
> On Sat, 29 Dec 2007 15:39:28 -0800, Daryl McCullough wrote:
> > Newberry says...
>
> >>Looking at your particular example;
> >>42: The sentence 42 is not true.
> >>56: The The sentence 42 is not true.
>
> >>The second instance actually says:
> >>56: "The The sentence 42 is not true" is not true.
>
> > Let's get rid of the extra "The".
>
> > 42. Sentence number 42 is not true.
> > 56. Sentence number 42 is not true.
>
> > Sentence 42 is the *same* sentence as sentence 56. And there are *no*
> > ambiguous referring expressions. So if 42 is meaningless, then so is 56.
>
> I think that what he's saying is that because "sentence number 42" is the
> name of a sentence then we can substitute the actual sentence for its
> name, in sentence number 56, giving:
>
> Sentence number 42 is not true, is not true.
>
Yes, I am saying that. I am saying that
56. Sentence number 42 is not true.
CAN be translated as
56'. "Sentence number 42 is not true" is not true.
56 and 56' are saying the SAME thing. I do not know if we can say
42. "Sentence number 42 is not true" is not true
It sounds incoherent.
> I think that somehow strengthens the case that sentence number 42
is
> indeed meaningless. Not sure exactly why.
Because it shows that 42 and 56 mean two different things.
Newberry
>
> > - Show quoted text -
>
> But this is an indexical sentence. Tokenism is the claim that even non
> indexical expression-tokens (or occurrences, perhaps) of the same
> expression-type may be logically non equivalent.
>
> I think you're on the right way. Please, let me suggest again a
> distinction between sentences (as syntactic objects) and propositions
> (the semantic objects sentences use to express according to some
> linguistic codes). This can show, for instance, that there is no
> strict self-reference in Gödel sentence.
>
> The distinction between expression-types and expression-tokens is
> necessary as well.
>
> Let me add that, while this kind of solution seems appealing and
> intuitive, it is by no means fully developed and clarified. E.g. why
> and when is self-reference impossible?
>
> Consider the three following sentences:
>
> A) this sentence has five words
>
> B) this sentence is false
>
> C) all propositions are either true or false
>
> All of them seem to be self-referential in some way but only B) seems
> pathological.
>
> Why
A sentence is meaningful if it is a picture of a possible fact. A
meaningful sentence it true if it corresponds to reality, false
otherwise. For example the apple I am looking at is green. But it is
possible that it would be red. Therefore the sentence "The apple is
red" is meaningful but false.
A: "The apple is green" <---> The apple is green
B: "The apple is red" <---> The apple is green
C: "The Good is identical" <---> [blank]
Sentences themselves are facts and we can make other sentences about
them. Thus we have
K: "A is true" <---> {"The apple is green" <---> The apple is green}
L: "B is false" <---> {"The apple is red" <---> The apple is green}
M: "C is meaningless" <---> {"The Good is identical" <---> [blank]}
In order to discover if
D: "This sentence is false"
is true we have to compare it with reality. That is we have to check
if D is false.
A1: "This sentence is false" <---> "This sentence is false" is false
But the fact on the right side is an attribute of a sentence. We need
to observe if "This sentence is false" is false. So again we have to
compare D with reality. This gets us into an infinite loop and the
comparison with reality cannot be done. D is not a picture of a
possible fact. So we have:
M1: "D is meaningless" <---> {"This sentence is false" <---> [blank]}
The situation is different with
E: "This sentence has five words"
We can very easily compare E with reality. The sentence itself is a
fact and all we have to do is to is to count the words in it. E refers
to syntactic features of itself that can be readily verified and
there is no infinite loop. So we have:
K1: " 'This sentence has five words' is true" <---> {"This sentence
has five words" <---> "This sentence has five words" has five words}
Now let us consider
F: "This sentence is meaningless."
How do we compare it with reality. The sentence says that F <-->
[blank]. Is this the case? To find that out we need to compare F <-->
[blank] with reality, so we have to interpret F. The interpretation of
F is F <--> [blank]. We again have infinite regress.
M2: "F is meaningless" <---> {"This sentence is meaningless." <--->
[blank]}
The important observation is that F is not saying anything. In
particular it is not saying that it is meaningless. M2 does say it but
a sentence cannot say about itself that it is meaningless. In order to
see that let's try a variant of M
M': M' is meaningless <---> { M' is meaningless <---> [blank]}
There is no need to say that the two different strings "M' is
meaningless" are NOT two different instances but the same instance
unfolded. We now observe that M' cannot be the case. The "M' is
meaningless" on the left is meaningful but the one on the right is
not. M' is the root of the extended liar fallacy. It assumes both at
the same time that the sentence is meaningful and that it is not.
Aatu Koskensilta
> But we actually do have a contradiction. The axioms of PA are
> manifestly true. truth cannot be inconsistent with itself, therefore
> PA is consistent. This is just as manifestly true as the axioms
> themselves. Now Goedel's sentence tells the truth if PA is consistent.
> But it is manifestly true that PA is consistent. Therefore
>
> Ex (P(x,#F)) -> F (a)
>
> is just as manifestly true as the rest of the axioms. But if PA is
> consistent then the entire hierarchy
> T, T' = T + G, T" = T' + G' ... is consistent. Therefore if we can
> construct
>
> Ex (P_ultimate(x,#F)) -> F (1)
>
> that can prove the entire hierarchy it is also true. But (1) results
> in a contradiction.
What is "P_ultimate"? If it is just the provability predicate for the union
of the theories T, T', T'', ... there is no contradiction, and indeed we can
observe that this union is manifestly true, and go on to consider the theory
T_omega+1 obtained from the union by adding the Gödel sentence of the
theory, and so on -- so P_ultimate is not quite ultimate. A beautiful
exposition of this and related issues is to be found in Torkel Franzén's
_Inexhaustibility -- a non-exhaustive treatment_.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> A sentence is meaningful if it is a picture of a possible fact. A
> meaningful sentence it true if it corresponds to reality, false
> otherwise.
Positively Wittgensteinian. Alas, all this talk about correspondence to
reality and so on is meaningless twaddle, according to Tractatus.
Newberry
wrote:
> On 2007-12-30, in sci.logic, Newberry wrote:
>
> > A sentence is meaningful if it is a picture of a possible fact. A
> > meaningful sentence it true if it corresponds to reality, false
> > otherwise.
>
> Positively Wittgensteinian.
It is also common sense.
> Alas, all this talk about correspondence to
> reality and so on is meaningless twaddle, according to Tractatus.
I use common sense, not regurgitated Tractatus. I think the talk about
correspondence to reality is meaningful.
But let's assume that it is not. How does it follow that "This
senetnce is meaningless" is meaningful? And what does it mean?
>
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
Newberry
wrote:
42 is self-referential, 56 is not. So 42 has a property that 56 does
not. I.e. the entences are different.
LauLuna
>
> - Show quoted text -
There is no P_ultimate. It would imply the existence of a largest
constructive ordinal. As Aatu Koskensilta says, this is the
inexhaustibility phenomenon. Inexhaustibility prevents the existence
of a paradox.
I tend to think that the inexhaustibility phenomenon is the essential
teaching of paradoxes.
Regards
Newberry
wrote:
>
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
>
> - Show quoted text -
This is what was said in another thread:
>> Now, if you want to go for the whole ball of wax and
>> come up with a theory T_ultimate with the following
>> property:
>> T_ultimate proves
>> Ex (P_ultimate(x,#F)) -> F
>> There is no such theory T_ultimate except for an
>> inconsistent theory.
>This is interesting stuff. Where can I read about it? How do you
>construct P_ultimate(x,y)?
Well, as I said, it's inconsistent, so it's not really that
interesting. But you can do this:
Let Pr(x,y,z) be defined so that it holds if and only
if x is a code for an r.e. theory T, and y is a code for a
proof p, and z is a code for a formula S, and p is a proof
of S from the axioms of T.
If #T0 is the code for an r.e. set of axioms in the language
of PA, then let f(#T0) be the code for the theory T1 where
the axioms of T1 consists of the axioms of T0 plus the "soundness"
schema for T0, which is, for every formula S in the language of
PA,
Ex Pr(#T0,x,#S) -> S
Now we just use the fixed point theorem for r.e. sets
to come up with a theory T extending PA such that
#T and f(#T) code the same r.e. set of axioms.
Here's what the fixed point theorem for r.e. sets says:
If x is any natural number, then let W_x be the r.e. set
coded by x (or the empty set, if x doesn't code anything).
Then if f is any recursive function from naturals to naturals,
then there is a natural number n such that
W_n = W_{f(n)}
http://groups.google.com/group/sci.logic/browse_frm/thread/30b550f50acc7451?scoring=d&q=P_ultimate&
Jim Burns
The expressions [2] and [1],
1/(x-1) [2] (note: x = 2)
1/(x-1) [1] (note: x = 1)
have exactly the same characters in the same order, since
the additional information noted is not part of the
expressions. I am tempted to say that they therefore
have the same syntax.
However, in English, the same expression, character
for character, placed in different contexts can have
clearly different syntaxes. A good example that comes
to mind (Groucho Marx:) "Time flies like an arrow;
fruit flies like a banana." Certainly, the goal
in mathematics is to be completely unambiguous,
very much unlike English, but I can't say, myself,
whether we've arrived at our goal.
In its most common context, the expression "1/(x-1)"
carries an often-tacit assumption that x <> 1.
What does it mean to me to have contradictory
assumptions to deal with? Not much, unfortunately.
If I fall back upon my classroom experience, all I
can remember is that this is a situation to be avoided.
Is there a /syntactic/ difference between having
a contradiction and not having one? I guess I
don't know what is and is not syntax well enough
to be sure. I am willing to be educated on the point.
I think what the poster you're responding to may have meant
that the expression "1/0" can be determined to be
forbidden by purely mechanical means (hence, "syntactic").
Given a definition of syntax that broad, I would stick
my neck out and say that, yes, the syntax of the expression
"1/(x-1)" is different when x = 2 from when x = 1.
Jim Burns
LauLuna
wrote:
> LauLuna says...
>
> >Assume there is a collection C of syntactic marks or types (expressing
> >levels, interpretations or whatever is required for disambiguation) by
> >which natural language can be so extended that any expression gets
> >logically disambiguated. C should be a part of a grammar able to
> >disambiguate all expressions; as such C must be nameable. Now consider
>
> >LC 'this sentence is meaningless in all levels/interpretations in C'
>
> That shows that the collection C cannot be expressed in natural language.
> It doesn't show that C does not exist.
How could a set of symbols belonging to a grammar (i.e. an algorithm
generating formulas) be unnameable in natural language?
Anyway, it seems the Liar sentence can be denied a univocal
propositional content for two main different reasons:
1) Because its subject fails to denote anything (the 'attempted' self-
reference is impossible).
2) Because the natural language truth predicate is ambiguous.
I'd say the first is a rather Russellian approach while the second is
a Tarskian-Quinean one.
As I see it, the difficulties the first approach appears to encounter
(namely, the apparent existence of successful self-references) get
solved by the distinction between sentences and propositions, and the
qualified rejection of self-reference, which states that no
proposition refers to itself but some propositions do refer to the
sentences that express them.
In contrast, the second approach seems to me to be unbearably
counterintuitive.
Regards
Charlie-Boo
On Dec 24, 6:11 pm, Newberry <newberr...@gmail.com> wrote:
> Let P be the sentence "This sentence is meaningless." Is it true or
> false?
It is false.
> It should not be difficult to answer.
That's one thing you've got right.
> When we attempt to compare
> "This sentence is meaningless" with reality we find that it is meaningless.
No it isn't. It's false.
> Case B: P is false.
> If P is false then it is not the case that it is meaningless. It is
> the opposite of what it claims. This is a contradiction.
No, that is a false statement, verifying that it is false. (Mistake #
1)
> Therefore P is not false.
Yes it is. It is false because it is not meaningless, but in fact is
false.
> Case C: P is meaningless.
> If P is meaningless then nothing is the case. There is no
> contradiction.
Yes there is. The contradiction is that you assume it's meaningless,
but then it is true not meaningless. (Mistake # 2)
> In a three valued logic (T, F, M) we conclude that P is meaningless.
No, it is false. (Overall mistake.)
You've got so many mistakes here - all concerning evaluating simple
propositional calculus wffs - that once again we have the sorry
situation where idiotic college professors such as Ulrich totally miss
all of them and give the whole error-ridden proof a pass, calling it
"lovely" but "irrelevant".
No, it is not irrelevant either. A sequence of disconnected false
statements has neither relevance nor irrelevance.
C-B
> It is not correct to say that if the sentence is meaningless than what
> it SAYS is true. This argument assumes that it becomes TRUE half way
> through the argument and then it IS THE CASE that it is meaningless.
> Thus the sentence confirms our initial assumption that it was
> meaningless. But if it stays meaningless all the time it confirms
> nothing.
>
> Clearly, if the sentence does not have any meaning then it does not
> have the meaning that it is meaningless.
>
> This gives us the basic insight that all self-referential, paradoxical
> sentences, including possibly Goedel's sentence, are probably
> meaningless.
Charlie-Boo
> On Mon, 24 Dec 2007 15:11:50 -0800 (PST), Newberry
>
> <newberr...@gmail.com> wrote:
> >Let P be the sentence "This sentence is meaningless." Is it true or
> >false? [...]
>
> >This gives us the basic insight that all self-referential, paradoxical
> >sentences, including possibly Goedel's sentence, are probably
> >meaningless.
>
> You're jumping a bit from one example to _all_ such sentences.
He's saying that there's an analogy.
> But much more important: There's nothing _literally_ self-referetial
> about "Godel's sentence"
If a Godel sentence is not self-referential, then what in the world
is?
By not understanding or saying anything in particular about his
argument, you make the fallacy of not distinguishing it from similar
proofs and your argument (lacking any premise relating to his
argument) applies to them as well. You "Refute the Problem" by
declaring in effect that all such proofs of this type are "not self-
referential" and so the notion of "self-referential" has no instances
and itself becomes essentially meaningless.
> your lovely anlysis is irrelevant there.
It is neither lovely nor irrelevant. It is riddled with errors and is
meaningless. But I see that you have a problem figuring out things
that are meaningless already - like P or your referring to something
erroneous as"lovely" but "irrelevant."
> The sentence in question is just an ordinary assertion about positive
> integers, with no problem whatever regarding what it means,
He didn't say there were any problems - he showed (erroneously) that
it can be fairly easily evaluated (although his many errors whizzed
right past you.)
> any more than there's a problem with "If n and m are even positive
> integers then n + m is even."
>
> ************************
>
> David C. Ullrich
Daryl McCullough
>
>On Dec 29, 5:14=A0pm, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> LauLuna says...
>>
>> >Assume there is a collection C of syntactic marks or types (expressing
>> >levels, interpretations or whatever is required for disambiguation) by
>> >which natural language can be so extended that any expression gets
>> >logically disambiguated. C should be a part of a grammar able to
>> >disambiguate all expressions; as such C must be nameable. Now consider
>>
>>
>> That shows that the collection C cannot be expressed in natural language.
>> It doesn't show that C does not exist.
>
>How could a set of symbols belonging to a grammar (i.e. an algorithm
>generating formulas) be unnameable in natural language?
Basically what I think you are proposing is a hierarchy of
types, where each type can be associated with a recursive
ordinal. The collection of all recursive ordinals is not
a recursive set. So there is no algorithm to determine
whether an ordinal is recursive or not.
>Anyway, it seems the Liar sentence can be denied a univocal
>propositional content for two main different reasons:
>
>1) Because its subject fails to denote anything (the 'attempted' self-
>reference is impossible).
There are two parts to the Liar sentence, the subject and
the predicate:
Part 1: "This sentence"
Part 2: "is not true"
There is no problem with part 1, at least of we say that
a sentence need not be semantically meaningful to be a
sentence (it only needs to be syntactically correct).
So the problem is with part 2.
>2) Because the natural language truth predicate is ambiguous.
>I'd say the first is a rather Russellian approach while the second is
>a Tarskian-Quinean one.
>
>As I see it, the difficulties the first approach appears to encounter
>(namely, the apparent existence of successful self-references) get
>solved by the distinction between sentences and propositions, and the
>qualified rejection of self-reference, which states that no
>proposition refers to itself but some propositions do refer to the
>sentences that express them.
>
>In contrast, the second approach seems to me to be unbearably
>counterintuitive.
The "Russellian" approach by itself doesn't work unless you
*also* adopt the Tarskian position that there is no truth
predicate. Peano Arithmetic has no sentences that refer to
themselves, but if you introduce a truth predicate
T(x) with the interpretation that
T(x) <-> x is the code for a true sentence
then the system becomes inconsistent. So it seems to
me that the Tarskian approach is necessary, whether it
is counterintuitive or not.
Daryl McCullough
>
>On Dec 29, 8:02=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>42 is self-referential, 56 is not. So 42 has a property that 56 does
>not. I.e. the entences are different.
That's just wrong. A sentence is a sequence of characters.
56 is the *same* sentence as sentence 42.
Perhaps this is a matter of terminology. You seem to be
conflating sentences with sentence tokens. That's fine,
but then we need another term to mean the equivalence
class of all tokens.
Let's define a string of characters to be "truish" if
it is syntactically identical to some true sentence.
So if
P and Q are two sentences, and
P has the same sequence of characters as Q,
and P is true.
then
Q is truish.
Conversely, if Q is *not* truish, then no sentence
with the same sequence of characters as Q can be true.
That's just a definition. Now let's suppose we
have a numbered list of sentences in which sentences
42 and 53 are the following:
42. Sentence 42 is not truish.
.
.
.
53. Sentence 42 is not truish.
Is 42 "truish" or not? If not, then by definition of
"truish", no sentence with the same sequence of characters
can be true. So, in particular, 53 cannot be true. But
53 just says that 42 is not truish!
Your tokenism does *not* work.
Daryl McCullough
>
>On 2007-12-29, in sci.logic, Newberry wrote:
>> But we actually do have a contradiction. The axioms of PA are
>> manifestly true. truth cannot be inconsistent with itself, therefore
>> PA is consistent. This is just as manifestly true as the axioms
>> themselves. Now Goedel's sentence tells the truth if PA is consistent.
>> But it is manifestly true that PA is consistent. Therefore
>>
>> Ex (P(x,#F)) -> F (a)
>>
>> is just as manifestly true as the rest of the axioms. But if PA is
>> consistent then the entire hierarchy
>> T, T' = T + G, T" = T' + G' ... is consistent. Therefore if we can
>> construct
>>
>> Ex (P_ultimate(x,#F)) -> F (1)
>>
>> that can prove the entire hierarchy it is also true. But (1) results
>> in a contradiction.
>
>What is "P_ultimate"? If it is just the provability predicate for the union
>of the theories T, T', T'', ... there is no contradiction,
No, T_ultimate can be defined as a fixed point of the following
operation on theories extending PA:
T --> F(T)
where F(T) = that theory whose axioms consist of all the
axioms of T, plus the additional axiom schema
(Ex Pr_T(x,#Phi)) -> Phi
where Pr_T(x,y) is the formalization in PA of the claim
that x is a Godel code for a proof of the sentence whose
Godel code is y, and #Phi means the Godel code of sentence
Phi.
The operation
T --> F(T)
considered as an operation on r.e. sets has a fixed point,
T_ultimate, which satisfies
T_ultimate |- (Ex Pr_T(x,#Phi)) -> Phi
Unfortunately, applying Godel's theorem, we can show that
T_ultimate is inconsistent.
Daryl McCullough
>Yes, I am saying that. I am saying that
>
>56. Sentence number 42 is not true.
>
>CAN be translated as
>
>56'. "Sentence number 42 is not true" is not true.
Follow through on your reasoning! If you can replace
"Sentence number 42" by its quotation, then you can
also replace "Sentence number 56" by its quotation.
So we have the following equivalences:
Sentence number 56 is not true <->
"Sentence number 42 is not true" is not true <->
Sentence number 56' is true
So even though you claim that sentence 56' means the same
thing as sentence 56, we've proved that
Sentence number 56' is true <-> Sentence number 56 is not true
Your tokenism does *not* resolve the paradox!
Chris Menzel
<stevend...@yahoo.com> said:
> ...
> The operation
>
> T --> F(T)
>
> considered as an operation on r.e. sets has a fixed point,
> T_ultimate, which satisfies
>
> T_ultimate |- (Ex Pr_T(x,#Phi)) -> Phi
Should that be
T_ultimate |- (Ex Pr_T_ultimate(x,#Phi)) -> Phi ?
Newberry
Newberry
wrote:
> Newberry says...
>
>
>
> >On Dec 29, 8:02=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
> >wrote:
> >42 is self-referential, 56 is not. So 42 has a property that 56 does
> >not. I.e. the entences are different.
>
> That's just wrong. A sentence is a sequence of characters.
That is certainly incorrect. The sequence has to be syntactically
correct and has to have a meaning.
Newberry
wrote:
Easy. If it is the same sequence as a true sequence then it is truish.
Otherwise it is not. Which true sequence is it equivalent to?
Newberry
wrote:
> Newberry says...
>
> >Yes, I am saying that. I am saying that
>
> >56. Sentence number 42 is not true.
>
> >CAN be translated as
>
> >56'. "Sentence number 42 is not true" is not true.
>
> Follow through on your reasoning! If you can replace
> "Sentence number 42" by its quotation, then you can
> also replace "Sentence number 56" by its quotation.
I did not get it. 56 is not referred to anywhere.
In 56: "Sentence number 42 is meaningless" we are substituting
"Sentence number 42 is meaningless" for "sentence number 42."
Daryl McCullough
>
>On Dec 31, 6:37=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> Let's define a string of characters to be "truish" if
>> it is syntactically identical to some true sentence.
>> So if
>>
>> P and Q are two sentences, and
>> P has the same sequence of characters as Q,
>> and P is true.
>>
>> then
>>
>> Q is truish.
>>
>> Conversely, if Q is *not* truish, then no sentence
>> with the same sequence of characters as Q can be true.
>>
>> That's just a definition. Now let's suppose we
>> have a numbered list of sentences in which sentences
>> 42 and 53 are the following:
>>
>> 42. Sentence 42 is not truish.
>> .
>> .
>> .
>> 53. Sentence 42 is not truish.
>>
>> Is 42 "truish" or not?
>
>Easy. If it is the same sequence as a true sequence then it is truish.
>Otherwise it is not. Which true sequence is it equivalent to?
That's what I'm asking you. *Is* there a sentence that is syntactically
the same as 42, but is true? If the answer is "yes", that leads to a
contradiction. If the answer is "no", that also leads to a contradiction.
So your tokenism doesn't resolve the paradox.
Newberry
I normally do not respond to abusive posts but this one is really
funny. Hahahahaha.
>
>
>
> > It is not correct to say that if the sentence is meaningless than what
> > it SAYS is true. This argument assumes that it becomes TRUE half way
> > through the argument and then it IS THE CASE that it is meaningless.
> > Thus the sentence confirms our initial assumption that it was
> > meaningless. But if it stays meaningless all the time it confirms
> > nothing.
>
> > Clearly, if the sentence does not have any meaning then it does not
> > have the meaning that it is meaningless.
>
> > This gives us the basic insight that all self-referential, paradoxical
> > sentences, including possibly Goedel's sentence, are probably
> > meaningless.- Hide quoted text -
Daryl McCullough
>
>On Dec 31, 7:00=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> Newberry says...
>>
>> >Yes, I am saying that. I am saying that
>>
>> >56. Sentence number 42 is not true.
>>
>> >CAN be translated as
>>
>> >56'. "Sentence number 42 is not true" is not true.
>>
>> Follow through on your reasoning! If you can replace
>> "Sentence number 42" by its quotation, then you can
>> also replace "Sentence number 56" by its quotation.
>
>I did not get it. 56 is not referred to anywhere.
But we certainly *can* refer to it. Sentence 56 is
the sentence "Sentence number 42 is not true". So
let Sentence 57 be the following:
57: Sentence number 56 is not true.
But sentence number 56 is just "Sentence number 42 is not true".
So 57 is logically equivalent to 57':
57': "Sentence number 42 is not true" is not true.
But that sentence is logically equivalent to sentence 56.
So sentence 56 is logically equivalent to sentence 57, which
claims that sentence 56 is not true. Your tokenism did *not*
resolve the paradox.
Daryl McCullough
>
>On Dec 31, 3:46=A0am, Charlie-Boo <shymath...@gmail.com> wrote:
>> On Dec 24, 6:11=A0pm, Newberry <newberr...@gmail.com> wrote:
>>
>> > Let P be the sentence "This sentence is meaningless." Is it true or
>> > false?
>>
>> It is false.
Charlie-Boo is right. There is no paradox with "This sentence is
meaningless". It is perfectly meaningful *and* false. That's not
a contradiction.
To get a strengthened liar's paradox, you need either
"This sentence is either meaningless or false"
or, more simply,
"This sentence is not true"
(with the interpretation that there are two ways for a sentence
to not be true: (1) it might be false, or (2) it might be meaningless.)
Daryl McCullough
>
>On 31 Dec 2007 06:44:11 -0800, Daryl McCullough
><stevend...@yahoo.com> said:
>> ...
>> The operation
>>
>> T --> F(T)
>>
>> considered as an operation on r.e. sets has a fixed point,
>> T_ultimate, which satisfies
>>
>> T_ultimate |- (Ex Pr_T(x,#Phi)) -> Phi
>
>Should that be
>
> T_ultimate |- (Ex Pr_T_ultimate(x,#Phi)) -> Phi ?
Yes, that's what I mean.
Actually, to know that T_ultimate exists, the easy way
is to use Godel's theorem to show that T_ultimate is
inconsistent, and there is only one inconsistent theory.
However, I think there is a proof that T_ultimate exists
without relying on Godel's theorem. There's probably a
recursion-theory theorem along the following lines:
Let f be any total recursive function. Then there is a
natural number n such that n and f(n) are codes for the
same r.e. set. (I think that's one version of the fixed-point
theorem.)
So we can let f(x) be the function that takes a code for
the axioms of T and returns a code for the axioms of F(T).
Then the n that is a fixed point of f(x) will be a code
for the axioms of T_ultimate.
Newberry
wrote:
> Newberry says...
>
>
>
> >On Dec 31, 3:46=A0am, Charlie-Boo <shymath...@gmail.com> wrote:
> >> On Dec 24, 6:11=A0pm, Newberry <newberr...@gmail.com> wrote:
>
> >> > Let P be the sentence "This sentence is meaningless." Is it true or
> >> > false?
>
> >> It is false.
>
> Charlie-Boo is right. There is no paradox with "This sentence is
> meaningless". It is perfectly meaningful *and* false. That's not
> a contradiction.
Yes, LauLuna pointed this out long time ago. But that's beside the
point.
Newberry
wrote:
> Newberry says...
>
>
>
>
>
>
>
> >On Dec 31, 7:00=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
> >wrote:
> >> Newberry says...
>
> >> >Yes, I am saying that. I am saying that
>
> >> >56. Sentence number 42 is not true.
>
> >> >CAN be translated as
>
> >> >56'. "Sentence number 42 is not true" is not true.
>
> >> Follow through on your reasoning! If you can replace
> >> "Sentence number 42" by its quotation, then you can
> >> also replace "Sentence number 56" by its quotation.
>
> >I did not get it. 56 is not referred to anywhere.
>
> But we certainly *can* refer to it. Sentence 56 is
> the sentence "Sentence number 42 is not true". So
> let Sentence 57 be the following:
>
> 57: Sentence number 56 is not true.
>
> But sentence number 56 is just "Sentence number 42 is not true".
> So 57 is logically equivalent to 57':
>
> 57': "Sentence number 42 is not true" is not true.
>
> But that sentence is logically equivalent to sentence 56.
>
> So sentence 56 is logically equivalent to sentence 57,
But it is not. You made this substitution:
57': "Sentence number 42 is not true"[56] is not true.
I made this substitution:
56'. "Sentence number 42 is not true"[42] is not true.
and we know that 42 and 56 have different meanings.
> which
> claims that sentence 56 is not true. Your tokenism did *not*
> resolve the paradox.
>
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
Aatu Koskensilta
> Actually, to know that T_ultimate exists, the easy way
> is to use Godel's theorem to show that T_ultimate is
> inconsistent, and there is only one inconsistent theory.
As you later surmise, the obvious way to obtain an axiomatisation of
T_ultimate is by using the recursion theorem, as noted in e.g.
http://cs.nyu.edu/pipermail/fom/2007-June/011631.html.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
R. Srinivasan
> On Dec 25, 2:47 pm, John Jones <jonescard...@aol.com> wrote:
>
>
>
>
>
> > On Dec 24, 11:11�pm, Newberry <newberr...@gmail.com> wrote:
>
> > > Let P be the sentence "This sentence is meaningless." Is it true or
> > > Philosophicus says: "In order to tell whether a picture is true or
> > > false we compare it with reality." [2.223] When we attempt to compare
> > > "This sentence is meaningless" with reality we find that it is not
> > > comparable with anything. It is not a picture of a fact; it is
> > > meaningless.
>
> > > We can analyze the situation further:
> > > Case A: P is true.
> > > If P is true. Then it is the case that it is meaningless. But then it
> > > cannot be true. This is a contradiction. Therefore P is not true.
>
> > > Case B: P is false.
> > > If P is false then it is not the case that it is meaningless. It is
> > > is not false.
>
> > > Case C: P is meaningless.
> > > If P is meaningless then nothing is the case. There is no
> > > meaningless.
>
> > > It is not correct to say that if the sentence is meaningless than what
> > > it SAYS is true. This argument assumes that it becomes TRUE half way
> > > through the argument and then it IS THE CASE that it is meaningless.
> > > Thus the sentence confirms our initial assumption that it was
> > > meaningless. But if it stays meaningless all the time it confirms
> > > nothing.
>
> > > Clearly, if the sentence does not have any meaning then it does not
> > > have the meaning that it is meaningless.
>
> > > This gives us the basic insight that all self-referential, paradoxical
> > > sentences, including possibly Goedel's sentence, are probably
> > > meaningless.
> > > false?
>
> > sentence is, nor even specified what sentence you refer to by 'it'.
>
>
> "This" in the sentence "This sentence is meaningless" obviously refers
> to the sentence in quotation marks.
>
>
That is not obvious at all. I agree with John Jones. To see what the
problem is, consider the following thought experiment:
Lat us say Mr. X blurts out "This sentence is....." and exactly at
that point he drops dead. Can you now let us know what is the sentence
that Mr. X had in mind? The answer is we cannot be sure at all. All we
can say is that if Mr. X had no sentence in mind at the time of his
death, he was attempting to attribute some property to a non-existent
sentence. If the sentence itself is non-existent, then any attributed
property to it (e.g. "meaninglessness") is also non-existent.
The problem here is that we human beings can parse sentences only when
their words are read sequentially, which means that the informal
notion of "time" is important in understanding what sentences mean.
Keeoing this in mind, let us re-formulate the sentence in question in
an *exactly* equivalent manner as follows:
"This sentence, which has not yet been defined, is meaningless"
Now we can clearly agree that the above sentence attempts to attribute
the property of "meaninglessness" to a non-existent sentence. Which
makes no sense whatsoever. Yet what is wrong with the above
formulation? If a human being who utters the above sentence drops dead
after uttering "defined", (s)he has clearly not yet defined any
sentence whatsoever, even under the assumption that "This" in the
above sentence points to the same sentence being constructed.
Conclusion: When somebody utters "This sentence is blah blah blah..."
then s(he) should have a *constructed* sentence in mind the moment
"sentence" is uttered. Then and only then is it legal to attribute
properties like "meaningful" or "meaningless" to that sentence.
In order to make sense of "This sentence is meaningless", the sentence
referred to must be taken as having a Platonic esistence, independent
of our self-referential attempt at "definition". The problem is that
Godel's reasoning enables us to formalize such Platonically existing
objects (e.g. "sentence", "theory") as formal mathematical objects of
some universe. This leads to paradox.
In NAFL also sentences are taken to be pre-existing, when formulating
theories. Theories are also pre-existing objects. E.g. "Every
proposition in the language of a NAFL theory is either provable of
refutable or undecdiable in that theory" is an absolute (Platonic)
truth in NAFL, as it is in classical logic. But the important
distinction is that NAFL does not accept that the notion of "sentence"
or "theory" itself can be formalized as an object of the universe.
Sentences and theories are used to make assertions *about* objects of
the universe and cannot themselves be objects of the same universe.
They are essentially "meta-mathematical" objects whose existence
cannot be formalized within NAFL theories. The way this conclusion
comes about is via the axiomatic nature of NAFL truth for formal
sentences in the languages of NAFL theories, as opposed to the
Platonic nature of classical truth.
For the NAFL truth definition,see the recent sci.logic thread "FOL/
Intuitionistic logic versus NAFL. Part 1. Failure of non-
contradiction" at the following link:
I will shortly be starting another thread (Part 2), which explains the
definition of NAFL theories.
Regards, RS
Marshall
>
> Lat us say Mr. X blurts out "This sentence is....." and exactly at
> that point he drops dead. Can you now let us know what is the sentence
> that Mr. X had in mind? The answer is we cannot be sure at all. All we
> can say is that if Mr. X had no sentence in mind at the time of his
> death, he was attempting to attribute some property to a non-existent
> sentence.
Suppose Mr. X had a sentence in mind, but because of some
drug-induced state instead said "False this is sentence." What
does that tell us about the meaning of some other sentence?
Nothing. Even if the other sentence we wish to consider is
a reordering of the words of the first sentence.
> The problem here is that we human beings can parse sentences only when
> their words are read sequentially, which means that the informal
> notion of "time" is important in understanding what sentences mean.
Syntax does not require time. It just requires order.
> Keeoing this in mind, let us re-formulate the sentence in question in
> an *exactly* equivalent manner as follows:
>
> "This sentence, which has not yet been defined, is meaningless"
>
> Now we can clearly agree that the above sentence attempts to attribute
> the property of "meaninglessness" to a non-existent sentence.
I do not agree that the above sentence does not exist.
> Conclusion: When somebody utters "This sentence is blah blah blah..."
> then s(he) should have a *constructed* sentence in mind the moment
> "sentence" is uttered.
A different syntax could put the word "sentence" at the end,
and then your objection goes away. Your objection depends
on English word order.
Marshall
Daryl McCullough
>
>On Dec 31, 8:57=A0am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>But it is not. You made this substitution:
>57': "Sentence number 42 is not true"[56] is not true.
>
>I made this substitution:
>56'. "Sentence number 42 is not true"[42] is not true.
You did not. You wrote:
56': "Sentence number 42 is not true" is not true.
I wrote:
57': "Sentence number 42 is not true" is not true.
They are the *same* sentence. Any claim that 56'
is true, while 57' is false is just too ridiculous
to continue to discuss.
Newberry
wrote:
There are plenty of examples in English where a grammatically correct
sequence has more than one possible interpretation. The ambiguity is
usually resolved by context. Are you saying that
This sentence is not true. (1)
and
"This sentence is not true" is not true. (2)
are saying the same thing because they both refer to
"This sentence is not true"?
Charlie-Boo
> Newberry wrote:
>
> > On Dec 25, 6:01 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> > > On Mon, 24 Dec 2007 15:11:50 -0800 (PST), Newberry
> > One of those number is the Goedel number of the sentence ITSELF.
>
> Under a different coding scheme the same sentence does not refer
> to itself.
Yes, and interpreted as being written in French (instead of German),
Godel's Theorem (sic) is gibberish.
C-B
> --
> hz- Hide quoted text -
Daryl McCullough
>There are plenty of examples in English where a grammatically correct
>sequence has more than one possible interpretation. The ambiguity is
>usually resolved by context. Are you saying that
>
>This sentence is not true. (1)
>
>and
>
>"This sentence is not true" is not true. (2)
>
>are saying the same thing because they both refer to
>
>"This sentence is not true"?
Yes, that's the essence of the liar paradox, that
a sentence and its negation seem to be paraphrases
of each other.
Look, the ambiguity of reference is *not* the
issue here. Once again, let's try this exercise:
Define a string of symbols to be "truish" if it
is the same string of symbols as some true sentence.
Now, let sentence number 42 be the sentence:
42. Sentence number 42 is not truish.
Is sentence number 42 truish? If the answer is "yes",
that leads to a contradiction. If the answer is "no",
that also leads to a contradiction.
It's the same contradiction as the Liar paradox, except
instead of asking whether the liar sentence is true, we
instead ask whether sentence 42 is truish. So "tokenism"
doesn't save you from the paradox.
Charlie-Boo
> here is what Torkel Franzen says about self-
> reference:
>
> QUOTE:
> But sentences constructed in the proof that every arithmetical
> property P has a provable fixpoint are self-referential in a stronger
> sense: they are sentences A of the form
>
> There is an m such that m has the property P and property Q
>
> where it is provable in PA that the only number that has the property
> P is the Goedel number of the sentence itself. It is in this sense
> that the sentence A "says of itself it has property Q."
> END OF QUOTE [p. 45]-
But there are arithmetical properties P for which more than one number
has that property, so it should not be "provable in PA that the only
number that has the property P is the Goedel number of the sentence
itself."
Let's switch P and Q around a bit to fix that and show that any such Q
has a fixed point.
Say that the only number that has property P is m: P(X) == X=m
so,
A expresses (eX)P(X)^Q(X)
which is (eX)X=m ^ Q(X)
which is Q(m)
So the above (fixed) is logically equivalent to the Fixed Point
Theorem. How is this "self-referential in a stronger sense"? Does
Torkel think it's a new theorem?
C-B