This sentence is not true
Newberry
This sentence is not true (1)
Richard L. Kikham in "Theories of Truth" (p. 293) argues as follows.
"If it is neiter true nor false (and thus not true), then it is
precisely what it claims to be, so it is true." I wonder if this
argument is correct. I would argue that if the sentence is meaningless
(and thus not true) than it does not "claim" anything.
We can argue as follows. Let us abbreviate the sentence (1) as S. Then
T(S) -> S
S -> -S (S asserts its own negation)
S & -S contradiction
F(S) Reductio ad absurdum
F(S) -> -S
-S -> S (S asserts its own negation)
S & -S contradiction
T(S) Reductio ad absurdum
But nothing follows from
M(S)
So I do not see how we can conclude
M(S) -> T(S) ??
I do not think that it follows from S's interpretation because its
interpretation is blank.
David Marcus
> Let us consider the "strengthened" liar paradox:
>
> This sentence is not true (1)
>
> Richard L. Kikham in "Theories of Truth" (p. 293) argues as follows.
> "If it is neiter true nor false (and thus not true), then it is
> precisely what it claims to be, so it is true." I wonder if this
> argument is correct.
Obviously not (since not true normally means false).
> I would argue that if the sentence is meaningless
> (and thus not true) than it does not "claim" anything.
>
> We can argue as follows. Let us abbreviate the sentence (1) as S. Then
>
> T(S) -> S
> S -> -S (S asserts its own negation)
> S & -S contradiction
> F(S) Reductio ad absurdum
>
> F(S) -> -S
> -S -> S (S asserts its own negation)
> S & -S contradiction
> T(S) Reductio ad absurdum
>
> But nothing follows from
> M(S)
> So I do not see how we can conclude
> M(S) -> T(S) ??
> I do not think that it follows from S's interpretation because its
> interpretation is blank.
The best explanation of the Liar paradox that I've seen is in "Semantic
Paradoxes as Equations" by Lan Wen, Mathematical Intelligencer, Vol. 23,
Number 1, 2001, pp. 43-48: "Solution to the Liar paradox (Informal
version). It is (the translation of) the assumption of existence of a
solution that causes contradiction in the Liar paradox."
--
David Marcus
aatu.kos...@xortec.fi
> Newberry wrote:
> > Let us consider the "strengthened" liar paradox:
>
> > This sentence is not true (1)
>
> > Richard L. Kikham in "Theories of Truth" (p. 293) argues as follows.
> > "If it is neiter true nor false (and thus not true), then it is
> > precisely what it claims to be, so it is true." I wonder if this
> > argument is correct.
>
> Obviously not (since not true normally means false).
The argument is just fine. Not true means just that, not true, and if
we allow the possibility that some sentences are neither true nor
false - for example because they're just random sequences of letters -
then not true does not mean, and is not equivalent to, "false".
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
LauLuna
> Let us consider the "strengthened" liar paradox:
>
> This sentence is not true (1)
>
> Richard L. Kikham in "Theories of Truth" (p. 293) argues as follows.
> "If it is neiter true nor false (and thus not true), then it is
> precisely what it claims to be, so it is true." I wonder if this
> argument is correct. I would argue that if the sentence is meaningless
> (and thus not true) than it does not "claim" anything.
>
Well, I would say that Kikham is plainly wrong, because once we have
proved that (1) has no truth value, we have proved that it is unable
to assert anything.
The consequence is that when we, on the basis of the precedent
reasoning, conclude that (1) is not true, we are uttering a sentence
that is not equivalent to (1), since our sentence is able to state
what (1) cannot state.
You can make both sentences absolutely identical as linguistic
expressions:
(1) '(1) is not true'
and once we've proven that (1) has no truth value, we assert:
(2) '(1) is not true'
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.
So, we must understand that (2) does not speak about the sentence-type
common to (1) and (2) but about the sentence-token named '(1)'.
I think these conclusions are inevitable in consistent bivalent logic.
Regards
G. Frege
>
> Let us consider the "strengthened" liar paradox:
>
> This sentence is not true. (1)
>
I guess the idea here is the following. Normally we would say that a
declarative sentence is either true or false (has the truth value t or
f). Now confronted with the liar paradox we might just come up with
the idea that there is a 3rd truth value 0, meaning: neither true nor
false.
So the truth value of (1) might be t or f or 0 then.
>
> Richard L. Kikham in "Theories of Truth" (p. 293) argues as follows.
> "If it is neither true nor false (and thus not true), then it is
> precisely what it claims to be, so it is true."
>
Here he's using Tarski's T-schema:
"X" is true iff X.
In this case we have:
(1) is true iff (1) is not true. (2)
A contradiction.
Or, using truth values: Assume (1) has truth value 0, i.e. is neither
true nor false. Hence (1) is not true, but then (1) is true by (2).
Contradiction!
>
> I wonder if this argument is correct.
>
Sure.
>
> I would argue that if the sentence is meaningless
> (and thus not true) than it does not "claim" anything.
>
Sounds reasonable. But wait a second. Look at the sentence:
This sentence is not true. (*)
Do you think, just looking at (reading) this sentence, that it is
"meaningless"? Why? Is there something wrong with the grammar? Or does
it use a word which is meaningless? (Obviously not!) It seems that it
is rather clear what (*) claims, actually it claims that the sentence
(*) is not true. Simple as that. But *is* is true that (*) is not
true? Or is it false? Or...?
>
> ???
>
F.
--
E-mail: info<at>simple-line<dot>de
Daryl McCullough
>
>On Feb 5, 3:04 am, "Newberry" <newbe...@ureach.com> wrote:
>> Let us consider the "strengthened" liar paradox:
>>
>> This sentence is not true (1)
>>
>> Richard L. Kikham in "Theories of Truth" (p. 293) argues as follows.
>> "If it is neiter true nor false (and thus not true), then it is
>> precisely what it claims to be, so it is true." I wonder if this
>> argument is correct. I would argue that if the sentence is meaningless
>> (and thus not true) than it does not "claim" anything.
>>
>
>Well, I would say that Kikham is plainly wrong, because once we have
>proved that (1) has no truth value, we have proved that it is unable
>to assert anything.
That's why it's a paradox. By one line of reasoning, the sentence
is meaningless, while by another line of reasoning it is true.
>The consequence is that when we, on the basis of the precedent
>reasoning, conclude that (1) is not true, we are uttering a sentence
>that is not equivalent to (1), since our sentence is able to state
>what (1) cannot state.
>
>You can make both sentences absolutely identical as linguistic
>expressions:
>
>(1) '(1) is not true'
>
>and once we've proven that (1) has no truth value, we assert:
>
>(2) '(1) is not true'
>
>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.
>
>So, we must understand that (2) does not speak about the sentence-type
>common to (1) and (2) but about the sentence-token named '(1)'.
>
>I think these conclusions are inevitable in consistent bivalent logic.
I don't like this resolution very much. Whether a sentence is
true or not should (according to the way "true" is usually used)
depend only on identifying (A) what is the subject of the sentence,
and (B) what is it saying about that subject. In other words, if
we simplify a claim to consist of a subject plus a predicate, then
a sentence is meaningful if it is clear what the subject of the
sentence is, and it is clear what is being claimed about that
subject. Contrapositively, a sentence should be meaningless
only if it lacks a clear subject or lacks a clear predicate.
Two sentences with the same subject and the same predicate
should have the same truth value (or should be equally
meaningless).
Of course, the same syntactic sentence can have different meanings
in different contexts; for example, "Today is Monday" will be true
when uttered one day but false when uttered another day. But that's
easily explained by the fact that the word "Today" denotes different
subjects depending on when it is uttered.
In contrast, your two sentences (1) and (2) clearly have the
same subject (namely, sentence (1)) and are clearly attempting
to make the same claim about it (that it is not true). So it
seems arbitrary and ad hoc to call one "true" and the other
"not true".
Anyway, we can always introduce a new concept, "reliably true".
A sentence is reliably true if every token of that sentence is
true in your sense. Then we can form the sentence
(1) Sentence (1) is not reliably true.
Clearly, (1) cannot be reliably true on pain of contradiction.
So we conclude
(2) Sentence (1) is not reliably true.
But Sentence (2) is the same sentence as Sentence (1). So
Sentence (2), while it may be true, can't be *reliably*
true. Clearly, this conclusion holds for every token
of sentence (1): That token is true, but not reliably true.
But by definition of "reliably true", if every token is true,
then the sentence is reliably true. So the paradox returns.
--
Daryl McCullough
Ithaca, NY
David Marcus
> On 5 Feb, 07:50, David Marcus wrote:
> > Newberry wrote:
> > > Let us consider the "strengthened" liar paradox:
> >
> > > This sentence is not true (1)
> >
> > > Richard L. Kikham in "Theories of Truth" (p. 293) argues as follows.
> > > "If it is neiter true nor false (and thus not true), then it is
> > > precisely what it claims to be, so it is true." I wonder if this
> > > argument is correct.
> >
> > Obviously not (since not true normally means false).
>
> The argument is just fine. Not true means just that, not true, and if
> we allow the possibility that some sentences are neither true nor
> false - for example because they're just random sequences of letters -
> then not true does not mean, and is not equivalent to, "false".
What is your criterion for deciding which sentences don't have a truth
value? The sentence above is not a random sequence of letters. I'm going
to stick to two-valued logic myself.
--
David Marcus
george
wrote:
> LauLuna says...
> That's why it's a paradox. By one line of reasoning, the sentence
> is meaningless, while by another line of reasoning it is true.
That's ridiculous. If it were actually meaningless, then no
harm could come of calling it true. If it has no meaning then
its truth will not have any further consequences that could
cause contradictions. More to the point, though, it CAN'T POSSIBLY
be meaningless. It is precisely in virtue of the facts that its terms
ARE KNOWN to mean what THEY mean, and that they are combined
in a WELL-defined and CORRECTLY grammatical way, that is
leading to the paradox; in other words, the sentence is paradoxical
BECAUSE OF and IN VIRTUE of its meaning; therefore you cannot
POSSIBLY be insisting that the paradox arises because the sentence
is meaningLESS; it is paradoxical beCAUSE it is meaningFUL.
> >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. Your first objection will probably be "No; 'today' IS
indexical." But that is just stupid. All attempts to get serious
about purging indexicality just wind up confirming that that is
impossible. THERE IS NO SUCH THING as a non-indexical
sentence IF you are trying to talk about the factual world.
If you are instead talking about a purely abstract/timeless
model via logical language, then again, THE SYMBOLS in
that language cannot have any meanings except through
axioms that THEMSEVLES CONSTITUTE the "logical context"
that plays THE SAME role as the real-world context in interpreting
sentences about the real world. Pun arguably INTENDED, one
must ALWAYS a FRAME of REFERENCE (both in physics AND
semantics).
> >So, we must understand that (2) does
> > not speak about the sentence-type
> >common to (1) and (2) but about the sentence-token named '(1)'.
That winds up being silly, too.
The point is NOT, as you said, that
> >different tokens
> >of a same non indexical sentence
CAN
> > have different logical values
It is rather that Sentences simpliciter DON'T HAVE
"logical values" (i.e. truth-values). Truth-values have
ALWAYS been PROPERLY ascribed ONLY to the tokens/utterances,
OR RATHER, to the "propositions" that are the Fregean senses
of the utterances, IN context. And you ALWAYS *need* a context
to "complete" the Fregean sense.
DMC again:
> Of course, the same syntactic sentence
> can have different meanings
> in different contexts; for example, "Today is Monday" will be true
> when uttered one day but false when uttered another day.
> But that's
> easily explained by the fact that the word "Today"
> denotes different
> subjects depending on when it is uttered.
That is a SIMILARITY, NOT a difference.
"This sentence", despite being officially classed as a demonstrative
rather than an indexical, is EVERY BIT as indexical as "today".
"This sentence" refers to a DIFFERENT sentence depending
on what sentence it IS UTTERED *in*, just as surely as "Today"
refers to a different day depending what day it is uttered on.
>
> In contrast, your two sentences (1) and (2) clearly have the
> same subject (namely, sentence (1)) and are clearly attempting
> to make the same claim about it (that it is not true).
But since "real-language" sentences CAN'T BE true
or false ANYwho, DESPITE the fact that we often speak
AS THOUGH they could, that is moot.
> Anyway, we can always introduce a new concept, "reliably true".
Again, NOT merely "can", but MUST.
> A sentence is reliably true if every token of that sentence is
> true in your sense. Then we can form the sentence
>
> (1) Sentence (1) is not reliably true.
>
> Clearly, (1) cannot be reliably true on pain of contradiction.
> So we conclude
>
> (2) Sentence (1) is not reliably true.
>
> But Sentence (2) is the same sentence as Sentence (1).
It IS NOT! JEEzus! Sentence (2) is labeled *2*!
Sentence (1) is labeled (1)! This CAN MATTER!
> So Sentence (2), while it may be true, can't be *reliably*
> true. Clearly, this conclusion holds for every token
> of sentence (1): That token is true, but not reliably true.
Sorry, you've (embarrassingly) just made a mistake with a concept
of your own invention. NO TOKENS ARE EVER reliably true,
just as no sentences are ever true simpliciter. TOKENS get to
be true simpliciter and SENTENCES get to be reliably true (iff all
their tokens are justPLAINtrue). Talking about whether a token
is or isn't reliably true is just confusing yourself.
If you actually want to make context relevant then you could
try Keith Simmons' invocation of the "reflective context".
I had a short unproductive argument with Andrew Clifton
about that in this newsgroup in early 2004. Simmons' article
on Deflationary Truth and the Liar is in vol.28 No.5 of the
Journal of Philosophical Logic but I don't have a link to it.
george
> What is your criterion for deciding which sentences don't have a truth
> value?
Obviously, the ability to derive a contradiction from the assumption
that they DO have a truth-value. This might be coupled with the
assumption that every sentence MUST have one of two truth-values.
> The sentence above is not a random sequence of letters.
Far more importantly, it is a GRAMMATICAL sequence of words
with KNOWN meanings. Therefore, it has a meaning. But it
cannot consistently be ascribed a truth-value.
> I'm going to stick to two-valued logic myself.
So which of your two values are you giving to this sentence?
Obviously, you are NOT sticking to two-valued logic.
John Jones
> This sentence is not true (1)
Which sentence?
If you can answer that I'll listen to you.
David Marcus
Neither. It is like an equation without a solution. If we assume such an
equation has a solution, we can produce paradoxical results. See the
article in the Mathematical Intelligencer that I referenced.
> Obviously, you are NOT sticking to two-valued logic.
I think I am.
--
David Marcus
galathaea
be careful of kirkham
although he has a very good overview of the field
and touches on a number of directions
his descriptions can be quite lacking at times
and is generally quite muddled
particularly his section on tarski
i like him as a source of furthur research
he is bibliographic in that way
but if i find him muddled...
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
Chris Menzel
> Let us consider the "strengthened" liar paradox:
>
> This sentence is not true (1)
>
> Richard L. Kikham in "Theories of Truth" (p. 293)
Kirkham
Newberry
Well, I did not say it was the best book on the subject. I found a few
problems myself. But in this particular case he claims that Kripke
himself does not believe that his (Kripke's) semantics does not
resolve the "strengthened' liar. This point is not quite clear to me
either as i do not know how a truth value would get assigned to "This
sentence is not true" if it does not get assigned to "This sentence is
false" or "This sentence is true."
>
> -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
> galathaea: prankster, fablist, magician, liar- Hide quoted text -
>
> - Show quoted text -
Newberry
But you are saying that sentence does not have to have a truth value.
So there are three possibilities
1. S is true
2. S is false
3. S does not have atruth value
>
> --
> David Marcus- Hide quoted text -
Newberry
Figuring whether sentence (1) is true or false is very, very simple.
Just compare it with reality!
Newberry
I tend to agree with everything you have said.
>
> I think these conclusions are inevitable in consistent bivalent logic.
That is I was with you so far. Now you have lost me. What does
bivalent logic have to do with this? If you admit the possibility of
no truth value then you obviously have a third option.
>
> Regards
Daryl McCullough
>On Feb 5, 11:47 am, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> LauLuna says...
>> That's why it's a paradox. By one line of reasoning, the sentence
>> is meaningless, while by another line of reasoning it is true.
>
>That's ridiculous.
I'm sure your post had something else worth reading,
but I'm not really motivated to continue after that
beginning.
Daryl McCullough
>> > I'm going to stick to two-valued logic myself.
>>
>> So which of your two values are you giving to this sentence?
>
>Neither. It is like an equation without a solution. If we assume such an
>equation has a solution, we can produce paradoxical results. See the
>article in the Mathematical Intelligencer that I referenced.
It seems to me that you are saying that some sentences are true,
and some sentences are false, and some sentences are neither.
I count three possibilities there, not two...
aatu.kos...@xortec.fi
> aatu.koskensi...@xortec.fi wrote:
> > The argument is just fine. Not true means just that, not true, and if
> > we allow the possibility that some sentences are neither true nor
> > false - for example because they're just random sequences of letters -
> > then not true does not mean, and is not equivalent to, "false".
>
> What is your criterion for deciding which sentences don't have a truth
> value?
I have no criteria to offer. My point was simply that the strengthened
liar is used to show that simply stipulating that e.g. the liar is
neither true nor false does not "solve" the paradox, since we can then
consider the strenghtened liar, for which stipulating that it is
neither true nor false does not resolve anything.
For my personal, though not particularly original, take on the liar,
see
http://groups.google.com/group/sci.logic/msg/3881153713544eb4
Going through old posts I also found a post from Torkel Franzén
presenting similar ideas:
http://groups.google.com/group/sci.philosophy.tech/msg/
812d6cd9e493316b
LauLuna
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
I'll make two points:
1. Tokenism is absolutely inevitable in consistent bivalent logic: no
one has ever found a flaw in the argument based on my (1) and (2).
2. Tokenism need not be a solution to Liar-like paradoxes; most
probably it is only a part of the solution.
Rejecting tokenism on the basis that it does not provide a solution to
all semantic paradoxes is an only too frequent false move. What is
needed to reject tokenism is a refutation of the argument by which it
is introduced. This I have never seen.
Your account for when the meaning of a sentence is fixed is the
classical one. It has to be rejected because of the paradoxes. A same
non indexical sentence can have different logical values in virtue of
the logical context in which it is used. I have argued in another
thread that this happens because not all states of affairs are
available to be asserted by any 'observer' (using terminology from
physics) in any context. But this is not the question here.
As for your meta-paradox, Abo proposed a related version recently. I
could offer an attempt of solution but I prefer not to do it here for
I wish to underline here that we don't need to derive a solution to
all semantic paradoxes from tokenism in order to accept it. Perhaps
other day, other thread.
However, it is clear that (1) has meaning in some sense: it is not
like 'blah blah' in all respects. But the question is whether it is
able to assert some state of affairs as being the case. And the
obvious answer is that it cannot.
Best regards
LauLuna
What does
> bivalent logic have to do with this? If you admit the possibility of
> no truth value then you obviously have a third option.
Oh, no. That's a frequent confusion. I think Bivalence should be
understood as referred to propositions (semantical objects), not to
sentences (syntactical objects). Propositions have to be either true
or false, although sentences can be none of both. If paradoxical
sentences are not taken into account, it might not be catastrophical
to say 'sentences' instead of 'propositions' (and this is the origin
of the confusion), but once we've brought paradoxical sentences in, we
should assign truth value always to propositions (even if for short we
speak otherwise).
So, the existence of sentences that have no truth value does not
contradict Bivalence.
At least that's the way I use 'bivalent'.
Regards
Daryl McCullough
>1. Tokenism is absolutely inevitable in consistent bivalent logic: no
>one has ever found a flaw in the argument based on my (1) and (2).
I think I must have missed something. I don't see anything at all
compelling about your argument. You are saying that two different
tokens of the same sentence can have different truth values. I would
agree, but only because the same referring expression can be interpreted
differently in different contexts.
So there is a context-dependent mapping between sentence tokens and
propositions. Different tokens of the same sentence may be mapped to
different propositions.
However, I think that the normal way that "true" is used would say
that a token is true if and only if the proposition to which it is
mapped is true. I don't see how it makes any sense to say that two
identical sentences with the same interpretation should have different
truth values.
>2. Tokenism need not be a solution to Liar-like paradoxes; most
>probably it is only a part of the solution.
I don't see how it helps at all.
>Rejecting tokenism on the basis that it does not provide a solution to
>all semantic paradoxes is an only too frequent false move. What is
>needed to reject tokenism is a refutation of the argument by which it
>is introduced. This I have never seen.
No, all that is needed to reject tokenism is to note that it doesn't
*help* anything. It's an unnecessary, pointless complication.
David Marcus
Yes, there are three possibilities, but I'm not adding the third as a
possible truth value. That probably needs some explanation.
Let's try an analogy first. Consider the following equations.
x = 7,
y = y + 1.
The first clearly has a solution. The second clearly doesn't. However,
we don't say this means we need more numbers. We don't define rules
for how "no solution" can be added to other numbers or what its square
root is. We just say there is no number that is a solution of the
second equation.
Now, let's go back to logic. Here is an example from "Semantic
Paradoxes as Equations" by Lan Wen (Mathematical Intelligencer, Vol.
23, No. 1, 2001, pp. 43-48):
The Three Cards Paradox. Consider three cards with the following
sentences:
The sentence on the second card is true, and the sentence on the
third card is false.
Either the sentence on the first card is false, or the sentence on
the third card is true.
The sentences on the first and second cards are both true.
You can check that this is a paradox (Wen's article gives a proof).
The way that Wen developed this paradox is to start with the following
system of equations in Boolean algebra. I'll use "-" for not, "+" for
or, and "*" for and.
x = y * (-z),
y = (-x) + z,
z = x * y.
This Boolean system has no solution (Wen's article gives a proof).
The Liar paradox corresponds to the Boolean equation x = -x.
Let me quote from Wen's article:
"The reader may have noticed a clear resemblance between the Boolean
system and the Three Cards paradox. The difference is clear too: In
their statements, one has a phrase 'has no solution', the other does
not; in their arguments, one has a standard frame of proof by
contradiction, that is the "head' 'Assume there is a solution, we
derive the following contradictions' and the 'tail' 'This
contradiction proves there is no solution', while the other does not.
"In fact my analysis of Three Cards was just translation of the
Boolean problem into ordinary language, only I cut off the phrase 'has
no solution' from the statement, and cut off the head and tail from
the argument. As expected, the normal Boolean proof became a
mysterious argument that leads to contradiction apparently with no
reason, that is, a 'paradox'.
"However, removing the head does not affect the argument. Because the
head 'Assume there exists a solution' is merely an announcement for
the assumption. The actual use of this assumption takes place not in
the head, but in the body of the Boolean proof. Removing the tail does
not affect the argument either, because the argument has finished
already. Thus the solution to the Three Cards paradox must be
(informally) this:
"It is (the translation of) the assumption of existence of a
solution that cause contradiction in the Three Cards paradox. The
assumption is tacit and goes unnoticed.
"It is believed traditionally that Liar-like paradoxes are logically
different from Boolean problems. It is believed that in Boolean
'proofs by contradiction' one assumes existence of solution and hence
derives contradiction, but in Liar-like paradoxes one does not assume
anything, except some basic rules of language and logic, hence
contradictions must have some deep, yet unknown cause in our language
or logic. The Three Cards paradox shows this is not the case."
--
David Marcus
LauLuna
Again consider the argument:
let's call '(1)' to the following sentence-token:
(1) '(1) expresses no true proposition'
We prove in the usual way that (1) has no truth value; so we are
entitled to state the following sentence-token (2):
(2) '(1) expresses no true proposition'
which expresses a true proposition.
There you are: two tokens of a non indexical sentence having different
logical values under the same linguistic code.
Where is the flaw?
We should not conflate the case of indexical sentences with the case
of different tokens if the same NON INDEXICAL sentence having
different logical values according to a constant linguistic code.
Tokenism need not help at all for any purpose whatsoever. It is not a
tool. It is a theory supported by the argument above.
If things are that complicated, well, that's how they are.
Regards
aatu.kos...@xortec.fi
> Let's try an analogy first. Consider the following equations.
>
> x = 7,
> y = y + 1.
>
> The first clearly has a solution. The second clearly doesn't.
But what about sentences that talk about the possibility of
consistently assigning a truth values to sentences? The strengthened
liar strikes back:
This sentence can not be consistently assigned a truth value or it is
not true.
Suppose we assign to it the truth value false. Then, in fact, it is
true since the last disjunct is true. Suppose we assign it the truth
value true. Then the first disjunct must hold, in which case the
sentence certainly can't be true. So we conclude that, in fact, the
sentence can't be consistently assigned a truth value. But that's just
what the first disjunct states, so the sentence should be true, after
all.
John Jones
Which is 'this' sentence?
I don't think you are getting the significance of this. You have
severe problems with your referencing criteria.
David Marcus
> On 6 helmi, 21:11, David Marcus <DavidMar...@alumdotmit.edu> wrote:
> > Let's try an analogy first. Consider the following equations.
> >
> > x = 7,
> > y = y + 1.
> >
> > The first clearly has a solution. The second clearly doesn't.
>
> But what about sentences that talk about the possibility of
> consistently assigning a truth values to sentences? The strengthened
> liar strikes back:
>
> This sentence can not be consistently assigned a truth value or it is
> not true.
>
> Suppose we assign to it the truth value false. Then, in fact, it is
> true since the last disjunct is true. Suppose we assign it the truth
> value true. Then the first disjunct must hold, in which case the
> sentence certainly can't be true. So we conclude that, in fact, the
> sentence can't be consistently assigned a truth value. But that's just
> what the first disjunct states, so the sentence should be true, after
> all.
If you want to discuss such sentences, then you need a formal
translation into some system. Boolean algebra doesn't suffice, since you
can't translate "assigned a truth value". If you argue using informal
logic (as you did above), then you are ignoring your assumption that a
solution exists. You used this assumption right at the start: "Suppose
we assign to it the truth value false". If you add the assumption
"Suppose a solution exists" to the beginning of your agument, then
(assuming your argument works in whatever formal translation you are
using) your conclusion at the end will be "Contradiction. So, no
solution exists."
Have you read the paper I referenced? It contains more than the excerpt
I posted.
--
David Marcus
Daryl McCullough
>Again consider the argument:
>
>let's call '(1)' to the following sentence-token:
>
>(1) '(1) expresses no true proposition'
>
>We prove in the usual way that (1) has no truth value;
What do you mean, you "prove in the usual way"? What's
usually done is to show that such a sentence is *paradoxical*.
Such a sentence, together with some plausible rules for
the way the phrase "true proposition" works leads to a
contradiction. From this it follows that *something* is
wrong with those plausible rules.
The usual resolution that I know of is to deny that
there is a truth predicate for languages that include
the word "true". You seem to be assuming that there is
some way out that involves classifying sentences such as
(1) as having no truth value. I don't think that's the
way to go, and I don't think that it is at all the
"usual way".
The usual way to go is to jetison the idea of absolute
truth in favor of the concept of truth under an interpretation.
Whether a sentence is true or not depends on the interpretation
of the terms and predicates. The same sentence can be true
under one interpretation and false (or meaningless) under
another interpretation.
>so we are entitled to state the following sentence-token (2):
>
>(2) '(1) expresses no true proposition'
I don't agree that we are entitled to state that. What we are
entitled to state is just that (1), together with certain
plausible rules for reasoning about truth, leads to a contradiction.
>which expresses a true proposition.
>There you are: two tokens of a non indexical sentence having different
>logical values under the same linguistic code.
>
>Where is the flaw?
The flaw is that you are reasoning within an inconsistent system. From
an inconsistent system you can prove anything.
The liar sentence, together with the usual rules for dealing with
the phrase "is true" leads to a contradiction. You can repair things
by introducing restrictions on how the phrase "is true" can be used,
but you haven't shown that there is only one way to repair things.
And there isn't just one way, there are many different ways. Perhaps
you're right that by carefully distinguishing between different tokens
of the same sentence you can make a consistent logic of truth. But you
haven't proved that, and you certainly have proved that that is *necessary*.
>We should not conflate the case of indexical sentences with the case
>of different tokens if the same NON INDEXICAL sentence having
>different logical values according to a constant linguistic code.
>
>Tokenism need not help at all for any purpose whatsoever. It is not a
>tool. It is a theory supported by the argument above.
I don't see that it is supported by your argument at all.
David Marcus
> aatu.kos...@xortec.fi wrote:
> > But what about sentences that talk about the possibility of
> > consistently assigning a truth values to sentences? The
> > strengthened liar strikes back:
> >
> > This sentence can not be consistently assigned a truth value or
> > it is not true.
> >
> > Suppose we assign to it the truth value false. Then, in fact, it
> > is true since the last disjunct is true. Suppose we assign it the
> > truth value true. Then the first disjunct must hold, in which case
> > the sentence certainly can't be true. So we conclude that, in
> > fact, the sentence can't be consistently assigned a truth value.
> > But that's just what the first disjunct states, so the sentence
> > should be true, after all.
>
> If you want to discuss such sentences, then you need a formal
> translation into some system. Boolean algebra doesn't suffice, since
> you can't translate "assigned a truth value". If you argue using
> informal logic (as you did above), then you are ignoring your
> assumption that a solution exists. You used this assumption right at
> the start: "Suppose we assign to it the truth value false". If you
> add the assumption "Suppose a solution exists" to the beginning of
> your argument, then (assuming your argument works in whatever formal
> translation you are using) your conclusion at the end will be
> "Contradiction. So, no solution exists."
Instead of {0,1} for our Boolean algebra, let's use {0,1,2}, where we
will interpret 2 as meaning "no truth value". Define functions n (for
"no truth value") and f (for "false") by
n(0) = n(1) = 0,
n(2) = 1,
f(0) = 1,
f(1) = f(2) = 1.
Then one interpretation for the sentence above corresponds to the
equation
x = n(x) + f(x).
If x = 0, then this is
0 = n(0) + f(0)
= 0 + 1
= 1.
If x = 1, then this is
1 = n(1) + f(1)
= 0 + 0
= 0.
If x = 2, then this is
2 = n(2) + f(2)
= 1 + 0
= 1.
So, the equation has no solution.
Let's define
-2 = 2,
0 + 2 = 2,
1 + 2 = 2.
Another interpretation for the sentence corresponds to the equation
x = n(x) + (-x).
Now, x = 2 is the unique solution. I prefer this interpretation.
--
David Marcus
Newberry
> On 6 helmi, 21:11, David Marcus <DavidMar...@alumdotmit.edu> wrote:
>
> > Let's try an analogy first. Consider the following equations.
>
> > x = 7,
> > y = y + 1.
>
> > The first clearly has a solution. The second clearly doesn't.
>
> But what about sentences that talk about the possibility of
> consistently assigning a truth values to sentences? The strengthened
> liar strikes back:
>
> This sentence can not be consistently assigned a truth value or it is
> not true.
>
> Suppose we assign to it the truth value false. Then, in fact, it is
> true since the last disjunct is true. Suppose we assign it the truth
> value true. Then the first disjunct must hold, in which case the
> sentence certainly can't be true. So we conclude that, in fact, the
> sentence can't be consistently assigned a truth value. But that's just
> what the first disjunct states, so the sentence should be true, after
> all.
The disjunct does not state anything; it is meaningless. (Some say the
disjunct is not a proposition.)
>
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
Newberry
> David Marcus wrote:
>
> - Show quoted text -
So who is right? There are reasons sentences are neither true nor
false other than an equation does not have a solution. "The Absolute
enters into but is itself incapable of evolution" is meaningles
because it is not a picture of any possible fact. Truth values T, F, M
seem natural to me.
David Marcus
About what?
> There are reasons sentences are neither true nor
> false other than an equation does not have a solution. "The Absolute
> enters into but is itself incapable of evolution" is meaningles
> because it is not a picture of any possible fact.
Obviously. However, the point is that the fact that certain equations
don't have solutions resolves the Liar paradox and similar paradoxes.
--
David Marcus
Newberry
Whether there are two truth values or three. I thought it was the
point of your previous post.
>
> > There are reasons sentences are neither true nor
> > false other than an equation does not have a solution. "The Absolute
> > enters into but is itself incapable of evolution" is meaningles
> > because it is not a picture of any possible fact.
>
> Obviously. However, the point is that the fact that certain equations
> don't have solutions resolves the Liar paradox and similar paradoxes.
Obviously. The issue is that on the face of it the sentence itself
seems to confirm that there is no solution. That creates the
impression that it is true.
LauLuna
wrote:
> >Where is the flaw?
>
> The flaw is that you are reasoning within an inconsistent system. From
> an inconsistent system you can prove anything.
Oh, no. I'm reasoning about an (apparently) inconsistent system, not
within it. This is another frequent confusion employed to reject any
reasoning from (the consideration of) paradoxes. I'm reasoning within
usual logic.
> Whether a sentence is true or not depends on the interpretation
> of the terms and predicates. The same sentence can be true
> under one interpretation and false (or meaningless) under
> another interpretation.
Let me say this is obvious but, as far as I can see, irrelevant to the
question.
Well, all I have used is a Tarskian scheme adapted to the possibility
of a sentence being incapable of stating anything and taking into
account the possibility that primitive truth bearers be propositions:
(T-) A sentence S is true (or expresses a true proposition)
according to some linguistic code C iff (there is something S says
according to C and) what S says according to C is the case.
Assuming (T-) seems to you to be too much? The fact is that it is
usually assumed in logical and mathematical proofs for it is an
obvious adaptation of Tarski's scheme.
So, the alternatives are:
1. Either rejecting an interpretation of the truth predicate that is
long since widely used in Logic, Math and in common day life,
2. Or rejecting that (1) expresses a proposition and has a truth
value.
An easy choice, I'd say.
Anyway, it is evident that if we don't agree on the meaning of the
word 'true', there is no easy way to go on with our talk.
So, let's put it in conditional:
(3) if (T-) holds, then (1) has no truth value (= is neither true nor
false)
Would you say (3) can be proved? Let me add:
(4) if (T-) and (3) hold, then (2) is true
Would you say (4) can be proved?
If yes, I suppose you will accept that tokenism can be proven from
(T-) or any equivalent scheme.
That's all I'm longing for :-).
Regards.
Daryl McCullough
>
>On Feb 6, 10:39 pm, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>
>> >Where is the flaw?
>>
>> The flaw is that you are reasoning within an inconsistent system. From
>> an inconsistent system you can prove anything.
>
>Oh, no. I'm reasoning about an (apparently) inconsistent system, not
>within it.
Well, then what system are you reasoning *in*? Try writing your
argument as an informal proof, making it clear what axioms you
are assuming, what rules of inference.
>This is another frequent confusion employed to reject any
>reasoning from (the consideration of) paradoxes. I'm reasoning within
>usual logic.
No, you are not. The predicate "is true" is not part of usual logic.
Tarski proved that it *can't* be. What is standard (as far as I
understand) is to *either* banish the predicate "is true", or to
restrict it to a particular language. If you take the first approach,
then sentence (1) is not expressible. If you take the second approach,
then there is nothing paradoxical about sentence (1) in the first place:
(1) Sentence (1) is not true_L.
where L is some specified language. In that case, sentence (1) is not
true in language L (because true_L is not available inside language L)
but is true inside a larger language that extends L with the predicate
true_L.
>> Whether a sentence is true or not depends on the interpretation
>> of the terms and predicates. The same sentence can be true
>> under one interpretation and false (or meaningless) under
>> another interpretation.
>
>Let me say this is obvious but, as far as I can see, irrelevant to the
>question.
>
>Well, all I have used is a Tarskian scheme adapted to the possibility
>of a sentence being incapable of stating anything and taking into
>account the possibility that primitive truth bearers be propositions:
>
>(T-) A sentence S is true (or expresses a true proposition)
>according to some linguistic code C iff (there is something S says
>according to C and) what S says according to C is the case.
>
>Assuming (T-) seems to you to be too much?
I'm not objecting to your T-.
>So, the alternatives are:
>
>1. Either rejecting an interpretation of the truth predicate that is
>long since widely used in Logic, Math and in common day life,
I think that's the way to go. The informal way that "true" is
used is inconsistent. There is nothing much lost by restricting
"true" to something that is language-specific, rather than absolute.
In any particular non-paradoxical context, it is usually possible
to figure out which truth predicate is meant, and the meaning of
particular sentences involving "true" are not substantially affected
by the choice.
>So, let's put it in conditional:
>
>(3) if (T-) holds, then (1) has no truth value (= is neither true nor
>false)
Your T- only talks about truth within a "linguistic code C", but
your conclusion (3) makes no mention of such a code. So I would
consider (3) to be misleading. I would say, rather
if (T-) holds, and the word "true" is interpreted as true within
a linguistic code C, then (1) has no truth value in linguistic code C.
>Would you say (3) can be proved? Let me add:
>
>(4) if (T-) and (3) hold, then (2) is true
>
>Would you say (4) can be proved?
If you make the linguistic code explicit, then (1) can be proved!
(1) Sentence (1) has no truth value according to linguistic code C.
That's true, but only in an extended language that goes beyond linguistic
code C. Sentence (2) says the same thing, and is true in exactly the same
way as sentence (1).
>If yes, I suppose you will accept that tokenism can be proven from
>(T-) or any equivalent scheme.
No, I don't.
Daryl McCullough
>No, you are not. The predicate "is true" is not part of usual logic.
>Tarski proved that it *can't* be. What is standard (as far as I
>understand) is to *either* banish the predicate "is true", or to
>restrict it to a particular language.
I left out some other common approaches. One approach that left out
is the "deflationary" view of truth. According to this view, the
meaning of a sentence involving "true" is obtained by replacing
"X is true" by just X. So the meaning of
It is true that snow is white.
is just
Snow is white.
If such a replacement is not possible then the sentence is meaningless.
This sort of works, but not quite. If you want to apply that to a
construction such as
"Everything that Bob tells you is true"
then the replacement would be an infinite conjunction of the
form
"If Bob tells you that snow is white, then snow is white,
and if Bob tells you that it will rain, then it will rain,
..."
where the ... includes everything that Bob could possibly tell you.
The deflationary view of truth does *not* allow us to conclude that
a meaningless sentence is not true.
It is not true that griffles blanchify gracelessly.
means the same thing as
Griffles do not blanchify gracelessly.
so both sentences are equally meaningless.
David Marcus
> On Feb 6, 10:26 pm, David Marcus <DavidMar...@alumdotmit.edu> wrote:
> > Newberry wrote:
> > > On Feb 6, 5:42 pm, David Marcus <DavidMar...@alumdotmit.edu> wrote:
> > > > David Marcus wrote:
> > > > > aatu.koskensi...@xortec.fi wrote:
> > > > > > This sentence can not be consistently assigned a truth value or
> > > > > > it is not true.
> > > > Instead of {0,1} for our Boolean algebra, let's use {0,1,2}, where we
What do you mean "whether there are"?
English is not as precise as math. Once we decide how to formalize an
English sentence, then we can analyze the situation. For sentences in
mathematics, two truth values suffice. To resolve the Liar paradox, two
truth values suffice. Aatu's sentence talks about sentences not having
truth values. So, to analyze this sentence it is convenient to work in a
system that contains three "truth values".
> > > There are reasons sentences are neither true nor
> > > false other than an equation does not have a solution. "The Absolute
> > > enters into but is itself incapable of evolution" is meaningles
> > > because it is not a picture of any possible fact.
> >
> > Obviously. However, the point is that the fact that certain equations
> > don't have solutions resolves the Liar paradox and similar paradoxes.
>
> Obviously. The issue is that on the face of it the sentence itself
> seems to confirm that there is no solution. That creates the
> impression that it is true.
Which is why you have to translate the imprecise English into precise
mathematics to see what is going on. Once you do that, the paradox
vanishes. In this case (using my second interpretation), the paradox
comes about because we naively assume that for all x, 1 + x = 1.
However, 1 + 2 = 2. So, the sentence isn't true.
--
David Marcus
John Jones
> Let us consider the "strengthened" liar paradox:
>
marcia
> On Feb 5, 2:04?am, "Newberry" <newbe...@ureach.com> wrote:
>
> > Let us consider the "strengthened" liar paradox:
>
>
> Which is 'this' sentence?
Dazzling them with your intelligence, John?
Newberry
>
> - Show quoted text -
Anyway, how do you define -0 and -1 in your calculus? If -0 = 1 and -1
= 0 then not true is equvalent to false and the liar has not been
strengthened at all.
David Marcus
> = 0
Yes.
> then not true is equvalent to false
If you don't like this, then you can use my first interpretation above.
In that case the equation has no solution. Such is life: not all
equations have solutions.
> and the liar has not been strengthened at all.
So what? You're unhappy because there is no paradox?
All the Liar type paradoxes are basically the same. They implicitly
assume a solution exists when none does. We can do the same without all
the logic and English. Just assume x = x + 1 has a solution and subtract
x from both sides. Then 0 = 1, and you've got a paradox.
--
David Marcus
William Elliot
This sentence is this sentence.
That sentence is not this sentence.
R. Srinivasan
> On Feb 5, 2:04?am, "Newberry" <newbe...@ureach.com> wrote:
>
> > Let us consider the "strengthened" liar paradox:
>
>
> Which is 'this' sentence?
>
That is the problem. The moment we utter "This sentence ....." we
*must* have a sentence in mind and provide a *construction* for that
sentence. Then, and only then, can we make assertions about the truth
or falsity of that sentence. If this rule is adhered to, the liar
"sentence" is illegitimate and there is no paradox. When we say some
sentence is not true , then obviously we are making an assertion about
a *completed* sentence and "is not true" cannot be part of that
sentence.
But then what do we know compared to the "greats" who debate this
issue endlessley.
Regards, RS
R. Srinivasan
> On Wed, 7 Feb 2007, John Jones wrote:
> > On Feb 5, 2:04?am, "Newberry" <newbe...@ureach.com> wrote:
> > > Let us consider the "strengthened" liar paradox:
>
>
> > Which is 'this' sentence?
>
> This sentence is this sentence.
> That sentence is not this sentence.
>
sentence.
This sentence is not this sentence --> This sentence could be this
sentence
That sentence is not this sentence --> That sentence could be this
sentence
Regards, RS
William Elliot
this sentence could be this sentence.
Daryl McCullough
>That is the problem. The moment we utter "This sentence ....." we
>*must* have a sentence in mind and provide a *construction* for that
>sentence. Then, and only then, can we make assertions about the truth
>or falsity of that sentence. If this rule is adhered to, the liar
>"sentence" is illegitimate and there is no paradox.
That's certainly a resolution, but I don't think it really sheds
much light on the original paradox. With other sorts of properties
other than truth, we can certainly make sense of properties of
sentences that have not yet been uttered. There is no problem
whatsoever with the self-referential sentence
This sentence has five words.
As it was being typed, it may not have been clear what "this sentence"
referred to, but afterwards we can see that "this sentence" can consistently
be taken to refer to the entire sentence "This sentence has five words".
It doesn't cause any problems. Why is truth special?
Your resolution, which is that the word "true" can
only be applied to previously uttered sentences, is
a way to avoid the paradox, but it's an artificial
way. What, conceptually, is wrong with my saying
"When you meet with Dr. Smith tomorrow, don't believe
whatever he says about cold fusion. It won't be true."
Your resolution is just a particular instance of a more
general solution, which is that the use of "true" must
be well-founded. More generally, we can use an arbitrary
well-founded partial ordering < on sentences and stipulate that
a sentence may only meaningfully refer to the truth of
other sentences that occur "earlier" in the ordering.
You are using the time of utterance as the ordering,
but other approaches work just as well. For instance,
we can classify sentences as follows:
A level-0 sentence is one that does not refer to truth at all.
A level-1 sentence is one that only talks about the truth of
level-0 sentences.
A level-2 sentence is one that only talks about the truth of
level-1 sentences.
etc.
There is no paradox possible using sentences that are assigned
a level. It doesn't matter what order they are uttered.
The liar sentence "This sentence is not true" is a sentence
that cannot be given a level.
John Jones
> On Wed, 7 Feb 2007, John Jones wrote:
> > On Feb 5, 2:04?am, "Newberry" <newbe...@ureach.com> wrote:
> > > Let us consider the "strengthened" liar paradox:
>
>
> > Which is 'this' sentence?
>
> This sentence is this sentence.
> That sentence is not this sentence.
It doesn't work does it. You MUST use the clause 'this sentence' to
indicate and not to self-reference.
John Jones
I am the great you refer to. You have repeated my point on all this. I
have solved the liar paradox and I will get the credit please, not you
or some bearded old goat found in a reference.
R. Srinivasan
It is not a big deal anyway and probably many others have said this
before either of us. For the record, I did make a similar assertion in
a posting dated Sep 24, 2002 in sci.logic under the title "My
resolution of the Liar Paradox".
By the way, the person who started that thread titled "Strong liar"
was Logicist,who I believe is now Newberry.
Regards, RS
R. Srinivasan
> R. Srinivasan says...
>
> >That is the problem. The moment we utter "This sentence ....." we
> >*must* have a sentence in mind and provide a *construction* for that
> >sentence. Then, and only then, can we make assertions about the truth
> >or falsity of that sentence. If this rule is adhered to, the liar
> >"sentence" is illegitimate and there is no paradox.
>
> That's certainly a resolution, but I don't think it really sheds
> much light on the original paradox. With other sorts of properties
> other than truth, we can certainly make sense of properties of
> sentences that have not yet been uttered. There is no problem
> whatsoever with the self-referential sentence
>
> This sentence has five words.
>
> As it was being typed, it may not have been clear what "this sentence"
> referred to, but afterwards we can see that "this sentence" can consistently
> be taken to refer to the entire sentence "This sentence has five words".
> It doesn't cause any problems. Why is truth special?
>
objecting to
This sentence has five words (0)
Let this sentence be formalized in some sound theory T as the
sentence "P" and let "~P" be its negation. Formally, let us say that
P is provable in T, so we can clearly see that ~P must be false via a
refutation in T. So what does ~P express in English? It cannot be
It is not the case that this sentence has five words (1)
because (1) is a true statement; after all, we have laid down that
"this sentence" must refer to the sentence that it is part of.
On the other hand if P is taken to formalize:
"This sentence has five words" is a sentence having five words (2)
then we can easily see that P is true and ~P does have a meaningful
English equivalent, which we can clearly see to be false:
It is not the case that "This sentence has five words" is a sentence
having five words. (3)
The problem with (0) and (1) is, I believe, that they use the same
word "sentence" in two different senses. One is that "sentence" refers
to the sentence (0) or (1) and the other is that "sentence" is merely
a word whose meaning we do not care about; you can see that these two
senses are clearly separated in (2) and (3).
>
> Your resolution, which is that the word "true" can
> only be applied to previously uttered sentences, is
> a way to avoid the paradox, but it's an artificial
> way. What, conceptually, is wrong with my saying
> "When you meet with Dr. Smith tomorrow, don't believe
> whatever he says about cold fusion. It won't be true."
>
Here "true" only refers to whatever Dr. Smith might utter about cold
fusion. So you can view the above as a metamathematical assertion
about an unspecified sentence that Dr. Smith may utter about cold
fusion. It looks unobjectionable to me.
Regards, RS
Daryl McCullough
>> This sentence has five words.
>>
>> As it was being typed, it may not have been clear what "this sentence"
>> referred to, but afterwards we can see that "this sentence" can consistently
>> be taken to refer to the entire sentence "This sentence has five words".
>> It doesn't cause any problems. Why is truth special?
>>
>You know more about this than I do, but let me take a stab at
>objecting to
>
>This sentence has five words (0)
>
>Let this sentence be formalized in some sound theory T as the
>sentence "P" and let "~P" be its negation. Formally, let us say that
>P is provable in T, so we can clearly see that ~P must be false via a
>refutation in T. So what does ~P express in English?
You have to be careful here. If on Monday I say "It will not rain today"
and then on Tuesday say "It will rain today", then have I
contradicted myself? Of course not. The word "today" has a different
referent in those two sentences. Similarly, in the pair of sentences
(1) This sentence has five words.
and
(2) This sentence does not have five words.
does the second sentence contradict the first? Of course not,
because the phrase "this sentence" has a different referent
in the two cases.
Sentences such as "It will rain today" or "This sentence has five
words" have referring expressions "today" and "this sentence" whose
meanings are context dependent. To successfully negate a sentence
involving such an expression in a new context, you have to first
come up with alternative referring expression that works in the
new context. If on Monday I say "It will rain today" and then
I want to say the negation on Tuesday, I have to say "It did not
rain on Monday". Those two statements, even though they don't
look like negations, are negations in the semantic sense that
if the first sentence was true when it was spoken, then the
second sentence was false when it was spoken, and vice-versa.
To successfully negate "This sentence has five words" you have
to come up with a second sentence which is true if and only
if the first sentence is false. The closest we can come is
"'This sentence has five words' does not have five words"
John Jones
wrote:
> This sentence has five words.
>
> As it was being typed, it may not have been clear what "this sentence"
> referred to, but afterwards we can see that "this sentence" can consistently
> be taken to refer to the entire sentence "This sentence has five words".
This is an example where logicians fail at the source. If you built up
a formal system on your argument then it would be wrong at the
foundations.
You are using a different referential framework when you say "but
afterwards..." . You are using "this" from a different perspective.
So my objection still stands.
John Jones
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
It comes down to this. If we propose that self-reference is reference,
we reduce the meaning of a 'truth value' of the 'self-referencing'
statement beyond rational intelligibility.
John Jones
>
> - Show quoted text -
I don't think it is similar or otherwise. Your example refers to a
truncated sentence 'this sentence ...', while I argue for the complete
sentence 'this sentence abc'. You also do not say why 'this
sentence ...' cannot refer. If I have missed anything out let me know.
Newberry
they corresond to reality not based on solutions of Boolean equations.
For example when I want to assign a truth value to the sentence "The
grass is green" I look out of the window and check if the grass is
green. If so then the sentence is true. Which Boolean equation would
you solve to determine if "The grass is green" is true?
David Marcus
> Well, I think we assign truth values to sentences based on whether
> they corresond to reality not based on solutions of Boolean equations.
> For example when I want to assign a truth value to the sentence "The
> grass is green" I look out of the window and check if the grass is
> green. If so then the sentence is true. Which Boolean equation would
> you solve to determine if "The grass is green" is true?
Are you trolling?
--
David Marcus
William Elliot
> R. Srinivasan says...
>
> This sentence has five words.
>
That sentence has five words.
That sentence has five words.
That sentence has five words.
...
...
...
William Elliot
galathaea
this made me smile
i dont know why
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
R. Srinivasan
> I don't think it is similar or otherwise. Your example refers to a
> truncated sentence 'this sentence ...', while I argue for the complete
> sentence 'this sentence abc'. You also do not say why 'this
> sentence ...' cannot refer. If I have missed anything out let me know.
>
In my posting of Sep. 2002, I was arguing that when a sentence starts
with "This sentence....", the sentence referred to must have a prior
existence that is independent of what comes afterwards. In that
posting I tried to argue that the self-reference in a sentence like
"This sentence ...." is objectionable from the intuitionistic point of
view, since a human being who utters "This sentence....." cannot
possibly have a sentence in mind just after uttering "sentence". After
all, intuitionism specifies that truths are mental constructions. Now
I know that intuitionism only requires that the human being must
*eventually* be able to provide a construction for "sentence". Unlike
NAFL, intuitionism is not capable of formalizing the temporal nature
of truth.
The objection to self-reference must be formulated in my proposed
logic NAFL. The logical objection is that to formalize self-reference,
such as Godel's "This sentence is unprovable", one needs to quantify
over infinite entities which NAFL does not permit. The reason being
that if quantification over infinite entities is allowed, that leads
to undecidable propositions (say, in Peano arithmetic via Godel's
theorems) which will violate the NAFL truth definition.
>From the temporal point of view NAFL does demand that a *prior*
construction be available for "sentence" the moment one utters "This
sentence....". If the human mind fails to complete the rest of this
sentence, it is clear that "sentence" refers to nothing, and such a
non-constructive existence (even if only temporary) is not permitted
for entities like sentences in NAFL. But NAFL does permit non-
constructive existence in general, provided such existence does not
violate the NAFL truth definition.
By the way, "This sentence ...." can refer, at least informally. For
example, I can say
'Consider the sentence "The sun rises in the east". This sentence has
6 words.'
I don't see any problems with "This sentence .." in the above, at
least in an informal sense.
Regards, RS
R. Srinivasan
These sentences have five words.
So where are these sentences?
They're either here or there
Which's neither here nor there
William Elliot
> On Feb 9, 10:04 am, William Elliot <m...@hevanet.remove.com> wrote:
> > On Thu, 8 Feb 2007, Daryl McCullough wrote:
> > > R. Srinivasan says...
> >
> > > This sentence has five words.
> >
> > That sentence has five words.
> > That sentence has five words.
> > ...
> >
> This sentence has five words
> That sentence has five words.
> Those sentences have five words
> These sentences have five words.
> So where are these sentences?
> They're either here or there
> Which's neither here nor there
>
Where is Everywhere?
Is it here, is it there?
Is it up, is it down?
Is it left, is it right?
Is it anywhere?
Is it somewhere?
Is it nowhere?
Where is Everywhere?
And so the Unfinished Opera remains to this day unfinished.
It is an eternal story,
for if it ever came to an end, it would be finished.
aatu.kos...@xortec.fi
> > consistently assigning a truth values to sentences? The strengthened
> > liar strikes back:
>
> > This sentence can not be consistently assigned a truth value or it is
> > not true.
>
> > true since the last disjunct is true. Suppose we assign it the truth
> > value true. Then the first disjunct must hold, in which case the
> > sentence certainly can't be true. So we conclude that, in fact, the
> > sentence can't be consistently assigned a truth value. But that's just
> > what the first disjunct states, so the sentence should be true, after
> > all.
>
> If you want to discuss such sentences, then you need a formal
> translation into some system.
Why? In fact, in order to determine which system or theory is the
correct choice we'd need to resolve what sort of modes of reasoning
and basic principles concerning truth it would contain. And that we
can't do before "resolving", in whatever way, the instance of the
(strenghtened) liar we're considering.
> Boolean algebra doesn't suffice, since you
> can't translate "assigned a truth value". If you argue using informal
> logic (as you did above), then you are ignoring your assumption that a
> solution exists.
Where did I rely on that assumption? I just went through the three
possibilities: no solution exists, there exists a solution assigning
the sentence the truth value 'true', there exists a solution assigning
the sentence the truth value 'false'. All apparently lead to
paradoxical conclusion.
> You used this assumption right at the start: "Suppose
> we assign to it the truth value false". If you add the assumption
> "Suppose a solution exists" to the beginning of your agument, then
> (assuming your argument works in whatever formal translation you are
> using) your conclusion at the end will be "Contradiction. So, no
> solution exists."
Right. But then according to any normal understanding the sentence is
true given that that's exactly what the first disjunct states.
> Have you read the paper I referenced?
No. Did you read the post I gave a link to, expounding the view that
the liar demonstrates the "indefinite extensibility" of truth?
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
John Jones
>
> - Show quoted text -
OK. I can see why you state the following and also accept that NAFL
cannot allow it.
> such as Godel's "This sentence is unprovable", one needs to quantify
> over infinite entities which NAFL does not permit.
It looks like infinite entities are proposed because by accomodating
ALL sentences we can eventually reference 'this sentence'.
But the infinite entity proposal is first unnecessary, and second, not
possible. It is unnecessary because we cannot identify 'this sentence'
by its mere referencing itself. A self-reference is not identifiable.
Second, the infinite entity proposal cannot be made because it relies
on a model of a hybrid of reference and self-reference. This is
because the infinite entity proposal is a normal referencing sentence
that, unfortunately, identifies "all" referencing sentences and self-
referencing sentences as equivalent regarding their referencing
ability, and this is not true.
So while NAFL may well show that we cannot quantify over infinite
entities, we do not need to invoke infinite entities if we want to
show that the Godel self-reference sentence is not supportable.
R. Srinivasan
> cannot allow it.
>
> > such as Godel's "This sentence is unprovable", one needs to quantify
> > over infinite entities which NAFL does not permit.
>
> It looks like infinite entities are proposed because by accomodating
> ALL sentences we can eventually reference 'this sentence'.
>
do you *formalize* it, i.e., how do you translate it into the language
of a first-order theory (say, Peano Arithmetic)? This is what Godel
achieved. The Godel sentence for PA is a purely arithmetical sentence
and can be thought of as a translation of the informal statement "This
sentence is unprovable". But in order to achieve this translation
Godel had to quantify over functions -- see for example the diagonal
argument given in the recent sci.logic thread "Goedel's proof" started
by Newberry. In other words, Godel had to consider infinitely many
functions fq(n), where each function f1(n), f2(n), ... is an infinite
entity. This is what NAFL objects to -- you cannot invoke an argument
that quantifies over infinite entities like "functions", i.e., you
cannot invoke infinitely many infinite entities like "functions" in a
legitimate NAFL proof.
Classically, one takes the stand that quantification over infinite
entities is legitimate. Since Godel translated self-referential
propositions into arithmetic using such quantification (and other
legitimate techniques), it follows that self-referential propositions
are classically legitimate.
Whereas in NAFL, one takes the stand that said quantification over
infinite entities is an infiintary operation and illegitimate. Since
self-referential propositions can only be translated into the language
of first-order theories using such an illegitimate technique, it
follows that self-referential propostions are illegitimate in NAFL. In
fact quantification over infinite entities is a self-referential
operation from the NAFL point of view.
>
> But the infinite entity proposal is first unnecessary, and second, not
> possible. It is unnecessary because we cannot identify 'this sentence'
> by its mere referencing itself. A self-reference is not identifiable.
>
> Second, the infinite entity proposal cannot be made because it relies
> on a model of a hybrid of reference and self-reference. This is
> because the infinite entity proposal is a normal referencing sentence
> that, unfortunately, identifies "all" referencing sentences and self-
> referencing sentences as equivalent regarding their referencing
> ability, and this is not true.
>
> So while NAFL may well show that we cannot quantify over infinite
> entities, we do not need to invoke infinite entities if we want to
> show that the Godel self-reference sentence is not supportable.
>
The *formal* Goedel sentence, for, say, PA, is purely arithmetical and
as such, is a legitimate sentence of PA. What you say is not
supportable is the informal version of the Goedel sentence in the form
"This sentence is unprovable". You apparently are giving a direct
argument for why this is not a legitmate sentence, even in informal
discourse. Whereas I am saying that in order to *formalize* this
sentence in classical first order predicate logic (i.e, translate it
into the language of first-order theories), you do need quantification
over infinite entities like functions, which makes it illegitmate from
the NAFL point of view.
Informally, my objection to "This sentence is not true" is that when
we utter "sentence", we have no construction in mind for any sentence.
This only comes later, and the delay is also objectionable from the
NAFL point of view.
Regards, RS
George Dance
> On 5 Feb, 07:50, David Marcus wrote:
>
> > Newberry wrote:
> > > Let us consider the "strengthened" liar paradox:
>
>
> > > "If it is neiter true nor false (and thus not true), then it is
> > > precisely what it claims to be, so it is true." I wonder if this
> > > argument is correct.
>
> > Obviously not (since not true normally means false).
>
> The argument is just fine.
Read it again. The 'sentence' says that it is not true, but
(according to Kikham) is is true. Are you saying that it's both true
and not true?
> Not true means just that, not true, and if
> we allow the possibility that some sentences are neither true nor
> false - for example because they're just random sequences of letters -
> then not true does not mean, and is not equivalent to, "false".
>
If you think that 'random sequences of letters' are 'sentences,' then
that explains why you're hopelessly confused by the whole subject.
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Why not take W's advice sometime?
Daryl McCullough
>No. The informal statement is "This sentence is unprovable". But how
>do you *formalize* it, i.e., how do you translate it into the language
>of a first-order theory (say, Peano Arithmetic)? This is what Godel
>achieved. The Godel sentence for PA is a purely arithmetical sentence
>and can be thought of as a translation of the informal statement "This
>sentence is unprovable". But in order to achieve this translation
>Godel had to quantify over functions -- see for example the diagonal
>argument given in the recent sci.logic thread "Goedel's proof" started
>by Newberry.
That doesn't sound correct. Goedel's translation doesn't quantify over
functions. There are several parts to Goedel's translation:
(1) Associating natural numbers with the formulas of PA.
(2) Defining a formula in PA (call it Theorem(x)) such that
for each natural number n, PA proves Theorem([n]) if and only
if n is the code of a theorem of PA. (where [n] means the
numeral corresponding to number n.
(3) Proving the fixed point lemma.
This says that for every formula Phi of PA, there is a
natural number n that is the code for a formula Psi such
that PA proves Psi <-> Phi' where Phi' is the result of
substituting [n] for all free variables occurring in Phi.
(4) Applying the fixed point lemma to the formula not Theorem(x)
gives Godel's theorem.
>In other words, Godel had to consider infinitely many
>functions fq(n), where each function f1(n), f2(n), ... is an infinite
>entity.
No, he doesn't do that.
>Informally, my objection to "This sentence is not true" is that when
>we utter "sentence", we have no construction in mind for any sentence.
That's an inessential feature of the liar paradox. Instead of using
the phrase "this sentence", you can introduce fixed points in some
other way. For example,
If S is any string, define the "quotation" of S to be the result of
replacing all occurrences of '\' in S by '\\', and replacing all
occurrences of '"' in S by '\"', and then placing '"' at the beginning
and end. For example, the quotation of
hello
is
"hello"
and the quotation of that is
"\"hello\""
and the quotation of that is
"\"\\\"hello\\\"\""
Now define the "fixed point term" for any string S to be
the string "The fixed point of" followed by the quotation
of S. So the fixed point term of
hello
is
The fixed point of "hello"
Finally, define the fixed point of any string S to be the
result of replacing all occurrences of the string
"[fill in the blank]" in S by the fixed point term of S.
So the fixed point of
hello
is just
hello
(since "hello" doesn't contain the string "[fill in the blank]") while
the fixed point of
Hello, [fill in the blank].
is
Hello, The fixed point of "Hello, [fill in the blank].".
So far, there is nothing *meaningful* about any of these operations.
They are just operations on meaningless strings that produce other
meaningless strings. However, now consider the string, (call it
L0)
[fill in the blank] is not true.
This is still a meaningless string. But now let's take the
fixed point of L0. It's the following string (call it L).
The fixed point of "[fill in the blank] is not true." is not true.
L appears to be a meaningful sentence. It seems to be claiming
that the fixed point of some particular string is not true.
That string is actually the string L0. So L seems to be claiming
that the fixed point of L0 is not true. But the fixed point of L0
is L. So L seems to be claiming that L itself is not true.
This is accomplished without actually using the construction
"This sentence".
LauLuna
> Daryl McCullough says...
>
> >No, you are not. The predicate "is true" is not part of usual logic.
> >Tarski proved that it *can't* be. What is standard (as far as I
> >understand) is to *either* banish the predicate "is true", or to
> >restrict it to a particular language.
>
> I left out some other common approaches. One approach that left out
> is the "deflationary" view of truth. According to this view, the
> meaning of a sentence involving "true" is obtained by replacing
> "X is true" by just X. So the meaning of
>
> It is true that snow is white.
>
> is just
>
> Snow is white.
>
> If such a replacement is not possible then the sentence is meaningless.
>
> This sort of works, but not quite. If you want to apply that to a
> construction such as
>
> "Everything that Bob tells you is true"
>
> then the replacement would be an infinite conjunction of the
> form
>
> "If Bob tells you that snow is white, then snow is white,
> and if Bob tells you that it will rain, then it will rain,
> ..."
>
> where the ... includes everything that Bob could possibly tell you.
>
> The deflationary view of truth does *not* allow us to conclude that
> a meaningless sentence is not true.
>
> It is not true that griffles blanchify gracelessly.
>
> means the same thing as
>
> Griffles do not blanchify gracelessly.
>
> so both sentences are equally meaningless.
>
> --
> Daryl McCullough
> Ithaca, NY
Oh, let's make the use of a code C explicit wherever you wish.
Then we have that according to code C and to T- (1) has no truth
value, where (1) is still:
'(1) expresses no true proposition'
Then, we have too that (1) expresses no true proposition, and this
token I called '(2)'.
It seems that you say that (2) does not obey the linguistic code C
we've used to interpret (1) and pronounce it non-propositional.
But it does. The conclusion of the argument is that there is a code C
(let's say English), such that if we apply C to (1), (1) turns out to
express no proposition and if we apply C to (2), (2) turns out to
express a true proposition.
Our move has been to interpret (1) as though it said what (2)
effectively says and to deduce that, under that interpretation, (1)
expresses no true proposition. So our reasoning is based on giving the
same interpretation to the two tokens of the same sentence and then
concluding that under that interpretation one expresses no proposition
while the other expresses a true proposition.
So, it is not a shift from one code or interpretation to another, but
a change in the logical context what makes the difference between (1)
and (2).
Now, it is clear that I am not reasining within an inconsistent system
but, if anything, about it, the burden of finding a flaw in my
reasoning falls on your part. I'm clumsy building up axiomatic
systems. But it is not fair to demand 'exactly what are your axioms
and inference rules in that argument?' in order to assess the
argument.
For, even if I would present one system, you could then ask what my
axioms and rules have been in the construction of my system, and so
on.
So, what is exactly the flaw now?
Regards
aatu.kos...@xortec.fi
> On Feb 5, 7:17 am,aatu.koskensi...@xortec.fi wrote:
> > The argument is just fine.
>
> Read it again. The 'sentence' says that it is not true, but
> (according to Kikham) is is true. Are you saying that it's both true
> and not true?
No, I'm not saying anything about the truth or falsity of the
strenghtened liar. Kirkham's argument is just fine in the sense that
it shows that if we accept the T-scheme for sentences containing the
word 'true' a contradiction follows.
> If you think that 'random sequences of letters' are 'sentences,' then
> that explains why you're hopelessly confused by the whole subject.
For my edification, perhaps you could mention a few examples of this
hopeless confusion manifesting?
> Why not take W's advice sometime?
Good idea - give it a try!
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
Chris Menzel
> ...The 'sentence' says that it is not true, but (according to Kikham)
Kirkham. Richard Kirkham.
LauLuna
> As it was being typed, it may not have been clear what "this sentence"
> referred to, but afterwards we can see that "this sentence" can consistently
> be taken to refer to the entire sentence "This sentence has five words".
> It doesn't cause any problems. Why is truth special?
I think the difference is that 'has five words' is a predicate for a
sentence (a syntactical object, a piece of objective world), while 'is
true' is a predicate for a proposition and propositions are not
always objective stuff in the objective world; reference to
propositions implies reference to thoughts and a thought cannot think
itself.
It could be argued that 'expresses no true proposition' is a predicate
for sentences and gives rise all the same to paradoxes of self-
reference. The cause is that this second predicate forces anyway to
consider propositions.
While all we have to do is considering objects in the world, like
strings of symbols, there will be no paradox. That may be the clue.
Regards
> For instance,
> we can classify sentences as follows:
>
> A level-0 sentence is one that does not refer to truth at all.
> A level-1 sentence is one that only talks about the truth of
> level-0 sentences.
> A level-2 sentence is one that only talks about the truth of
> level-1 sentences.
> etc.
>
> There is no paradox possible using sentences that are assigned
> a level. It doesn't matter what order they are uttered.
> The liar sentence "This sentence is not true" is a sentence
> that cannot be given a level.
We all know what the problem with those levels is; sentences like:
'no sentence can attribute truth to a sentence of its own level'
can't be given a level.
Whatever levels we propose, we go beyond them. So our theory can never
become completely explicit.
There are things about which we can speak but about which we cannot be
quite explicit.
Regards
R. Srinivasan
> R. Srinivasan says...
>
> >No. The informal statement is "This sentence is unprovable". But how
> >do you *formalize* it, i.e., how do you translate it into the language
> >of a first-order theory (say, Peano Arithmetic)? This is what Godel
> >achieved. The Godel sentence for PA is a purely arithmetical sentence
> >and can be thought of as a translation of the informal statement "This
> >sentence is unprovable". But in order to achieve this translation
> >Godel had to quantify over functions -- see for example the diagonal
> >argument given in the recent sci.logic thread "Goedel's proof" started
> >by Newberry.
>
> That doesn't sound correct. Goedel's translation doesn't quantify over
> functions. There are several parts to Goedel's translation:
>
> (1) Associating natural numbers with the formulas of PA.
> (2) Defining a formula in PA (call it Theorem(x)) such that
> for each natural number n, PA proves Theorem([n]) if and only
> if n is the code of a theorem of PA. (where [n] means the
> numeral corresponding to number n.
> (3) Proving the fixed point lemma.
> This says that for every formula Phi of PA, there is a
> natural number n that is the code for a formula Psi such
> that PA proves Psi <-> Phi' where Phi' is the result of
> substituting [n] for all free variables occurring in Phi.
>
said something similar to what you are saying. The difference in our
perceptions occurs at the above point. The objection is to
quantification over formulae like Phi with one or more free variables
(i.e., "For every formula Phi..." is objectionable in NAFL).
Classically, you view a formula Phi(x) with a free variable as a
finite entity, in the purely syntactical sense. In NAFL however, a
free variable x that ranges over, say, the natural numbers, has a
specific value when, and only when, the human mind specifies a value
for x. Thus x could have the value 10 only when the human mind
specifies a value (temporarily) of 10 for x, in which case Phi would
take on the value Phi(10). If no value is specified for x, x and
Phi(x) are considered to be in a superpositon state of all possible
values (i.e., <x=0, Phi(x)=Phi(0)> & <x=1, Phi(x)=Phi(1)> & .....),
whereas classically, Phi(x) is merely an uninterpreted syntactical
(and finite) entity. The superposition state is to be interpreted as
"The human mind has specified no value for x". Thus in NAFL, Phi(x),
for an unspecified x, is to be viewed as an infinite entity, something
like the class of all ordered pairs {(0,Phi(0)}, (1,Phi(1)), ...}. So
it is a function as far as NAFL is concerned. The bottom line is NAFL
will not permit you to quantify over these formulae.
In NAFL, the notion of provability is not formalizable, precisely
because you cannot quantify over formulae treated as purely
syntactical entities. An NAFL theory is a metamathematical object, not
formalizable as an object within NAFL theories. You may find this
unpalatable, but that is the way NAFL works.
This superposition state is similar to the Schrodinger cat's
superposition state of all possible classically permitted states,
namely, "alive and dead", when the cat's (classical) state is not
accessible (while it is in the box). The cat has a specific classical
state only when the box is opened and the human mind perceives that
state, according to the NAFL interpretation.
>
> (4) Applying the fixed point lemma to the formula not Theorem(x)
> gives Godel's theorem.
>
> >In other words, Godel had to consider infinitely many
> >functions fq(n), where each function f1(n), f2(n), ... is an infinite
> >entity.
>
> No, he doesn't do that.
>
Newberry essentially stated the argument in terms of the fixed point
lemma and your "formulae" are my "functions".
OK, let me think about this a little more and see if I can formulate
an objection from the NAFL point of view.
Regards, RS
John Jones
> On Feb 9, 4:54 pm, "John Jones" <jonescard...@aol.com> wrote:
> > It looks like infinite entities are proposed because by accomodating
> > ALL sentences we can eventually reference 'this sentence'.
> No. The informal statement is "This sentence is unprovable". But how
> do you *formalize* it, i.e., how do you translate it into the language
> of a first-order theory (say, Peano Arithmetic)?
I don't see how you can translate what you cannot identify. You cannot
identify the nature of the act of reference in the sentence 'this
sentence..' simply by pointing to the black marks that constitute the
signifier.
This is important. How could Godel identify a self-reference? By
looking at the marks made on the page? The 'string'? or by invoking an
all-inclusive framework that could accomodate self-reference?Surely
not. And there was no other recourse available to him. Certainly the
idea that Peano arithmetic could sort it out begged the question of
how to identify a self-reference.
>The Godel sentence for PA is a purely arithmetical sentence
> and can be thought of as a translation of the informal statement "This
> sentence is unprovable".
As I say, you can't translate what can't be identified - I can still
ask 'which is "this" sentence'.
>But in order to achieve this translation
> Godel had to quantify over functions -- see for example the diagonal
> argument given in the recent sci.logic thread "Goedel's proof" started
> by Newberry. In other words, Godel had to consider infinitely many
> functions fq(n), where each function f1(n), f2(n), ... is an infinite
> entity. This is what NAFL objects to -- you cannot invoke an argument
> that quantifies over infinite entities like "functions", i.e., you
> cannot invoke infinitely many infinite entities like "functions" in a
> legitimate NAFL proof.
>
> Classically, one takes the stand that quantification over infinite
> entities is legitimate. Since Godel translated self-referential
> propositions into arithmetic using such quantification (and other
> legitimate techniques), it follows that self-referential propositions
> are classically legitimate.
I've suggested how this is not possible. How is Godel going to
'translate' or identify the unidentifiable? While I have no doubt that
NAFL works here, Godel's quantification ploy was, nevertheless, not
legitimate in the first place - it was not put to the service of
anything.
> > So while NAFL may well show that we cannot quantify over infinite
> > entities, we do not need to invoke infinite entities if we want to
> > show that the Godel self-reference sentence is not supportable.
> The *formal* Goedel sentence, for, say, PA, is purely arithmetical and
> as such, is a legitimate sentence of PA. What you say is not
> supportable is the informal version of the Goedel sentence in the form
> "This sentence is unprovable". You apparently are giving a direct
> argument for why this is not a legitmate sentence, even in informal
> discourse. Whereas I am saying that in order to *formalize* this
> sentence in classical first order predicate logic (i.e, translate it
> into the language of first-order theories), you do need quantification
> over infinite entities like functions, which makes it illegitmate from
> the NAFL point of view.
There must be a schism between language and arithmetic if 'this
sentence ...' can be represented in arithmetic.
> Informally, my objection to "This sentence is not true" is that when
> we utter "sentence", we have no construction in mind for any sentence.
> This only comes later, and the delay is also objectionable from the
> NAFL point of view.
Yes, I know. I thought that there is a problem here with introducing
'time' and order of presentation in your argument, however, in that
someone may claim that these are psychologistic intrusions.
Daryl McCullough
>Oh, let's make the use of a code C explicit wherever you wish.
>
>Then we have that according to code C and to T- (1) has no truth
>value, where (1) is still:
>
>'(1) expresses no true proposition'
If "true" is relative to an interpretation C (I don't know why you
are using the word "code" here), then the meaning of (1) is
not defined until we have specified C. If that's your intent,
then substitute some other meaningless word, say "bloogle".
(1) expresses no bloogle proposition.
>Then, we have too that (1) expresses no true proposition
You haven't specified C, so you haven't specified what "true
proposition means", so you *can't* conclude that (1) expresses
no true proposition any more than you can conclude that it expresses
no bloogle proposition.
>and this token I called '(2)'.
Until you specify C, then the meaning of (2) is no more defined
than the meaning of (1). They have the *same* meaning, as far as
I can tell.
I consider (1) and (2) to be equally meaningless until you've
specified the domain of the truth predicate being used.
>It seems that you say that (2) does not obey the linguistic code C
>we've used to interpret (1) and pronounce it non-propositional.
>
>But it does. The conclusion of the argument is that there is a code C
>(let's say English),
English is a *language*, not a semantics. Calling something a true
sentence of English isn't saying anything at all. We know what it
means for *some* sentences of English to be true, but we don't have
a *complete* definition of truth of an English sentence.
>such that if we apply C to (1), (1) turns out to
>express no proposition and if we apply C to (2), (2) turns out to
>express a true proposition.
I don't think so. Give an explicit definition of "true"
such that (2) is true under that definition and (1) is not.
There is no definition of "true English sentence". There
is only a notion of true for limited domains. We can talk
about truths of arithmetic, or truths of weather forecasting
or truths of baseball, but "true English sentence" is not
a well-defined concept. And neither is "true English sentence token".
>Our move has been to interpret (1) as though it said what (2)
>effectively says and to deduce that, under that interpretation, (1)
>expresses no true proposition. So our reasoning is based on giving the
>same interpretation to the two tokens of the same sentence and then
>concluding that under that interpretation one expresses no proposition
>while the other expresses a true proposition.
>
>So, it is not a shift from one code or interpretation to another, but
>a change in the logical context what makes the difference between (1)
>and (2).
>
>Now, it is clear that I am not reasining within an inconsistent system
Well, you haven't made your steps in reasoning explicit, so there's
no way to say for certain, but I believe, to the contrary that you
are reasoning within an inconsistent system.
>but, if anything, about it, the burden of finding a flaw in my
>reasoning falls on your part.
I can't *prove* that you're being inconsistent unless you
give more detail as to what your rules of inference are, but
*if* you have a concept of "true sentence token" then in terms
of that concept, you can define the concept of "true string"
via
A string is true if it syntactically has the form of a
sentence and if every token of that sentence is true.
A string is not true if it fails to have the form of a
sentence, or if it has the form of a sentence and some
tokens of that sentence are not true.
Once we have a concept of "true string", then we can form
the original liar paradox
(1) String (1) is not true.
>I'm clumsy building up axiomatic systems.
>But it is not fair to demand 'exactly what are your axioms
>and inference rules in that argument?' in order to assess the
>argument.
Based on the details that you've provided, I think that your
argument makes no sense. It's possible that my assessment would
change if I knew more about your axioms and rules of inference,
but my tentative assessment is that it is nonsense.
>For, even if I would present one system, you could then ask what my
>axioms and rules have been in the construction of my system
Why in the world would I ask that?
>So, what is exactly the flaw now?
The flaw is that you are implicitly using axioms and rules, but
you aren't saying what they are. You are using the word "true"
without defining what it means to you or what rules of inference
apply to it. You can't say "I'm just using it in the usual way"
because the usual way that "true" is used is *inconsistent*. So
if you want me to believe that your system is consistent, then
you have to say what that system is.
John Jones
As I said before I think, you cannot present a self-reference by
referencing the marks it makes. That is a ploy that fails to work.
Daryl McCullough
>I have had this discussion before with Aatu Koskensilta also and he
>said something similar to what you are saying. The difference in our
>perceptions occurs at the above point. The objection is to
>quantification over formulae like Phi with one or more free variables
>(i.e., "For every formula Phi..." is objectionable in NAFL).
Then sorry, NAFL sounds useless.
Daryl McCullough
>Now, it is clear that I am not reasining within an inconsistent system
Why is that clear? Your system seems very similar to other systems
that are known to be inconsistent. What reason is there for believing
that it is consistent?
David Marcus
> On 6 Feb, 22:53, David Marcus wrote:
> > aatu.koskensi...@xortec.fi wrote:
> > > But what about sentences that talk about the possibility of
> > > consistently assigning a truth values to sentences? The strengthened
> > > liar strikes back:
> >
> > > This sentence can not be consistently assigned a truth value or it is
> > > not true.
> >
> > > Suppose we assign to it the truth value false. Then, in fact, it is
> > > true since the last disjunct is true. Suppose we assign it the truth
> > > value true. Then the first disjunct must hold, in which case the
> > > sentence certainly can't be true. So we conclude that, in fact, the
> > > sentence can't be consistently assigned a truth value. But that's just
> > > what the first disjunct states, so the sentence should be true, after
> > > all.
> >
> > If you want to discuss such sentences, then you need a formal
> > translation into some system.
>
> Why?
Because English is not precise.
> In fact, in order to determine which system or theory is the
> correct choice we'd need to resolve what sort of modes of reasoning
> and basic principles concerning truth it would contain. And that we
> can't do before "resolving", in whatever way, the instance of the
> (strenghtened) liar we're considering.
I don't follow. I gave two interpretations for the English sentence.
Depending on which you meant when you wrote the sentence, we can decide
what the resolution is. The fact that I can give more than one
interpretation shows that the English version is not (by itself)
precise.
> > Boolean algebra doesn't suffice, since you
> > can't translate "assigned a truth value". If you argue using informal
> > logic (as you did above), then you are ignoring your assumption that a
> > solution exists.
>
> Where did I rely on that assumption? I just went through the three
> possibilities: no solution exists, there exists a solution assigning
> the sentence the truth value 'true', there exists a solution assigning
> the sentence the truth value 'false'. All apparently lead to
> paradoxical conclusion.
Even if you are allowing three possibilities, then it is still possible
to have an equation with no solution. This doesn't surprise me.
> > You used this assumption right at the start: "Suppose
> > we assign to it the truth value false". If you add the assumption
> > "Suppose a solution exists" to the beginning of your agument, then
> > (assuming your argument works in whatever formal translation you are
> > using) your conclusion at the end will be "Contradiction. So, no
> > solution exists."
>
> Right. But then according to any normal understanding the sentence is
> true given that that's exactly what the first disjunct states.
What is a "normal understanding"? Is it one of the two interpretations I
gave or something else? If it is my interpretation #1, then the equation
has no solution. If it is my interpretation #2, then the problem is that
1 + 2 does not equal 1, so you can't tell the truth of a disjunction by
looking at just the first disjunct.
> > Have you read the paper I referenced?
>
> No. Did you read the post I gave a link to, expounding the view that
> the liar demonstrates the "indefinite extensibility" of truth?
Yes. After reading what you wrote a couple of times, I still prefer Lan
Wen's resolution. How does your notion of indefinite extensibility
resolve Wen's Three Cards Paradox?
Wen's resolution for the paradoxes seems so simple: we are making an
implicit assumption that a solution exists. So, the Liar paradox is
fundamentally no different from assuming an equation has a solution when
it doesn't or a series converges when it doesn't. It seems more
paradoxical because it is a logical paradox in English. But, once we
analyze it mathematically, the similarity is revealed.
--
David Marcus
R. Srinivasan
wrote:
subjective opinions. The reasons for the restrictions that NAFL places
upon classical reasoning are precisely the same reasons that NAFL
justifies the quantum superpositon and entanglement principles.
Classical logic is, in my opinion, more or less "useless' in trying to
make sense of quantum mechanics (in particular , "weird" aspects like
quantum superposition and quantum entanglement). NAFL has the
potential to provide a theory covering all aspects of quantum
mechanics in a single logic, whereas in conventional QM you need to
jump logics (need to use classical logic for stuff requiring real
analysis, while quantum logic becomes necessary for stuff like
superposition, entanglement, etc.).
Regards, RS
David Marcus
Just use Bohmian mechanics or similar systems that don't suffer from the
vagueness and illogic of the Copenhagen version. We shouldn't use bad
physics to justify the need for new logic.
--
David Marcus
Daryl McCullough
>I kind of expected this reaction. "Useful" and "useless" are
>subjective opinions.
I would say, rather, that they are relative to the purpose
you have in mind. For the purpose of analyzing a domain
(such as theorem proving, in doing metamathematics), you need
a theory that is able to quantify over the objects of the
domain. If you can't do that, then you need to use a different
system.
>The reasons for the restrictions that NAFL places
>upon classical reasoning are precisely the same reasons that NAFL
>justifies the quantum superpositon and entanglement principles.
But what does that have to do with the current thread? We're
not talking about quantum mechanics.
David Marcus
> Wen's resolution for the paradoxes seems so simple: we are making an
> implicit assumption that a solution exists. So, the Liar paradox is
> fundamentally no different from assuming an equation has a solution when
> it doesn't or a series converges when it doesn't. It seems more
> paradoxical because it is a logical paradox in English. But, once we
> analyze it mathematically, the similarity is revealed.
I forgot to mention the set theory paradoxes where we assume a set
satisfying a certain sentence exists when it doesn't.
--
David Marcus
Newberry
Now try the fourth possibility: the sentence is meaningless.
>
> > You used this assumption right at the start: "Suppose
> > we assign to it the truth value false". If you add the assumption
> > "Suppose a solution exists" to the beginning of your agument, then
> > (assuming your argument works in whatever formal translation you are
> > using) your conclusion at the end will be "Contradiction. So, no
> > solution exists."
>
> Right. But then according to any normal understanding the sentence is
> true given that that's exactly what the first disjunct states.
>
> > Have you read the paper I referenced?
>
> No. Did you read the post I gave a link to, expounding the view that
> the liar demonstrates the "indefinite extensibility" of truth?
>
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
>
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus- Hide quoted text -
David Marcus
> On Feb 9, 2:21 am, aatu.koskensi...@xortec.fi wrote:
> > Where did I rely on that assumption? I just went through the three
> > possibilities: no solution exists, there exists a solution assigning
> > the sentence the truth value 'true', there exists a solution assigning
> > the sentence the truth value 'false'. All apparently lead to
> > paradoxical conclusion.
>
> Now try the fourth possibility: the sentence is meaningless.
That's really the third possibility. And, yes, it works.
--
David Marcus
R. Srinivasan
wrote:
Godel's incompleteness theorems require:
\begin{quote}
1) Associating natural numbers with the formulas of PA.
(2) Defining a formula in PA (call it Theorem(x)) such that
for each natural number n, PA proves Theorem([n]) if and only
if n is the code of a theorem of PA. (where [n] means the
numeral corresponding to number n.
(3) Proving the fixed point lemma.
This says that for every formula Phi of PA, there is a
natural number n that is the code for a formula Psi such
that PA proves Psi <-> Phi' where Phi' is the result of
substituting [n] for all free variables occurring in Phi.
(4) Applying the fixed point lemma to the formula not Theorem(x)
gives Godel's theorem.
\end{quote}
In step (1), you have associated a natural number with each formula of
PA, which already amounts to quantifying over all formulas of PA. In
step (3), the fixed point lemma is a statement that invokes "every
formula" of PA, and presumably this lemma is translated into PA as a
formula of PA. So we presumably have *a* formula of PA that quantifies
over *all* formulas of PA. In effect we have a formula of PA that
makes an assertion about itself. This is as self-referential as it
gets.
The point I was making was that the reasons NAFL provides for
rejecting this formulation as infinitary are also the exact same
reasons why some puzzling quantum phenomena, such as quantum
superposition and entanglement, are justified in NAFL. So in NAFL we
lose much of classical infinitary reasoning, but that doesn't make
NAFL "useless" -- gains *could* appear elsewhere, assuming that a
consistent framework for quantum mechanics can be provided in NAFL,
(which is far from obvious at this stage).
Regards, RS
R. Srinivasan
physicists do not seem to accept it. I believe it has notions of
"particle" and "trajectory" that are as weird as superposition and
entanglement in the Copenhagen intepretation. I have to study Bohmian
mechanics further to see if its basic principles are compatible with
NAFL. I believe it requires instantaneous action at a distance that is
"real" in some sense. Whether such a notion of "reality" is compatible
with NAFL is a matter to be studied.
Regards, RS
R. Srinivasan
> On Feb 9, 12:44?pm, "R. Srinivasan" <sradh...@in.ibm.com> wrote:
>
> > On Feb 9, 4:54 pm, "John Jones" <jonescard...@aol.com> wrote:
> > > It looks like infinite entities are proposed because by accomodating
> > > ALL sentences we can eventually reference 'this sentence'.
> > No. The informal statement is "This sentence is unprovable". But how
> > do you *formalize* it, i.e., how do you translate it into the language
> > of a first-order theory (say, Peano Arithmetic)?
>
> I don't see how you can translate what you cannot identify. You cannot
> identify the nature of the act of reference in the sentence 'this
> sentence..' simply by pointing to the black marks that constitute the
> signifier.
>
> This is important. How could Godel identify a self-reference? By
> looking at the marks made on the page? The 'string'? or by invoking an
> all-inclusive framework that could accomodate self-reference?Surely
> not. And there was no other recourse available to him. Certainly the
> idea that Peano arithmetic could sort it out begged the question of
> how to identify a self-reference.
>
> >The Godel sentence for PA is a purely arithmetical sentence
> > and can be thought of as a translation of the informal statement "This
> > sentence is unprovable".
>
> As I say, you can't translate what can't be identified - I can still
> ask 'which is "this" sentence'.
>
justify self-reference. If you reject self-reference, you have to
reject the translation as well. NAFL identifies the translation
procedure with infinitary reasoning and hence unaccpetable in NAFL for
logical reasons.
>
> >But in order to achieve this translation
> > Godel had to quantify over functions -- see for example the diagonal
> > argument given in the recent sci.logic thread "Goedel's proof" started
> > by Newberry. In other words, Godel had to consider infinitely many
> > functions fq(n), where each function f1(n), f2(n), ... is an infinite
> > entity. This is what NAFL objects to -- you cannot invoke an argument
> > that quantifies over infinite entities like "functions", i.e., you
> > cannot invoke infinitely many infinite entities like "functions" in a
> > legitimate NAFL proof.
>
> > Classically, one takes the stand that quantification over infinite
> > entities is legitimate. Since Godel translated self-referential
> > propositions into arithmetic using such quantification (and other
> > legitimate techniques), it follows that self-referential propositions
> > are classically legitimate.
>
> I've suggested how this is not possible. How is Godel going to
> 'translate' or identify the unidentifiable? While I have no doubt that
> NAFL works here, Godel's quantification ploy was, nevertheless, not
> legitimate in the first place - it was not put to the service of
> anything.
>
with each formula of PA. This quantification over "all" formulas of PA
is what NAFL objects to. Godel's ploy effectively allows you to have a
formula of PA (which makes assertions about all natural numbers)
interpreted as making assertions about all formulas of PA, and hence
about the said formula itself. This is self-reference, and amounts to
unacceptable infinitary reasoning in NAFL.
Regards, RS
Daryl McCullough
>stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
>\begin{quote}
>1) Associating natural numbers with the formulas of PA.
>(2) Defining a formula in PA (call it Theorem(x)) such that
>for each natural number n, PA proves Theorem([n]) if and only
>if n is the code of a theorem of PA. (where [n] means the
>numeral corresponding to number n.
>(3) Proving the fixed point lemma.
>This says that for every formula Phi of PA, there is a
>natural number n that is the code for a formula Psi such
>that PA proves Psi <-> Phi' where Phi' is the result of
>substituting [n] for all free variables occurring in Phi.
>(4) Applying the fixed point lemma to the formula not Theorem(x)
>gives Godel's theorem.
>\end{quote}
>
>In step (1), you have associated a natural number with each formula of
>PA, which already amounts to quantifying over all formulas of PA.
Well, not exactly. The establishment of a connection between natural
numbers and formulas is not logical but procedural. Godel essentially
gives an algorithm for computing the number associated with each
formula. Do you have doubts about such an algorithm? You think
that the algorithm described doesn't cover all possible formulas?
>In step (3), the fixed point lemma is a statement that invokes "every
>formula" of PA, and presumably this lemma is translated into PA as a
>formula of PA.
No, it's not. It's a theorem *about* PA, not a theorem *in* PA.
Godel is proving facts about PA. Of course, *if* you agree that
Godel's algorithm succeeds in setting up a correspondence between
natural numbers and formulas, then it is possible to prove an
analogous lemma within PA.
>So we presumably have *a* formula of PA that quantifies
>over *all* formulas of PA.
No, we don't. PA only has quantification over natural numbers.
>In effect we have a formula of PA that makes an assertion about
>itself.
Not exactly. We have formulas in PA that make assertions about
natural numbers. But then we (because we have set up a correspondence
between natural numbers and formulas) can interpret these statements
as saying something about formulas.
>The point I was making was that the reasons NAFL provides for
>rejecting this formulation as infinitary are also the exact same
>reasons why some puzzling quantum phenomena, such as quantum
>superposition and entanglement, are justified in NAFL.
As I said, if NAFL cannot be used to study formulas and proofs,
then it cannot be used for metamathematics. That's why I said it
was useless. Maybe it's not useless for quantum mechanics, but it
is useless for many things that people wish to use mathematics for.
>So in NAFL we lose much of classical infinitary reasoning,
>but that doesn't make NAFL "useless"
It makes it useless for *some* things. So why bring up NAFL
when we are discussing a subject that NAFL is incapable of
shedding light on?
>-- gains *could* appear elsewhere, assuming that a
>consistent framework for quantum mechanics can be provided in NAFL,
I don't understand why you think that there is a tradeoff involved.
NAFL is useless for studying proof theory, so don't use it for that.
NAFL may be useful for studying quantum phenomena, so you can use
it for that. There is no reason to use one formalism for all purposes.
Newberry
Not according to Aatu and Kirkham. The sentence is meaningless implies
that a truth value cannot be consistently assigne to it (if you take
the position that "meaningless" is NOT a truth value) but not the
other way around. Stating that this sentence cannot be consistently
assigned a truth value appears to be exactly what the first conjunct
is saying. But when we start with the assumption that the sentence is
meaningless then it is saying nothing.
It is fashionable nowdays to ignore the notion of meaninglessness. Not
sure what the difficulty with it is.
>> This sentence can not be consistently assigned a truth value or it is not true. <<
>
> --
> David Marcus
R. Srinivasan
wrote:
> R. Srinivasan says...
>
>
>
> >stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> >\begin{quote}
> >1) Associating natural numbers with the formulas of PA.
> >(2) Defining a formula in PA (call it Theorem(x)) such that
> >for each natural number n, PA proves Theorem([n]) if and only
> >if n is the code of a theorem of PA. (where [n] means the
> >numeral corresponding to number n.
> >(3) Proving the fixed point lemma.
> >This says that for every formula Phi of PA, there is a
> >natural number n that is the code for a formula Psi such
> >that PA proves Psi <-> Phi' where Phi' is the result of
> >substituting [n] for all free variables occurring in Phi.
> >(4) Applying the fixed point lemma to the formula not Theorem(x)
> >gives Godel's theorem.
> >\end{quote}
>
> >In step (1), you have associated a natural number with each formula of
> >PA, which already amounts to quantifying over all formulas of PA.
>
> Well, not exactly. The establishment of a connection between natural
> numbers and formulas is not logical but procedural. Godel essentially
> gives an algorithm for computing the number associated with each
> formula. Do you have doubts about such an algorithm? You think
> that the algorithm described doesn't cover all possible formulas?
>
claims to accomplish a mapping between formulas of PA and natural
numbers. This mapping can only be expressed formally by quantifiation
(for each formula X there exists a unique natural number n such
that....)
>
> >In step (3), the fixed point lemma is a statement that invokes "every
> >formula" of PA, and presumably this lemma is translated into PA as a
> >formula of PA.
>
> No, it's not. It's a theorem *about* PA, not a theorem *in* PA.
> Godel is proving facts about PA. Of course, *if* you agree that
> Godel's algorithm succeeds in setting up a correspondence between
> natural numbers and formulas, then it is possible to prove an
> analogous lemma within PA.
>
is important because that is what justifies the claim that the
incompleteness theorems follow from purely arithmetical reasoning.
>
> >So we presumably have *a* formula of PA that quantifies
> >over *all* formulas of PA.
>
> No, we don't. PA only has quantification over natural numbers.
>
assertions about all formulas of PA. Surely this is what the
translation into PA is all about.
>
> >In effect we have a formula of PA that makes an assertion about
> >itself.
>
> Not exactly. We have formulas in PA that make assertions about
> natural numbers. But then we (because we have set up a correspondence
> between natural numbers and formulas) can interpret these statements
> as saying something about formulas.
>
>
> >The point I was making was that the reasons NAFL provides for
> >rejecting this formulation as infinitary are also the exact same
> >reasons why some puzzling quantum phenomena, such as quantum
> >superposition and entanglement, are justified in NAFL.
>
> As I said, if NAFL cannot be used to study formulas and proofs,
> then it cannot be used for metamathematics. That's why I said it
> was useless. Maybe it's not useless for quantum mechanics, but it
> is useless for many things that people wish to use mathematics for.
>
> >So in NAFL we lose much of classical infinitary reasoning,
> >but that doesn't make NAFL "useless"
>
> It makes it useless for *some* things. So why bring up NAFL
> when we are discussing a subject that NAFL is incapable of
> shedding light on?
>
as defined by NAFL. Does NAFL provide a new definition of finitism? I
think so.
>
> >-- gains *could* appear elsewhere, assuming that a
> >consistent framework for quantum mechanics can be provided in NAFL,
>
> I don't understand why you think that there is a tradeoff involved.
> NAFL is useless for studying proof theory, so don't use it for that.
> NAFL may be useful for studying quantum phenomena, so you can use
> it for that. There is no reason to use one formalism for all purposes.
>
finitary reasoning and what doesn't, at least from the NAFL point of
view. If some reasoning does cross the boundaries, then can we be
cetain that such reasoning is "safe"? This is an issue that needs
further study.
Regards, RS
David Marcus
> On Feb 9, 11:27 pm, David Marcus <DavidMar...@alumdotmit.edu> wrote:
> > Just use Bohmian mechanics or similar systems that don't suffer from the
> > vagueness and illogic of the Copenhagen version. We shouldn't use bad
> > physics to justify the need for new logic.
>
> I don't know much about Bohmian mechanics, but the majority of
> physicists do not seem to accept it.
This says more about physicists than it does about the theory. Also,
this is changing. More physicists are accepting it as they throw off the
nonsense they learned in school.
> I believe it has notions of
> "particle" and "trajectory" that are as weird as superposition and
> entanglement in the Copenhagen intepretation.
Huh? What is weird about particles having trajectories? Of course, the
trajectories are not classical, but if they were, we wouldn't need
quantum mechanics.
What do your quotes on "particles" and "trajectories" mean?
> I have to study Bohmian
> mechanics further to see if its basic principles are compatible with
> NAFL. I believe it requires instantaneous action at a distance that is
> "real" in some sense.
Bell proved that this is required of any theory that matches experiment.
I suggest you start your study by reading Bell's book. Here are some
other things that you can read:
http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/weingold.htm
http://www.davidmarcus.com/Articles/PhilosophicalBaggage.pdf
http://www.davidmarcus.com/Articles/DemiseOfLocalReality.pdf
--
David Marcus
David Marcus
> On Feb 9, 7:39 pm, David Marcus <DavidMar...@alumdotmit.edu> wrote:
> > Newberry wrote:
> > > On Feb 9, 2:21 am, aatu.koskensi...@xortec.fi wrote:
> > > > Where did I rely on that assumption? I just went through the three
> > > > possibilities: no solution exists, there exists a solution assigning
> > > > the sentence the truth value 'true', there exists a solution assigning
> > > > the sentence the truth value 'false'. All apparently lead to
> > > > paradoxical conclusion.
> >
> > > Now try the fourth possibility: the sentence is meaningless.
> >
> > That's really the third possibility. And, yes, it works.
>
> Not according to Aatu and Kirkham.
That's between you and Aatu.
> The sentence is meaningless implies
> that a truth value cannot be consistently assigne to it (if you take
> the position that "meaningless" is NOT a truth value) but not the
> other way around.
Obviously "meaningless" is not T or F. Is that what you mean by "NOT a
truth value"?
> Stating that this sentence cannot be consistently
> assigned a truth value appears to be exactly what the first conjunct
> is saying.
Sure.
> But when we start with the assumption that the sentence is
> meaningless then it is saying nothing.
I can't tell if you are agreeing with me or disagreeing.
> It is fashionable nowdays to ignore the notion of meaninglessness. Not
> sure what the difficulty with it is.
I've never followed fashion.
--
David Marcus
R. Srinivasan
> R. Srinivasan wrote:
> > On Feb 9, 11:27 pm, David Marcus <DavidMar...@alumdotmit.edu> wrote:
> > > Just use Bohmian mechanics or similar systems that don't suffer from the
> > > vagueness and illogic of the Copenhagen version. We shouldn't use bad
> > > physics to justify the need for new logic.
>
> > I don't know much about Bohmian mechanics, but the majority of
> > physicists do not seem to accept it.
>
> This says more about physicists than it does about the theory. Also,
> this is changing. More physicists are accepting it as they throw off the
> nonsense they learned in school.
>
> > I believe it has notions of
> > "particle" and "trajectory" that are as weird as superposition and
> > entanglement in the Copenhagen intepretation.
>
> Huh? What is weird about particles having trajectories? Of course, the
> trajectories are not classical, but if they were, we wouldn't need
> quantum mechanics.
>
> What do your quotes on "particles" and "trajectories" mean?
>
understanding. In the Copenhagen interpretation, the superposition of
trajectories, etc. need not be real. But the non-locality in Bohmian
mechanics corresponds to reality. I could be wrong here.
>
> > I have to study Bohmian
> > mechanics further to see if its basic principles are compatible with
> > NAFL. I believe it requires instantaneous action at a distance that is
> > "real" in some sense.
>
> Bell proved that this is required of any theory that matches experiment.
> I suggest you start your study by reading Bell's book. Here are some
> other things that you can read:
>
> http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/weingold.htm
http://www.davidmarcus.com/Articles/PhilosophicalBaggage.pdf
http://www.davidmarcus.com/Articles/DemiseOfLocalReality.pdf
>
Regards, RS
David Marcus
> On Feb 10, 9:34 pm, David Marcus <DavidMar...@alumdotmit.edu> wrote:
> > What do your quotes on "particles" and "trajectories" mean?
>
> Non-local nature of particles is what is problematic, according to my
> understanding. In the Copenhagen interpretation, the superposition of
> trajectories, etc. need not be real. But the non-locality in Bohmian
> mechanics corresponds to reality. I could be wrong here.
The motion of a given particle is determined by the wave function, and
the latter is a function of the positions of all particles (including
those far away). However, each particle has a definite position at all
times.
Bell proved that any theory that matches experiment must be non-local.
The Copenhagen interpretation manages to obscure the non-locality by
being vague.
--
David Marcus
Daryl McCullough
>> Stating that this sentence cannot be consistently
>> assigned a truth value appears to be exactly what the first conjunct
>> is saying.
>
>Sure.
>
>> But when we start with the assumption that the sentence is
>> meaningless then it is saying nothing.
>
>I can't tell if you are agreeing with me or disagreeing.
Calling it meaningless is *not* a resolution. Let's go through
it once again. Let the following sentence be Sentence 1:
Sentence 1 is not true.
You want to say that it is meaningless, or that it's "equation
has no solution" or whatever. So let's write that down:
Sentence 1 is meaningless.
If it's meaningless, then it certainly is not true, right? So
we conclude:
Sentence 1 is not true.
But that conclusion *is* Sentence 1. So which is it, a correct
conclusion, or a meaningless one?
David Marcus
> David Marcus says...
>
> >> Stating that this sentence cannot be consistently
> >> assigned a truth value appears to be exactly what the first conjunct
> >> is saying.
> >
> >Sure.
> >
> >> But when we start with the assumption that the sentence is
> >> meaningless then it is saying nothing.
> >
> >I can't tell if you are agreeing with me or disagreeing.
>
> Calling it meaningless is *not* a resolution. Let's go through
> it once again. Let the following sentence be Sentence 1:
>
> Sentence 1 is not true.
>
> You want to say that it is meaningless, or that it's "equation
> has no solution" or whatever. So let's write that down:
>
> Sentence 1 is meaningless.
>
> If it's meaningless, then it certainly is not true, right? So
> we conclude:
>
> Sentence 1 is not true.
>
> But that conclusion *is* Sentence 1.
You say that "meaningless" is not true. Also, false is not true. So,
define g by
g(0) = g(2) = 1,
g(1) = 0.
Then your first sentence translates into math as
x = g(x).
This equation has no solution.
> So which is it, a correct conclusion, or a meaningless one?
Your argument proves that the equation (whose translation into English
you started with) has no solution. So, your argument is a correct proof
by contradiction, and there is no paradox.
Not all equations have solutions. Assuming all equations have solutions
leads to paradox. Just as assuming all series converge or all predicates
define sets leads to paradox. Translating the equations into English
obscures the fact that there is no solution. You feel that it is just
English, so it must make sense. But, this isn't true. Why should it be?
--
David Marcus
John Jones
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
OK.
(but would anyone like what we wote?)
John Jones
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
OK.
(but would anyone like what we wrote?)
Daryl McCullough
>
>Daryl McCullough wrote:
>> Calling it meaningless is *not* a resolution. Let's go through
>> it once again. Let the following sentence be Sentence 1:
>>
>> Sentence 1 is not true.
>>
>> You want to say that it is meaningless, or that it's "equation
>> has no solution" or whatever. So let's write that down:
>>
>> Sentence 1 is meaningless.
>>
>> If it's meaningless, then it certainly is not true, right? So
>> we conclude:
>>
>> Sentence 1 is not true.
>>
>> But that conclusion *is* Sentence 1.
>
>You say that "meaningless" is not true. Also, false is not true. So,
>define g by
>
> g(0) = g(2) = 1,
> g(1) = 0.
>
>Then your first sentence translates into math as
>
> x = g(x).
>
>This equation has no solution.
>> So which is it, a correct conclusion, or a meaningless one?
>
>Your argument proves that the equation (whose translation into English
>you started with) has no solution.
The statement that I started with is *exactly* the statement that
I proved. Let's start with a modified sentence from the usual Liar
paradox. Let Sentence 1 be
The equation for the truth value of Sentence 1 has no solution.
Your conclusion is that the equation for the truth value of
sentence 1 has no solution. So your conclusion is *exactly*
the sentence you started with.
>So, your argument is a correct proof
>by contradiction, and there is no paradox.
You don't think it is a paradox that the *conclusion* of a proof
is a statement that we had just agreed was meaningless?
>Not all equations have solutions. Assuming all equations have solutions
>leads to paradox.
In this case, assuming that it *doesn't* have a solution leads
to a paradox, as well. I *didn't* assume that the equation for
the truth value of sentence 1 had a solution.
>Translating the equations into English
>obscures the fact that there is no solution. You feel that it is just
>English, so it must make sense. But, this isn't true. Why should it be?
You are confused about what the paradox is. The paradox is that
your line of reasoning leads to a conclusion which is a statement
you had previously dismissed as meaningless.