Incompatibility of Computationalism and Consistent Bivalent Logic
LauLuna
please, take a look at the following.
DEFINITION 1: consistent bivalent logic (CBL) is the claim that
propositions are either true or false, but not both, and that nothing
else has a truth value.
DEFINITION 2: let (G) be the following sentence:
no correct cognitive behavior proves (G)
PROPOSITION 1: if CBL holds, (G) has no truth value.
JUSTIFICATION
(G) cannot be proven: if (G) can be proven, then (G) is true and then
(G) cannot be proven. So, no correct cognitive behavior proves (G).
If (G) expresses a proposition, (G) says that (G) cannot be proven;
then (G) is true, because what it says is true, and false, because we
have correctly proved it above.
So, (G) expresses no proposition and therefore has no truth value.
DEFINITION 3: computationalism (CP) is the claim that any cognitive
behavior is a Turing computation.
PROPOSITION 2: if CBL holds, the set of all correct cognitive behaviors
is a well defined set.
JUSTIFICATION: any cognitive behavior consists of the explicit or
implicit assertion of some set of propositions, so that its correctness
is equivalent to the simultaneous truth of all those propositions;
according to CBL, the truth value of a proposition is always a well
defined state of affairs; consequently, according to CBL, the
correctness or incorrectness of a cognitive behavior is always a well
defined state of affairs.
PROPOSITION 3: if CP and CBL both hold, then (G) has a truth value.
JUSTIFICATION
Assume CP. Then, any correct cognitive behavior C is a Turing
computation (from definition 3) and then (G) asserts that none of a
well defined (from proposition 2) class of Turing computations proves
(G); since this is a well defined physical state of affairs, that has
either to be or not to be the case, (G) has a truth value.
PROPOSITION 4: if CBL holds, CP is false.
JUSTIFICATION: from proposition 1 and proposition 3 by the deduction
theorem.
Note that the kernel of the argument is the fact that we cannot produce
a paradox about the behavior of a pure physical device but we can do it
by referring to cognitive behaviors.
I wait for your comments.
Thanks
abo
> I know that arguing about paradoxical sentences is a bit too bold but,
> please, take a look at the following.
>
>
> DEFINITION 1: consistent bivalent logic (CBL)
Actually, in the pages of sci.logic, CBL is well-known to be
Charlie-Boo Logic. ;-) But let's consider your definition instead.
>is the claim that
> propositions are either true or false, but not both, and that nothing
> else has a truth value.
>
> DEFINITION 2: let (G) be the following sentence:
>
> no correct cognitive behavior proves (G)
>
> PROPOSITION 1: if CBL holds, (G) has no truth value.
>
> JUSTIFICATION
>
> (G) cannot be proven: if (G) can be proven, then (G) is true and then
> (G) cannot be proven. So, no correct cognitive behavior proves (G).
I don't see why this follows. First, you would agree that (G) proves
(G); so one can prove (G) if one has (G) as axiom, or if one has a
sillly rule of deduction which allows one simply to assert (G). So
obviously you don't mean "prove" in the sense of "prove from axioms". I
don't think you mean "prove with no axioms" because then you aren't
able to prove anything. (In which case, (G) isn't provable and (G) is
true). Do you mean "prove in first-order logic" (without any other
axioms)? No, I guess not, because your reasoning depends on what
"correct" means, and so one would have to formulate at least one
non-logical axiom which explains what "correct" means. Anyway, to make
a long story short, it is not clear to me what you mean by "prove".
Also, I'd strongly suggest you stop with this "no truth value"
business. Every statement has one of two truth values: true or
not-true. If you are saying there is a third truth value or a gap,
then you are simply dividing "not-true" into "false" statements and the
rest (the ones with the third value or the gap). By itself this is
definitional and so by itself it would be extremely surprising to solve
any problem. At the least you need to explain what you mean by
"false".
>
> If (G) expresses a proposition, (G) says that (G) cannot be proven;
> then (G) is true, because what it says is true, and false, because we
> have correctly proved it above.
>
> So, (G) expresses no proposition and therefore has no truth value.
Leaving aside the fact that (G) is unclear because your meaning of
"prove" is unclear (well unclear to me, at least), whether or not a
sentence expresses a proposition depends strongly on what you mean by
"proposition". Since you haven't provided any explanation, this kind
of reasoning is far too quick. Again, (G) at least prime facie has the
truth value "true" or "not-true".
>
> DEFINITION 3: computationalism (CP) is the claim that any cognitive
> behavior is a Turing computation.
This is an extremely strong thesis, which is clearly false by the
normal meaning of "Turing computation," namely "computation of a Turing
Machine." I'm not a Turing Machine, because Turing Machines don't have
fingers, and I don't have an infinite tape. OK that's a rather
facetious reply, but I think there's enough truth in it to mention -
because again it's not really clear to me what you mean.
>
> PROPOSITION 2: if CBL holds, the set of all correct cognitive behaviors
> is a well defined set.
No problem with Prop 2.
>
> JUSTIFICATION: any cognitive behavior consists of the explicit or
> implicit assertion of some set of propositions, so that its correctness
> is equivalent to the simultaneous truth of all those propositions;
> according to CBL, the truth value of a proposition is always a well
> defined state of affairs; consequently, according to CBL, the
> correctness or incorrectness of a cognitive behavior is always a well
> defined state of affairs.
>
> PROPOSITION 3: if CP and CBL both hold, then (G) has a truth value.
>
> JUSTIFICATION
>
> Assume CP. Then, any correct cognitive behavior C is a Turing
> computation (from definition 3) and then (G) asserts that none of a
> well defined (from proposition 2) class of Turing computations proves
> (G); since this is a well defined physical state of affairs,
A Turing machine does not exist physically.
LauLuna
For 'proving X' I mean concluding X or coming to believe X on correct
grounds, but not necessarily within a particular formal system.
Not every sentence is a statement: the Liar sentence isn't. (G) is a
sentence but obviously fails to yield a statement; so it has no truth
value. 'Gap' is no truth value in consistent bivalent logic, but
discussing this is discussing about names, that are always a matter of
convention. The conclusion that (G) expresses no proposition is quite
straightforward, and so is the conclusion that it has no truth value.
You should remember I'm arguing within CBL as defined.
Demanding definitions for 'correct' and 'false', which most probably
are primitive terms, seems no sensible requirement.
As my definition of CBL states, propositions are the sole kind of
objects that can be true or false, and this suffices for the present
purpose.
For the identity of a cognitive behavior the number of fingers of the
one who performs it is irrelevant. Nevertheless, most computationalists
or mechanists do believe there are Turing computations that are carried
out by objects endowed with hands and fingers. Computationalism is the
claim that whenever we conclude and assert propositions we are
implementig a Turing computation; this is a usual definition of
computationalism. However, I don't intend to discuss the strength of
the thesis but its compatibility with consistent bivalent logic as
defined.
A Turing machine can indeed be implemented by a physical device; but
the point is that any behavior of a Turing machine is always equivalent
to the behavior of some possible physical device and thus to some
possible physical event. So, there is no way of producing a paradox by
talking about the behavior of well defined class of Turing machines, no
more than by talking about car performances or about the speed of
elevators.
I hope I've put it a bit clearer.
Thanks again
abo ha escrito:
abo
LauLuna wrote:
> Thanks a lot for your response. I try to answer to your objections.
>
> For 'proving X' I mean concluding X or coming to believe X on correct
> grounds, but not necessarily within a particular formal system.
Sorry, but I have no idea what "correct grounds" mean. Is this just a
soundness condition, ensuring that if one "proves" X, then X is true?
In that case both: (G) is not true and no correct cognitive behaviour
proves (G).
>
> Not every sentence is a statement: the Liar sentence isn't.
I'm afraid this is not very explanatory. If not every sentence is a
statement, you need to explain what a statement is and why some
sentences are not statements.
> (G) is a
> sentence but obviously fails to yield a statement; so it has no truth
> value.
I don't know why you think the Liar isn't a statement, so I don't know
why you think G isn't.
> 'Gap' is no truth value in consistent bivalent logic, but
> discussing this is discussing about names, that are always a matter of
> convention. The conclusion that (G) expresses no proposition is quite
> straightforward, and so is the conclusion that it has no truth value.
Do you agree that (G) is not true?
> You should remember I'm arguing within CBL as defined.
>
> Demanding definitions for 'correct' and 'false', which most probably
> are primitive terms, seems no sensible requirement.
Try going over your argument with "not true" replaced with "false." Do
you think it still works?
<snip>
>
> Thanks again
>
My pleasure!
David C. Ullrich
>I know that arguing about paradoxical sentences is a bit too bold but,
>please, take a look at the following.
>
>
>DEFINITION 1: consistent bivalent logic (CBL) is the claim that
>propositions are either true or false, but not both, and that nothing
>else has a truth value.
>
>DEFINITION 2: let (G) be the following sentence:
>
> no correct cognitive behavior proves (G)
This is not a definition of G. It can't be, because "G"
appears in the supposed definition: If one doesn't know
what G is then one still doesn't know what G is after
reading the supposed definition.
Not just hypothetical - I _don't_ know what you mean by
"G" here.
>PROPOSITION 1: if CBL holds, (G) has no truth value.
>
>JUSTIFICATION
Before you can justify a statement about G you need to
tell us what G _is_.
************************
David C. Ullrich
abo
David C. Ullrich wrote:
> On 4 Dec 2006 09:52:48 -0800, "LauLuna" <laurea...@yahoo.es> wrote:
>
> >I know that arguing about paradoxical sentences is a bit too bold but,
> >please, take a look at the following.
> >
> >
> >DEFINITION 1: consistent bivalent logic (CBL) is the claim that
> >propositions are either true or false, but not both, and that nothing
> >else has a truth value.
> >
> >DEFINITION 2: let (G) be the following sentence:
> >
> > no correct cognitive behavior proves (G)
>
> This is not a definition of G. It can't be, because "G"
> appears in the supposed definition: If one doesn't know
> what G is then one still doesn't know what G is after
> reading the supposed definition.
>
> Not just hypothetical - I _don't_ know what you mean by
> "G" here.
I assumed this is what LL meant:
Let (G) be the following sentence:
no correct cognitive behavior proves this sentence
Charlie-Boo
> LauLuna wrote:
> > I know that arguing about paradoxical sentences is a bit too bold but,
> > please, take a look at the following.
> >
> >
> > DEFINITION 1: consistent bivalent logic (CBL)
>
> Actually, in the pages of sci.logic, CBL is well-known to be
> Charlie-Boo Logic. ;-)
No. You need to check the literature. Where does it say that? CBL is
Computationally Based Logic. Regular Logic represents propositions
that a given proposition is true or provable. A Computationally Based
Logic is a meta system that represents what can be represented.
ZFC et. al. don't realize that you need a different system for metamath
than for math outside of metamath. Predicate Calculus is fine for
Math, but not for metamath. ZFC can be expressed in about 1/10 the
concepts and characters using any Computationally Based Logic.
C-B
Charlie-Boo
LauLuna wrote:
> I know that arguing about paradoxical sentences is a bit too bold but,
> please, take a look at the following.
>
>
> DEFINITION 1: consistent bivalent logic (CBL) is the claim that
> propositions are either true or false, but not both, and that nothing
> else has a truth value.
>
> DEFINITION 2: let (G) be the following sentence:
>
> no correct cognitive behavior proves (G)
>
> PROPOSITION 1: if CBL holds, (G) has no truth value.
>
> JUSTIFICATION
>
> (G) cannot be proven: if (G) can be proven, then (G) is true and then
> (G) cannot be proven. So, no correct cognitive behavior proves (G).
>
> If (G) expresses a proposition, (G) says that (G) cannot be proven;
> then (G) is true, because what it says is true, and false, because we
> have correctly proved it above.
No. Conventional wisdom says that it can't be formally proven (it is
not a theorem) in the system, but that we proved it in metamath, a
different system. (Godel said this explicitly in his 1931 paper.)
More recent developments (e.g. CBL, which was not discovered until
after Godel's time) show us that you (and Godel) did NOT prove the
theorem. You proved that SOUND => THAT THEOREM.
Now, can you formalize the above in the logic?
C-B
Grouchy
Here's a link to a paper by Jamie Tappenden that touches on some of the
issues:
Charlie-Boo
Here's an even better paper on CBL:
C-B
BTW There really is no reason for people to debate the Liar Paraodox
(or any of a number of other paradoxes.) It is simply 3 conditions
being placed on a system that are mutually incompatible. These 3
conditions are also at play in Russell's Paradox, Turing's proof of the
unsolvability of the Halting Problem, and many others. Basically,
there is an analogous result for any base, which is to say, any
two-place relationship (not all relationships are relations.)
LauLuna
Liar is propositional and hence has a truth value.
I've shown how to proceed to the effect wrt (G).
Of course, if one assumes that whatever is not true has to be false,
then my argument fails, does not even start off.
But, how could we support that claim?
LauLuna
defined term!
Regards
Charlie-Boo ha escrito:
LauLuna
David C. Ullrich ha escrito:
> >DEFINITION 2: let (G) be the following sentence:
> >
> > no correct cognitive behavior proves (G)
>
> This is not a definition of G. It can't be, because "G"
> appears in the supposed definition: If one doesn't know
> what G is then one still doesn't know what G is after
> reading the supposed definition.
>
> Not just hypothetical - I _don't_ know what you mean by
> "G" here.
>
On the contrary, it is a perfectly sound definition because the second
'(G)' stays there as a simple symbol string and needs no further
definition, while the first '(G)' is a denoting term.
When one produces a sentence and names it, then one is defining what
the name refers to by simply presenting the sentence as a string of
symbols. To know what (G) is you just have to look at.
Regards
LauLuna
from the meaning of 'proving in system S', 'proving in metasystem MS' '
.
Formal systems perform no more than capturing our intuitive ways of
proving and, as we know from Goedel's theorem, each of them only
captures a proper part of the possible proving resources. So, when we
refer to proving in general, as I do (since no formal background is
referred to), we are not speaking about formal proofs in any system.
I must insist on the fact that my argument that (G) is paradoxical is
quite straightforward. I see no need to convert it into a formal one.
But there is goes; it is trivial once one has defined the suitable
predicates and assumed obvious rules:
V = has a truth value
P = is provable
~ = negation
1. V(G) <-> ((G) <-> ~P(G)) from the definition of (G)
2. ~V(G) -> ~P(G)
3. V(G) Assumption
4. (G) <-> ~P(G)
5. P(G) Assumption
6. (G)
7 ~P(G)
8. ~P(G)
9. V(G) -> ~P(G)
10. ~P(G) from 2, 9
11. V(G) Assumption
12. (G) <-> ~P(G)
13. P(G) <-> P(~P(G))
14. P(~P(G)) from 13, reflecting on it.
15. P(G)
16. ~V(G) reductio 10, 15
Nevertheless, the proof that (G) is paradoxical relies on a previous
proof that (G) is not provable (step 13), and this suggest we are
dealing here with two levels of proof. But note that this is the core
of the argument: (G) is paradoxical because the concept of 'proving' is
not well defined for general cognitive behaviors as regards (G) (this
is independent of Goedel's theorem and other topics on formal systems)
while it cannot be so if all cognitive behaviors are equivalent to
mechanical behaviors.
Regards
Charlie-Boo ha escrito:
LauLuna
Grouchy ha escrito:
LauLuna
Charlie-Boo ha escrito:
> BTW There really is no reason for people to debate the Liar Paraodox
> (or any of a number of other paradoxes.) It is simply 3 conditions
> being placed on a system that are mutually incompatible. These 3
> conditions are also at play in Russell's Paradox, Turing's proof of the
> unsolvability of the Halting Problem, and many others. Basically,
> there is an analogous result for any base, which is to say, any
> two-place relationship (not all relationships are relations.)
Could you elaborate this?
Thanks for the link.
LauLuna
abo
LauLuna wrote:
> Every one knows how to get a contradiction from the assumption that the
> Liar is propositional and hence has a truth value.
First, not "every one knows" this.
Secondly, you also get a contradiction, using the usual reasoning, from
the strengthened Liar (in your case, "This sentence is false or does
not have a truth value.")
>
> I've shown how to proceed to the effect wrt (G).
>
> Of course, if one assumes that whatever is not true has to be false,
> then my argument fails, does not even start off.
>
> But, how could we support that claim?
I'm not trying to support this claim. I have no problem with someone
who wants to divide the non-true into two distinct classes, false and
X. The problem I have is that you are doing this with absolutely no
explanation and therefore it is difficult to evaluate your argument,
which depends crucially on the distinction between false and X. I will
repeat my question to you: do you agree that (G) is not true? (I am
not asking - I repeat - whether or not (G) is false. Is it *not
true*?) Please answer the question.
Basically you are arguing that "no Turing computation proves (G)" is
true or false and that "no cognitive behaviour proves (G)" is not
either true or false, and because of this difference, it can be
concluded that cognitive behaviour is not a Turing computation.
Suppose we change "false" to "not true". Then both "no Turing
computation proves (G)" is true or not true, and "no cognitive
behaviour proves (G)" is true or not true, and so there is no
difference, and so you wouldn't be able to conclude that cognitive
behaviour is not a Turing computation. I point out to you this (what I
take to be) obvious fact so that you understand that your argument
depends crucially on what "false" means.
I'm fairly confident that when the dust settles (I'm having to see
through a lot of dust here, so give me credit) your argument probably
falls down here. You state "since this is a well defined physical
state of affairs, that has either to be or not to be the case." Now
one could presumably equally assert, "G is or is not the case" without
any assumption at all, and in particular any assumption about well
defined physical states of affair or otherwise. That is, either G or
not G, whether or not there are physical states of affair involved.
LauLuna
It is obvious that one can derive from CBL (consistent bivalent logic
:-) ) that the (Strengthened) Liar has no truth value.
I conjecture you have in mind the 'metaparadox' of the Strengthened
Liar, that I think Tyler Burge proposed and that have painstakingly
been considered by Haim Gaifman and Lawrence Goldstein among others.
Let me reproduce it here:
(1) (1) is not true
Clearly, according to CBL, (1) has no truth value; hence we are
entitled to assert:
(2) (1) is not true
We can say we now have an almost standard solution to this
'metaparadox' within CBL: two different tokens of the same sentence can
possess different logical values. This is due to Gaifman and Goldstein.
Consider that to assert (2) an assessment of (1) has to be previously
accomplished, which evidently cannot be done, on pain of circularity,
while uttering (1). A token of a sentence can be used to express a
proposition in some logical contexts and not in others. The thought
which (2) is able to express or convey cannot be behind (1), for the
distinctive feature of (1) is its being uttered before anyone has ever
uttered (1); after this the logical context pushes us up to (2).
This amounts to saying that the state of affairs '(1) is not true' is
not 'always' available as such to be asserted; the possibility depends
on the logical context or logical level at which an utterer of the
sentence in both (1) and (2) stands.
Of course, this is a trait that only some paradoxical sentences possess
according to CBL.
Quite the same for (G). There are some tokens of (G) that express no
proposition and others that express a true one, as you can see in my
argument. I hope I've answered your question now.
This can only happen if the state of affairs that (G) cannot be proven
is not 'always' available as such to be expressed and asserted. This
cannot happen if that state of affairs reduces to the behavior of some
Turing machines of a well defined class of them: algorithmic behaviors
are, by their very definition, always (at least potentially)
objectified for all observers; and so are arithmetical truths or sets
of arithmetical truths. Always according to CBL.
This is the point of the argument.
I believe that the ultimate root of this situation lies in the fact
that intentional (mental) states are not always (not even potentially)
available objects, in fact they are not even potentially available for
themselves, because no intentional act can be its own intentional
object. So, we find here a definite difference between intentional acts
and mechanical behaviors, and this is the background of the argument.
As you have most probably gathered, this relates to (Husserl's)
Phenomenology, that I've found is relatively ignored, even by
logicians, outside of continental Europe (I'd be glad to be wrong about
this!).
I guess I would have raised fewer objections from you (or perhaps
different objections) if I had only introduced a tokenist version of
(G) in my argument. I wanted to give as simple a version as possible,
but in order to fill it up, just take (G) as a sentence-token, not as a
sentence.
As for the sentence (G'):
no Turing computation proves (G')
it could not be paradoxical, it is false. But consider the sentence
(G''):
no CORRECT Turing computation proves (G'')
I have good reasons to believe that the set of correct Tuirng
computations is ill defined or empty according to CBL. This makes
things a bit harder than you put them. But this does not affect the
argument, it just suggests another way of getting the same result.
Regards
abo ha escrito:
LauLuna
I wrote:
>The thought
> which (2) is able to express or convey cannot be behind (1), for the
> distinctive feature of (1) is its being uttered before anyone has ever
> uttered (1); after this the logical context pushes us up to (2).
This is not what I meant, I beg your pardon.
I meant that the distinctive feature of (1), and the class of tokens of
that same sentence that are logically equivalent to (1), is that they
are uttered while the utterer has not yet (1) available as an object in
order to accomplish an assessment of it.
This seems to make sense just now.
Sorry
David C. Ullrich
>
>David C. Ullrich ha escrito:
>
>> >DEFINITION 2: let (G) be the following sentence:
>> >
>> > no correct cognitive behavior proves (G)
>>
>> This is not a definition of G. It can't be, because "G"
>> appears in the supposed definition: If one doesn't know
>> what G is then one still doesn't know what G is after
>> reading the supposed definition.
>>
>> Not just hypothetical - I _don't_ know what you mean by
>> "G" here.
>>
>
>On the contrary, it is a perfectly sound definition because the second
>'(G)' stays there as a simple symbol string and needs no further
>definition, while the first '(G)' is a denoting term.
Um. If in fact you meant two different things you should not
use the same notation for both of them. And if in fact the
two mean different things then whatever you imagine your
point to have been evaporates.
It's as though I said "let x = 2. Now let x = 4" and then
deduced that there was an inconsistency in mathematics
because I'd shown that 2 = 4.
>When one produces a sentence and names it, then one is defining what
>the name refers to by simply presenting the sentence as a string of
>symbols. To know what (G) is you just have to look at.
>
>Regards
************************
David C. Ullrich
abo
> It is obvious that one can derive from CBL (consistent bivalent logic
> :-) ) that the (Strengthened) Liar has no truth value.
>
> I conjecture you have in mind the 'metaparadox' of the Strengthened
> Liar, that I think Tyler Burge proposed and that have painstakingly
> been considered by Haim Gaifman and Lawrence Goldstein among others.
> Let me reproduce it here:
>
> (1) (1) is not true
>
> Clearly, according to CBL, (1) has no truth value; hence we are
> entitled to assert:
>
> (2) (1) is not true
>
> We can say we now have an almost standard solution to this
> 'metaparadox' within CBL: two different tokens of the same sentence can
> possess different logical values. This is due to Gaifman and Goldstein.
Consider "All tokens of this sentence are not true."
>
> Consider that to assert (2) an assessment of (1) has to be previously
> accomplished, which evidently cannot be done, on pain of circularity,
> while uttering (1). A token of a sentence can be used to express a
> proposition in some logical contexts and not in others. The thought
> which (2) is able to express or convey cannot be behind (1), for the
> distinctive feature of (1) is its being uttered before anyone has ever
> uttered (1); after this the logical context pushes us up to (2).
>
> This amounts to saying that the state of affairs '(1) is not true' is
> not 'always' available as such to be asserted; the possibility depends
> on the logical context or logical level at which an utterer of the
> sentence in both (1) and (2) stands.
Is "All tokens are true or not true" true?
Unless there are conditions on what the logical context or logical
level is which makes an assertion unavailable to be asserted, this is
not especially helpful. It's basically saying, "Because the statement
causes trouble, let's forget about it."
Anyway, it seems to me this solution depends on the universe of
statements being closed. But it doesn't work for the unending
sequence,
"The next token is not true"
"The next token is not true"
"The next token is not true"
etc.
Which tokens in this sequence are not true?
>
> Of course, this is a trait that only some paradoxical sentences possess
> according to CBL.
>
> Quite the same for (G). There are some tokens of (G) that express no
> proposition and others that express a true one, as you can see in my
> argument. I hope I've answered your question now.
Well sort of. You should of shifted the dust now - swept the dirt
under the carpet, so to speak. I have no real idea when a sentence
will have tokens with different truth values - except (for the moment
anyway) your say so. This isn't satisfactory.
>
> This can only happen if the state of affairs that (G) cannot be proven
> is not 'always' available as such to be expressed and asserted.
> This
> cannot happen if that state of affairs reduces to the behavior of some
> Turing machines of a well defined class of them: algorithmic behaviors
> are, by their very definition, always (at least potentially)
> objectified for all observers; and so are arithmetical truths or sets
> of arithmetical truths. Always according to CBL.
>
> This is the point of the argument.
I'm afraid that's a little too fast to me.
>
> I believe that the ultimate root of this situation lies in the fact
> that intentional (mental) states are not always (not even potentially)
> available objects, in fact they are not even potentially available for
> themselves, because no intentional act can be its own intentional
> object. So, we find here a definite difference between intentional acts
> and mechanical behaviors, and this is the background of the argument.
If that's the background of the argument, then why are not you just
making it directly? Mental states are not available, mechancial
behaviours are; therefore mental states are not mechanical. Why go
through (G) - it only confuses the issue. Do you think the following
is a good argument: I have thoughts; Turing computations clearly do
not have thoughts; etc. ??
>
> As you have most probably gathered, this relates to (Husserl's)
> Phenomenology, that I've found is relatively ignored, even by
> logicians, outside of continental Europe (I'd be glad to be wrong about
> this!).
I'm not the counterexample! Basically we seem to be able to make
assertions without having a complete idea of their "object" all the
time. "Everything Nixon said about Watergate is not true" can be
asserted - and surely *was* asserted in some form or other by the
veritable man on the street - without knowing in detail everything that
Nixon said about Watergate.
>
> I guess I would have raised fewer objections from you (or perhaps
> different objections) if I had only introduced a tokenist version of
> (G) in my argument. I wanted to give as simple a version as possible,
> but in order to fill it up, just take (G) as a sentence-token, not as a
> sentence.
>
> As for the sentence (G'):
>
> no Turing computation proves (G')
>
> it could not be paradoxical, it is false. But consider the sentence
> (G''):
>
> no CORRECT Turing computation proves (G'')
>
> I have good reasons to believe that the set of correct Tuirng
> computations is ill defined or empty according to CBL. This makes
> things a bit harder than you put them. But this does not affect the
> argument, it just suggests another way of getting the same result.
Sorry this is much too fast for me.
LauLuna
am not giving a same name to different things, I'm just naming a
sentence (a symbol string) and then exhibiting the symbol string at
issue; if the name of the sentence appears among the symbols of the
string, that entails no contradiction.
Consider: let (H) be the sentence:
'nobody has ever written (H)'
Well, (H) is simply false and nothing is ill defined.
Regards
LauLuna
>I have no real idea when a sentence
> will have tokens with different truth values - except (for the moment
> anyway) your say so. This isn't satisfactory.
The Strengthened Liar metaparadox shows pretty well how it works: you
try to assign some sentence truth values in the ordinary way; it can
happen that you are obliged to fail; the failure entitles you to assert
a token of the initial sentence, and there you are.
>Consider "All tokens of this sentence are not true."
If you embed into the sentence the level distinction used in the
previous solution, we most probably have to introduce a new
distinction: levels of tokens. Certainly, no token of your sentence is
true, but it cannot be asserted by means of one of them. Why? I would
say the sentence is ill formed because it does not distinguish levels
of tokens: it tries to encompass the whole hierarchy of tokens
provoking a kind of Burali-Forti paradox. Of course, we can correctly
assert 'no token of the sentence "all tokens of this sentence are not
true" is true'.
Anyway, this does not change the fact that (G) has some non
propositional tokens and some propositional ones, and that this could
not happen if it were a sentence about mechanical behaviors.
>Is "All tokens are true or not true" true?
> Unless there are conditions on what the logical context or logical
> level is which makes an assertion unavailable to be asserted, this is
> not especially helpful. It's basically saying, "Because the statement
> causes trouble, let's forget about it."
I don't quite understand you. But nobody turns a blind eye: it is only
shown that a same sentence can be sometimes used to make a statement
and sometimes it cannot. I think this is pretty clear according to CBL.
Please, do not forget that I'm reasoning under the hypothesis of CBL,
for I am only proposing an incompatibility result.
>If that's the background of the argument, then why are not you just
> making it directly? Mental states are not available, mechancial
> behaviours are; therefore mental states are not mechanical. Why go
> through (G) - it only confuses the issue. Do you think the following
> is a good argument: I have thoughts; Turing computations clearly do
> not have thoughts; etc. ??
I have indeed taken a trip a bit too far to the phenomenological
background of the whole matter, but this was only an attempt of
explanation, it is not the result itself. I think, the result stands on
its own. As far as I can see, things are as it puts them. Trying to go
deeper to get an insight into the roots of the phenomenon, that's
really something else.
So, we should perhaps limit ourselves to discussing whether CBL really
entails what I say it entails about (G), namely, that it cannot always
have a trurh value, and whether at the same time CBL+Computationalism
entail that (G) always depicts a well defined state of affairs.
Please, tell me: do you agree that (G) cannot always be given a truth
value? Do you agree that Turing machine behaviors are always well
defined states of affairs?
And no, a kind of Chinese room argument is not enough to establish an
incompatibility result. Certainly, Turing machines have no FUNCTIONAL
mental features, for they can be thoroughly described without any
reference to mental states. Certainly minds have mental features but,
do they have FUNCTIONAL mental features?
Thanks a lot for your comments.
On 10 dic, 15:34, "abo" <dkfjd...@yahoo.com> wrote:
> LauLuna wrote:
> > It is obvious that one can derive from CBL (consistent bivalent logic
> > :-) ) that the (Strengthened) Liar has no truth value.
>
> > I conjecture you have in mind the 'metaparadox' of the Strengthened
> > Liar, that I think Tyler Burge proposed and that have painstakingly
> > been considered by Haim Gaifman and Lawrence Goldstein among others.
> > Let me reproduce it here:
>
> > (1) (1) is not true
>
> > Clearly, according to CBL, (1) has no truth value; hence we are
> > entitled to assert:
>
> > (2) (1) is not true
>
> > We can say we now have an almost standard solution to this
> > 'metaparadox' within CBL: two different tokens of the same sentence can
> > possess different logical values. This is due to Gaifman and Goldstein.Consider "All tokens of this sentence are not true."
>
>
>
> > Consider that to assert (2) an assessment of (1) has to be previously
> > accomplished, which evidently cannot be done, on pain of circularity,
> > while uttering (1). A token of a sentence can be used to express a
> > proposition in some logical contexts and not in others. The thought
> > which (2) is able to express or convey cannot be behind (1), for the
> > distinctive feature of (1) is its being uttered before anyone has ever
> > uttered (1); after this the logical context pushes us up to (2).
>
> > This amounts to saying that the state of affairs '(1) is not true' is
> > not 'always' available as such to be asserted; the possibility depends
> > on the logical context or logical level at which an utterer of the
> > sentence in both (1) and (2) stands.Is "All tokens are true or not true" true?
>
> Unless there are conditions on what the logical context or logical
> level is which makes an assertion unavailable to be asserted, this is
> not especially helpful. It's basically saying, "Because the statement
> causes trouble, let's forget about it."
>
> Anyway, it seems to me this solution depends on the universe of
> statements being closed. But it doesn't work for the unending
> sequence,
> "The next token is not true"
> "The next token is not true"
> "The next token is not true"
> etc.
>
> Which tokens in this sequence are not true?
>
>
>
> > Of course, this is a trait that only some paradoxical sentences possess
> > according to CBL.
>
> > Quite the same for (G). There are some tokens of (G) that express no
> > proposition and others that express a true one, as you can see in my
> > argument. I hope I've answered your question now.Well sort of. You should of shifted the dust now - swept the dirt
> under the carpet, so to speak. I have no real idea when a sentence
> will have tokens with different truth values - except (for the moment
> anyway) your say so. This isn't satisfactory.
>
>
>
> > This can only happen if the state of affairs that (G) cannot be proven
> > is not 'always' available as such to be expressed and asserted.
> > This
> > cannot happen if that state of affairs reduces to the behavior of some
> > Turing machines of a well defined class of them: algorithmic behaviors
> > are, by their very definition, always (at least potentially)
> > objectified for all observers; and so are arithmetical truths or sets
> > of arithmetical truths. Always according to CBL.
>
> > This is the point of the argument.I'm afraid that's a little too fast to me.
>
>
>
> > I believe that the ultimate root of this situation lies in the fact
> > that intentional (mental) states are not always (not even potentially)
> > available objects, in fact they are not even potentially available for
> > themselves, because no intentional act can be its own intentional
> > object. So, we find here a definite difference between intentional acts
> > and mechanical behaviors, and this is the background of the argument.If that's the background of the argument, then why are not you just
> making it directly? Mental states are not available, mechancial
> behaviours are; therefore mental states are not mechanical. Why go
> through (G) - it only confuses the issue. Do you think the following
> is a good argument: I have thoughts; Turing computations clearly do
> not have thoughts; etc. ??
>
>
>
> > As you have most probably gathered, this relates to (Husserl's)
> > Phenomenology, that I've found is relatively ignored, even by
> > logicians, outside of continental Europe (I'd be glad to be wrong about
> > this!).I'm not the counterexample! Basically we seem to be able to make
> assertions without having a complete idea of their "object" all the
> time. "Everything Nixon said about Watergate is not true" can be
> asserted - and surely *was* asserted in some form or other by the
> veritable man on the street - without knowing in detail everything that
> Nixon said about Watergate.
>
>
>
>
>
>
>
> > I guess I would have raised fewer objections from you (or perhaps
> > different objections) if I had only introduced a tokenist version of
> > (G) in my argument. I wanted to give as simple a version as possible,
> > but in order to fill it up, just take (G) as a sentence-token, not as a
> > sentence.
>
> > As for the sentence (G'):
>
> > no Turing computation proves (G')
>
> > it could not be paradoxical, it is false. But consider the sentence
> > (G''):
>
> > no CORRECT Turing computation proves (G'')
>
> > I have good reasons to believe that the set of correct Tuirng
> > computations is ill defined or empty according to CBL. This makes
> > things a bit harder than you put them. But this does not affect the
> > argument, it just suggests another way of getting the same result.Sorry this is much too fast for me.- Ocultar texto de la cita -- Mostrar texto de la cita -- Ocultar texto de la cita -- Mostrar texto de la cita -
abo
LauLuna wrote:
> abo wrote:
>
> >I have no real idea when a sentence
> > will have tokens with different truth values - except (for the moment
> > anyway) your say so. This isn't satisfactory.
>
> The Strengthened Liar metaparadox shows pretty well how it works: you
> try to assign some sentence truth values in the ordinary way; it can
> happen that you are obliged to fail; the failure entitles you to assert
> a token of the initial sentence, and there you are.
>
> >Consider "All tokens of this sentence are not true."
>
> If you embed into the sentence the level distinction used in the
> previous solution, we most probably have to introduce a new
> distinction: levels of tokens. Certainly, no token of your sentence is
> true, but it cannot be asserted by means of one of them. Why? I would
> say the sentence is ill formed because it does not distinguish levels
> of tokens: it tries to encompass the whole hierarchy of tokens
> provoking a kind of Burali-Forti paradox. Of course, we can correctly
> assert 'no token of the sentence "all tokens of this sentence are not
> true" is true'.
1/ If you're willing to use hierarchies, then why go the route of
tokens? Simply say that truth is hierarchal and be done with it, at
least knowing that you are in the good company of Bertrand Russell and
a lot of others. I'm not saying I like the truth-is-hierarchal
solution, but it seems that the introduction of tokens is simply more
complicated, if in the end you are forced to talk about hierarchies any
way.
2/ If you are disallowing perfectly normal sentences as ill-formed,
then you might as well just say "This sentence is not true" is
ill-formed and be done with it. That is, this talk of tokens seems to
be the introduction of a vocabulary (tokens) which in fact solves
nothing. At the end of the day, there is still an ad hoc claim about
which kind of assertions are permissible and which are not - and what
determines impermissibility is whether there is paradox.
>
> Anyway, this does not change the fact that (G) has some non
> propositional tokens and some propositional ones, and that this could
> not happen if it were a sentence about mechanical behaviors.
Perhaps. Your argument, I think you would agree, flies pretty close to
the Liar's Paradox. So it is normal that I am skeptical that you are
not simply deriving your conclusion from the contradiction which comes
from the Paradox (in which case you can, of course, conclude anything).
So, while technically you are right, I think people will find your
argument convincing only if you first really do have a convincing
solution to the Liar and the Strengthened Liar. And, I'm afraid, for
the moment anyway, I'm not convinced. (I should be candid that I think
I *do* know the solution, and since yours does not co-incide with mine,
I think it is incorrect.)
>
> >Is "All tokens are true or not true" true?
> > Unless there are conditions on what the logical context or logical
> > level is which makes an assertion unavailable to be asserted, this is
> > not especially helpful. It's basically saying, "Because the statement
> > causes trouble, let's forget about it."
>
> I don't quite understand you. But nobody turns a blind eye: it is only
> shown that a same sentence can be sometimes used to make a statement
> and sometimes it cannot. I think this is pretty clear according to CBL.
> Please, do not forget that I'm reasoning under the hypothesis of CBL,
> for I am only proposing an incompatibility result.
I accept CBL and I don't agree at all that a sentence can sometimes be
used to make a statement and sometimes it cannot, unless you're using
"statement" in a different sense than what I'm used to. At the least
this is not a claim which can be made without some kind of argument;
"pretty clear" doesn't cut it, at least for me.
>
> >If that's the background of the argument, then why are not you just
> > making it directly? Mental states are not available, mechancial
> > behaviours are; therefore mental states are not mechanical. Why go
> > through (G) - it only confuses the issue. Do you think the following
> > is a good argument: I have thoughts; Turing computations clearly do
> > not have thoughts; etc. ??
>
> I have indeed taken a trip a bit too far to the phenomenological
> background of the whole matter, but this was only an attempt of
> explanation, it is not the result itself. I think, the result stands on
> its own. As far as I can see, things are as it puts them. Trying to go
> deeper to get an insight into the roots of the phenomenon, that's
> really something else.
>
> So, we should perhaps limit ourselves to discussing whether CBL really
> entails what I say it entails about (G), namely, that it cannot always
> have a trurh value, and whether at the same time CBL+Computationalism
> entail that (G) always depicts a well defined state of affairs.
>
> Please, tell me: do you agree that (G) cannot always be given a truth
> value? Do you agree that Turing machine behaviors are always well
> defined states of affairs?
To be honest, I think the notion of "correct cognitive behaviour" is
still a little too unclear to me before I give a definitive answer. I
do think "This statement is false" and "This statement is not true" are
both (always) not true. So by analogy, I imagine at the end of the day
I would say that G does (always) have a truth value: it is not true.
I think that Turing machine behaviors - when these are physical
machines in the real world, which is what you apparently mean - are
just as well defined states of affairs as cognitive behaviour, no more,
no less.
>
> And no, a kind of Chinese room argument is not enough to establish an
> incompatibility result. Certainly, Turing machines have no FUNCTIONAL
> mental features, for they can be thoroughly described without any
> reference to mental states. Certainly minds have mental features but,
> do they have FUNCTIONAL mental features?
>
> Thanks a lot for your comments.
>
Again, my pleasure!
David C. Ullrich
wrote:
>As far as I can see your comment has nothing to so with my argument. I
>am not giving a same name to different things, I'm just naming a
>sentence (a symbol string) and then exhibiting the symbol string at
>issue; if the name of the sentence appears among the symbols of the
>string, that entails no contradiction.
Make up your mind. Yesterday you said
>On the contrary, it is a perfectly sound definition because the second
>'(G)' stays there as a simple symbol string and needs no further
>definition, while the first '(G)' is a denoting term.
Now today you say that the second (G) does indeed denote the
entire sentence? Then my original comment becomes valid again:
You have not said what (G) is, because if the reader doesn't
know he still doesn't know after reading your "definition"
of (G).
************************
David C. Ullrich
LauLuna
'(J) contains the word "the" '
Are we to believe that you do not know whether (J) is true or false?
LauLuna
I do think the tokenist position implies a hierarchy but not Russell's
or Tarski's. In particular I 'd say it is not a hierarchy of truth
predicates but of acts of thinking; the hierarchy seems to me
ultimately arising from phenomenological facts.
> 2/ If you are disallowing perfectly normal sentences as ill-formed,
> then you might as well just say "This sentence is not true" is
> ill-formed and be done with it. That is, this talk of tokens seems to
> be the introduction of a vocabulary (tokens) which in fact solves
> nothing. At the end of the day, there is still an ad hoc claim about
> which kind of assertions are permissible and which are not - and what
> determines impermissibility is whether there is paradox.
It happens I have no reason to believe that 'this sentence is not true'
is ill formed.
Nevertheless, you should not require from me a complete solution of the
Liar and its sequels in order to discuss my (much simpler) argument.
> Perhaps. Your argument, I think you would agree, flies pretty close to
> the Liar's Paradox. So it is normal that I am skeptical that you are
> not simply deriving your conclusion from the contradiction which comes
> from the Paradox (in which case you can, of course, conclude anything).
> So, while technically you are right, I think people will find your
> argument convincing only if you first really do have a convincing
> solution to the Liar and the Strengthened Liar. And, I'm afraid, for
> the moment anyway, I'm not convinced. (I should be candid that I think
> I *do* know the solution, and since yours does not co-incide with mine,
> I think it is incorrect.)
Yes, as a psychological matter of fact it is probably true that arguing
from paradoxes seems audacious and unconvincing to most people.
> To be honest, I think the notion of "correct cognitive behaviour" is
> still a little too unclear to me before I give a definitive answer. I
> do think "This statement is false" and "This statement is not true" are
> both (always) not true. So by analogy, I imagine at the end of the day
> I would say that G does (always) have a truth value: it is not true.
>
>
> I think that Turing machine behaviors - when these are physical
> machines in the real world, which is what you apparently mean - are
> just as well defined states of affairs as cognitive behaviour, no more,
> no less.
You said you accept CBL but you actually do not: not-true is no truth
value in CBL as defined. Surely 'dffgghjjkkllññ' is also not-true.
And I wasn't asking from you to compare Turing computations and
cognitive behaviors to their well-definedness. But I undestand you
simply are reluctant to grant there might a difference between them.
I can easily understand your skepticism. However (excuse me if I'm
wrong) sometimes it looks like you're trying to dodge the result at any
price. Even if it is so, which I do not affirm, (or perhaps precisely
because of it) your objections are highly useful because you seem to be
probing my argument in search of any weakness and this is always
helpful.
For my part, I confess I'm influenced by philosophical (pre)conceptions
in this topic and I would probably do the same with respect to any
argument favoring computationalism.
I find it intriguing that you do not accept that CBL implies tokenism.
Please, re-consider the argument concerning
(1) (1) is not true
(2) (1) is not true
and tell me where you find the flaw.
Best regards
abo
LauLuna wrote:
>
> You said you accept CBL but you actually do not: not-true is no truth
> value in CBL as defined. Surely 'dffgghjjkkllññ' is also not-true.
I think there are two truth values: truth and non-truth. Since CBL
carries with it the demand that truth values must be restricted to
statements, then the two truth values are: truths and statements which
are not true. And here I'm using statement in the wide sense, so that
the Liar and the Strengthened Liar are clearly statements.
>
> And I wasn't asking from you to compare Turing computations and
> cognitive behaviors to their well-definedness. But I undestand you
> simply are reluctant to grant there might a difference between them.
>
> I can easily understand your skepticism. However (excuse me if I'm
> wrong) sometimes it looks like you're trying to dodge the result at any
> price. Even if it is so, which I do not affirm, (or perhaps precisely
> because of it) your objections are highly useful because you seem to be
> probing my argument in search of any weakness and this is always
> helpful.
>
> For my part, I confess I'm influenced by philosophical (pre)conceptions
> in this topic and I would probably do the same with respect to any
> argument favoring computationalism.
>
> I find it intriguing that you do not accept that CBL implies tokenism.
> Please, re-consider the argument concerning
>
> (1) (1) is not true
>
> (2) (1) is not true
>
> and tell me where you find the flaw.
As I said somewhere else, you can prove anything with the Liar, using
normal reasoning. The Liar produces a contradiction, and from that you
can conclude whatever you want - tokenism, a hierarchy of truths, Santa
Claus. The difficulty comes when one considers whether the solution
is itself coherent. I don't think tokenism is. First, the solution of
tokenism is prime facie unbelievable, because (1) and (2) are both
making the same assertion of the same thing. They do seem to mean the
same thing, hence they should not, at first sight anyway, differ in
their truth value. Secondly, tokenism does not IMHO provide a solution
to an unending sequence of "The next token is not true."
LauLuna
abo ha escrito:
> LauLuna wrote:
> >
> > You said you accept CBL but you actually do not: not-true is no truth
> > value in CBL as defined. Surely 'dffgghjjkkllññ' is also not-true.
>
> I think there are two truth values: truth and non-truth. Since CBL
> carries with it the demand that truth values must be restricted to
> statements, then the two truth values are: truths and statements which
> are not true. And here I'm using statement in the wide sense, so that
> the Liar and the Strengthened Liar are clearly statements.
Clearly we are giving the name 'CBL' to different things; I thought we
both accepted the definition I gave in the first post.
In order to derive whatever you like from a contradiction you must
previously assume that contradiction as true. I have never done such a
thing in my argument nor is it implicit in the reasoning about (1) and
(2), nor has any tokenist done it for all I know.
I think tokenism demands a clear distinction between sentences
(syntactical objects) and propositions (semantical objects). It is
evident that sentences as mere strings of symbols cannot be the primary
truth value bearers; it is evident that different sentences like 'snow
is white', 'der Schnee ist weiss', etc. express the same proposition.
So, the distinction seems straightforward.
Propositions, as semantical objects, are related to thoughts as
intentional objects of propositional attitudes. When I first meet:
(1) (1) is not true
I easily come to know that no one could express a thought by means of
that token: nobody can really think 'what I'm thinking just now is
false'.
Once I've accomplished this, I am able for the first time to express
part of it through the same sentence in (1) but by means of another
utterance, now performed in a quite new mental state only rendered
possible by my previous assessment of (1); then I utter (2).
So, the essential is that the thought that lies behind (2) could never
have been behind (1). This is why (1) expresses no proposition while
(2) expresses a true one. The act of thought which would correspond to
(1) is phenomenologically impossible. That's why the fact that (1) is
not true is not (yet) available to be asserted by the one who utters
(1), i. e. by anyone who hasn't previously evaluated (1).
This is not so hard, only counterintuitive at first sight; easy to
accept after one has clearly distinguished sentences from propositions;
rather obvious then, I'd say.
As for the infinite kindred liars... I guess none will ever be able to
give a detailed and complete solution to all paradoxes once and
forever, because it seems new paradoxes can always be built starting
from any proposed solution. One can only offer a general orientation as
I think the phenomenological approach does. But this is no reason to
reject what seems evidently gained up to the moment.
Nevertheless, I have some ideas about Yablo's and Sorensen's paradoxes.
Perhaps a question for another thread or another day.
Regards
abo
LauLuna wrote:
>
> I think tokenism demands a clear distinction between sentences
> (syntactical objects) and propositions (semantical objects). It is
> evident that sentences as mere strings of symbols cannot be the primary
> truth value bearers; it is evident that different sentences like 'snow
> is white', 'der Schnee ist weiss', etc. express the same proposition.
> So, the distinction seems straightforward.
No problem with the idea that propositions rather than sentences are
the bearers of truth.
>
> Propositions, as semantical objects, are related to thoughts as
> intentional objects of propositional attitudes. When I first meet:
>
> (1) (1) is not true
>
> I easily come to know that no one could express a thought by means of
> that token: nobody can really think 'what I'm thinking just now is
> false'.
>
> Once I've accomplished this, I am able for the first time to express
> part of it through the same sentence in (1) but by means of another
> utterance, now performed in a quite new mental state only rendered
> possible by my previous assessment of (1); then I utter (2).
>
> So, the essential is that the thought that lies behind (2) could never
> have been behind (1). This is why (1) expresses no proposition while
> (2) expresses a true one. The act of thought which would correspond to
> (1) is phenomenologically impossible. That's why the fact that (1) is
> not true is not (yet) available to be asserted by the one who utters
> (1), i. e. by anyone who hasn't previously evaluated (1).
I had a certain sympathy for this kind of argument once. However, it
seems to me now that it depends on an unrealistic idea about what
propositions may be. I or someone else can claim, "Everything that
Nixon said about Watergate isn't true," without having in mind every
statement that Nixon said about Watergate. I can claim that the
proposition on the back of a piece of paper is not true, without
knowing what the proposition is. And while the reason for the claim
may change once I know what the proposition is, it doesn't actually
change the claim. So while (2) may have a different justification or
benefit from more information or analysis than (1) does, the actual
claim (that (1) is not true) is the same in both cases (1) and (2).
Truth should not depend on justification or information available, only
on the claim itself; and tokenism seems to go against this.
> This is not so hard, only counterintuitive at first sight; easy to
> accept after one has clearly distinguished sentences from propositions;
> rather obvious then, I'd say.
>
> As for the infinite kindred liars... I guess none will ever be able to
> give a detailed and complete solution to all paradoxes once and
> forever, because it seems new paradoxes can always be built starting
> from any proposed solution. One can only offer a general orientation as
> I think the phenomenological approach does. But this is no reason to
> reject what seems evidently gained up to the moment.
>
> Nevertheless, I have some ideas about Yablo's and Sorensen's paradoxes.
> Perhaps a question for another thread or another day.
I agree - another thread and another day !
>
> Regards
David C. Ullrich
wrote:
>Let us call '(J)' the following sentence:
>
[*] '(J) contains the word "the" '
That's not a sentence, because "(J)" is not an English word.
If the "(J)" in the sentence itself is supposed to refer to
what you think is the sentence (J) then you have not said
what (J) is.
Because before reading (*) I don't know what (J) is, and
reading (*) doesn't help with that, because the supposed
definition of (J) includes the string "(J)", and I don't
know what that means.
>Are we to believe that you do not know whether (J) is true or false?
Believe what you want. I can't know whether (J) is true or false
until I know what (J) _is_.
************************
David C. Ullrich