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LauLuna
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 Posted: Tue Jul 22, 2008 12:01 pm Post subject: Intensional Operators Introduction in Hypothetical Reasoning |
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I post this here hoping someone can help me clarify a matter I haven>t
found sufficiently explained in my books and links. It goes on the
introduction of the necessity and provability operators in
hypothetical reasoning.
Let p and q be any propositions; let 'N' be the necessity operator,
let '|-' mean logical derivability and let '|=' mean logical
consequence. Consider these two similar and evidently incorrect
deductions:
A)
1. p -> q assumption
2. p assumption
3. q
4. |-q
5. Nq necessitation 4
6. p -> Nq
7. (p->q) -> (p->Nq)
B)
1. p -> q assumption
2. p assumption
3. q
4. p|-q 2-3
5. p|=q logical derivability implies logical consequence
6. (p->q) -> (p|=q)
Surely, the flaw lies in the introduction of the intensional operators
under uncancelled assumptions. But an apparently similar derivation is
used in the proof of Fitch>s theorem (or paradox, as you prefer):
http://plato.stanford.edu/entries/fitch-paradox/ step (8). So I wonder
whether intensional logics usually prohibit this kind of reasoning and
how they manage to separate this from other sound operator
introduction cases.
Thanks |
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Jesse F. Hughes
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| Posted: Tue Jul 22, 2008 6:42 pm Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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LauLuna <laureanoluna@yahoo.es> writes:
[quote]Surely, the flaw lies in the introduction of the intensional operators
under uncancelled assumptions. But an apparently similar derivation is
used in the proof of Fitch>s theorem (or paradox, as you prefer):
http://plato.stanford.edu/entries/fitch-paradox/ step (8).
[/quote]
Perhaps I misunderstand what you mean, but step (8) occurs after the
discharge of the assumption (4) in step (7), no?
So the necessitation in (8) is not under any uncanceled assumptions as
far as I can see.
--
"I don>t know why I live in a world with so many supposed
mathematicians who are all so dumb AND rude. Why oh why couldn>t
someone like Gauss or Dedekind still be around? Shoot, I>d even take
someone like Hardy at this point." -- James S Harris compromises |
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William Elliot
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| Posted: Wed Jul 23, 2008 1:54 pm Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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On Tue, 22 Jul 2008, LauLuna wrote:
[quote]I post this here hoping someone can help me clarify a matter I haven>t
found sufficiently explained in my books and links. It goes on the
introduction of the necessity and provability operators in
hypothetical reasoning.
Let p and q be any propositions; let 'N' be the necessity operator,
let '|-' mean logical derivability and let '|=' mean logical
consequence. Consider these two similar and evidently incorrect
deductions:
A)
1. p -> q assumption
2. p assumption
3. q
4. |-q
[/quote]
Unwarranted. All you get is
p -> q, p |- q.
[quote]5. Nq necessitation 4
6. p -> Nq
7. (p->q) -> (p->Nq)
B)
1. p -> q assumption
2. p assumption
3. q
4. p|-q 2-3
[/quote]
No, p -> q, p | q.
[quote]5. p|=q logical derivability implies logical consequence
6. (p->q) -> (p|=q)
Surely, the flaw lies in the introduction of the intensional operators
under uncancelled assumptions. But an apparently similar derivation is
used in the proof of Fitch>s theorem (or paradox, as you prefer):
http://plato.stanford.edu/entries/fitch-paradox/ step (8). So I wonder
whether intensional logics usually prohibit this kind of reasoning and
how they manage to separate this from other sound operator
introduction cases.
Thanks
[/quote] |
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LauLuna
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| Posted: Thu Jul 24, 2008 5:25 pm Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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On Jul 22, 3:42 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
[quote]LauLuna <laureanol...@yahoo.es> writes:
Surely, the flaw lies in the introduction of the intensional operators
under uncancelled assumptions. But an apparently similar derivation is
used in the proof of Fitch>s theorem (or paradox, as you prefer):
http://plato.stanford.edu/entries/fitch-paradox/step (8).
Perhaps I misunderstand what you mean, but step (8) occurs after the
discharge of the assumption (4) in step (7), no?
So the necessitation in (8) is not under any uncanceled assumptions as
far as I can see.
--
"I don>t know why I live in a world with so many supposed
mathematicians who are all so dumb AND rude. Why oh why couldn>t
someone like Gauss or Dedekind still be around? Shoot, I>d even take
someone like Hardy at this point." -- James S Harris compromises
[/quote]
Yes, but I yake we are there still under the assumption that something
will ever be unknown, namely p in:
(1) p & ~Kp
Necessitation in step (8):
(8) N ~K(p & ~Kp)
depends on (1) being true.
So, when we state (8), we are under the assumption that (1) holds.
Note that the paradox is expressed by these formulae:
p & ~Kp -> N ~K(p & ~Kp)
p & ~Kp |- N ~K(p & ~Kp)
If something is true but will never be known, then there is something
true that cannot be known.
Regards |
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LauLuna
Guest
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| Posted: Thu Jul 24, 2008 5:29 pm Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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On Jul 23, 10:54 am, William Elliot <ma...@hevanet.remove.com> wrote:
[quote]On Tue, 22 Jul 2008, LauLuna wrote:
I post this here hoping someone can help me clarify a matter I haven>t
found sufficiently explained in my books and links. It goes on the
introduction of the necessity and provability operators in
hypothetical reasoning.
Let p and q be any propositions; let 'N' be the necessity operator,
let '|-' mean logical derivability and let '|=' mean logical
consequence. Consider these two similar and evidently incorrect
deductions:
A)
1. p -> q assumption
2. p assumption
3. q
4. |-q
Unwarranted. All you get is
p -> q, p |- q.
[/quote]
Why? I haven>t yet discharged the assumptions but under the
assumptions I have indeed derived q.
[quote]5. Nq necessitation 4
6. p -> Nq
7. (p->q) -> (p->Nq)
B)
1. p -> q assumption
2. p assumption
3. q
4. p|-q 2-3
No, p -> q, p | q.
[/quote]
Why? Note that I haven>t discharged 1. So, I state p|-q under the
assumption of 1. That>s why the writing is still displaced to the
right.
Regards |
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Balthasar
Guest
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| Posted: Thu Jul 24, 2008 10:56 pm Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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On Thu, 24 Jul 2008 10:25:56 -0700 (PDT), LauLuna
<laureanoluna@yahoo.es> wrote:
[quote]
Yes, but I yake we are there still under the assumption that something
will ever be unknown, namely p in:
(1) p & ~Kp
Necessitation in step (8):
(8) N ~K(p & ~Kp)
depends on (1) being true.
No. It doesn>t. Steps (4) - (9) are an independent from (1).[/quote]
I see that Lemmon>s device to denote dependence would be a good thing
here.
B.
--
"For every line of Cantor>s list it is true that this line does not
contain the diagonal number. Nevertheless the diagonal number may
be in the infinite list." (WM, sci.logic) |
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Balthasar
Guest
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| Posted: Thu Jul 24, 2008 10:58 pm Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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On Thu, 24 Jul 2008 10:29:48 -0700 (PDT), LauLuna
<laureanoluna@yahoo.es> wrote:
[quote]
1. p -> q assumption
2. p assumption
3. q
4. |-q
Unwarranted. All you get is
p -> q, p |- q.
Why? I haven>t yet discharged the assumptions but under the
assumptions I have indeed derived q.
Because "p -> q, p |- q" denotes JUST THAT, while "|- p" denotes that p[/quote]
has been derived from NO assumptions.
B.
--
"For every line of Cantor>s list it is true that this line does not
contain the diagonal number. Nevertheless the diagonal number may
be in the infinite list." (WM, sci.logic) |
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Balthasar
Guest
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| Posted: Thu Jul 24, 2008 10:59 pm Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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On Tue, 22 Jul 2008 05:01:13 -0700 (PDT), LauLuna
<laureanoluna@yahoo.es> wrote:
[quote]A)
1. p -> q assumption
2. p assumption
3. q
4. |-q
Well, I>m not familiar with this system, but for me at step 4 we would[/quote]
have:
p -> q, p |- q.
[quote]
5. Nq necessitation 4
Which would only be allowed in case of |- p. But we have (as just[/quote]
mentioned) p -> q, p |- q.
[quote]
B)
1. p -> q assumption
2. p assumption
3. q
4. p|-q 2-3
Same, same. Here we clearly just have[/quote]
p -> q, p |- q.
B. |
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Jesse F. Hughes
Guest
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| Posted: Fri Jul 25, 2008 4:02 am Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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LauLuna <laureanoluna@yahoo.es> writes:
[quote]Yes, but I yake we are there still under the assumption that something
will ever be unknown, namely p in:
(1) p & ~Kp
[/quote]
As Balthasar says, the argument (4) - (9) does not depend on (1) - (3)
at all. It>s an independent proof.
--
Jesse F. Hughes
"You shouldn>t hate Mother Mathematics."
-- James S. Harris |
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LauLuna
Guest
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| Posted: Mon Jul 28, 2008 12:28 am Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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On Jul 25, 1:02 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
[quote]LauLuna <laureanol...@yahoo.es> writes:
Yes, but I yake we are there still under the assumption that something
will ever be unknown, namely p in:
(1) p & ~Kp
As Balthasar says, the argument (4) - (9) does not depend on (1) - (3)
at all. It>s an independent proof.
--
Jesse F. Hughes
"You shouldn>t hate Mother Mathematics."
-- James S. Harris
[/quote]
Yes, you and Balthasar are right. I had misread it. Thanks. |
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LauLuna
Guest
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| Posted: Mon Jul 28, 2008 12:29 am Post subject: Re: Intensional Operators Introduction in Hypothetical Reaso |
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On Jul 24, 7:58 pm, Balthasar <nomail@invalid> wrote:
[quote]On Thu, 24 Jul 2008 10:29:48 -0700 (PDT), LauLuna
laureanol...@yahoo.es> wrote:
1. p -> q assumption
2. p assumption
3. q
4. |-q
Unwarranted. All you get is
p -> q, p |- q.
Why? I haven>t yet discharged the assumptions but under the
assumptions I have indeed derived q.
Because "p -> q, p |- q" denotes JUST THAT, while "|- p" denotes that p
has been derived from NO assumptions.
B.
--
"For every line of Cantor>s list it is true that this line does not
contain the diagonal number. Nevertheless the diagonal number may
be in the infinite list." (WM, sci.logic)
[/quote]
I see. Thanks once more. |
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