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Topic: Question about proof by contradiction
Replies: 92   Last Post: Mar 15, 2008 12:59 AM

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Marshall

Posts: 1,936
Registered: 8/9/06
Re: Question about proof by contradiction
Posted: Mar 10, 2008 1:46 PM
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On Mar 9, 7:00 pm, magi...@math.berkeley.edu (Arturo Magidin) wrote:
>
> >> Proof by contradiction can be formalized as
>
> >> (P -> (A and not(A))) -> not(P).
[...]
> The proof in question, in fact, does not even use proof by
> contradiction. It has the form
>
> (P -> not(P)) -> not(P).
>
> This is not a proof by contradiction.


Does the second one have a name?

Superficially, the two formulas appear similar. Certainly they
evoke a similar sense. Is there any sense in which we might
consider an antecedent as being in-scope in the consequent,
which would allow us to connect the two? Or is the relationship
a superficial structural similarity merely?

Does anything interesting happen if we transform them somewhat?

(P -> (A and not(A))) -> not(P)
(P -> false) -> not(P).
(not(P) or false) -> not(P)
not(P) -> not(P)

Well, I seem to have destroyed the formula's essential nature
by these manipulations. How did THAT happen? Apparently
truth-value-preserving transformations don't preserve some things
that aren't truth values.


Marshall


Date Subject Author
3/9/08
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goodchild.trevor@gmail.com
3/9/08
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magidin@math.berkeley.edu
3/10/08
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LudovicoVan
3/10/08
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Aatu Koskensilta
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Aatu Koskensilta
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LudovicoVan
3/10/08
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Aatu Koskensilta
3/10/08
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LudovicoVan
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LudovicoVan
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3/9/08
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goodchild.trevor@gmail.com
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Aatu Koskensilta
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goodchild.trevor@gmail.com
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LudovicoVan
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Aatu Koskensilta
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magidin@math.berkeley.edu
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Aatu Koskensilta
3/10/08
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magidin@math.berkeley.edu
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Aatu Koskensilta
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Herman Jurjus
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Aatu Koskensilta
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LudovicoVan
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goodchild.trevor@gmail.com
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goodchild.trevor@gmail.com
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LudovicoVan
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magidin@math.berkeley.edu
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Marshall
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G. Frege
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3/11/08
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Marshall
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magidin@math.berkeley.edu
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G. Frege
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3/13/08
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magidin@math.berkeley.edu
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magidin@math.berkeley.edu
3/12/08
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pn4039@albany.edu
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Jack Markan
3/12/08
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G. Frege
3/13/08
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pn4039@albany.edu
3/13/08
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magidin@math.berkeley.edu
3/13/08
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G. Frege
3/14/08
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pn4039@albany.edu
3/14/08
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G. Frege
3/14/08
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magidin@math.berkeley.edu
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3/13/08
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3/13/08
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Marshall
3/13/08
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quasi
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3/12/08
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Read Re: Question about proof by contradiction
LudovicoVan
3/13/08
Read Re: Question about proof by contradiction
LudovicoVan
3/13/08
Read Re: Question about proof by contradiction
G. Frege
3/13/08
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LudovicoVan
3/11/08
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G. Frege
3/15/08
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3/9/08
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goodchild.trevor@gmail.com
3/10/08
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G. Frege
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LudovicoVan
3/10/08
Read Re: Question about proof by contradiction
G. Frege
3/10/08
Read Re: Question about proof by contradiction
G. Frege
3/10/08
Read Re: Question about proof by contradiction
LudovicoVan
3/10/08
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Aatu Koskensilta
3/10/08
Read Re: Question about proof by contradiction
LudovicoVan
3/12/08
Read Re: Question about proof by contradiction
pn4039@albany.edu
3/12/08
Read Re: Question about proof by contradiction
G. Frege
3/13/08
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LudovicoVan
3/13/08
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Marshall
3/13/08
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LudovicoVan
3/12/08
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Jack Markan
3/10/08
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G. Frege
3/10/08
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3/10/08
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3/10/08
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3/10/08
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G. Frege

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