Simple yet Profound Metatheorem
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From: H. J. Sander Bruggink on 16 Dec 2005 08:47 sradhakr wrote: > H. J. Sander Bruggink wrote: > >>sradhakr wrote: >> >>>Barb Knox wrote: >>> >>> >>>>In article <1134713007.253164.75100(a)z14g2000cwz.googlegroups.com>, >>>>"sradhakr" <sradhakr(a)in.ibm.com> wrote: >>>> >>>>>Any "proof" of ~(P&~P) that you produce from contradictory >>>>>premises is not a valid proof in these logics. >>>> >>>>Eh? I've given a perfectly valid Intuitionistic proof. On what grounds >>>>do you object to it (if you do)? >>>> >>> >>>*Any* proposition can be proven in intiuitionistic logic if you start >>>with the premise P&~P. >> >>Ok, please produce, in the same way, a proof of the >>proposition P&~P, then. >>(I mean "P&~P", *not* "(P&~P) -> (P&~P)".) >> > > That is precisely the point of my objection. If the claimed "proof" of > ~(P&~P) is allowed to go through, then just about any proposition, > including P&~P should also be provable. Fine, show me that proof of P&~P, then. > > In other words, the claimed proof shows that from the hypothesis P&~P > one can conclude ~(P&~P). But from the hypothesis P&~P one can conclude > any proposition, so what is the basis for the claimed proof? Yes, for any proposition Q, you can prove (P&~P) -> Q. What's your point? > > >>[snip more nonsense] >> > > ?????? > If you have something meaningful to say, say it. Otherwiise just keep > shut. I apologize. It was not nice of me to call you nonsense "nonsense". :-) groente -- Sander
From: sradhakr on 16 Dec 2005 08:56 H. J. Sander Bruggink wrote: > sradhakr wrote: > > H. J. Sander Bruggink wrote: > > > >>sradhakr wrote: > >> > >>>Barb Knox wrote: > >>> > >>> > >>>>In article <1134713007.253164.75100(a)z14g2000cwz.googlegroups.com>, > >>>>"sradhakr" <sradhakr(a)in.ibm.com> wrote: > >>>> > >>>>>Any "proof" of ~(P&~P) that you produce from contradictory > >>>>>premises is not a valid proof in these logics. > >>>> > >>>>Eh? I've given a perfectly valid Intuitionistic proof. On what grounds > >>>>do you object to it (if you do)? > >>>> > >>> > >>>*Any* proposition can be proven in intiuitionistic logic if you start > >>>with the premise P&~P. > >> > >>Ok, please produce, in the same way, a proof of the > >>proposition P&~P, then. > >>(I mean "P&~P", *not* "(P&~P) -> (P&~P)".) > >> > > > > That is precisely the point of my objection. If the claimed "proof" of > > ~(P&~P) is allowed to go through, then just about any proposition, > > including P&~P should also be provable. > > Fine, show me that proof of P&~P, then. > > > > > In other words, the claimed proof shows that from the hypothesis P&~P > > one can conclude ~(P&~P). But from the hypothesis P&~P one can conclude > > any proposition, so what is the basis for the claimed proof? > > Yes, for any proposition Q, you can prove (P&~P) -> Q. > What's your point? > That therefore no valid proof of ~(P&~P) should start with the hypothesis P&~P. Understand this point before you hit the keyboard again. > > > > > >>[snip more nonsense] > >> > > > > ?????? > > If you have something meaningful to say, say it. Otherwiise just keep > > shut. > > I apologize. It was not nice of me to call you nonsense > "nonsense". :-) > ???? Read what I wrote above. Regards, RS
From: Torkel Franzen on 16 Dec 2005 09:39 "H. J. Sander Bruggink" <bruggink(a)phil.uu.nl> writes: > In other words, the claimed proof shows that from the hypothesis P&~P > one can conclude ~(P&~P). But from the hypothesis P&~P one can conclude > any proposition, so what is the basis for the claimed proof? Your comments are based on a confusion. The assumption P&~P is discharged in the derivation. Ex falso quodlibet is not used.
From: Torkel Franzen on 16 Dec 2005 09:41 "sradhakr" <sradhakr(a)in.ibm.com> writes: > In other words, the claimed proof shows that from the hypothesis P&~P > one can conclude ~(P&~P). But from the hypothesis P&~P one can conclude > any proposition, so what is the basis for the claimed proof? Your comments are based on a confusion. The assumption P&~P is discharged in the derivation. Ex falso quodlibet is not used. (Apologies for earlier misattributed version.)
From: sradhakr on 16 Dec 2005 09:58
Torkel Franzen wrote: > "sradhakr" <sradhakr(a)in.ibm.com> writes: > > > In other words, the claimed proof shows that from the hypothesis P&~P > > one can conclude ~(P&~P). But from the hypothesis P&~P one can conclude > > any proposition, so what is the basis for the claimed proof? > > Your comments are based on a confusion. The assumption P&~P is > discharged in the derivation. Ex falso quodlibet is not used. > > (Apologies for earlier misattributed version.) Ex falso quodlibet is not directly used, I agree. But EFQ *could* be used to deduce just about whatever we want from P&~P. So an assertion of P&~P as a hypothesis is in principle the same as asserting an arbitrary proposition Q. So is it surprising that Q could in particular, be ~(P&~P)? You might argue that the actual proof doesn't run this way. But what, precisely, is the "absurdity" that you deduce from P&~P, in order to conclude ~(P&~P) in the claimed proof? The fact that you can deduce an arbitrary proposition from P&~P? If so, that is a tacit use of EFQ, and invalidates the conclusion of ~(P&~P) from the same hypothesis P&~P, or so I claim. Let me know what you think. Regards, RS |