Re: Set Theorists use EXTENSIONALITY ..... Logic Programmers UNIFY!

Article View: sci.logic

Article #715

#715

From: Jan Burse
Date: Thu, 13 Jun 2013 07:54

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Graham Cooper schrieb:
> ~tail <- ~head
>
> directly follows from
>
> head <- tail

The above cannot is contraposition, see also:
http://en.wikipedia.org/wiki/Contraposition

It cannot turn positive information only:

    p(1).

Into positive and negative information:

    forall X(p(X) <-> X=1).

The contraposition cannot add negative information.
It might only yield logical variants of the input
theory, but it might not perform some completion.

Contraposition is also an equivalence, in classical
logic, because of double negation elimination:

    P -> Q   =>   ~Q -> ~P
             =>   ~~P -> ~~Q
             <=>   P -> Q

So its actualy:

   P -> Q   <=>   ~Q -> ~P

You can try to apply it to the fact p(1). Frist
represent the fact as it is done in Prolog:

     p(1) <- true.

Then apply the contraposition:

     ~true <- ~p(1).

No negative information added! It is still that
we cannot derive, assuming nats as domain:

     P |- ~p(2)   /* not derivable */

Bye

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From: Jan Burse <janburse@fastmail.fm>
Newsgroups: sci.logic,comp.lang.prolog,aus.computers
Subject: Re: Set Theorists use EXTENSIONALITY ..... Logic Programmers UNIFY!
Date: Thu, 13 Jun 2013 09:54:29 +0200
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