Re: The limit of all computations

On Mon, May 28, 2012 at 10:37:53AM +0200, Bruno Marchal wrote:
>
> On 28 May 2012, at 04:00, Russell Standish wrote:
>
> >On Sun, May 27, 2012 at 06:20:29PM +0200, Bruno Marchal wrote:
> >>
> >>On 27 May 2012, at 12:15, Russell Standish wrote:
> >>>I still don't follow. If I have proved a is true in some world, why
> >>>should I infer that it is true in all worlds? What am I missing?
> >>
> >>I realize my previous answer might be too long and miss your
> >>question. Apology if it is the case.
> >>
> >>Here is a shorter answer. The idea of proving, is that what is
> >>proved in true in all possible world. If not, a world would exist as
> >>a counter-example, invalidating the argument.
> >
> >I certainly missed that. Is that given as an axiom?
>
> That would be a meta-axiom in a theory defining what is logic. But
> that does not exist. It is just part of what logic intuitively
> consists in.

Well, I can tell you, it is not intuitive! Perhaps there is some
background understanding that is missing.

> Logicians are not interested of truth or interpretation of
> statements. They are interested in validity. What sentences follow
> from what sentences, independently of interpretations, and thus true
> in all possible worlds.
>
>
>
> >It seems like that
> >would be written p -> []p.
>
> This means that if p then p is provable. "p -> Bp", if B = provable,

[]p means (primarily) true in all worlds. In Kripke semantics, it is
relativised to mean true in all accessible worlds.

The meaning of provability is a different interpretation.

>
>
> >
> >When I say p is true in a world, I can only prove that p is true in
> >that world.
>
> I don't think so. If p is true, that does not mean you can prove it,
> neither in your world, nor in some other world.

p may be true, but if I don't know it (or can't prove it), I shouldn't be
asserting it :).

>
>
> >I am mute on the subject of whether p is true in any other
> >world (unless I can use an axiom like the above).
>
> By the logicians notion of proof, if you prove a proposition, it is
> true in all worlds/model/interpretation.
>

Even if the proof relied upon some facet that may or may not be true
in all worlds?

>
> >
> >In what class of logics would such an axiom be taken to be true.
>
> All.
>
>

--

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Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au
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