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.`

`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,`

`is completeness (with the meaning of completeness = its meaning in`

`incompleteness). This is false in non rich theory (by the fact that`

`their are non rich) and false in rich theory, by the fact that rich`

`theory obeys to the incompleteness theorem. So, it is true for rare`

`exception (like the first order theory of real numbers) which is not`

`rich (not sigma_1 complete).`

`Take the proposition (a v b) in propositional logic. Take the world`

`{(a t), (b, f)}, i.e. the world with a true, and b false. Let p = (a v`

`b). This provides a counter-example to p -> Bp. p is true in that`

`world (because a v b is true if a is true), yet it is not provable,`

`because it is false in some other world, like the world with both a`

`and b false.`

`Or take p = Dt. Dt -> BDt contradicts immediately the second`

`incompleteness theorem which says that Dt -> ~BDt.`

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.`

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.`

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

All.

(Ofcourse it is true in classical logic, but there is only one "world"there).

`In classical propositional logic, a world is just anything to which we`

`attach a valuation t, or f, to the atomic proposition, p, q r, ...`

`This makes 2^aleph_zero worlds. A world can be identified with a`

`function from {p, q, r, ...} to {t, f}.`

`In first order logic, worlds can be identified with interpretations,`

`or models. All first order theories have many models. In fact for any`

`cardinal, there is a model having that cardinal. The number of worlds`

`exceeds the cardinals nameable in set theory.`

Bruno

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