On 13 Jun 2012, at 00:38, Russell Standish wrote:

On Tue, Jun 12, 2012 at 08:17:38PM +0200, Bruno Marchal wrote:

On 12 Jun 2012, at 00:47, Russell Standish wrote:

On Thu, Jun 07, 2012 at 01:33:48PM +0200, Bruno Marchal wrote:

In fact we have p/p for any p. If you were correct we would have []p
for any p.

This is what I thought you said the "meta-axiom" stated?

How else do we get p/[]p for Kripke semantics?


Because if p is true in all worlds, then []p is true in all worlds
OK?

No.  I didn't say that. p means p is true in a world. p true in all
worlds would be written []p.

But in logic, if p appears in a deduction, p is true in all worlds.

Take as example a formalization of classical propositional calculus. The axioms have to be tautologies, and so are true in all worlds (valuations, interpretation). The modus ponens concerves tautologicalness, so all theorems (the formula appearing in the deduction) are true in all worlds.

And p/[]p means that if p is true in all worlds (like if it is proved) then []p is true in all worlds.

If you want to mean that p is true in a world, or the actual world, you can say it, but not in deduction. Usually you will name that world, by saying that p is true in alpha, at some meta level.

Bruno




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