On 6/13/2012 12:14 AM, Bruno Marchal wrote:


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.

You mean if p is a tautology. It may be a deduction from premises that are not true in all worlds. Is Russell thinking of p="Socrates is mortal" while you're thinking of p="If all men are mortal and Socrates is a man then Socrates is mortal."



Take as example a formalization of classical propositional calculus. The axioms have to be tautologies,

An axiom is just a proposition taken to be true for purposes of inference. Why can't "Socrates is a man" be an axiom?

Brent

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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